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Robust control design for systems subject to ellipsoidal uncertainty

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Robust control design for systems subject to ellipsoidal uncertainty
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vi, 126 leaves : ill. ; 29 cm.

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Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 121-125).
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Vita.
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by Harry Michael Mahon.

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ROBUST CONTROL DESIGN FOR SYSTEMS SUBJECT TO ELLIPSOIDAL UNCERTAINTY













By

HARRY MICHAEL MAHON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1998














ACKNOWLEDGMENTS


I would like to express my gratitude to my advisor Oscar Crisalle for his guidance and support during my studies at the University of Florida. The latitude he allowed enabled me to investigate many interesting topics in the field of advanced control that were not directly related to my dissertation. He also allowed me the freedom to complete a master's degree in electrical engineering, and was a source of encouragement when the extra degree seemed too daunting a task.

I wish to thank Professors Spyros Svoronos, Haniph Latchman, Thomas Bullock and Rich Dickinson for serving on my supervisory committee, and for all the wisdom they have imparted to me over the past few years through courses and discussions.

I would like to thank the National Science Foundation for financial support under grant number CTS-9502936.

I thank my colleagues Kostas Hrissagis, V. R. Basker, Rick Gibbs, Jon Engelstad, Tony Dutka, Chuck Baab, and Serkan Kincal for their friendship and support. Much of what I have learned during my stay is due to discussions in our lab, especially with Kostas and Basker. Many other friends in Gainesville have been sources of support and inspiration and I thank you all. I would like to thank Florence Kristy Mei-Ann Doo specifically. She has been an amazing source of inspiration to me for the past seven years. I am without words to thank her enough for all that she has done. Finally I wish to thank my family for the continuous support and love that they have given me throughout my life. Without them, I would not have been able to accomplish this work.


ii















TABLE OF CONTENTS


ACKNOW LEDGMENTS....................................................................... H

A B ST R A C T ......................................................................................v

CHAPTERS

1 INTRODUCTION...........................................................................1
1.1 M otivation .........................................................................1
1.2 O bjectives ...................................................................... 4
1.3 Structure of the Dissertation..................................................5

2 STABILITY ANALYSIS FOR ELLIPSOIDAL SYSTEMS........................7
2.1 Introduction .......................................................................7
2.2 Stability Analysis - Continuous Time ..........................................8
2.2.1 Ellipsoidal Uncertainty........................................... 8
2.2.2 Closed Loop Analysis ...............................................10
2.2.3 Robust Stability Analysis..........................................14
2.2.4 Computation of the Parametric Robust Stability Margin.........18
2.3 Stability Analysis - Discrete Time ..............................................19
2.4 Design Example for Ellipsoidal Systems ....................................25
2.4.1 Plant Model and Ellipsoidal Uncertainty........................25
2.4.2 Controller Design................................................. 28
2.4.3 Discussion...........................................................31
2.5 Conclusions.....................................................................32

3 STABILITY MARGIN CALCULATION FOR ELLIPSOIDAL SYSTEMS ........33
3 .1 Introduction .......................................................................33
3.2 Construction of F(s)...........................................................33
3.3 Robust Stability Testing Using F(s).........................................38
3.4 Discrete Time Stability Analysis ................................................40
3 .5 E x am p le ............................................................................44
3.6 C onclusions..................................................................... 48


4 ROBUST PREDICTIVE CONTROL DESIGN FOR ELLIPSOIDALLY
UNCERTAIN SYSTEMS................................................................49
4 . 1 Introduction .......................................................................49
4.2 Nominal Predictive Control Design .........................................54


iii









4.3 Nominally Stabilizing Controller Parameterization ........................58
4.4 Robust Predictive Control Design ...........................................61
4.4.1 Constraint Testing and Objective Function Value Computation. 70
4.5 Robust Control Design With Steady State Disturbance Rejection ..........73
4.6 Inclusion of Unstructured Uncertainty......................................75
4 .7 E xam ples ........................................................................ 77
4.7.1 Example 1..........................................................77
4 .7 .2 E xam ple 2 .............................................................83
4.8 Design Equations for Nominal Predictive Control........................ 88
4.9 C onclusions..................................................................... 90

5 ANALYTIC SOLUTION TO A LIMITING SYNTHESIS PROBLEM.............91
5 .1 Introduction .......................................................................9 1
5.2 Problem Statement..............................................................91
5.3 Robustness Analysis .............................................................94
5.4 Robust Synthesis ..............................................................96
5.5 Continuous Time Limiting Case .............................................98
5 .6 E x am p le ............................................................................100
5.7 C onclusions .......................................................................10 1

6 FUTURE DIRECTIONS ...................................................................103
6.1 Ellipsoidal Systems with Delays................................................103
6.2 Performance Constraints.........................................................103
6.3 LMI-based Robust Control Design.............................................104

A P PE N D IX .......................................................................................105
1. Proof of Lemma 2.2..............................................................105
2. Proof of Lemma 3.1..............................................................107
3. Proof of Theorem 3.2........................................................... 109
4. Proof of Theorem 3.3............................................................112
5. Proof of Lemma 4.1..............................................................116
6. Proof of Theorem 5.3............................................................118

R E FE R E N C E S ...................................................................................12 1

BIOGRAPHICAL SKETCH ...................................................................126


iv














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy




ROBUST CONTROL DESIGN FOR SYSTEMS SUBJECT TO ELLIPSOIDAL UNCERTAINTY



By


Harry Michael Mahon

May 1998



Chairman: Dr. Oscar D. Crisalle Major Department: Chemical Engineering


A robust control design methodology is presented for systems subject to ellipsoidal uncertainty. Both the robust analysis problem and the robust synthesis problems are considered, and both discrete and continuous systems are considered. For continuous systems subject to ellipsoidal uncertainty, a frequency dependent rational function is constructed to measure the stability margin at each frequency. This rational function may have multiple minima, therefore a frequency search must be carried out to find the overall stability margin of the system.

An alternative method for testing robust stability is developed that is based on constructing a stable transfer function whose frequency response magnitude is equal to the stability margin of the system under consideration. The overall stability margin is then found as the H. norm of this transfer function. The construction of this transfer function


v








requires performing two spectral factorizations. This method is applicable to both discrete and continuous systems.

A robust control synthesis procedure for systems subject to ellipsoidal uncertainty is also developed. This method consists of constructing a fixed order Youla parameter to achieve a robustly stable closed loop for the specified level of uncertainty. The appropriate coefficients of the Youla parameter are found through a constrained quasi-convex optimization. The design methodology is applicable to any nominally stabilizing controller, but the specific case of a predictive controller is considered in this work. The resulting controller is a predictive controller that is robust with respect to real parameter variations, and it also retains the nominal servo performance of the original predictive controller. The design procedure is easily modified to allow the incorporation of integral action into the robust controller, guaranteeing the offset-free rejection of asymptotically constant disturbances. It is also possible to incorporate a slightly restricted form of unstructured uncertainty into the design.


vi














CHAPTER 1
INTRODUCTION



1.1 Motivation


The control engineer always faces the problem of uncertainty in the plant model used for control design. This uncertainty arises from numerous sources, including unmodeled or neglected dynamics, use of lumped parameter models, and variation of model parameter values. To compensate for the fact that the model used is not an exact representation of the actual system to be controlled, it is common practice to design a controller that will yield acceptable performance for a set or family of plants that includes the specific (nominal) model used. If the true system is modeled well by some plant in this family, not necessarily the nominal model, then the controller will adequately regulate the true system. Robust control is the process of designing a controller for a set of plants rather than a single model. This design technique requires the solution of two closely related problems: the robust analysis problem and the robust synthesis problem.

The problem of testing whether or not a closed-loop system remains stable when subject to perturbation, called the robust analysis problem, has been studied since the earliest days of feedback control theory. Indeed, much of the pioneering work of this field, such as that done by Nyquist (1932) on feedback amplifiers, considered system stability subject to parameter variation. The classical stability margins-the gain margin and the phase margin-are consequences of this concern about stability robustness of closed-loop systems. However, the only available analytic tool for testing the stability of a characteristic polynomial was the Routh-Hurwitz criterion. The stability conditions derived from the criterion yielded quite unwieldy algebraic inequalities, even for a small number of


I






2

uncertain parameters, so it appeared the task of deriving succinct robust stability conditions might be insurmountable. Less attention was given to this problem as the field of optimal control matured. The linear quadratic regulator (LQR) design developed in the early 1960s seemed to be capable of producing controllers that "guaranteed" good stability properties. However, Doyle (1978) demonstrated that the stability "guarantee" disappears when the LQR is implemented as output feedback. This result emphasized not only the importance of solving the robust analysis problem, but also of solving the robust synthesis problem, that is, actually designing a controller that could guarantee stability for a family of plants.

The stability analysis of systems subject to real parametric uncertainty began to receive renewed attention with the result of Kharitonov (1979) on the stability of interval polynomials. This important result states that the stability of a general characteristic polynomial with coefficients that vary independently within real intervals can be checked by testing the stability of only four specific polynomials, regardless of the original polynomial order. Suddenly, it appeared that analytic solutions might be available for many types of real parametric uncertainty. Interval systems received the lion's share of the resulting interest. Several strong results, such as the sixteen plant theorem of Barmish et al. (1992), and the generalized Kharitonov theorem due to Chapellat and Bhattacharayya (1989) followed, but these restrict the allowable controller structure, and represent only analysis results. A more general exact method for analysis of real parametric uncertainty is given by deGaston and Safonov (1988), however, this approach requires a large numerical effort for computation of the stability margin. It is noted that all of these methods assume that the allowable parameter variations are independent.

For systems whose parameters are interdependent, fewer results are available for the computation of the stability margin. Soh et al. (1985) present a method for calculating the largest stability hypersphere directly in the coefficient space. This technique is applicable to continuous-time or discrete-time characteristic polynomials. Biernacki et al. (1987) extend this result to computing the stability hypersphere directly in the plant parameter space, for






3

multiple input or multiple output systems. Also of particular significance is the approach of Tsypkin and Polyak (1991) who provide an elegant frequency domain method for testing the robust stability of a continuous-time characteristic polynomial whose coefficients could vary within an f -ball. Similarly, Guzzella et al. (1991) develop a method for analyzing the robust stability of systems subject to ellipsoidal (weighted f2) uncertainty.

The ellipsoidal uncertainty description is a specific class of parametric uncertainty that assumes the nominal transfer function model is of fixed order but has real coefficients that lie inside an ellipsoid in the coefficient space. This uncertainty description is ideally suited to the case where parameter estimation techniques are used to determine the coefficients of a nominal plant transfer function and also a parameter covariance matrix. The matrix can be used to construct an ellipsoid that describes, in statistical terms, the expected values of the plant transfer function coefficients. The works of Fogel and Huang (1982), Belfonte and Bona (1985), and Belfonte et al. (1990) consider algorithms where parameter estimates are constrained to lie inside ellipsoidal domains. The direct incorporation of the identification information into the uncertainty description is highly desirable. The focus of this dissertation is the robust control of systems subject to ellipsoidal uncertainty.

The robust synthesis problem was first studied in detail for the case of unstructured uncertainty. Here, the true system is modeled as a nominal transfer function matrix subject to additive or multiplicative perturbation. The only information known about the perturbation is a norm bound. In this case, the robust analysis problem is straightforward to solve. For unstructured uncertainties the powerful results of H. theory (Doyle et al. 1989) were developed to solve the robust synthesis problem, for both the single-input, single-output (SISO) and multi-input, multi-output (MIMO) cases.

In most situations, more information than simply a norm bound is known about the uncertainty affecting the system; this is referred to as structured uncertainty. Ideally, any available information about the uncertainty should be incorporated into the control design to reduce conservatism. This added information often makes both the robust analysis and






4

synthesis problems difficult to solve explicitly. Perhaps the strongest and most complete result is the structured singular value (g) analysis method introduced by Doyle (1982), and the almost identical multivariable stability margin (kM) method introduced by Safonov (1982). For any linear plant subject to uncertainty, the p-analysis method involves constructing a fictitious nominal system transfer matrix and a block diagonal matrix whose blocks correspond to the actual uncertainty affecting the system. Then, the value of t corresponds to the smallest uncertainty that will destabilize the system. This value is also referred to as the stability margin of the system. Unfortunately, the exact computation of t is NP hard in general (Braatz et al. 1994), and usually a convex upper bound is computed instead. The p synthesis procedure uses the upper bound to compute a controller that is a local minimizer of the structured singular value. Since the optimization involved is not convex in both of the variables involved, a local optimum is the best that can be guaranteed. The presence of real scalar uncertainty blocks complicates both the robust analysis and robust synthesis problems. A convex upper bound is available for the computation of R with real parameter uncertainty (Young 1994), and a controller design procedure employing this upper bound has been proposed (Young 1996), but it will also, in general, find a local optimum. The generalized structured singular value (Chen et al. 1994a, Chen et al. 1994b) is a metric similar to t but allows for different norms to be used on the real and complex blocks of the uncertainty matrix, thereby allowing an analysis of more general uncertainty descriptions. Braatz and Crisalle (1997) have extended the generalized structured singular value to include ellipsoidal uncertainty descriptions. However, there are no controller design methodologies based on the generalized structured singular value.




1.2 Objectives


The first goal of this dissertation is the development of a new method for testing the robust stability of ellipsoidally uncertain systems. A method for the computation of the






5

robust stability margin for ellipsoidal systems is developed that employs a bisection search instead of a frequency search. Two spectral factorizations are performed to construct a stable, real-rational transfer function whose magnitude corresponds to the system's stability margin. The stability margin is found as the H. norm of this transfer function, allowing a bisection search to be used to compute the stability margin. The method is also applicable to both continuous time and discrete time systems.

The second goal is to develop a robust controller synthesis method for ellipsoidal systems. A control design method is presented that is applicable to any ellipsoidally uncertain system with a stabilizing nominal controller. The design of a robust controller based on a nominal predictive controller is considered specifically in this dissertation. Predictive control is a class of control designs that uses knowledge of the future set point values to explicitly predict the future plant output, and compute a control law that will drive the output as close as possible to the desired set point. Predictive control is wellestablished in process industries (Seborg 1994) mainly due to its flexibility and (relative) simplicity. However, few results are available for the robustness analysis of predictive controllers, especially for the case of real parametric uncertainty. Therefore, the design procedure presented here addresses a relevant topic in robust control.





1.3 Structure of the Dissertation


The dissertation is organized as follows. In Chapter 2 the problem of robust analysis of ellipsoidally uncertain systems is introduced. A necessary and sufficient condition for robust stability of continuous time systems subject to ellipsoidal uncertainty is derived, using frequency domain analysis. The condition is similar to that proposed for discrete systems by Guzzella et al. (1991). The problems associated with this method are discussed, and a design example is presented to illustrate the possibility of controller design based on the proposed robust stability condition.






6

Chapter 3 presents a new method for stability analysis of ellipsoidal systems. The method consists of constructing a stable transfer function whose frequency response magnitude is equivalent to the stability margin of the uncertain system. This transfer function is constructed from knowledge of the nominal plant, a nominally stabilizing controller, and the matrix describing the uncertainty ellipsoid. Two spectral factorizations are performed to determine the numerator and denominator polynomials of the transfer function. The Nyquist robust stability margin can then be found as the H. norm of the transfer function. The method is applicable to both discrete and continuous systems.

In Chapter 4, the construction of a controller robust with respect to ellipsoidal uncertainty is developed. The underlying method is due to Rantzer and Megretski (1994), and is applicable to continuous or discrete systems. In this chapter, the discrete case is considered, and further, the nominal controller for the system is assumed to be a predictive controller. Robust controllers that retain the nominal performance of the predictive controller are derived. The proposed method is extended so that the resulting robust controller exhibits integral action, assuring offset-free rejection of asymptotically constant disturbances.

Chapter 5 considers a limiting-case robust control synthesis problem for ellipsoidal systems. The plant is assumed to be first-order and the controller considered is a static gain. This case is of interest because the analysis of Chapter 2 is not strictly applicable and also because it is possible to directly synthesize a maximally robust controller for the system.

Chapter 6 discusses the future directions of this dissertation. Several possible extensions of this work are described, along with a discussion of their importance, and possible methods for solution of these problems are given.















CHAPTER 2
STABILITY ANALYSIS FOR ELLIPSOIDAL SYSTEMS


2.1 Introduction


This chapter discusses methods for stability analysis of ellipsoidally uncertain systems. For a continuous time system subject to ellipsoidal uncertainty, a necessary and sufficient condition for robust stability is derived. For discrete time systems, the necessary and sufficient condition of Guzzella et al. (1991), is discussed for comparison. In both the discrete and continuous time cases, the stability condition is that the magnitude of a realrational function of frequency be less than one. The convex nature of the uncertainty regions in the frequency domain allows an analytic form for this function to be derived. This function can have multiple local maxima and minima; a frequency sweep is required to find the global maxima.

A necessary and sufficient condition for the robust stability of a continuous time system subject to ellipsoidal uncertainty is given in Section 2.2. The results of this section are continuous time counterparts of the results given in Guzzella et al. (1991), although an independent derivation is provided. Section 2.3 contains a brief review of the discrete time results of Guzzella et al. (1991), highlighting the differences between the continuous time and discrete time cases. Section 2.4 presents an example that uses the stability margin defined in the previous sections to compute a robust controller for a water heating system, and the performance of this controller is compared to controllers designed using a nominal performance measure.


7






8


2.2 Stability Analysis - Continuous Time A derivation of the robust stability analysis problem for ellipsoidally uncertain continuous time systems is given in this section. In subsection 1, the general ellipsoidal uncertainty structure is introduced. The closed-loop frequency domain uncertainty regions that arise due to the ellipsoidal uncertainty in the plant are discussed in subsection 2, along with the derivation of a graphical stability test for ellipsoidally uncertain systems. Subsection 3 contains the necessary and sufficient conditions for robust stability of the continuous time ellipsoidal uncertainty problem. Subsection 4 discusses the relationship between the parametric stability margin and the Nyquist robust stability margin.


2.2.1 Ellipsoidal Uncertainty


The general plant considered in this analysis is the linear, strictly proper plant

P(s) = b,_1s~I +...+ b _ B(s) (2.1)
s" +an-is +...+ao A(s)
which is represented by the coefficient vector b[a,, ... ao bli (2.2)

The values of the coefficients in this vector are uncertain, however, a nominal value of the vector is assumed known. The actual value of the parameter vector is represented by the nominal value plus an additive perturbation p = p0 + 8p (2.3)

This perturbation vector is constrained to lie in an ellipsoid 'E in the parameter space. This ellipsoid is defined by a positive definite, symmetric matrix Q, such that Ep = {8P e 92n 5pTQ-~PSp6 1} (2.4)

It is noted that the numerator and denominator polynomials of (2.1) can be expressed as a nominal polynomial and a perturbation polynomial as follows B0(s)+ AB(s) (2.5)
A0(s)+ AA(s)
where






9


AA(s)=8a,_is" +...+6aO
and

AB(s)= 8bjs"~ +...+ 8bo

The coefficients of these perturbation polynomials are captured in the vector 8p 8p = [8a,- ... 8aO b1-1 ... 8bo e "2n (2.6)

An example of an ellipsoidal uncertainty region for a two parameter system is shown in Figure 2.1.


ao

aoO - - - Cs


p0



Figure 2.1. Ellipsoidal uncertainty region Figure 2.2. General feedback interconnection


Ellipsoidal uncertainty descriptions arise naturally in parameter identification techniques where the uncertainty ellipsoid is associated with the parameter-error covariance matrix. The covariance matrix can be used to define a matrix Qp that corresponds to a certain confidence level for the parameters. Ellipsoidal uncertainty descriptions are commonly encountered in chemical engineering applications where model parameters are found by fitting experimental data using linear or nonlinear regression techniques. Such uncertainty descriptions have been adopted in several studies, including Biernacki et al. (1987) where ellipsoidal domains are used for the analysis of systems characterized with weighted perturbation bounds.

Ellipsoidal parametric uncertainty models also appear in various other contexts. For example, the work of Agarwal and Bonvin (1989) on Kalman filtering provides a means for estimating the covariance matrix. The publications by Fogel and Huang (1982),






10


Belfonte and Bona (1985), and Belfonte et al. (1990) discuss ellipsoidal outer-bounding algorithms which produce parameter estimates where the uncertainty is also constrained to lie inside ellipsoidal domains of the form (2.4).


2.2.2 Closed Loop Analysis

Consider the problem of analyzing the robust stability of a feedback loop containing a fixed controller and a plant subject to ellipsoidal uncertainty. Figure 2.2 shows the general feedback structure adopted, where C(s) is a proper controller given by C(s)= M ".n.+ O P(s) (2.7)
s" + (Xm-1s"-l +... + (Xo ((s)
and is represented by the coefficient vector c c =[an-I ... oo -..- e 912m+1 (2.8)

The plant P(s) is given by (2.1) and subject to the ellipsoidal uncertainty described by equations (2.3) and (2.4). It is assumed that the plant order is at least as large as the controller order, i.e., n > m. Furthermore, it is also assumed that the index k := n+m satisfies k 2. The case of k = 1, corresponding to a first order plant (n = 1) and a constant controller (m = 0), is a limiting case that is discussed in detail in Chapter 5. The characteristic polynomial of the feedback loop G(s)= G (s)+ AG(s) (2.9)
is the sum of nominal and perturbation polynomials, where
0 0 1()0(s k 0 k-1 0
G (s)= c(s)A0(s) + g(s)B Sk k- +... + go (2.10)
is the monic nominal characteristic polynomial and the perturbation polynomial is
AG(s) = x(s)AA(s) + P(s)AB(s) gk-Is -1+... + (2.11)

The coefficients of the characteristic polynomial define the characteristic vector g=[gk-1 ... go]T E 9k (2.12)

It is straightforward to show that
g = S'p+ c

where Sc is the Sylvester matrix for the controller C(s), and has the form






11


1 0 ... 0 P111 0 ... 0 0r-1 1 0 ni-I O -n - 0

SC= O 1 Po PiEl kx2n
0 -n.-1 0 - ni-1


0 Xo 0 Po
and
Z =[a -1 ... (Yo 0 ... O]T 91k The nominal value of the characteristic vector is generated when the plant parameter vector assumes its nominal value, i.e., g0 =Sp0 +eZ= gk1 gk.-2 go (2.13)

The difference between the actual and nominal values of the characteristic vector is given by the perturbation term

8g:= g - g0 = S p+ Z-S p0 - S = (P - 0)Sc8p (2.14)

The elements of 8g are the coefficients of the polynomial AG(s) given in equation (2.11). It has been shown in Guzzella et al. (1991) that if 8p is constrained to lie in the set Ep, then 5g lies in the ellipsoidal uncertainty region Eg = 15g e 12n 8gT Q-Ig 11 (2.15)
where
Qg =S'QPS T E kxk (2.16)

is a positive definite, symmetric matrix.

Throughout the remainder of this chapter it is assumed that the controller has been chosen so that it stabilizes the nominal system. This implies that G (s) corresponds to a Hurwitz polynomial, i.e., all of the zeros of GO(s) have negative real part. Therefore, the problem of testing if a given controller robustly stabilizes the system becomes the problem of testing whether any allowable AG(s) produces a non-Hurwitz G(s). This problem is analyzed in the frequency domain.






12


The image of the nominal polynomial Go(s) as s varies along the imaginary axis is

G(jo'))=(g90 -g 2+...) + Agfo - go +..1: s(0) + j'1'((0) (2.17) Defining the vector

[(FO):= r 1
_(o = ;( )

it follows that

'(O)= WT (O)g + t(c) where
2 0 1 [ T 1() W -. 0 li0) w R(E) 9j2xk (2.18)
0 0 o 0 w (o) The vector t(o) is given by t(o) = [(-1)k /2 e 2 if k is even

and
t(o) = [0 (-)(k-I)/2 k T 2 if k is odd

The image of the polynomial AG(s) as s varies along the imaginary axis is given by

AG(jo) = (8g0 - 82W +. ..)+ j(6g1o - 8g3 +...):- 6TR(O)+ j6TI(o) (2.19) and can also be expressed as the vector


TR(0)) WT(o)8g Guzzella et al. (1991) show that as 8p takes on all possible values inside the parameter ellipsoid 'Ep, the region that AG(jo) traces out, for (o # 0, is an ellipse 'EO in the complex plane. This ellipse is described as Ew = {,(o) E 2 TT(CO)Ql (2.20)
where the matrix

QO = WT (W)QgW(W) e y2x2 (2.21)

has full rank for all nonzero frequencies. At (o = 0, the nominal polynomial is simply G0(j0)= go (2.22)






13


and the uncertainty region in the complex plane degenerates into a line segment defined by


'E={6g 91 6gl 4 } (2.23)

where qk,k is the (k,k) element of the matrix Qg defined in (2.16) and k = n + m. Uncertainty regions for a non-zero frequency o and for co = 0 are shown in Figure 2.3.

The frequency domain uncertainty set is called a value set, or template, and plays a major role in the stability analysis of ellipsoidal systems. This analysis, as well as the classical Nyquist stability criterion, is based on a result from complex variable theory called the principle of the argument.

The principle of the argument states that as a complex-valued function F(s) is evaluated along a closed, simple curve in the s-plane, along which F(s) has no zeros or poles, and interior to which F(s) is analytic, then the net change in angle is A arg{F(s)} = 21t(n P - nZ)

where nz and nP are the number of zeros and poles, respectively, of F(s) inside the contour.

The principle of the argument can be used to test the stability of the closed loop of Figure 2.2 by applying the principle to the characteristic polynomial G(s). If the stability region is taken to be the open left half of the complex plane, and the contour is the classical Nyquist contour, traversed in the clockwise sense, then the polynomial G(s) has no zeros in the right half plane if and only if

A arg{G(s)} = 0

since a polynomial has no finite poles. The above angle change is the net change as the Nyquist contour is traversed, and is the sum due to the semi-circular part and the imaginary axis part. The change in angle due to the semicircular portion of the contour is -kn. This follows from the fact that on this part of the contour s = reji where r -4 oc and O varies from -1 to - Thus
G(s) -> rkeIke






14


and the net change in angle of G(s) is -k. Therefore, it follows that the change in angle of G(jo) as o varies from -. to 0 must be kit for the closed loop to be stable. Since the coefficients of G(s) are real, the plot of G(jo) on (-o,0] is symmetric to the plot on [0,--), so the change in arg{G(jo)} on [0,o) must be - . This means that the plot of G(jo) must encircle the origin k times in the counterclockwise direction. This important result is
4
summarized in Lemma 2.1

Lemma 2.1 A polynomial G(s) of degree k has all of its roots in the open left half
k
plane if and only if the plot of G(jo) encircles the origin of the complex plane - times in
4
the counterclockwise direction as co varies from 0 to c.

A very comprehensive discussion of the applications of the principle of the argument to stability testing, including a proof of the Routh-Hurwitz criteria, is given in Porter (1968).


2.2.3 Robust Stability Analysis

A robust stability test can be constructed from the result of Lemma 2.1. The nominal characteristic polynomial G 0(s) is assumed to be stable, and therefore the plot of G 0 jo0) has the correct number of encirclements for stability, as described by Lemma 2.1. Figure 2.3 shows the uncertainty ellipses E at several frequencies. The band that is swept out by these ellipses is called a Nyquist envelope. Each point in a particular ellipse represents an allowable characteristic polynomial frequency response evaluated at that particular frequency, so the envelope represents the frequency response plots G(jo) for all allowable characteristic polynomials. If the envelope does not include the origin, then all the frequency response plots have the same number of encirclements of the origin as the nominal polynomial, and thus all allowable characteristic polynomials are stable. However, if the Nyquist envelope contains or touches the origin, then at least one allowable characteristic polynomial has a different number of encirclements of the origin






15


than the nominal polynomial and is unstable. Therefore, the system is robustly stable if and only if the value sets exclude the origin at all frequencies.

The need for the value sets to exclude zero can also be explained in terms of the root locations of the polynomial G(s). The nominal polynomial G0(s) has all its roots in the left half-plane (LHP). As the parameter vector p assumes all possible values in 'E, the roots of G(s) move. In order for the roots to travel into the right half-plane (RHP), they must cross the imaginary axis. The possibility of the roots moving due to degree dropping has been ruled out by specifying that G(s) is monic. Continuity arguments can be used to establish that if any G(s) has roots in the RHP, then an allowable G(s) exists that has at least one root on the imaginary axis. However, this implies that G(jco) = 0 for an allowable G(s) which in turn means that the origin must be included in the value set at that frequency. Therefore, all allowable characteristic polynomials have roots in the LHP if and only if the origin is excluded from every value set.

Testing this zero exclusion condition can be accomplished using the critical direction method (Latchman et al. 1997), modified here for the analysis of polynomials. Consider the following quantities:

i. The stability segment is the line segment joining the nominal curve G 0 (jo) and the origin. Note that the length of the stability segment is simply Go (O) ii. The critical direction is defined as the direction of the stability segment, and is characterized by the unit vector

d(jo) - o)
G (jo)

iii. The critical perturbation radius

Pc (o)= maxIa e 9 1 Go(jo)+ ad(jo) e-E; These quantities are illustrated in Figure 2.3. It is noted that the only part of the value set important for robust stability is that part that lies along the direction to the critical point. This is the fundamental premise of the critical direction method.






16


Im{ G(jw)}
E*)


stability Pc0w)
segment



ICP~jo))lRe(Gojo)})




Figure 2.3. Stability analysis quantities As stated before, the closed loop is unstable if any value set includes the origin. The value sets do not include the origin provided the critical perturbation radius is smaller than the length of the stability segment at every frequency. Therefore, a necessary and sufficient condition for robust stability is pC(o) < Go (jO) Vo E [0,oo) (2.24)

Equation (2.24) is particularly useful when an analytic form for pc(o) is available. As Lemma 2.2 shows, the ellipsoidal nature of the value sets allows such a form to be derived.

Lemma 2.2 Suppose the nominal characteristic polynomial G0 (jo) given by equation (2.17) is subject to ellipsoidal uncertainty described by equations (2.20) and (2.23). Then the critical perturbation radius is P(O)= (o)112 0 > 0 (2.25a)
T 1 (0) Q = 0 (225b

PC (O) = qk,k CO = 0 (2.25b)


Proof. The proof of Lemma 2.2 is given in the Appendix.






17


By definition of r((o), and from equation (2.22), it follows that Go (jco) =l|1(W)1 0 > 0 (2.26a)


G (JO) 0= 0 0 = 0 (2.26b)

The results of Lemma 2.2 and equation (2.26) can be used to write the necessary and sufficient conditions for robust stability given in (2.24) as I'CO12 < T(O) 2 > 0



qkk < go = 0

or they can be cast as

I rW < I > 0
TT(0))Q _(W


< 1 w =0


The Nyquist robust stability margin kN introduced in Latchman et al. (1997) for rational transfer matrices can be reformulated for the case of polynomial systems. Define the frequency-dependent Nyquist robust stability margin as


TT -r)l (
kNq(0)= (2.27)
01=


and define the Nyquist robust stability margin as kN = sup kN(o) (2.28)
(00



Theorem 2.1 Suppose the ellipsoidally uncertain plant P(s) described by (2.1)-(2.4) and the controller C(s) described by (2.7) are joined in unity negative feedback as shown in






18


Figure 2.2, with the nominal system stable. Then the system is robustly stable if and only if kN < 1.

Proof: By definition of both Nyquist robust stability margins and Lemma 2.2, it is immediately apparent that the necessary and sufficient condition given in (2.24) is satisfied if and only if kN < 1. V

Although equation (2.27) gives a closed-form expression that can be used to test robust stability, the frequency-dependent Nyquist robust stability margin is a rational function of frequency that may have local minima. This implies that the result of Theorem 2.1 must be tested by performing a frequency search to find the value of the Nyquist robust stability margin. While each point of the search requires little computational effort, it is more difficult to determine the range of frequencies on which to test robust stability. The results of Chapter 3 provide a method for testing robust stability that does not require a frequency sweep, and allows the value of kN to be calculated through a bisection search.


2.2.4 Computation of the Parametric Robust Stability Margin


The parametric stability margin is defined as the minimum expansion (or contraction) of the uncertainty region in the parameter space required to bring the system to the edge of stability. For ellipsoidal uncertainty, the parameter space uncertainty region is contracted or expanded by multiplying the matrix Qp by a scalar factor c. From equation (2.16), it follows that multiplying the matrix Qp by a scalar ox results in Qg changing to cQg. The new uncertainty region in the characteristic space is thus cuEg 8g y2n 8gT (Qg)~Ig

Using this definition of a scaled uncertainty set, the parametric stability margin for ellipsoidally uncertain systems can be expressed as cc =min c(co)
W)






19


where

cx(o)=minfc I G(jo,8g)=O 6gecvEg}
aE91

The relationship between the parametric stability margin and the frequency-dependent Nyquist robust stability margin is detailed in the following lemma.

Lemma 2.3 Suppose the ellipsoidally uncertain plant P(s) described by (2.1)-(2.4) and the controller C(s) described by (2.7) are joined in unity negative feedback as shown in Figure 2.2, with the nominal system stable. Then the parametric stability margin is
* N 2
cx = kb2

Proof. Let the frequency that defines kN be denoted *. Then kN = kN


Multiplying the matrix Qp by a scalar results in Qg changing to cxQg and also Qw changing to aQw. Thus, the system with the scaled uncertainty satisfies

-T9 -- IT~ -- V
kN =kN()= = cXkN
F'ET(TT((O*)QWOLO)W)

Since the factor (x scales all the values of the frequency-dependent Nyquist robust stability margin, the maximum for the scaled system will occur at the same frequency as that for the original system. The parametric stability margin is defined as the minimum value of (x that destabilizes the system, which corresponds to kN = 1. Therefore, kN = 1 = [W*N or

*kN 2 v


2.3 Stability Analysis - Discrete Time


The stability analysis of discrete time ellipsoidally uncertain systems is very similar to that for continuous time systems. In this section, the few differences between the two cases are highlighted, as the complete derivation can be found in Guzzella et al. (1991).






20


In the discrete time formulation of the ellipsoidal stability analysis problem, the general linear plant considered is given by P(Z) b,11zn- +...+b0 _ B(z) (2.29)
Zt?+ an-1z n-1 +... +ao A(z) and the controller is

C(z)= 'niz" +... + ._ P(z) (2.30)
z"M+ anl1z n-I +... + UX0 c(z) The plant parameter vector p is exactly as defined in (2.2) and the uncertainty description is still given by (2.3) and (2.4). Also, the controller vector c is given by (2.8). The characteristic polynomial for the discrete time system is described by equations (2.9)(2.11) with the Laplace variable s replaced by the discrete time variable z. The characteristic polynomial degree remains k:= n + m. Furthermore, the uncertainty region Eg is given by equation (2.15) and the uncertainty matrix Qg is given by (2.16).

The primary difference between the two cases is the frequency domain analysis of the characteristic polynomial. The image of the nominal characteristic polynomial G (z) as z varies along the unit circle is given by

GO(ejw) = (go +... + cos[(k - 1)w]g_1 + cos[ko])


+j(sin[o]go +... + sin[(k - l)o]g_1 + sin[ko]) (2.31) and the image of the perturbation polynomial is AG(e3") = (8go +... + cos[(k - 1)o]gk-1) + j(sin[o]gk +... + sin[(k - 1)o]gk-1) (2.32)

These polynomials can be expressed as the vectors Re{G0(eJ*" -l t~
Re = G~j)j. :=R = W Tg +
Im {GO(eO" _ wr1

and
Re [ AG(ej)I._ [ I = WTg Im IAG(ej")1 _- s'r,






21


where
t = [cos(ko) sin(ko)]T e 32 and
WT:y [cos[(k - 1)o] - cos[o] 1 [w(o) 2xk
_sin[(k - l)0] . sin[o] oj [w (o)As 8p takes on all values inside Ep, the uncertainty region AG(e1") that is generated around GO(O) at every frequency in the open interval (0,7r) is an ellipse ' = 1{& e 92 TQ-I6T[5l} (2.33)

where
F A~o)B~co)iF W~())QgW (O) ~C)QgWj(O))] [A(o) B(o)) ~ _w (O)Q gW R(O) ( )T (2.34)
B((o) D((o) w ((O)Qg((O) W, (0)Qg(O)1 The matrix QO is rank deficient at the two frequencies 0 and iT. At these points, the uncertainty regions are line segments

EI = 6TR(t) E= 16TR(0)1 WI(O)QgWR(O)} (2.35)


= 0tE) e 91 16TR(70) WR(jT)QgWRWt) (2.36)

The plot of all possible characteristic polynomials G(eJW) for o e [ir,2n] is symmetric about the real axis to the plot for o e [0,7c], thus only the range o e [0,7t] need be considered.

The principle of the argument is used to deduce the root locations of the polynomial G (z) by analyzing the number of encirclements of the origin of the plot of G0 (do) as (o varies between 0 and n. The only difference from the continuous time case is that now the stability region is the interior of the unit circle IzI = 1. The counterpart of Lemma 2.1 for discrete time systems is stated below.






22


Lemma 2.4 [Guzzella et al. 1991] A polynomial G(z) of order k has all its roots inside the unit circle if and only if the number of counterclockwise encirclements of the origin performed by the plot of G(elW) as o varies from 0 to 27r is equal to k. The nominal characteristic polynomial is assumed to be a Schur polynomial, i.e., it has all its roots in the unit disc lzl

Im

GO(e(*) GO'nc






P Re

E(T) 'E40)

Figure 2.4. Nominal curve GO(ei(), value sets 'Em at frequencies o = 0, o*, it, the stability segment for o*, and the critical perturbation radius pc(o*)






23


The robust stability condition for the discrete time case is based on the knowledge of the explicit form for the critical perturbation radius, which is given in Lemma 2.5

Lemma 2.5 Suppose the nominal characteristic polynomial GO(e o) given by equation (2.31) is subject to ellipsoidal uncertainty described by equations (2.33), (2.35) and (2.36). Then the critical perturbation radius is


petIo)=)2L
TT (o)Q T(O) p'(O) = W (O)QgWR(O) P(7) = W (1r)QgWR('r)


Proof: The proof of Lemma 2.5 is entirely analogous is therefore omitted.


o e (0,R) (2.37a) 0 = 0 (2.37b)


(2.37c)


to the proof of Lemma 2.2 and


The value sets 'E do not include the origin provided the critical perturbation radius is smaller than the length of the stability segment at every frequency. Therefore, a necessary and sufficient condition for robust stability is pe(o) < GO(eI) e [0, n] (2.38)

Lemma 2.6 Suppose the conditions of Lemma 2.5 hold. Then the condition (2.38) is equivalent to the three conditions


1 <1

wT (0)Qw()
WR(O)QgWR(O) <1
t'R(O)J
T T

WR(t)QgWRW< 1
ItR(lr)J


0e (O,2T) (2.39a) O = 0 (2.39b)



(0 = 7t (2.39c)






24


Proof: The form of p,(co) is given in Lemma 2.5. For o e (0,7E), GO(e'W) = N1r02 by definition. For the extreme points o = 0 and o = xT, tr(a) = 0, and GO(ej) = ITR(O)I. Plugging these values into (2.38) yields (2.39) immediately. V

Defining the frequency-dependent Nyquist robust stability margin as

o e (0,i7)

w (o)Qjw (0)
k(0 WTR(0)QgWR(0) C) 0 (2.40)
1tR(0)l

W (7)QgWR(7r)
ITR(7t) ()7

and the Nyquist robust stability margin as kN = sup kN(0)
We[O,it]

then the necessary and sufficient condition for robust stability of ellipsoidally uncertain systems is given in Theorem 2.2.

Theorem 2.2 Suppose the ellipsoidally uncertain plant P(z) described by (2.29), (2.2)-(2.4) and the controller C(z) described by (2.30), are joined in unity negative feedback, with the nominal system stable. Then the system is robustly stable if and only if kN < 1 (2.41)


Proof. Lemma 2.6 shows that the satisfaction of the three conditions (2.39) is the necessary and sufficient condition for robust stability. However, the quantities on the left hand side of (2.39) are identical to those on the right hand side of (2.40). It follows immediately that the necessary and sufficient condition for robust stability is that kN(co) < 1 for o E [O,], or kN < 1, which is equation (2.41). V






25


2.4 Design Example for Ellipsoidal Systems


In this section, a simple water-heating control system with a discrete PI controller is used for illustrating the analysis and synthesis methods introduced in this chapter. The nominal plant parameters are used to tune three candidate PI controllers using standard techniques, and then the robustness of each controller is analyzed. It is verified that controllers cannot be robustified from knowledge of the nominal process alone. Finally, a robust control design based on the results of Section 3 is realized via numerical optimization.


2.4.1 Plant Model and Ellipsoidal Uncertainty


The two-tank system with recycle shown in Figure 2.5 is used as a model for illustrating the robust analysis and synthesis techniques. The control objective is to maintain the temperature of the second tank (T2) at a desired set point by manipulating the power (R) delivered by the heater located in the first tank. The actuation on the heater is performed through a zero order sample-and-hold element. The only available measurement is temperature T2.

F1 F2
T. F1 = 0.050 m3/min R T F1+F2 F2 = 0.150 m3/min
p = 1000 kg/m3
Cp= 4286 J/kg-C
T2 FI+F2 F1 V =1 m3

Figure 2.5. Two mixing tanks arranged in cascade with recycle stream.


The actual plant model is generated by carrying out an energy balance in each tank. Assuming that the liquid volume V remains constant and equal in both tanks one arrives at the representation
T2(s) 2 k, (2.42)
R(s) T22s +2Ts+k2






26

where
k1 0.07
(F1 + F2)pCp kW


k2= F1 -0.25
F, + F2
V
- = V= 5 min
F + F2

The actuation on R is carried out by means of a sample-and-hold element with sampling period T= 100 sec. The corresponding discrete transfer function representation, the actual plant, is then
T2(z) _ 0.0031z + 0.0025 (2.43)
R(z) z2 - 1.4932z + 0.5134 The identification of the nominal plant model is carried out using a standard leastsquares parameter identification method. The input-output data needed for identifying the nominal plant G(z; pO) and the uncertainty model Ep is generated by applying a pseudorandom binary input to the heater and recording the temperature response.

In the simulation it is assumed that the temperature data are acquired by a thermocouple which perturbs the measurements with an additive white Gaussian noise with variance (y2=4.5 10-3 C2. A total of 125 samples are gathered using a sampling period T=100 sec. The collected data is then regressed using the ARX model


T2(Z)= 2 b1z + b0 R(z) + e(Z) z + a1z + a0

where e(z) is assumed to be a white-noise sequence of unknown variance. Hence, the plant model and parameter vector for this representation are, respectively, b1z + o
P(z;p)= Z2 + a1z + ao , n = 2 (2.44)

and
p=[a, ao b, bo]T

The least-squares procedure minimizes the functional J(p) = (y-Hp)T(y-Hp), where y is a vector of temperature measurements and H is the regression matrix (Draper and






27


Smith 1981). The minimum is realized by the solution vector po=(HTH)-IHTy. For the two-tank plant under study the solution is pO =[-1.1871 0.2087 0.0028 0.0038]T

yielding the identified nominal plant model o 0.0028z + 0.0038 (2.45)
Z - 1.1871z + 0.2087
An ellipsoidal uncertainty model is readily available as a by-product of the leastsquares identification technique. A standard result (Draper and Smith 1981; Crisalle and Bonvin 1991) states that a 100(1-a)% joint confidence region for the parameter estimate po is given by the ellipsoidal domain given in (2.4) Ep = {6p e 9y2n 8PT Q-6p l}
with
Qp = 2nFIs2 (2.46)

In the above equations, 2n is the dimension of parameter vector pO, k is the number of sample points used in the identification, F = F(2n, k-2n; (x) is a factor characterizing the 1-cx quantile of an F distribution of order (2n,k-2n), s2=(t-Hp)T(t-Hp)/(k-2n) is an estimate of the noise variance, and 1s2=(HTH)-1s2 is an estimate of the covariance matrix for the parameter error. The statistical interpretation of the parameter uncertainty region provides a sound theoretical justification for the ellipsoidal uncertainty description.

An ellipsoidal uncertainty model for the two-tank process is obtained by substituting n = 2, along with the result s2 = 1.14 10-2C2 and the numerical values for matrix , in (2.46) to get
1.26x10-2 -1.23x10 2 -1.00x10 5 3.51x10Q = F -1.23x10 2 1.27x10 2 8.51x10-6 -3.62x10-5 (2.47)
-1.00x10 5 8.51x10-6 4.25x10-' -3.41x10-8 3.51x10 -3.62x10 -3.41x10-8 5.17x10-7






28


Hence, using the above form for the matrix Qp in (2.4) represents an ellipsoidal uncertainty model which can be interpreted as a region with a confidence level of 100(1-X)% depending on the value of the F-factor chosen.

The fundamental uncertainty information is in fact contained in matrix I whose eigenvectors determine the principal directions of the ellipsoid 'Ep. The remaining scalar factors adjust the volume of the ellipsoid without perturbing the principal directions. Setting the F factor equal to 2.37 leads to a 95% confidence region; however, it is also possible to choose the value of the F factor to encompass a family of uncertainties of larger or smaller scope. We adopt the value F=10 for our design, thus requiring that the controller be able to stabilize a family of plants belonging to a relatively large ellipsoidal region.


2.4.2 Controller Design


The controller is a discrete proportional-integral (PI) compensator C(z;c) = pIZ + = i Do], i=

whose associated Sylvester matrix Sc and vector Z are

1 0 Pi 0~
SC = -1 1 00 P, and [-l 0 0]T
0 -1 0 POFor the nominal plant and uncertainty description just described, three PI controllers are synthesized using a conventional tuning method. The designs proposed, denoted Cl, C2, and C3, are characterized by the tuning parameters f31 and PO shown in Table 2.1. The control parameters for design CI are found using the estimated model (2.45) and the tuning settings 1i=Kc(1+0.5T/ti) and o0=-Kc, where Kc and Ti are the control gain and integralmode constants of a corresponding analog PI controller (Seborg et al. 1989). In order to find appropriate constants for the analog controller, the step-response of (2.42) is first






29


approximated by a first-order lag and a pure delay equal to one half of the sampling period. Parameters Kc and t1 are then determined from the correlations minimizing the ITAE (Seborg et al. 1989).

Designs C2 and C3 are intended refinements to CI based on the observation that the

parameter for the latter is near a nominal stability limit. In fact, when Po=-98, a nominally stable loop is obtained for 98
10
---- C(a)
C2
-... C3
5-------------- ----C*

0
0 0.2 0.4 0.6 0.8 1
Time [hours]

1 0 -(-b-)




0 -5
0 0.2 0.4 0.6 0.8 1
Time [hours]


Figure 2.6. Closed-loop responses of conventional controllers Cl, C2, and C3, and of the optimally robust controller C* to a 5 0C step change in setpoint. (a) Nominal response, (b) actual response.


The stability robustness of all three candidate controllers is then analyzed using the critical direction technique. Values for the robustness parameter kN are calculated by means






30


of an exhaustive numerical search using a frequency increment A(o=0.01. The result obtained is given in the last column of Table 2.1. It is concluded that the loop involving C 1 is robustly stable (kN<1), but that neither controller C2 or C3 is robustly stable. In particular, controller C3 produces an unstable loop when it is used to control the actual plant, as shown in Figure 2.6b. Remarkably, in this example the intuitive idea of displacing parameter P away from the stability boundary leads to the loss of robustness in C2 and C3 because the tuning adjustments disregard the parametric uncertainties present.

From Theorem 2.2 it follows that a controller may be considered optimally robust if it is capable of producing the smallest possible value of kN. Hence, the problem of robust synthesis reduces to the following optimization: c* = argmin kN(c) (2.48)
cE 0

where c* is the optimal controller, 0 represents the set of all controllers which lead to stable loops with the nominal plant, and kN(c) is the robustness parameter (2.41) for a given controller c.

The constrained optimization problem (2.48) is nonlinear and nondifferentiable. In this study the synthesis problems are solved by exhaustive numerical search over the control space. This approach is viable only because of the low dimensionality of the control space. For higher dimensions this approach would rapidly become computationally intractable due to the combinatorial explosion in the number of grid points. The exhaustive search is carried out by discretizing the space of control parameters with a grid of size Ajo=Aj1=0.5, and then calculating the values of kN(c) at every grid point. Grid points corresponding to controllers that are not nominally stable are discarded, thereby gaining execution speed.

The last row of Table 2.1 shows the numerically-determined optimal values for the control and robustness parameters for design C*. The optimal design is robust since it satisfies the constraint kN < 1, and it achieves the smallest value of kN (0.501). The






31


optimal design stabilizes the nominal and actual plants, as is confirmed from the stepresponses shown in Figures 2.6a and 2.6b.


Table 2.1. Control and robustness parameters for conventional controllers
Cl, C2, and C3, and optimally robust controller C*.


Controller Controller parameters Stability margin
Design P IP0 _kN

C1 101.0 -98.0 0.842

C2 115.0 -98.0 1.260

C3 127.0 -98.0 2.041

C* 31.0 -30.5 0.501



Figure 2.7 shows several contour plots of the functional kN(c) as a function of the controller parameters 0o and 1. The contour kN(c) = 1 defines the boundary of robust stability. The optimal design is remarkably close to the nominal stability boundary, making it very sensitive to perturbations in the control parameters. For this reason, it may not be desirable to design a controller to optimize the stability margin alone.




2.4.3 Discussion


The performance in servo-response tests is documented in Figures 2.6a and 2.6b for the various designs studied. Only the nominal and actual plants are considered. The figure shows that controller C* produces low overshoot levels and minor transient oscillations. However, caution must be exercised when attempting to generalize performance observations from this example because the optimally robust controller has been designed ignoring performance constraints.







32


It is important to remark that functional kN(c) is not convex with respect to the control vector c. Profiles calculated using very fine grids also reveal that the functionals are non smooth. As a consequence, gradient-based optimization techniques are not suitable substitutes for the exhaustive search method.


200 180 160 140 120 100 80 60 40 20

0


-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 PO


0


Figure 2.7. Contour plot on the (Po, P i) plane of the functional kN(c).





2.5 Conclusions



This chapter presents a necessary and sufficient condition for the robust stability of a continuous time system subject to ellipsoidal uncertainty. The condition is similar to that given in Guzzella et al. (1991), for discrete time systems. The test requires a frequency search to be performed. In order to avoid the problems associated with such a search, a new analysis method that does not require a frequency search is presented in Chapter 3.


- 3












.6
. . . . . . . . . . . . . . . . .

- . . . . . . . . . . . . . . . . . . . . . . . . . .















CHAPTER 3
STABILITY MARGIN CALCULATION FOR ELLIPSOIDAL SYSTEMS


3.1 Introduction


The goal of this chapter is the development of a robust stability test for ellipsoidally uncertain systems that does not require a frequency search. It is shown that the necessary and sufficient condition of Theorem 2.1 can be transformed into a condition of the form ||F(s)IL < 1

along with an auxiliary condition for (o = 0. The function F(s) is a stable, minimumphase, strictly proper transfer function. The infinity norm condition requires only the checking of the eigenvalues of a certain Hamiltonian matrix, and thus avoids a frequency sweep. Furthermore, computing the actual value of the norm can be done using a bisection search, thereby guaranteeing a prespecified level of accuracy. Section 3.2 describes the construction of the transfer function F(s). The new method of robust stability analysis using this transfer function is developed Section 3.3. The formulation of the appropriate transfer function F(z) for discrete systems is given in Section 3.4. An example is given in Section 3.5 to illustrate the proposed method.



3.2 Construction of F(s)


To test the necessary and sufficient condition for robust stability established in Theorem 2.1, the function kN(o) must be evaluated over the frequency range [0,oo). This frequency sweep is computationally expensive and is inherently imprecise since a finite number of frequency points must be used. An alternative is to construct a stable transfer


33






34


function whose frequency response is equal to the function kN (o) for positive frequencies. Then the maximum of kN (o) for positive frequencies can be found as the infinity norm of this transfer function. Actually, as a first step in this construction, the square of the function kN ((o) will be fitted to make the mathematical developments more tractable. From the resulting form, the appropriate F(s) can be easily constructed.

The function kN ((o) is defined in Chapter 2 as

> 0
j TT(O)Q~(O))(3.1) kN0qk
0 k. ( = 0


where

[Re IG 0U(O) (o2
R(e) := I 912 (3.2)


and

Q O [A(o)) B((o) W(O)QgWR(O) WR(W)QgWi(O) (3.3)
B(o) D(o)) IWT(O)QgWR(O) I(oQw;(O)

The vectors wR(O) and w(o) are defined in (2.18).

Define h(o) to be the square of the function that defines kN(co) for positive frequencies

h(o) = (3.4)

Even though Q-1 is not defined at o = 0, a limiting value of h(o) as (o -> 0 is well defined, but this limiting value is not equal to kN(O) in general. Substituting equations (3.2) and (3.3) into (3.4) yields

h(o) = A(oj)D(c)- B 2(tti (3.5)
A(O))T (0) + D(co)x 2 ~-2B(O)rR((O)T;((O) The objective is to find a stable transfer function F(s) that satisfies






35


IF(jo)I2 = h(o)) (3.6)

for non-zero frequencies. To facilitate construction of such an F(s), Lemma 3.1 states useful properties of the function h(o).

Lemma 3.1 The function h((o) is a finite, non-negative, even, rational function of (o, with numerator degree 4k-8 and denominator degree 4k-4.

Proof: The proof of Lemma 3.1 is given in the Appendix. V

Lemma 3.1 is used in the proof of Theorem 3.1, which describes a factorization of h(o) used to compute F(s).

Theorem 3.1 The function h(o) has a factorization of the form

h(o) = 2 (3.7)

where the polynomials
mX MY
X(s) = xrsr and Y(s) = XYrsr
r=0 r=0

have no zeros in the closed right half plane (RHP), have real coefficients, and mx = 2k-4 and my = 2k-2.

Proof: The proof is based on a result from spectral factorization, given by Rozanov (1967), which states that every non-negative, even, rational function of a scalar variable o has a factorization of the form (3.7) where X(s) and Y(s) have no zeros in the open RHP and have real coefficients. Lemma 3.1 shows that h(co) satisfies the conditions of Rozanov, and thus has such a factorization. The degree of the numerator of h(io) is 4k-8 and this is twice the degree of X(s). Similarly, the denominator degree of h(o) is 4k-4, and this is twice the degree of Y(s). Furthermore, since h(o) is finite and non-zero for all (o, it follows that X(s) and Y(s) have no zeros in the closed RHP. V


If the function F(s) is constructed as






36


F(s) = X(s) (3.8)
Y(s)
then

IF(jio) = X(jo)X(-jo) _ X(jO) 2 = h(o) (3.9)
Y(je)Y(-jo) Y(jw)

and (3.6) is satisfied. It remains to calculate the coefficients of the polynomials X(s) and Y(s) so that (3.7) is satisfied. The first step is to express h(O) as a ratio of polynomials in jo so that the coefficients of X(jo) and Y(jo), and thus of X(s) and Y(s), can be found.

Theorem 3.2 The function h(o) defined in (3.5) is equivalent to

h(Ao) = (jo)(3.10)
A(jo (0)+ D(jw)tR(j0) -2B(to)~tR(jong(j0) where the coefficients of the polynomials A(jo), B(jo), 6(jAO), tR(jo) and t(jo) depend only on the nominal coefficient vector gO and the matrix Qg. The polynomials ~R(j') and t;(j)are given by mR k k even
t(jo)- ~gi(jo)2e with m = 2 (3.lla)
O 2R (k - 1) k odd

and
k keven
~t(j) 0= g j2o-()(jo) with m, = {1k2 (3.1lb)
(k 1 k odd

0 0
where gf are the elements of the vector gO defined in (2.15), except for gk 1. The polynomials A(jo), B(jo), and D(jo) are given by

"r k-2 k even
A(j) = Xa2,(jO)2P with nA = jk kodd (3.12a)
/=0
and
n B
(jo) = 2-I)(jo)(21 with nB = k-I (3.12b)
f=1
and
nc k-1 k even
)(j(o) = Xd2t(jo)2f with nD =k2 kodd (3.12c)
e=1






37


Furthermore, the coefficients of the polynomials A(jo), q(k-2h),(k-2e+2h)


-mA)(k-2h),(k-2e+2h) h=("-MA)


b(2f-l)


B(jo), and 6(jco) are given by

O C! e:!mA (3.13a)
(MA + 1)5 e! nA


q(k -2h),(k-2t+2h+l)
m0
Y 9(k-2h),(k-2f+2h+1)
h=(P-ml)


(3. 13b)


q(k-2h+1),(k-2e+2h-1) 1D d2e = h (3.13c)
I (k-2h+l),(k-2P+2h-1) (MD + 1) 5 nD h=(P-MD+l)

respectively, where qij represents the (ij) element of the matrix Qg, and the indices mA, mD, m, and m2 are given by mA = nA

mD = 2(nD + 1) k k even { (k -1) k odd and

-(k - 2) k even M2 2
{ (k-1) kodd


Proof The proof of Theorem 3.2 is given in the Appendix.

Theorem 3.2 gives an expression that can be used to compute the function h(o) as a rational function of jo. However, the coefficients of the polynomials X(s) and Y(s) are the desired quantities. It is not possible to extract a simple expression for the coefficients of


(m + 1): t :5 nB






38


X(s) and Y(s) in terms of the variable c, however. Therefore, the coefficients of these polynomials must be found numerically.

The problem of computing the coefficients of the stable polynomials X(s) and Y(s) given the coefficients of the polynomials IX(j0)2 = X(joo)X(-jo) and Y(jo)12 = Y(jo)Y(-jo) is a spectral factorization problem. Efficient algorithms for solving this problem, for discrete or continuous systems, are given in Jezek and Kucera (1985) and Kucera (1979). The algorithms presented in Jezek and Kucera (1985) are iterative, but are quadratically convergent, and can be used to compute the desired polynomial coefficients to a user-specified tolerance.



3.3 Robust Stability Testing Using F(s)


It follows from Section 3.2 that F(s) defined by equation (3.8) is strictly proper, stable and minimum phase. Now, since h(o) is taken to be the square of the function that defines x(0), it follows that

F(j) = X(j) X(jo)X(-jco) = i(w) = x(o) (3.14)
Y(jo) Y(jO)Y(-jo)for nonzero frequencies, implying that sup x(O) = ||F(s)ll_ (3.15)
o>O
Equations (3.15) and (3.1) for o = 0 imply that kN = max kN(o) = max |IF(s)J. qkk (3.16)
g9001

so that testing the condition of Theorem 2.1 is equivalent to testing the two conditions |IF(s)JL < 1 (3.17a)

k < (3.17b)
g0 1






39


The function F(s) is a strictly proper, stable transfer function, so the algorithm proposed by Boyd et al. (1989) can be used to compute |IF(s)l,_. This algorithm requires a state-space realization of F(s) as input. The basic idea behind the algorithm is that if a strictly proper, stable, real-rational transfer function H(s) has the realization H(s)= C(sI - A)-'B (3.18)
then

IIH(s)I| y
if and only if the matrix


y _-[CTC -AyBBT]

has at least one purely imaginary eigenvalue. This implies IIH(s)ll < y if and only if My has no purely imaginary eigenvalues. The function F(s) will have a state-space realization of the form given in equation (3.18) since it is strictly proper. F(s) is stable also, so the algorithm can be used to test the condition (3.17a) by checking if the matrix M := A BBT] (3.19)
- CTC -AThas any purely imaginary eigenvalues. The actual value of ||F(s)L, is found through a bisection search on 7. Upper and lower bounds for |IF(s)l| in terms of the realization {A,B,C} are given in Boyd et al. (1989). A method for testing whether or not My has any imaginary eigenvalues without actually computing the eigenvalues is also given. Such a method could be used to avoid any numerical problems computing eigenvalues. The robust stability test that has been developed is summarized below. Robust Stability Testing Algorithm

1) Obtain the nominal plant, the uncertainty matrix Qp, and a nominally stabilizing

controller C(s).

2) Construct the vector go and the matrix Qg.






40


3) Compute the coefficients of the polynomials A(j(o), 5(jo), D(jo), tR(jo) and

~I(jw) given in (3.11)-(3.13).

4) Construct the function h(o)) given in equation (3.10).

5) Perform spectral factorizations to find X(s) and Y(s) with roots in the left half of the

complex plane.

6) Construct F(s) and obtain {A,B,C} such that F(s) = C(sI-A)-1B.

7) Calculate |IF(s)|ll using the algorithm of Boyd et al. (1989).

8) Robust Stability Test: Test the conditions given in (3.17a) and (3.17b).


The advantage of this method for testing robust stability is that it avoids having to construct a frequency plot or perform a grid search. Testing condition (3.17a) requires only checking the eigenvalues of one Hamiltonian matrix constructed from the state-space matrices of F(s). Calculating the coefficients of the polynomials used to find F(s) is straightforward. Furthermore, computing the actual value of the norm in (3.17a) can be done using a bisection search based on the algorithm mentioned above. Thus, the stability margin of the system can be obtained to any desired degree of accuracy.


3.4 Discrete Time Stability Analysis


The necessary and sufficient condition for robust stability for the discrete time case is pc(W) < GO(ejW) V o e [0, t] (3.20)

where pc(o) is given by equation (2.37). This condition is equivalent to the three conditions

< 1 (3.21a)



WR()QgWR(O) < 1 (3.21b)
tR(O)i






41


WR(7TQgWR(t) < 1 (3.21c)
tR(n)l

As in the continuous time case, the endpoint conditions (3.2 1b) and (3.2 1c) will be left to test separately. Defining h(o) as h(o) = (3.22)

then it follows from the definitions in Chapter 2 that


h(o) = A(o)D(o) - B 2() (3.23)
A(o)T (o) +D(o)TR() - 2B(o)TR(o)T1(0) The goal is to construct a stable transfer function F(z) that satisfies F(ej") = h(o) (3.24)

Invoking the discrete time version of Theorem 3.1 given in Rozanov (1967), it follows that h(o) has a factorization of the form
2
X(eJW)
h(o) = 2 (3.25)
Y(el")

where the polynomials
mX my
X(z) = YXrZr and Y(z) = yrZr
r = 0 r = 0
have all their zeros in the open unit circle. The problem is treated in a manner similar to that given in Section 3.2, except now the function h(o) is rewritten as a function of e .

Theorem 3.3 The function h(co) defined in (3.22) is equivalent to

h (w) = A(ej()f)(ejW) - 2 (jo) (3.26)
A(ejw)t2 (ejo)+ D(ejW)t2(ejw)- 2B(ejw)tR(ejO)c;(e'O) ( where the coefficients of the polynomials A(ejw), B(ejW), D(ejo), tR(ei") and tI(eW) depend only on the nominal coefficient vector g0 and the matrix Qg. The polynomials tR(el") and tI(e1") are given by






42


t (e) = (e + e-g(e)
(=0


k
~ ejw) = ig eJ'*j - e-jrW) i=0


(3.27a)


(3.27b)


where g9 are the elements of the vector gO defined in (2.13), except for go :=1. The polynomials A(ej"), B(ejo), and D(ej0)are given by


k-I A(ej"o) = 4q kk + 2jqt,, f=1


and


2k-2
+ lar(ejr
r=1


+ e-jrw)


2k-2
B(ejW) = Xbr(ejrco - e-jrw)
r=1


k-I
5(el") =
'=1 I


(3.28a)


(3.28b)


(3.28c)


2k-2
+ Xdr(ejro + e-jrw)
r=1


The coefficients ar, br, and dr are determined by the magnitude of r. Define the quantities


k
= 2q2k-r-l,
t=k-(m -1)


k-r
; j 2qf,r+f
f=1


(3.29)


where m, = (r/2) if r is even and m, = (r-1)/2 if r is odd. Now, if 1 r k-1, then the coefficients are given as


a r= "qkmkn. + U + V
U + V)


br= {qk .k. + +



dr ={qk-m2,k-m2 + C - U
(Y - I


r even r odd r even r odd r even


(3.30a)


(3.30b)


(3.30c)


r odd


Finally, if k r 2k-2, then


and


and






43


M2 -1
qm2,m2 + I2q,2k-r-t r even
ar = br = dr= 1"2 ,=1 (3.31)

12q ,2k-r-P r odd

where m2 = k - (r/2) if r is even and m2 = k - (r+1)/2 if r is odd.

Proof: The proof of Theorem 3.3 is given in the Appendix. V

When h(co) is computed according to equation (3.26), the result is

4k-4 4k-4
Y ne(ejo~ + e-j() e-j(4k-4)w~ In,,(ej'4-) + ei(f4k+4)o)}
h(w) = '=O - '=0
4k-2 4k-2
d,(ei"' + e-j() e-j(4k-2)o 1: d(e('+4k-2w + e-j( t-4k+2)o
t=o t= 0

where nt and dt are the coefficients that result from equations (3.27)-(3.31). This is equivalent to
8k-8

h(co) = e 2jow P=O
8k-4 . (3.32)

t=0
for appropriately calculated fi, and d,. As a result of the fact that many of the polynomial coefficients given by (3.28)-(3.31) are identical up to sign, many of the coefficients in (3.32) are exactly zero. The number of terms lost is the same for both the numerator and denominator, and is equal to 4(k-1). Removing these terms gives a reduced form for h(co) 4k-2
1ie e0'e

h(uo) -e2
h~)=4k22 .w (3.33)

1=0
The polynomials X(e^W) and Y(e W) are found by performing a spectral factorization using the results of Jezek and Kucera (1985) in a manner similar to the continuous time case. The final polynomials X(z) and Y(z) have degree 2k-1, however, the first coefficient of X(z) is identically zero.






44


The transfer function that results in the discrete case is of comparable order to that obtained in the continuous case, and this order is roughly twice that of the nominal characteristic polynomial. In both cases, the transfer function is found by constructing a stable transfer function from the vector of coefficients of the nominal characteristic polynomial and the elements of the matrix describing the uncertainty ellipsoid.



3.5 Example


In this section, an example is presented to illustrate the ideas introduced in this chapter. Consider the following nominal polynomial

GO(s) = s4 +5.8600s3 +9.3954s2 +6.0126s +5.3237 (3.34)

represented by go, and the associated uncertainty matrix Qg:

~5.86001 ~2.0425 2.3648 1.7252 1.26031
o 9.3954 2.3648 4.9454 3.5362 1.4123
g = 6.0126 Q9 = 1.7252 3.5362 4.6202 2.2330
5.3237] _1.2603 1.4123 2.2330 3.6206]

The polynomial (3.34) has roots at s = {-3.60, -2.00, -0.13 0.85j}. Using the equations from Theorem 3.2 and the algorithm of Jezek and Kucera (1985), the spectral factors X(s) and Y(s) for this system are found to be X(s) = 2.12s4 +6.37s3 +9.58s2 +7.84s+3.43 (3.35)
and

Y(s) = 1.43s6 +6.92s5 +21.7s4 +34.2s3 +34.9s2 + 23.3s + 10.9 (3.36) The degree of the nominal polynomial in equation (3.34) is k = 4; it is easily verified that the degrees of X(s) and Y(s) match the specifications of Theorem 3.1. The roots of X(s) are s = {-1.02 0.60j, -0.48 0.96]), and those of Y(s) are s = {-1.30 2.22j,

-0.99 0.63j, -0.13 0.90j}, showing that both polynomials are Hurwitz.






45


Figures 3. la and 3. lb show plots of the frequency-dependent Nyquist robust stability margin vs. frequency for the frequency range o e [0, 10]. It is obvious that kN(O) is discontinuous at w = 0 for this example, and kN(O) = 0.357420. Using equation (3.16), the value of kN is found to be kN = 0.971650, rounded to six digits, and since the condition (3.17) is satisfied, the polynomial is robustly stable. The plot in Figure 3.la is constructed by evaluating (3.1) over the frequency range (o e [0, 10] using A0 = 0.0 1. The plot of the frequency response magnitude of X(s)/Y(s), evaluated at the same frequency points, is shown in Figure 3.1b. Figures 3.lc and 3.ld show a comparison of the results of the two methods. The maximum absolute difference between the two curves is less than 4xl0-6, while the maximum percentage difference is slightly greater than

0.0004%.

The results of the frequency sweep method are shown in Table 3.1 for different values of Ao. The computed value of kN (rounded to six digits) as well as the number of floating point operations needed by a simple MATLAB routine are shown. It is interesting to note that the result for Aw = 0.3 is quite good, and in fact, much better than the result for Ao = 0.2. This is due to the fact that the maximum of kN(co) occurs almost exactly at S= 0.9. This point will be evaluated in the sweep using Ao = 0.3, yielding a much better estimate of kN than for the sweep using Ao = 0.2.

Table 3.1. Floating point computations needed for a frequency search.

Resolution Aco for the frequency sweep

0.3 0.2 0.1 0.01

kN 0.971648 0.787774 0.971648 0.971648

flops 14,425 21,858 43,718 437,206



















6


8


Frequency

(b)


Frq u e ncy


Frequencv


Frequency


6


6


6


8


8


8


Figure 3.1.


Plot of the Nyquist robust stability margin kN((O) VS. frequency a.
(a) Frequency sweep method. (b) Proposed method. (c) Percentage error between the two methods. (d) Absolute error between the two methods.


1


46


0.5


z


(a)


0


2


0


1


~0.5 U-


4


0
Y 104


2


1








1


4


L
0

-0


(c)


6F


r


0
x 10-6


2


0








0








0








0


4


I.
0
I.
U .0


-


1


(d)


0


2


4


1


-2

-4


-4






47


Tables 3.2 and 3.3 show the results for the method proposed in this paper. In Table 3.2, the number of floating point computations needed to find both spectral factors is listed. These numbers represent the computations of a simple MATLAB program implementing the algorithm given in Jezek and Kucera (1985). The tolerance is taken to be the sum of squared errors between the generated coefficients of X(jo)X(-jo) or Y(jco)Y(-jo) and the actual values of the coefficients. This tolerance must be satisfied by both polynomials X(s) and Y(s). The number of iterations of the algorithm in Jezek and Kucera (1985) required to attain this tolerance is also given in Table 3.2. For the computations presented in Table 3.3, the value 10-4 was used for the tolerance. Table 3.2. Floating point computations and iterations needed to compute X(s) and Y(s).

Tolerance for computing spectral factors

_1 10-2 10-4 10-6 10-9

iterations (X/Y) 4/6 4/6 5/7 5/7

flops 3682 3682 4337 4337


Table 3.3 shows the number of floating point computations needed to find IIF(s)IL. given X(s) and Y(s). The tolerance values listed are the relative tolerances required for termination of the algorithm used to compute IIF(s)IL., i.e., the estimated value of IIF(s)IL. is guaranteed to be within IIF(s)L. of the true value. It is very interesting to note that the number of computations required to find the spectral factors X(s) and Y(s) is small relative to the computation required to find IIF(s)IL. This is desirable, as the termination criterion for the spectral factorization algorithm can be stringent without seriously affecting the number of overall calculations needed to compute IIF(s)IL. Thus, the computational burden of the proposed method is primarily dictated by the desired numerical accuracy of the Nyquist robust stability margin.






48


Table 3.3. Floating point computations needed for computing IIF(s)IL.

Relative tolerance for computing IIF(s)IIL

0.1 0.01 0.001 0.00001

lF(s) 86 0.917423 0.973195 0.971801 0.971659

flops 1 181,669 268,629 358,405 573,145


For a tolerance as low as E = 0.001 used in computing the infinity norm, the computational burden for the proposed algorithm (358,405 + 3,682 = 362,087 flops) is lower than that for the frequency sweep using Ao = 0.01 (437,206 flops). If the relative tolerance level is increased to F = 0.00001, then the computational burden (573,145 + 3,682 = 576,827 flops) surpasses that of the frequency sweep using Ao = 0.01. However, the proposed algorithm affords a guaranteed level of accuracy in the final result that the frequency sweep cannot.



3.6 Conclusions


In this chapter, a new robust stability test for ellipsoidal systems is presented. This test is based on construction of a strictly-proper, stable transfer function whose infinity norm is equivalent to the stability margin of the system. The proposed test avoids performing a frequency search, and can calculate the stability margin using a bisection search.














CHAPTER 4
ROBUST PREDICTIVE CONTROL DESIGN FOR ELLIPSOIDALLY UNCERTAIN SYSTEMS


4.1 Introduction


The issue of designing controllers to robustly stabilize a system subject to real parametric uncertainty has received much attention in recent years. Since many physical uncertainties are most directly and effectively modeled by real parametric uncertainty, it is of great practical interest to include real uncertainty in the design of robust controllers. Unfortunately, the inclusion of real uncertainties greatly complicates both the robust control analysis and synthesis procedures. However, there are cases of interest where the uncertainty structure is such that the analysis and synthesis problems simplify considerably. This chapter considers such a case and establishes a procedure to design a robust predictive controller for a system subject to ellipsoidal parametric uncertainty.

The standard method for analyzing the effect of uncertainty on a system is to construct a fictitious feedback loop where all the uncertainties are pulled out in a block-diagonal A matrix and all the known components, including the nominal plant, the controller, and any weights, are lumped into an M matrix. The robust control analysis problem is, given a fixed M and a set structure for A, find the smallest A that destabilizes the system. The structured singular value (Doyle 1982), t, is a stability margin defined by the size of the smallest destabilizing A as

p(M) := {min d(A) I det(I - AM) = 0}1where g(M) = 0 if no allowable A destabilizes the system. The robust control synthesis problem is, given a description of all possible As, design a controller that stabilizes the


49






50


closed loop for any allowable A. It is the structure of the A matrix that determines the difficulty associated with solving the analysis and synthesis problems.

The simplest uncertainty structure is where the matrix A is a full complex block, and only a norm bound on the whole matrix is known. This case is referred to as unstructured uncertainty, since no internal structure for the A matrix is known (or assumed). In this case, both the analysis and synthesis problems simplify considerably. The structured singular value for a full complex block A is simply p.(M) = d(M). Furthermore, a controller is robustly stable if it generates an M matrix that satisfies UIMILO < |lAlli. Thus, H. design methods (Doyle et al. 1989), (Glover and Doyle 1988) can be used to find the optimal controller, or any sub-optimal controller that stabilizes the closed loop.

In many cases of practical interest, the block-diagonal A matrix contains real and complex (possibly repeated) scalar blocks as well as norm bounded complex uncertainty blocks. If A contains real blocks as well as complex blocks, p(M) is referred to as mixed p. In general, the calculation of mixed p is NP hard (Braatz et al. 1994), (Braatz 1996), (Poljak and Rohn 1993). For both the purely complex and mixed cases, convex upper bounds for g exist (Doyle 1982), (Fan et al. 1991), and these are the basis for controller synthesis methods. The complex g-synthesis control design procedure attempts to find a controller minimizing p for the closed loop by iterating between computing an optimal scaling matrix D appearing in the upper bound and computing an H. controller K for this D, hence the name D-K iteration (Doyle 1985). The procedure is not jointly convex in D and K so there is no guarantee that the resulting controller is globally optimal, but it will be a locally optimal choice. The process of controller design for mixed p is similar to that for complex p, but now another scaling matrix is involved in the optimization, leading to the so-called D,G-K iteration method (Young 1996). This optimization is not jointly convex in D, G and K, so it may not yield the globally optimal controller.

In the structured singular value framework, the individual elements of the A matrix are assumed to vary independently. Often it is more realistic to allow interactions between the






51


uncertainties. One practical example is the ellipsoidal uncertainty description where the allowable coefficients of the system transfer function are restricted to lie inside an ellipsoid in the plant parameter vector space. This corresponds to a A matrix with real diagonal elements that is constrained by a weighted two-norm bound. Ellipsoidal uncertainty descriptions arise naturally in parameter identification techniques where the uncertainty ellipsoid is associated with the parameter-error covariance matrix. Ellipsoidal uncertainty descriptions are commonly encountered in chemical engineering applications where model parameters are found by fitting experimental data using linear or nonlinear regression techniques (Fogel and Huang 1982), (Belfonte and Bona 1985), (Belfonte et al. 1990). Methods for the computation of stability margins for ellipsoidal systems are given by several authors such as Guzzella et al. (1991), Biernacki et al. (1987), and Tsypkin and Polyak (1991). Biernacki et al. (1987) outline an iterative robust controller design procedure based on expanding the parameter space stability hyperellipsoid until it covers the allowable uncertainty region, but this procedure is not a convex optimization. The generalized structured singular value (Chen et al 1994a, Chen et al 1994b) is an extension of the structured singular value that takes into account interactions of the system uncertainties. The complex uncertainties, real uncertainties and complex block uncertainties are grouped together, and each group is norm bounded in an appropriate manner. Therefore, this is a more general framework than the mixed pi description. Recently, Braatz and Crisalle (1997) recast the ellipsoidal uncertainty problem in terms of the generalized structured singular value, allowing the straightforward inclusion of complex uncertainties in the stability margin calculation. However, there are no currently available controller design methods based on the generalized structured singular value.

In some specialized cases, such as when the uncertain parameters appear in an affine manner in the numerator and denominator of a single input, single output system transfer function, the M matrix that results is rank one. Then the structured singular value is equal to its convex upper bound (Young 1994), (Young and Doyle 1996), and can be computed






52


directly. The generalized structured singular value can also be computed in closed form (Chen et al. 1994b), (Braatz and Crisalle 1997). It is important to note that the restriction of the M matrix to be rank one does not depend on the number of uncertain parameters, but rather the manner in which they enter the system description. For the case of a rank one M matrix, Rantzer and Megretski (1994) derive a convex necessary and sufficient condition for robust stability that is used as the basis for design of a fixed-order Youla parameter whose coefficients are found using convex or quasi-convex optimization. The uncertainty description considered in this chapter yields a rank one M matrix, and the method of Rantzer and Megretski (1994) is used to compute an appropriate Youla parameter (Youla et al. 1976) that defines a robust predictive controller.

Predictive control is a model-based control methodology that has found wide acceptance in industry. This is because predicitive control offers good performance, is easy to understand and formulate, and can accommodate input/output process constraints. The industrial success of the predictive control techniques is apparent by the variety of commercial predictive controllers that are available to the chemical processing industry through specialized vendors. Seborg (1994) reports that in oil refineries and petrochemical plants around the world, there are hundreds, perhaps thousands, of predictive controllers employed.

It is possible to design predictive controllers using different plant representations, including finite impulse response (FIR) models, transfer function models and state space models. FIR based schemes include Dynamic Matrix Control (DMC) (Cutler and Ramaker 1980), Model Algorithmic Control (Mehra et al. 1979), and the quadratic DMC formulation of Garcia and Morshedi (1986). These methods are applicable only to stable plants, and often require large model orders, typically retaining 30 to 40 impulse response coefficients. Transfer function models are applicable to both stable and unstable plants, and lead to lower order representations. The well-known Generalized Predictive Control (GPC) technique (Clarke et al. 1987) and the MUSMAR approach (Greco et al. 1984) are






53


examples of transfer function based predictive control formulations. Kwon and Pearson (1977) and Muske and Rawlings (1993) present state space formulations for predictive control. This large body of literature constitutes a rich source of knowledge to support the design and analysis of predictive controllers.

There is currently a large amount of research focusing on the issue of stability of predictive control designs when the plant model is uncertain, such as the work of Zafiriou (1990) and Genceli and Nikolaou (1993). There is an interest in the research community to revisit the predictive control design techniques with the intention of including robustness features that guarantee stability or adequate performance when the plant model is uncertain. One interesting example is the robust quadratic DMC design including hard constraints studied by Zafiriou (1990). This work uses a contraction mapping first proposed by Economou (1985) to derive time-domain conditions for robust stability with respect to uncertainty in the impulse response coefficients of the nominal model, but this approach involves a very large numerical computation effort. Genceli and Nikolaou (1993) propose an analysis and synthesis method for predictive controllers based on FIR models, including constraints and using a linear cost functional. These authors use a parametric model uncertainty description that bounds the maximum deviations allowed for each pulseresponse coefficient, and obtain a sufficient condition for robust closed loop stability.

The robustness of predictive controllers designed using transfer function representations is receiving increasing attention in the literature. Kouvaritakis et al. (1992) propose an alternative approach to GPC that employs a precompensator to stabilize the plant before the predictive design is carried out. The Q-parameterization procedure popularized by Youla (Youla et al. 1976) is employed in order to construct a final controller that is robust with respect to unstructured perturbations. The authors state rigorous necessary and sufficient conditions for robust stability; however, the approach proposed for synthesizing robust controllers is an approximate albeit practical scheme. The method consists of using polynomial or fixed-order transfer function approximations for






54


the Youla parameter, and least-squares methods to identify the parameters of the robust design. Hrissagis et al. (1996) present a direct method for designing a predictive controller that is robust with respect to unstructured perturbations. In this approach, an appropriate Youla parameter is explicitly computed by solving a model-matching problem.

This chapter presents a method for designing controllers that are robust with respect to real parametric uncertainty. The technique is based on the results of Rantzer and Megretski (1994), and therefore relies on using fixed-order approximations for the Youla parameter in the controller parameterization. The nominal predictive controller is designed using wellestablished methods and the nominal servo performance is retained by the robust controller. Furthermore, the design technique can be easily modified to incorporate integral action in the robust controller, allowing for the rejection of asymptotically constant disturbances.

The chapter is organized in the following manner. The next section discusses the design of a predictive controller of the GPC type for a nominal transfer function plant model. The third section details the parameterization of all nominally stabilizing controllers through the use of the Youla parameter Q. The robust predictive control design based on the ellipsoidal uncertainty description is discussed in the fourth section. The fifth section contains the modifications of the design procedure required to incorporate integral action into the robust controller. A further modification to allow the inclusion of unstructured uncertainty is discussed in the sixth section. An example is given in the seventh section to illustrate the proposed design method. The design equations used to construct the nominal predictive controller are given in the final section.



4.2 Nominal Predictive Control Design

Predictive control is a control methodology that is well documented in the literature. In particular, a wealth of knowledge is available to resolve crucial design issues such as nominal closed-loop stability, and parameter tuning (Lambert 1987; Mohtadi 1987).






55


Predictive controllers are usually implemeted by executing at every sampling instant an algorithm that solves a quadratic optimization problem. For analysis purposes, it is desirable to represent the algorithmic controller in terms of transfer functions, allowing the utilization of classical z-domain tools for analyzing stability and performance. This section presents a brief review of the analysis technique discussed by Crisalle et al. (1989), which casts an algorithmic predictive control law of the GPC type into a form involving transfer function operators. The resulting nominal controller is used as the basis for the design of a robust controller.

Consider the nominal process model
A(z) y(z)= B(z) u(z) (4.1)

where y(z) and u(z) are the process output and input, respectively, and A(z) and B(z) are the coprime polynomials

A(z) = z" + an-lzn-I +... + a0 (4.2)

B(z)= bnz"m + bi-izm I +...+ b0 (4.3)

of order n and m, respectively, where n>m. Predictive control involves the selection of future control moves that minimize the quadratic cost functional N N14
J(t) = [r(t + i) - y(t + ilt)] + X EX[Au(t +i)]2 (4.4)
i=1 i=O

where { r(t+i) } is the sequence of future values of the set point, {y(t+ilt) } is the sequence of predicted future values of the output, { Au(t+i) } is the sequence of future control increments, A is the move-suppression parameter used to penalize excessive control energy, and parameters NV and Nu are the prediction and control horizons, respectively. The optimal control move, the u(t) that minimizes the functional J(t) for the prescribed set point sequence { r(t) }, is found by differentiating (4.4) with respect to the control moves, equating the result to zero, and solving for u(t). The predictive control law can also be cast in terms of transfer function operators as (Crisalle et al. 1989)






56


R(z) S(z) (5
u(z)= T(z)r(z) - n y(z) (4.5)

where

R(z)= z +z + ro (4.6)

S(z) sIZn + snz-1 +... + SO (4.7)

T(z)=tN, Z NY + tN -N -1 +...+tz (4.8)


These operators satisfy
R(1) = 0 (4.9)

and
T(I) = S(1) (4.10)

The coefficients of the moving-average polynomial S(z), the regressor polynomial R(z), and the set-point advancement polynomial T(z) are functions of the tuning parameters N, Na, and A, and of the model polynomials A(z) and B(z). Note that the predictive control law (4.5) includes an integrator if equation (4.9) is satisfied. A block-diagram representation of the predictive control structure is shown in Figure 4. la. Specific equations for the polynomials (4.6) - (4.8) are given in Section 4.8; further details of the derivation can be found in Crisalle et al. (1989). A formulation equivalent to (4.5) is also derived in McIntosh et al. (1991).

Note that the transfer functions operating on u(z) and y(z) in the nominal predictive controller (4.5) are biproper and of order n, the order of the nominal plant model. Note also that the set-point advancement polynomial T(z) is of degree equal to the prediction horizon N,,. Since N, n is a common tuning prescription (Clarke et al. 1987), the order of T(z) may exceed the order of R(z), making the control law noncausal with respect to the set-point signal. This noncausality is a natural consequence of the inclusion of future values of the set point in (4.4). Figure 4.la shows that T(z) acts on the set point to






57


produce the intermediate signal v(z)=T(z)r(z), which has the simple time-domain representation

v(t)=tN, r(t+N ,)+tN,.--r(t+NY-l)+...+tIr(t+l) (4.11)

It is useful to remark that the nominal model (4.1) and the functional (4.4) are simpler versions of more elaborate formulations that improve the design performance at the expense of added complexity. Enhancements of the predicitve control law presented above, such as the inclusion of a lower prediction horizon parameter (Clarke et al. 1987), the addition of a weighted end-point term in (4.4) to guarantee stability for arbitrary parameter choices (Kwon and Byun 1989; Demircioglu and Clarke 1993), and the use of a filtered set point, can be accommodated within the framework proposed in this chapter through obvious modifications.

Figure 4. la illustrates the closed loop system established when the nominal predictive controller (4.5) is connected to the process (4.1). In addition to the set point signal r(t), the figure also shows an additive output disturbance signal d(t). Note that the servo dynamics of the closed loop are fully characterized by the equations [A(z)R(z) + B(z)S(z)] y(z) = znB(z)T(z) r(z) (4.12)

[A(z)R(z) + B(z)S(z)] u(z) = znA(z)T(z) r(z) (4.13)

Therefore, the stability of the closed loop for a given nominal predictive controller is contingent on the location of the roots of the characteristic polynomial A(z)R(z) + B(z)S(z). Furthermore, due to the presence of the integral action (4.9) in the controller and to the gain equality (4.10), the closed loop dynamics described by (4.12) are guaranteed to realize zero offset in the servo response. The integrator also guarantees perfect steady-state disturbance rejection for all disturbance signals that reach a constant steady-state. These desirable performance characteristics of the nominal controller will be preserved in the robust predictive controller designed in the following sections.






58


d
(a)

r v v1 e u B
T --__->0
+ R A


S

(b) d


r v t z" v1 e U B

A 2 +,Y- NQ A


X +MQ <

Figure 4.1. (a) Structure of a nominal predictive controller. (b) Structure
of the parameterized predictive controller including the Youla parameter Q(z).




4.3 Nominally Stabilizing Controller Parameterization


In this section, all nominally stabilizing controllers are parameterized in terms of the nominal predictive controller (4.5) and a transfer function Q(z) selected in the spirit of Wiener-Hopf design (Youla et al. 1976). However, a modification in the parameterization is introduced to achieve two important design requirements: (i) the parameterized controller must preserve the servo performance and the steady-state disturbance rejection properties of the nominal controller, and (ii) the parameterized controller must also be a predictive controller.

Consider a nominal predictive controller (4.5) that stabilizes the closed loop system (4.12) - (4.13). Provided the closed loop is stable, the nominal closed-loop characteristic polynomial


A*(z) = A(z)R(z) + B(z)S(z)


(4.14)






59


is a Schur polynomial of degree 2n. As a first step in the parameterization of all nominally stabilizing controllers, a coprime fractional representation of the nominal plant model (4.1) is constructed

PO =B(z) = (4.15)
A(z) M(z)

where N(z) and M(z) are proper and stable transfer functions that satisfy the Diophantine equation
N(z) X(z)+ M(z) Y(z) = 1 (4.16)

for some pair of stable and proper transfer functions X(z) and Y(z). (Note the use of italicized capital letters for transfer functions, while polynomials are designated with plain capital letters.) A suitable (M(z), N(z)) pair is readily derived from the nominal characteristic polynomial (4.14) as in Hrisaggis et al. (1996). First, the closed-loop characteristic polynomial is factored into the form A*(z) = Ai(z)A2(z), where both A1(z) and A2(z) are Schur polynomials of degree n. If A*(z) contains complex poles then A1(z) and A2(z) are constructed such that each complex-conjugate pair is contained in either Al(z) or A2(z) to ensure that each polynomial factor has only real coefficients. Both sides of (4.14) are then divided by the factored characteristic polynomial to obtain A(z)R(z) B(z)S(z) (4.17)
Al(z)A2(z) A1(z)A2(z)

Finally, stable and proper factorizations satisfying (4.16) are defined as M(z):= (z) N(z):= B(z) (4.18)
Al(z) Al(z)

and
SWz R(z)
X(z):= , Y(z):= R(Z) (4.19)
A2(z) A2(z)
This result allows the nominal predictive control law (4.5) to be written in the equivalent form
Y(z) u(z)= Z(z) r(z) - X(z) y(z) (4.20)


where






60


fl
Z(z):= Z T(z) (4.21)
A2(z)

The set of all solutions of (4.16) can be written in terms of the transfer functions (4.18) - (4.19) and a stable, proper transfer-function Q(z) through the well-known relations (Youla et al. 1976)

X'(z) = X(z) + M(z)Q(z) (4.22)

Y'(z) = Y(z) - N(z)Q(z) (4.23)

Therefore, the set of all stabilizing controllers with the structure (4.20) is parameterized in the form
[Y(z) - N(z)Q(z)] u(z) = Z(z) r(z) - [X(z) + M(z)Q(z)] y(z) (4.24) to yield the control structure shown in Figure 4.1b. Setting Q(z)=0 recovers the nominal predictive controller (4.20).

In contrast to the standard Youla parameterization approach, the transfer function X(z) + M(z)Q(z) appears in the feedback path of Figure 4. lb, instead of appearing in the control block immediately preceding the plant. This approach, adopted from Hrissagis et al. (1996), in conjunction with factorizations (4.18) and (4.19) that make use of the nominal closed-loop polynomial, introduces the following highly desirable properties in the parameterized input-output maps.


Property 4.1. The nominal control loop of Figure 4.la and the parameterized control loop of Figure 4. 1b have identical servo transfer functions y(z)/r(z) and u(z)/r(z).

Proof. The proposition is proved by carrying out block-diagram algebra on each figure to derive in both cases the servo transfer functions y(z)/r(z) and u(z)/r(z). +


Property 4.2. Given that the nominal controller (4.5) is a predictive controller, then the parameterized controller (4.24) is also a predictive controller.

Proof. If (4.5) is a predictive controller, then by definition it yields a control sequence {u(t)} that minimizes the predictive performance index (4.4) for any prescribed set-point






61


trajectory { r(t) }. For the given set point trajectory, it follows from Property 4.1 that the parameterized controller (4.24) will also produce the same control sequence due to the equality of the servo transfer function u(z)/r(z). It follows that the parameterized controller is also a predictive controller because it yields a control sequence that minimizes (4.4). +

Since any allowable parameter Q(z) yields the same servo transfer functions y(z)/r(z) and u(z)/r(z), the parameterized controller has the intrinsic capability of preserving the nominal servo performance. Also note that although the terms containing Q(z) effectively cancel out in the servo transfer functions, the transfer function e(z)/vl(z) = M(z)[Y(z) N(z)Q(z)] in Figure 4.1b is affine in Q(z), as in the standard Youla parameterization method.



4.4 Robust Predictive Control Design In order to incorporate uncertainty into the plant description, the nominal plant transfer function shown in Figure 4. 1b is now represented as the uncertain transfer function

P(Z) = B(z) + AB(z) _ b,,zm ...+ bo +8b,,.z" +...+6b0 (4.25)
nn-I
A(z) + AA(z) z" + _anIz"- +...+ ao +6aniz"- +...+6aO

where A(z) and B(z) are given in (4.2) and (4.3) and AB(z) = 8bZ'" +... + 8bo (4.26)

AA(z)= 8an-_iz- I +...+5aO (4.27)

The values of the coefficients of AA(z) and AB(z) are not known explicitly, however, the perturbation vector

5p=[5a,- ... 6aO b,n ... E 91'+"I (4.28)

composed of the coefficients of AA(z) and AB(z), is constrained to lie in an ellipsoid 'E, in the parameter space. This ellipsoid is defined by a positive definite, symmetric matrix Q, such that






62


'E, = 8P C9'+n+1 8pTQI6P:5 11


(4.29)


The objective of the robust control design is to select a suitable Youla parameter Q(z) such that the resulting controller stabilizes the family of plants described by E,.

The work of Rantzer and Megretski (1994) considers robust controller design for any system where the uncertainties can be extracted from the closed loop as shown in Figure 4.2, where it is noted that the signal w is scalar. This formulation immediately restricts this technique to rank one M-A problems. The affine uncertainty description in equation (4.25) results in a rank one M-A problem, so this formulation can be accommodated by the technique proposed by Rantzer and Megretski (1994). It is straightforward to transform the system shown in Figure 4. lb, now with an uncertain plant, into the form of Figure 4.2. This is done by first extracting the uncertainty vector 8p out of the plant block, and then constructing the appropriate closed loop transfer function Tzw from the uncertainty block output w to the uncertainty block input z. The first step in this process is achieved by first rewriting the plant block as shown in Figure 4.3.

8TG


w G


Figure 4.2. Feedback structure from
Rantzer and Megretski.


Figure 4.3. Augmented plant with
uncertainty vector 5p.


For the model given in equation (4.25), Figure 4.4 shows explicitly the internal structure of the augmented plant given in Figure 4.3. The closed loop shown in Figure 4.3 is given by


[z] = [G11(z) G12(Z) (4.30)
y_ G21(z) G22(Z)_ _uI






63

From Figure 4.4, it is straightforward to show that both Gi I(z) and G I2(z) are transfer function vectors with n+m+] rows while G21(z) and G22(z) are scalar transfer functions. The exact form for these transfer functions is given below.


G 1(z) =


n-I
z
A(z)


A(z)
0


0


G21(z) = -l


z 1B(z)

A(z) B(z)
A2( G12(z) A (z


Az)
B(z)


G22 (Z) =
A(z)


AznA(Z)




Az)


~~ [8a,-I - - - SaO 81bm - - -8bo



-Z








U +
A(z)


___Lk ,M


Figure 4.4. Augmented plant in detail. When the plant block in Figure 4. lb is replaced by that shown in Figure 4.3, the resulting closed loop is transformed to the form shown in Figure 4.5. The overall closed loop from w to z becomes T, where the transfer function vector Tz is given by


(4.3 1a) (4.3 1b)






64


T, = T + T2Q = {GI I-G12MXG21} + {-G12M2G21 Q (4.32)

The expressions in (4.32) for the transfer function vectors Ti(z) and T2(z) simplify to

-'R(z) - B(z)
2
AI (z)A2 (z) A (z)

R(z) B(z)

T(z) - A1(z)A2(z) and T2(z) = A1(z) (4.33)
z'"S(z) z 'A(z)
2
Al(z)A2(z) AI(z)

S(z) A(z)
2
Al(z)A2(z) _ A,(z)

Note that all elements of both Ti(z) and T2(z) are stable, strictly proper transfer functions, and that both Ti(z) and T2(z) have n+m+1 rows.












6pTT
w z
r V. e d lp f >
--> T ->---> G
A 2 + , Y- NQ


X +MQ

Figure 4.5. RST configuration with the augmented plant.




-- 1pT


Tzw


Figure 4.6. Closed loop from w to z.






65


In the controller design method of Rantzer and Megretski (1994), the approach is to express the Youla parameter Q(z) as a ratio of stable proper transfer functions $(z) and x(z) in the form

Q(z) = P(z)/c(z) (4.34)

The main result of Rantzer and Megretski (1994) is that the closed loop in Figure 4.6 is stable for perturbations 8p satisfying I8pl I if ct(z) and P(z) satisfy


Re{Ti (e")c(e"d) + T2(e1")P(eiO' )1d < Re Ja(eiO )} (4.35)

for all o r [O,n]. The norm used in measuring the size of 8p is any appropriate vector norm, and 161d is the dual norm to this norm (Luenberger, 1969).

Since the nominal servo performance of the predictive controller is preserved for any choice of Q(z), a possible method for choosing this parameter would be to maximize the stability robustness of the system with respect to the size of the uncertainty vector 8p. Note that if


Re{T(eiO)(eJi )+ T2(e")P(e")} < - Re Jx(ei')} for I6plI 1 (4.36) where y > 1, then


Re{Ti(eW)(X(eiO))+T(e")P(edo)} < Rejc(e")J for Iapl -y (4.37) Therefore, if cx(z) and 1(z) are selected to solve the optimization Re{7j(ei*)cx(eiO) + T2(eiO)P(eJW)d
min max (4.38)
a,$eRH. (oe[0,1] Reja(ed")J


subject to the constraint

Retc(eiO))>0 Vwo(e[O,iT] (4.39)

then the resulting closed loop will be robustly stable for the largest possible value of Ipl.






66


Since a(z) and 1(z) can be of any order (such that the resulting Q(z) is proper), this is an infinite dimensional optimization. This optimization is converted to a finite dimensional problem by restricting a(z) and P(z) to be of the form N 2N+1
(x,z) = 1 + XxkXk(z) and P(x,z) = XXkP W(z) (4.40)
k=1 k=N+I
where x = [xI -- X2N+I ]T E 12N+1. Then Q(z) is a rational transfer function of order N 2N+I
X xk13k(Z)
QN(Z) k=N+1 (4.41)
1 + Yxk (Z)
k=1

The functions 'Xk (z) and k (z) should be chosen such that the limiting QN (z) lim QN (Z) = lim {X, z)/c(xzM (4.42)
N-coo Ncan approximate any stable transfer function Q(z) arbitrarily well. Here, the basis functions ak (z) and Pk (z) are chosen as Xk(Z) = pk(Z) = -az+ (4.43)

where a is taken to be a real scalar that satisfies lal < 1. This choice constrains the basis functions to be all-pass functions. An interesting choice for a is simply a = 0, reducing the basis functions to the form

-k = Pk ( = Z-k (4.44)

Obviously, a ratio of polynomials in z-1 can recover any rational transfer function as N grows without bound.

Choosing basis functions of the form (4.43), the functions c(x,z) and $(x,z) become X(x,z) = 1 +-x az+1j and P(x,z) = x. 5+)1k-N+ (
k=1 ( - k=N+..Then, the closed loop in Figure 4.6 is stable for all allowable perturbations 8p (that is, all 5p satisfying ipl 1) if ct(x,z) and $(x,z) satisfy






67


ReJi(ei'W)oc(x,ei'"O) + T2(eJ'O)P(x,ei'O) d < Re Ja(x,e'W)} (4.46)

The constraint in equation (4.29) is 8pTQP18p 1. A new vector S can be defined

S := Q P (4.47)

where the symmetric positive definite matrix Q1,2 satisfies = QI/2Q/2 (4.48)

Then the vector S satisfies 1I142 1 for all allowable 8p. The block diagram shown in Figure 4.3 can be adjusted to use the new vector S as the perturbation vector, but a factor QI must be included in the plant matrix G(z), as shown in Figure 4.7. The new vector i is related to z simply by z = Q},2z. Therefore, the transfer vectors G I and G12 are changed to Q/ Gi and QI2G12, respectively. Plugging these new expressions into equation (4.31) yields

T = P TW = QPT, - QPT2Q (4.49)

Note that all the elements of the matrix Q are real, and that the dual norm to the Euclidean 2-norm is the Euclidean 2-norm itself. Therefore, equation (4.35) becomes

QI2 Re{T(ej')a(x,eJ")+ T2(ej')P(x,e'W) 2 ixex,e;) Vo e [0,] (4.50)






3|G
uz
U y
Figure 4.7. Augmented plant with new uncertainty vector S. Alternatively, the norm used to measure 6p can be defined as 15pI := Q P (4.51)

and then the dual norm in equation (4.35) is Ix = 2 (4.52)






68


which yields exactly the result shown in equation (4.50).

Equation (4.50) cannot be satisfied if ReJa(x,eW)I < 0 for any frequency. Therefore, provided that

min ReJcx(x,eiO)} > 0 (4.53)
WE[O,n]

then the closed loop is robustly stable (for a specific value of x) if Ql / 2 Re{IJ(eiw")(x,ei")+ T2(eiO)P(xei)} max 2 < 1 (4.54)
WE[O,n] Refaox'e )I

As outlined above, then the optimally robust controller (actually the optimal vector x) can be found by solving the optimization program

- Qg2 Re{Ti(eiw)a (x,e))+ T2(ew )P(x ,e ) 2 ( xO = arg min max 2- o I (4.55)
OE[OM,] ReJa(x,e")I

subject to

min Re Jc(x,e'")} > 0 (4.56)
O)E[O,It]

for larger and larger values of N. In practice, a value of N will be found such that the value a Q2 ReJTi (eiw)(x,ei")+ T2(e3)P(x,e) max P W 12 (4.57)
we[On] RefcL(x,e )

does not decrease appreciably as N is increased further. The optimal vector xOPt for this value of N is considered to describe the optimal controller. Alternatively, the value of N can be chosen to enforce a maximum degree condition on the resulting controller. The final form of the robust predictive controller is CN (Z) = X(z) + M(z)QN (Z) (4.58)
Y(z) - N(z)QN (z)






69


which will be biproper and of degree 2n+N if there are no common factors that cancel in forming CN(Z). Thus, a specific value for N can be chosen, and the "optimal" QN(Z) found for that value of N.

The use of basis functions of the form (4.45) results in the following desirable properties of the optimization (4.55) - (4.56) as outlined below.

Lemma 4.1. If the basis functions o(x,z) and (x,z) are given by equation (4.45), then the objective function

Q 2 ReT (ei")o(x,e")+ T2(eJW)P(x,eJ)2
((x) := max P)12 (4.59)
o)e[O,n] Rejc(x,e")}

is a quasi-convex function of x, and the constraint function $(x) := min ReJac(x,eW)} (4.60)
we[On]

is a concave function of x.

Proof The proof is given in the Appendix.

Any general optimization routine for quasi-convex functions requires gradients or subgradients of both the objective and constraint functions. It is noted that for any given frequency it is possible to write

Q' Re{Ti(e")c(x,ejw)+ T2(eJi')(x,ejw) 2 {A(o)x + b(O) 1 )
P )J112 P) 2 (4.61)
ReJac(x,eiw)} 1 + c(O)x

where the matrix A(o) and vector b(o) depend on T (ei*), T2 (ei"), and the pole of basis function, and vector c(o) depends on the pole of basis function. If the value (o =O* that satisfies


mQl,/2 Re{T, (ei" )c(x,e1() + T2 (ejm )P(x,ejo)} Q 112{A(co*)x + b(o*)}
max 2 (4.62)
We[On] Re Jc(x,e')) I+ cT(co*)x

is found, then an appropriate subgradient g,(x) for the objective function (4.59) is






70


d 2{A(o*)x + b(oi*)}
gS(x) 1 + c((*)x (4.63)


(Boyd and Barratt, 1991). Similarly, if the value co =co* that satisfies min ReJc(x,e")} = 1 + cT(o*)x (4.64)
0)E[0,nJ

is found, then an appropriate subgradient go(x) for the constraint function (4.60) is go(x) = d + cT(cO*)x) (4.65)
dx
To compute the subgradients for the constraint and objective functions, the extreme frequencies that satisfy equations (4.62) and (4.64) must be found. The details of this process are discussed in the following subsection.


4.4.1 Constraint Testing and Objective Function Value Computation


The optimization (4.55)-(4.56) can be performed using standard methods for constrained quasi-convex optimization, such as the ellipsoidal method outlined in Boyd and Barratt (1991). This method requires the value of the objective function and its subgradient if the current point is feasible (satisfies the constraint (4.56)), and the value of the constraint and its subgradient if the constraint is violated. As commented above, computation of either subgradient requires finding the frequency that satisfies (4.62) or (4.64). The case of the objective function is considered first, since the methods used are very similar to the methods introduced in Chapter 3.

If the constraint (4.56) is satisfied, then the function

QI/2 RefTi(eio)((x,eJW)+ T2(e0)P(xe ) 2
Re Jo(x,ei')} (4.66)


is non-negative for all o e [0, 7r]. Therefore, it is possible to find a stable rational transfer function whose magnitude matches the value of ((x) for all co e [0,it], using the methods of Chapter 3. Then the objective function value is found as the H. norm of this transfer






71


function, and it is also possible to compute the maximizing frequency so that the subgradient can be computed. There are several additional points that must be addressed before the method of Chapter 3 is used to find the transfer function fit to (4.66). The first is that only the real part of Ti(ejw)a(x,ew)+ T2(e'))p(x,e'") is retained. Since the quantity TI(ejW)ca(x,ejl)+T2(eO)P(x,ejW) is a transfer function vector, this is equivalent to computing the real part of each element. Given any general polynomial in z f(z)=f. znP+...+fo+...+f-nf Z~"" (4.67)

then the real part of this polynomial, evaluated along the unit circle, is

Reif(ejo) ) = fn cos(n Po) +... + fo + ...+ f-nn cos(nno) (4.68) and if it assumed that nn>np, then

Reif(e'") = fO + (LI + f1)cos(o) + . ..+(fn + f-n )cos(nP )+


f(-nP-1)cos((n +)Co)+...+ f n cos(noO) (4.69)

This is equivalent to the frequency response magnitude of the polynomial fr(z):=I fnn zn" +...+ f(_n I)Zni +1 (Ln + fn )z nP +...+ (fI + f1)z + fo +
2 -2 P_ 2 P P 2

(fI +...+ (f f I f +fn )z np + f(-n pi)Zp- +...+ I f 'n "(47
22 P P 2 P2 -n (4.70)

where it is noted that the frequency response of (4.70) is purely real by definition. Therefore, it is possible to compute the coefficients of a polynomial whose frequency response magnitude is equivalent to the real part of any individual element of the vector T(ejw)x(x,e'o)+ T2(ejw)o(x,ejw), and all of these elements can be stacked to forma matrix of coefficients. The effect of the matrix multiplication by Q1'2 is accounted for simply by multiplying the matrix of coefficients on the left by Q1 12. Each polynomial is then multiplied with itself to achieve the effect of the squaring involved in the two-norm, all of the coefficients corresponding to like powers are summed, and finally a spectral factor is






72


found for the numerator to account for taking the square root. Before this is done, a common denominator for the two-norm part and Retc(x,ejl)I is found and extracted. A spectral factor for Re (X(x,ejl)} is also found, but it is multiplied by itself since there is no need to take the square root of Re+c(x,ejW) .

The constraint (4.56) can be tested using the Kalman-Yakubovich-Popov lemma. The form of this lemma given in Rantzer (1993) is reproduced here:

Lemma 4.2.(Kalman-Yakubovich-Popov) Given G(s)=C(sI-A)-1B + D with A stable, the following statements are equivalent:

(i) G(s) + G*(s) < 0 for all Res> 0

(ii) There exists a positive definite, symmetric matrix P such that P 0]A B] +A CT][pO] < 0
0 I C D_ _BT DT _ I (4.71)

For a scalar G(s), condition (i) becomes Re(G(s)) < 0. For the discrete time case, the transfer function u(x,z) is transformed into continuous time by means of a bilinear transformation. Define

c(x,z) = j x,I+ := c'(x,s) (4.72)


and note that

(' x, I = C(x,z) (4.73)


If Re{a'(x,jo)} > 0 for o 0 implies Re ac'x(, Ie.6' > 0 for 6 e [0,r], then, since

Re a'{x, .0 = Re1ci(x,ej )1 (4.74)


Lemma 4.2 can be used to test Re{o'(x,jo)} > 0 instead of Re cL(x,ejw)}. Now I--e- _ 1+2jsinO-l .( sinO (4.75)
1+ e-10 2+2cosO I +cos(






73


( sine
so~ 1+ w os )' then as O varies from 0 to 7r, w varies from 0 to -o. Therefore, with
(I+ Cos 0 _e-i )
this definition of o, Re~j'(x,jo)} > 0 for o 0 implies Re I x e J > 0 for 0 e [0, t]. The transfer function 0'(x,s) is a scalar, but we wish to test Re(&(x,s)) > 0 so we take '(x,s) = -G(s). A state space description of a'(x,s) can be found easily in controller canonical form, and the state matrix A will be stable by definition of the basis functions chosen. Given the state space description of W'(x,s) as [A,B,C,D], Re( c'(x,s)) > 0 for all Re s 0 becomes equivalent to the existence of a positive definite, symmetric matrix P such that

P 0[ A B] + [A T CT][p ] < 0 (4.76)
0 1 _-C -D. BT _D T .0 1.

This condition is a linear matrix inequality (LMI) feasibility problem in the matrix variable P, and (4.76) can be checked using the LMI toolbox in Matlab. This formulation of the constraint is very useful in the continuous time version of this controller design method.



4.5 Robust Control Design With Steady State Disturbance Rejection

Offset-free regulation in the presence of asymptotically constant disturbances will not be attained in general from the robust controllers designed using the methods of the previous section. In this section, a simple modification derived in Hrissagis et al. (1996) is discussed that ensures that the robust controller obtained provides integral action. The nominal predictive controller (4.5) leads to the nominal regulation transfer function y(z) A(z)R(z)
d(z) A(z)R(z)+B(z)S(z) (4.77)
from which it follows that limy(t)=0 for step disturbances as well as for other disturbances with a constant steady state because R(l) = 0. On the other hand, the nominal regulation transfer function for the parameterized robust controller (4.24) is y(z) _ A(z)R(z) A(z)B(z) (4.78)
d(z) A(z)R(z)+B(z)S(z) A'(z)






74


Because the synthesis procedures described in the previous section do not necessarily yield a Youla parameter satisfying Q(l) = 0, the robust predictive controller may display unacceptable nominal regulation performance at the steady state, unless the nominal plant has an integrator (that is, A(l)=0) or is a self-regulating process (B(1) = 0).

Clearly, the robust predictive controller will attain perfect steady state disturbance rejection for all the plants belonging to the uncertain family (4.25) only if the Youla parameter has a zero gain, that is, Q(l) = 0. This gain constraint can be introduced in the robust predictive controller design through a simple modification of the factorizations (4.18) - (4.19). First, the integrator is extracted from the nominal predictive controller by writing R(z) = (z-l)R'(z), and then (4.17) is rewritten in the form A(z)(z - 1)R'(z) B(z)S(z)
Al(z)A2(z) A,(z)A2(z)

Introducing the modified coprime factorization (z - 1)A(z) B(z)(
zA,(z) A(z)

and

Z(z):= S(z) (z):=zR'(z) (4.81)
A2 (Z) A2(z)

leads to operators that satisfy the Diophantine equation N(z) X(z) + M(z) Y(z) =1. Note that this modified factorization is equivalent to augmenting the nominal plant with an integrator. Thus the modified plant output is -:= y so that (4.30) becomes z-1

[z' _ G11(z) G12(z) lw~
G21(z) G22(z)J u_ (4.82)

where

G21(z) - z G21(z) (4.83)
z-l
22(z) = Z G22(z) (4.84)

These modified transfer functions are then substituted into (4.32) resulting in






75


11 = G11 -G12MXG21 = GI - G2 Z-1MX z G21 = TI (4.85)
z (Z-M2 ( z-Z1)


2 = -G12 -G12 M G2 (4.86)
(Z ) Z-1) Z

It is now possible to solve for a parameter N (z) using the design procedure detailed in the previous section. After a solution is found, the Youla parameter QN(z) used in the parameterized predictive control structure of Figure 4. lb is constructed by re-associating the augmented-plant integrator with the controller to obtain QJ(Z) - Z1) &N(Z) (4.87)

The final robust predictive controller design for this case is obtained by substituting the Youla parameter (4.87) and the factorizations (4.18) - (4.19) into the structure (4.24). The resulting controller includes an integrator since (4.87) satisfies the zero-gain condition QN(i) = 04.6 Inclusion of Unstructured Uncertainty

High-frequency dynamics that are often neglected in system modeling can be represented in most cases as unstructured uncertainty. In this section, unstructured uncertainty is incorporated into the uncertainty description and a modified algorithm for controller design is presented.

The structural constraint on the uncertainty description in Figure 4.2 is that the signal w must be a scalar. Therefore it is possible to include an element A(z) along with the vector 8p in the uncertainty block, provided w remains a scalar. Figure 4.8 shows one possible combination. This structure is similar to the additive uncertainty model that appears in the literature in that if all the elements of 8p are zero, then the plant block is equivalent to the transfer function

P(z) -- + A(z) (4.88)
A(z)






76


or


P(z) = Po(z) + A(z)


U


I.,


n-I
A(z)




A(z)


(4.89)


Figure 4.8 Plant block with unstructured element A(z) included.


This is the standard additive uncertainty model provided |IA(z)II 1. Obviously, an appropriate stable, proper weighting function can be extracted from any stable A(z) as A(z) = W(z)A(z) so that A(z) 1.

When the elements of 8p are non-zero, then the general transfer function from u to y is P(Z) = B(z) + AB(z) A(z) A(Z) (4.90)
A(z) + AA(z) A(z) + AA(z) and a suitable weighting function can be extracted such that B(z) + AB(z) A(z) (4.91)
P(Z) = + ()()(.1
A(z) + AA(z) A(z) + AA(z) which satisfies A(z)|[ 1.

The inclusion of the unstructured element A(z) changes the stability condition (4.35) to the following form (Rantzer and Megretski, 1994)


[ 6an- - g6ao 5bm - - - bo A]


I __ Y -IN-


-~ B z






77


Q 1/2 Re{Ti(ej&)C(x,eo)+ T2(ej" )P(x,ej"o)} 2+


TiA(ej)a(x,ejo)+ T2(ejw)I(x,ej)2 < RecL(x,ejW)} (4.92)

8 8
for ) e [0,7r]. The transfer function vectors T (z) and T2 (z) are exactly the Ti(z) and T2(z) given in (4.33), while T (z) and T2 (z) are simply

T (z) = - W(z)M(z)X(z) and 2(z) = - W(z)M2 (z) (4.93)



4.7 Examples



4.7.1 Example 1

Consider the stable nominal plant model P(z) = 5(4.94) z -.4

with the corresponding nominal plant parameter vector po = [-.4 5]T. The matrix Qp defining the plant parameter uncertainty ellipsoid is F 0.3 -0.091
1 -0.09 1.0 _ (4.95)

The allowable uncertainty region described by this ellipse is shown in Figure 4.9. It is noted that the range ao e (-1,1) defines all stable plant models.

A nominal predictive controller is designed for the system using the tuning parameters NY = 1, Nu = 1, and X = 0, resulting in the following controller and prefilter polynomials R(z)= z -1 (4.96)

S(z)= 0.28z - 0.08 (4.97)

T(z) =0.20z (4.98)






78


6


5- p0

4


3

2

1


0
-1 -0.5 0 0.5 1
a0

Figure 4.9 Allowable uncertainty region for the example.

It is readily verified that these polynomials satisfy (4.10) - (4.11). The nominal closed loop polynomial is
A (z)= z2 (4.99)

and the polynomials Aj(z) and A2(z) are taken as A1(z)= z and A2(z) = z. For this example, the pole of the basis function is taken to be a = 0. It is straightforward to show that the perturbation 6p=[0.5 0.25]T (4.100)

yields 6pTQp16 = 0.9978 and is therefore allowable. The plant model resulting from this perturbation is

P(z)= 5.25 (4.101)
z +.I

The closed loop polynomial resulting from this plant model and the nominal controller is G(z)= z2 +0.57z - 0.52 (4.102)






79


which has a root outside the unit circle, showing that the nominal controller is not robustly stable.

Robust controllers with and without integral action are designed for the choice N = 1. The optimal vector x for the robust control design without integral action is xOpt =[-0.0008 -0.2002 0.0338]T (4.103)

The objective function value (4.59) for this vector is

-(xoPt) = 0.4683 (4.104)

The Youla parameter corresponding to xOPt is Q, (Z) :-0.2002z + 0.0338 (4.105)
z - 0.0008

The final form (equation 4.58) of the robust predictive controller for this Youla parameter is C1(z) = 0.0798z 2+0.0337z -0.0135 (4.106)
z2 +0.0002z - 0.1682

The modified design procedure, using equations (4.85) and (4.86) in the optimization (4.55) - (4.56), yields the optimal vector xOpt = [0.2448 - 0.0959 - 0.0332] (4.107)

which corresponds to the following Youla parameter Q1,i(Z) =(z-I -0.0959z - 0.0332 (4.108)
z ) z +0.2448

The final form (equation 4.58) of the robust predictive controller for this Youla parameter is

C1 (Z) =- 0.1841z3 +0.0896z2 -0.0115z - 0.0133 (4.109)
Ci()= 3 2419
z -0.2757z2 -0.5583z - 0.1660 The servo performance of the nominal predictive controller, along with both robust controllers, is shown in Figure 4.10. No uncertainty is assumed present in the plant model, i.e., 8p=O. A unit step change in the set point occurs at t = 0 and t = 50 sec. A step disturbance of magnitude 0.2 occurs at t = 12 sec. By design, both the nominal






80


1.4 1.2


0.8


0


0.6

0.4


0.2

0

-0.2


0


Figure 4.10


0.25


20


40


Time, t


60


80


100


Performance comparison of the three different control designs: the nominal controller(:), robust controller without integral action (-.) and with integral action (--).


0.21-


- 0.15

C 0.1
0
0.05

. 0
0
0
o -n w;


-0.1

-0.15


0


Figure 4.11


20


40


Time, t


60


80


100


Control actions produced by the three different control designs: the nominal controller(:), robust controller without integral action (-.) and with integral action (--).


-- - - - - - - - - - - -

-- - - - - - - -- - - - - - -





-L


I

- -.
-- ---- ------ ------ ------ ------ - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - -


.






81


1.5



1


=3

0


0.5


0


-0.5'
(


20


40


Time, t


60


80


100


Figure 4.12 Performance comparison of the different control designs for the perturbed
plant: the nominal controller(:), robust controller without integral action (-.)
and with integral action (--).


0.5





0
0


0



-0.5


0


20


40


Time, t


60


80


100


Figure 4.13 Control actions produced by the different control designs for the perturbed
plant: the nominal controller(:), robust controller without integral action (-.)
and with integral action (--).


... . . ......... . . . .
I.............
I..............

II1 ......... . ...........


. . a . .. .. .... ....... 1.... .. ... ... .. . .. .. .. . ..


- - -- -
:






82


predictive controller and the robust controller with integral action exhibit no steady-state offset, while the robust controller designed without integral action does produce offset. The disturbance rejection properties of the three designs are quite similar.

Figure 4.11 shows the control actions produced by the three control designs. Again, all three controllers show similar behavior, and it is noted that all three designs require only modest control actions in regulation and disturbance rejection for the nominal plant.

Figure 4.12 shows the response of the three predictive controllers when the plant is given by the model (4.101). As noted above, the nominal controller produces an unstable closed loop for this specific plant model, and the vertical range has been restricted to highlight the responses of the two stable designs. The robust controllers both produce stable closed loops with the perturbed plant, however, there are more noticeable transients after the set point changes. The disturbance rejection properties of the two robust designs are still quite good. The steady-state offset of the original robust control design is still very apparent.

Figure 4.13 shows the control actions produced by the three control designs in

controlling the perturbed plant. The nominal controller requires actions of larger magnitude than shown on this plot; the vertical range is truncated to highlight the control actions of the robust controllers.

It is of interest to find out if it is possible to design a predictive controller for the

nominal plant that is robustly stable. Table 4.1 shows the results of designing predictive controllers for different values of NY and Nu. The value of kN is given for the choices of NY and N, shown, and as discussed in Chapter 2, values less than one correspond to a robustly stable controller. For all the entries of Table 4.1, the move-suppresion parameter X is set equal to zero. For all the entries where Nu > 1 the predictive controller given in equations (4.96)-(4.98) is recovered, which is not robustly stable.






83


Table 4.1 Nyquist robust stability margin for nominal predictive controllers
designed using various NY and N, values.

NY \N 1 2 345

2 0.9165 1.0954

3 0.8595 1.0954 1.0954

4 0.8332 1.0954 1.0954 1.0954

5 0.8187 1.0954 1.0954 1.0954 1.0954


4.7.2 Example 2


The second example is the stable nominal plant model

P(Z)= 20.30z + 0.42 z + 1.20z + 0.54


with the corresponding nominal plant parameter vector po = [1.20 0.54 0.30 0.42IT. The matrix Qp for this example is


(4.110)


0.1000
0.0490 P =~ 0.0140
0.0105


0.0490 0.0700 0.0275 0.0130


0.0140 0.0105 0.0275 0.0130 0.1200 -0.0208
-0.0208 0.0600


A nominal predictive controller is designed for the system using the tuning parameters NY = 4, Nu = 2, and X = 0, resulting in the following controller and prefilter polynomials R(z)= z2 - 0.5402z - 0.4598 (4.112)


2
S(z) =0.2 197Z + l.5860z+0.591 1


T(z) = 1.3037Z4 - 0.8814z3 + 0.5 lOlz + 1.4643z The nominal closed loop polynomial is A*(z) = z4 +0.7257z3


(4.113) (4.114)


(4.115)


(4.111)






84


and the polynomials A1(z) and A2(z) are taken as A1(z) = z2+0.7257z and A2(z) = z2. Again, the pole of the basis function is taken to be a = 0. It is straightforward to show that the perturbation

6p=[0.07 -0.10 -0.27 0.10]T (4.116)

yields 8pTQ-, 1p = 0.9627 and is therefore allowable. The plant model resulting from this perturbation is

P(z)= 0.03z +0.52 (4.117)
z +1.27z+0.44

The closed loop polynomial resulting from this plant model and the nominal controller is
4 3 2
G(z) = z + 0.7364z - 0.5440z + 0.0208z + 0.1051 (4.118)

which has a root outside the unit circle, thus the nominal controller is not robustly stable.

Robust controllers with and without integral action are designed for the choice N = 2. The optimal vector x for the robust control design without integral action is

x t= [0.6397 0.3024 - 0.4932 - 2.6443 - 1.7309] (4.119)

The objective function value (4.57) for this vector is ((xOpt) = 0.8793 (4.120)

The Youla parameter corresponding to xOPt is

-0.4932z2 - 2.6443z -1.7309 (4.121)
z2 +0.6397z+0.3024

The final form (equation 4.56) of the robust predictive controller for this Youla parameter is

-0.2735z5 - 1.3502z - 2.2453z3 - 1.4338z2 - 0. 1335z +0.1297 (4.122)
C25)=4 34.22
z5 +0.9732z +0.5697z3 +0.8074z2 +0.2559z - 0.1009

The modified design procedure, using equations (4.72) and (4.73) in the optimization (4.53) - (4.54), yields the optimal vector

x0t - [0.5843 0.3764 0.0660 -1.0629 -0.8583]T (4.123)


which corresponds to the following Youla parameter






85


Z -1 0.0660z2 -1.0629z - 0.8583 (4.124)
(z) z2 +0.5843z+0.3764

The final form (equation 4.56) of the robust predictive controller for this Youla parameter is

0.2857z5 +0.8241z4 +1.7302z3 +2.5980z2 +2.0468z+0.6249 (4.125) z5 +0.7500z4 - 0.0561z3 - 0.3488z2 -0.8590z - 0.4861 Note that the degree of this controller is only 5, since two pole-zero cancellations at the origin occur when the controller is constructed.

The servo performance of the nominal predictive controller, along with both robust controllers, is shown in Figure 4.14. No uncertainty is assumed present in the plant model, i.e., 8p=O. A unit step change in the set point occurs at t = 0 and t = 50 sec. A step disturbance of magnitude 0.2 occurs at t = 12 sec. By design, both the nominal predictive controller and the robust controller with integral action exhibit no steady-state offset, while the robust controller designed without integral action does produce offset. The disturbance rejection properties of the three designs are quite similar.

Figure 4.15 shows the control actions produced by the three control designs. Again, all three controllers show similar behavior, and it is noted that for this example all three designs require more substantial control actions in regulation and disturbance rejection for the nominal plant.

Figure 4.16 shows the response of the three predictive controllers when the plant is given by the model (4.105). The nominal controller produces an unstable closed loop for this specific plant model, so the vertical range has been restricted to highlight the responses of the two stable designs. The robust controllers both produce stable closed loops with the perturbed plant, however, there are very noticeable oscillations after the set point changes. The steady-state offset of the original robust control design is still very apparent.






86


1.4


1.2f


0.8


0


0.6

0.4


0.2

0

-0.2


0


Figure 4.14




5 r-


4

C
0


C
0
0


4

3

2

1

0


-1
0


Figure 4.15


20


40


Time, t


60


80


100


Performance comparison of the three different control designs: the nominal controller(:), robust controller without integral action (-.) and with integral action (--).


- - - - - - - - - - - - - - - -


---------------

-T
A


-- - - - - - -- - - - - - - -- - - - - - - -


20


40


Time, t


60


80


100


Control actions produced by the three different control designs: the nominal controller(:), robust controller without integral action (-.) and with integral action (--).


-



- -,


-

-- - - - - - - - - - - - - - - - - - - -


I I


1






87


1 .


I


1I


0


0.5


0


-0.5'
0


20


40


Time, t


60


80


100


Figure 4.16 Performance comparison of the different control designs for the perturbed
plant: the nominal controller(:), robust controller without integral action (-.)
and with integral action (--).


5.


4.


0


C4 0'
Q


3


2


1


0
0


20


40


Time, t


60


80


100


Figure 4.17 Control actions produced by the different control designs for the perturbed
plant: the nominal controller(:), robust controller without integral action (-.)
and with integral action (--).


: : : : : 14: L . I
. . . . . . . . ..4

II 14444141 . . . . r,-l~t~ . . .........
11 1. 1 4 1.11 . . .
I or !Ii
v141 n W 4' I I ....
.4.' . . .. . .. . . . .. .. . ..
.4 . .. 444. . . . . . .. . . ....IlI t. . . - : !.! .!t
. .r . ....: ::
:17 R .. .. . .
.3l4~ .- !1
. 444 ..... r

.4 .... .....444 ....
. . . . .. .. . .. . a


. .. . . . . .





.. .. . . . . . . .. . . . . .. .
.............
.... .... Z,...........
. . . . .................. .


I I






88


Figure 4.17 shows the control actions produced by the three control designs in controlling the perturbed plant. The nominal controller requires actions of larger magnitude than shown on this plot; the vertical range is truncated to highlight the control actions of the robust controllers. The robust design with integral action requires smaller initial control actions than the standard robust design. Both designs require similar control efforts after the onset of the disturbance.

Table 4.2 shows the values of the Nyquist robust stability margin for several different predicitive controllers designed using the values of NY and N, shown. For all the designs, ? = 0. It is interesting to note that as with Example 1, only the designs with N0 = 1 are robustly stable. However, the Nyquist robust stability margin does not continually decrease with increasing N,, as is seen from the first column of Table 4.2. The entries marked with an X represent NY and Nu pairs that result in a controller that is not nominally stable.


Table 4.2 Nyquist robust stability margin for nominal predictive controllers
designed using various NY and Nu values.

Ny \ Nu 1 2 ][_1

2 0.8767 X

3 0.8372 3.2163 X

4 0.8701 2.1186 1.7620 X

5 0.8690 2.0701 1.7132 1.8914 X


4.8 Design Equations for Nominal Predictive Control

This section provides specific design equations used to synthesize a nominal predictive controller following the approach of Crisalle et al. (1989). An equivalent formulation is






89


given by McIntosh et al. (1991). The final design equations for the polynomials (4.6) (4.8) that appear in the predictive control law (4.5) are:



N)
R(z) = z"[ 1+ z~1 Xk F (z~ )j(1 -5' ) (4.126)



S(z) = z' k F (z- (4.127)


N,
T(z) = lk z' (4.128)
i=1

where the design operators Fi(z-1) and ['i(z-1) and the coefficients ki, i=1, 2, ..., Ny are determined from the process model according the following procedure. First, rewrite the nominal plant model (4.1) - (4.3) in the equivalent form AI(zJ ) y(z)= z I B-I(z ) u(z) (4.129)

involving inverse powers of z, where AI(z-1) and B(z'-) are related to (4.2) and (4.3) in an obvious manner and are of the form A-I(z~ )= 1+a-,Iz I+a-1,2Z +...+a-1,z (4.130)

B-I(z I)= b-,o +b,iJz1 +...+b1,1b 5, (4.131)

To obtain the design operators Fi(z-1), which are polynomials of degree n (the order of the plant), solve the set of Diophantine equations

Ej(zI )A(z- )A-I(zJ I)+ z~ F (z^ = 1, i = 1, 2, ..., NY (4.132) which also yields the intermediate polynomials Ei(z-1) of degree i-i. The second design operators, the polynomials T (z-1) of degree nb-1, are obtained by decomposing the product E;(z-)BI(z-1) in the form E (z )BI(z- I)=G(z- I)+z- (Z~ ) (4.133)






90


where polynomials Gi(z-1) of degree i-I are known as the dynamic polynomials, and are characterized by the fact that their coefficients are the sampled values of the step-response of the plant (2.6.10). Note that the polynomials Fi(z-1) are identically zero if nb equals 0. In turn, the coefficients of the dynamic polynomials are used to define the nonzero elements of the Toeplitz matrix GNU known as the truncated dynamic matrix, which contains only Nu columns. Finally, the coefficients ki , i=l, 2, ..., Ny are obtained as the components of the gain vector kT=[k, k2 ... kNy ], calculated from the expression kT = [1 0 - -- 0](GIN G N, + I) GNIN (4.134)



4.9 Conclusions

A design technique to synthesize robust predictive controllers for systems subject to ellipsoidal uncertainty has been presented. The technique uses a constrained quasi-convex optimization to determine the coefficients of a fixed order Youla parameter to robustly stabilize the system. The robust controller retains the nominal servo performance of the original predictive controller designed for the nominal system. A straightforward modification is given to allow the incorporation of integral action into the robust controller design. Furthermore, the design can accommodate unstructured uncertainty, as long as the problem remains equivalent to a rank one M-A problem.














CHAPTER 5
ANALYTIC SOLUTION TO A LIMITING SYNTHESIS PROBLEM



5.1 Introduction


This chapter presents an analytic solution to the limiting case of the robust control design problem discussed in Chapter 2. The analysis method developed in Chapter 2 is not strictly applicable to the limiting case of a first order plant and constant controller. This is the case where the characteristic polynomial is of degree 1, i.e., k := n + m = 1. For this case, an analytic solution to the robust synthesis problem is available.

This chapter is organized in the following manner. Section 2 presents the limiting case discrete time problem definition. Section 3 presents the results of the robustness analysis for the special case. These results are used to produce the solution to the robust synthesis problem, which is presented in Section 4. The continuous time limiting case robustness design is presented in Section 5. An example is presented in Section 6 that illustrates the different possible results of the robust synthesis procedure. Concluding remarks are presented in Section 7.



5.2 Problem Statement

Consider the closed loop system shown in Figure 5.1, where the compensator is a static-gain controller k e 91, with k # 0, and the plant is the discrete time system P(Z; P) = P(5.1) which is characterized by the parameter vector p = [x O]T.


91






92


k -.OJkP(z)

t
Figure 5.1. Feedback loop with plant P(z) and controller k. The plant parameter vector is modeled as p = pO + 6p (5.2)
where p0 = [XO P0]T , PO # 0, is a vector containing the known nominal values of the plant parameters, and the uncertainty 8p is an element of the set 'E := {8p e 912 6ppTQ-I p (5.3)
where
Q 11 q12 ]9122
q12 q22.
is a symmetric positive-definite matrix.

With the plant model (5.1) placed in the closed-loop of Figure 5.1, the characteristic polynomial is simply G(z) = z + g where g = a + ko. It is obvious that the closed loop will be robustly stable if and only if all possible values of g satisfy IgI < 1. Therefore, it must be established how the ellipsoidal uncertainty (5.3) affects the characteristic polynomial coefficient.

The relationship between the plant parameter vector p and the characteristic polynomial coefficient g is given by the linear map g = Sp where S = [1 k]. The nominal values of the plant parameters map to a nominal characteristic coefficient g0 = Sp0 or, explicitly, g0 = 0 + kP0. It follows that the uncertainty in the coefficient of the characteristic polynomial 8g = g - gO is of the form 6g = S6p (5.4)
where 8g is a real scalar. As 5p takes on all values in Ep, (5.4) generates a family of characteristic polynomials

G = { G(z) I g = g0 + 8g, 6g = S~p, 6p e 'EP} (5.5)






93


It is not difficult to show that when 8p belongs to the set Ep defined in (5.3), then 6g given by (5.4) lies in a continuous, real interval Eg that is symmetric with respect to zero. That is, Eg is of the form

E8 = I 8g e Y I g 8g*, 8g*e , 8g*> 0} (5.6)
The general theory of Guzzella et al. (1991) is not applicable to the limiting plant/controller pair considered in this chapter because the line-segment (5.6) is in fact a degenerate ellipse. Since the condition for robust stability is IgI < 1, the endpoint 8g* of the characteristic interval 'E is of particular significance to the analysis of robustness. The explicit form for 8g* is given in Theorem 5.1

Theorem 5.1 Consider the closed-loop system of Figure 5.1, with k e 91 and the family of plants P(z) described by (5.1)-(5.3). Then the uncertainty 8g in the coefficient of the characteristic polynomial lies in the interval Eg= {g e % I 1gl JSQS } (5.7)


Proof. Writing the endpoint of the interval 'Eg in (5.6) as 8g* = S6p* where 8p* some element of 'Ep, it follows by linearity that 8p* must lie on the boundary of the ellipse 'Ep, i.e., 8p* satisfies

8p*TQ- = 1 (5.8)

The value of 5g* is found by solving the optimization problem 6g* = max S~p (5.9)
8p E EP

where 'E is defined in (5.3). This constrained optimization problem is solved using the Lagrange multiplier method. Let

J(8p) = S6p - (8pTQ-I8P
2
be the Lagrangian, where X > 0 is the unknown multiplier. Equating to zero the derivative of J(8p) with respect to 5p and solving for the optimal value of 8p yields 8p* = XJIQST (5.10)






94


The Lagrange multiplier is readily obtained by substituting (5.10) into the constraint (5.8), yielding k2 = SQST. Substitution of (5.10) into 5g* = S~p* yields 8g* = k-ISQST (5.11)

from which it follows that

5g* = (SQST )1/2 (5.12)

The quadratic form SQST = qi I + 2q12k + q22k2 is a positive scalar for all values of k, since S is never equal to the null vector and Q is a positive-definite matrix. Therefore, the uncertainty 8g satisfies

16gl [SQST (5.13)

and the uncertainty region 'E given in (5.6) can be rewritten in the form 'Eg = g e 911 1g| : VSQS} V


It is of interest to note that if the continuous-time limiting case is considered, the resulting closed loop characteristic polynomial is exactly identical to the discrete time case except that the discrete variable z is replaced by the Laplace variable s. Therefore, the analysis of the previous section holds without modification except in this case the robust stability criterion is that g be positive for all values of the uncertainty 8g. The continuous time case will be considered further in Section 5.



5.3 Robustness Analysis

This section presents a criterion for testing whether a given controller k places the roots of all of the members of the characteristic polynomial family G inside the unit circle.

Theorem 5.2 Consider the closed-loop system of Figure 5.1, with k E 91 and the family of plants P(z) described by (5.1)-(5.3). Furthermore, let q = SQST. Then the closed loop system is robustly stable if and only if Ig0! + 'Fq <1 (5.14)




Full Text

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ROBUST CONTROL DESIGN FOR SYSTEMS SUBJECT TO ELLIPSOIDAL UNCERTAINTY By HARRY MICHAEL MAHON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998

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ACKNOWLEDGMENTS I would like to express my gratitude to my advisor Oscar Crisalle for his guidance and support during my studies at the University of Florida. The latitude he allowed enabled me to investigate many interesting topics in the field of advanced control that were not directly related to my dissertation. He also allowed me the freedom to complete a masterÂ’s degree in electrical engineering, and was a source of encouragement when the extra degree seemed too daunting a task. I wish to thank Professors Spyros Svoronos, Haniph Latchman, Thomas Bullock and Rich Dickinson for serving on my supervisory committee, and for all the wisdom they have imparted to me over the past few years through courses and discussions. I would like to thank the National Science Foundation for financial support under grant number CTS-9502936. I thank my colleagues Kostas Hrissagis, V. R. Basker, Rick Gibbs, Jon Engelstad, Tony Dutka, Chuck Baab, and Serkan Kincal for their friendship and support. Much of what I have learned during my stay is due to discussions in our lab, especially with Kostas and Basker. Many other friends in Gainesville have been sources of support and inspiration and I thank you all. I would like to thank Florence Kristy Mei-Ann Doo specifically. She has been an amazing source of inspiration to me for the past seven years. I am without words to thank her enough for all that she has done. Finally I wish to thank my family for the continuous support and love that they have given me throughout my life. Without them, I would not have been able to accomplish this work.

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TABLE OF CONTENTS ACKNOWLEDGMENTS ii ABSTRACT v CHAPTERS 1 INTRODUCTION 1 1.1 Motivation 1 1.2 Objectives 4 1.3 Structure of the Dissertation 5 2 STABILITY ANALYSIS FOR ELLIPSOIDAL SYSTEMS 7 2.1 Introduction 7 2.2 Stability Analysis Continuous Time 8 2.2.1 Ellipsoidal Uncertainty 8 2.2.2 Closed Loop Analysis 10 2.2.3 Robust Stability Analysis 14 2.2.4 Computation of the Parametric Robust Stability Margin 18 2.3 Stability Analysis Discrete Time 19 2.4 Design Example for Ellipsoidal Systems 25 2.4.1 Plant Model and Ellipsoidal Uncertainty 25 2.4.2 Controller Design 28 2.4.3 Discussion 31 2.5 Conclusions 32 3 STABILITY MARGIN CALCULATION FOR ELLIPSOIDAL SYSTEMS 33 3.1 Introduction 33 3.2 Construction of F(s) 33 3.3 Robust Stability Testing Using F(s) 38 3.4 Discrete Time Stability Analysis 40 3.5 Example 44 3.6 Conclusions 48 4 ROBUST PREDICTIVE CONTROL DESIGN FOR ELLIPSOIDALLY UNCERTAIN SYSTEMS 49 4.1 Introduction 49 4.2 Nominal Predictive Control Design 54 iii

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4 . 3 Nominally Stabilizing Controller Parameterization 58 4.4 Robust Predictive Control Design 61 4.4. 1 Constraint Testing and Objective Function Value Computation . 70 4.5 Robust Control Design With Steady State Disturbance Rejection 73 4.6 Inclusion of Unstructured Uncertainty 75 4.7 Examples 77 4.7.1 Example 1 77 4.7.2 Example 2 83 4.8 Design Equations for Nominal Predictive Control 88 4.9 Conclusions 90 5 ANALYTIC SOLUTION TO A LIMITING SYNTHESIS PROBLEM 91 5 . 1 Introduction 91 5.2 Problem Statement 91 5 . 3 Robustness Analysis 94 5.4 Robust Synthesis 96 5 . 5 Continuous Time Limiting Case 98 5.6 Example 100 5.7 Conclusions 101 6 FUTURE DIRECTIONS 103 6 . 1 Ellipsoidal Systems with Delays 103 6.2 Performance Constraints 103 6 . 3 LMI-based Robust Control Design 104 APPENDIX 105 1. Proof of Lemma 2.2 105 2. Proof of Lemma 3.1 107 3 . Proof of Theorem 3.2 109 4. Proof of Theorem 3.3 112 5. Proof of Lemma 4.1 116 6. Proof of Theorem 5.3 118 REFERENCES 121 BIOGRAPHICAL SKETCH 126 IV

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ROBUST CONTROL DESIGN FOR SYSTEMS SUBJECT TO ELLIPSOIDAL UNCERTAINTY By Harry Michael Mahon May 1998 Chairman: Dr. Oscar D. Crisalle Major Department: Chemical Engineering A robust control design methodology is presented for systems subject to ellipsoidal uncertainty. Both the robust analysis problem and the robust synthesis problems are considered, and both discrete and continuous systems are considered. For continuous systems subject to ellipsoidal uncertainty, a frequency dependent rational function is constructed to measure the stability margin at each frequency. This rational function may have multiple minima, therefore a frequency search must be carried out to find the overall stability margin of the system. An alternative method for testing robust stability is developed that is based on constructing a stable transfer function whose frequency response magnitude is equal to the stability margin of the system under consideration. The overall stability margin is then found as the Hoo norm of this transfer function. The construction of this transfer function v

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requires performing two spectral factorizations. This method is applicable to both discrete and continuous systems. A robust control synthesis procedure for systems subject to ellipsoidal uncertainty is also developed. This method consists of constructing a fixed order Youla parameter to achieve a robustly stable closed loop for the specified level of uncertainty. The appropriate coefficients of the Youla parameter are found through a constrained quasi-con vex optimization. The design methodology is applicable to any nominally stabilizing controller, but the specific case of a predictive controller is considered in this work. The resulting controller is a predictive controller that is robust with respect to real parameter variations, and it also retains the nominal servo performance of the original predictive controller. The design procedure is easily modified to allow the incorporation of integral action into the robust controller, guaranteeing the offset-free rejection of asymptotically constant disturbances. It is also possible to incorporate a slightly restricted form of unstructured uncertainty into the design. vi

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CHAPTER 1 INTRODUCTION 1.1 Motivation The control engineer always faces the problem of uncertainty in the plant model used for control design. This uncertainty arises from numerous sources, including unmodeled or neglected dynamics, use of lumped parameter models, and variation of model parameter values. To compensate for the fact that the model used is not an exact representation of the actual system to be controlled, it is common practice to design a controller that will yield acceptable performance for a set or family of plants that includes the specific (nominal) model used. If the true system is modeled well by some plant in this family, not necessarily the nominal model, then the controller will adequately regulate the true system. Robust control is the process of designing a controller for a set of plants rather than a single model. This design technique requires the solution of two closely related problems: the robust analysis problem and the robust synthesis problem. The problem of testing whether or not a closed-loop system remains stable when subject to perturbation, called the robust analysis problem, has been studied since the earliest days of feedback control theory. Indeed, much of the pioneering work of this field, such as that done by Nyquist (1932) on feedback amplifiers, considered system stability subject to parameter variation. The classical stability margins-the gain margin and the phase margin-are consequences of this concern about stability robustness of closed-loop systems. However, the only available analytic tool for testing the stability of a characteristic polynomial was the Routh-Hurwitz criterion. The stability conditions derived from the criterion yielded quite unwieldy algebraic inequalities, even for a small number of 1

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2 uncertain parameters, so it appeared the task of deriving succinct robust stability conditions might be insurmountable. Less attention was given to this problem as the field of optimal control matured. The linear quadratic regulator (LQR) design developed in the early 1960s seemed to be capable of producing controllers that "guaranteed" good stability properties. However, Doyle (1978) demonstrated that the stability "guarantee" disappears when the LQR is implemented as output feedback. This result emphasized not only the importance of solving the robust analysis problem, but also of solving the robust synthesis problem, that is, actually designing a controller that could guarantee stability for a family of plants. The stability analysis of systems subject to real parametric uncertainty began to receive renewed attention with the result of Kharitonov (1979) on the stability of interval polynomials. This important result states that the stability of a general characteristic polynomial with coefficients that vary independently within real intervals can be checked by testing the stability of only four specific polynomials, regardless of the original polynomial order. Suddenly, it appeared that analytic solutions might be available for many types of real parametric uncertainty. Interval systems received the lion's share of the resulting interest. Several strong results, such as the sixteen plant theorem of Barmish et al. (1992), and the generalized Kharitonov theorem due to Chapellat and Bhattacharayya (1989) followed, but these restrict the allowable controller structure, and represent only analysis results. A more general exact method for analysis of real parametric uncertainty is given by deGaston and Safonov (1988), however, this approach requires a large numerical effort for computation of the stability margin. It is noted that all of these methods assume that the allowable parameter variations are independent. For systems whose parameters are interdependent, fewer results are available for the computation of the stability margin. Soh et al. (1985) present a method for calculating the largest stability hypersphere directly in the coefficient space. This technique is applicable to continuous-time or discrete-time characteristic polynomials. Biemacki et al. (1987) extend this result to computing the stability hypersphere directly in the plant parameter space, for

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3 multiple input or multiple output systems. Also of particular significance is the approach of Tsypkin and Polyak (1991) who provide an elegant frequency domain method for testing the robust stability of a continuous-time characteristic polynomial whose coefficients could vary within an t? p -ball. Similarly, Guzzella et al. (1991) develop a method for analyzing the robust stability of systems subject to ellipsoidal (weighted l 2 ) uncertainty. The ellipsoidal uncertainty description is a specific class of parametric uncertainty that assumes the nominal transfer function model is of fixed order but has real coefficients that lie inside an ellipsoid in the coefficient space. This uncertainty description is ideally suited to the case where parameter estimation techniques are used to determine the coefficients of a nominal plant transfer function and also a parameter covariance matrix. The matrix can be used to construct an ellipsoid that describes, in statistical terms, the expected values of the plant transfer function coefficients. The works of Fogel and Huang (1982), Belfonte and Bona (1985), and Belfonte et al. (1990) consider algorithms where parameter estimates are constrained to lie inside ellipsoidal domains. The direct incorporation of the identification information into the uncertainty description is highly desirable. The focus of this dissertation is the robust control of systems subject to ellipsoidal uncertainty. The robust synthesis problem was first studied in detail for the case of unstructured uncertainty. Here, the true system is modeled as a nominal transfer function matrix subject to additive or multiplicative perturbation. The only information known about the perturbation is a norm bound. In this case, the robust analysis problem is straightforward to solve. For unstructured uncertainties the powerful results of Hoc theory (Doyle et al. 1989) were developed to solve the robust synthesis problem, for both the single-input, single-output (SISO) and multi-input, multi-output (MIMO) cases. In most situations, more information than simply a norm bound is known about the uncertainty affecting the system; this is referred to as structured uncertainty. Ideally, any available information about the uncertainty should be incorporated into the control design to reduce conservatism. This added information often makes both the robust analysis and

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4 synthesis problems difficult to solve explicitly. Perhaps the strongest and most complete result is the structured singular value (p) analysis method introduced by Doyle (1982), and the almost identical multivariable stability margin (1cm) method introduced by Safonov (1982). For any linear plant subject to uncertainty, the p-analysis method involves constructing a fictitious nominal system transfer matrix and a block diagonal matrix whose blocks correspond to the actual uncertainty affecting the system. Then, the value of p corresponds to the smallest uncertainty that will destabilize the system. This value is also referred to as the stability margin of the system. Unfortunately, the exact computation of |l is NP hard in general (Braatz et al. 1994), and usually a convex upper bound is computed instead. The p synthesis procedure uses the upper bound to compute a controller that is a local minimizer of the structured singular value. Since the optimization involved is not convex in both of the variables involved, a local optimum is the best that can be guaranteed. The presence of real scalar uncertainty blocks complicates both the robust analysis and robust synthesis problems. A convex upper bound is available for the computation of p with real parameter uncertainty (Young 1994), and a controller design procedure employing this upper bound has been proposed (Young 1996), but it will also, in general, find a local optimum. The generalized structured singular value (Chen et al. 1994a, Chen et al. 1994b) is a metric similar to p but allows for different norms to be used on the real and complex blocks of the uncertainty matrix, thereby allowing an analysis of more general uncertainty descriptions. Braatz and Crisalle (1997) have extended the generalized structured singular value to include ellipsoidal uncertainty descriptions. However, there are no controller design methodologies based on the generalized structured singular value. 1.2 Objectives The first goal of this dissertation is the development of a new method for testing the robust stability of ellipsoidally uncertain systems. A method for the computation of the

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5 robust stability margin for ellipsoidal systems is developed that employs a bisection search instead of a frequency search. Two spectral factorizations are performed to construct a stable, real-rational transfer function whose magnitude corresponds to the system's stability margin. The stability margin is found as the Hoo norm of this transfer function, allowing a bisection search to be used to compute the stability margin. The method is also applicable to both continuous time and discrete time systems. The second goal is to develop a robust controller synthesis method for ellipsoidal systems. A control design method is presented that is applicable to any ellipsoidally uncertain system with a stabilizing nominal controller. The design of a robust controller based on a nominal predictive controller is considered specifically in this dissertation. Predictive control is a class of control designs that uses knowledge of the future set point values to explicitly predict the future plant output, and compute a control law that will drive the output as close as possible to the desired set point. Predictive control is wellestablished in process industries (Seborg 1994) mainly due to its flexibility and (relative) simplicity. However, few results are available for the robustness analysis of predictive controllers, especially for the case of real parametric uncertainty. Therefore, the design procedure presented here addresses a relevant topic in robust control. 1.3 Structure of the Dissertation The dissertation is organized as follows. In Chapter 2 the problem of robust analysis of ellipsoidally uncertain systems is introduced. A necessary and sufficient condition for robust stability of continuous time systems subject to ellipsoidal uncertainty is derived, using frequency domain analysis. The condition is similar to that proposed for discrete systems by Guzzella et al. (1991). The problems associated with this method are discussed, and a design example is presented to illustrate the possibility of controller design based on the proposed robust stability condition.

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6 Chapter 3 presents a new method for stability analysis of ellipsoidal systems. The method consists of constructing a stable transfer function whose frequency response magnitude is equivalent to the stability margin of the uncertain system. This transfer function is constructed from knowledge of the nominal plant, a nominally stabilizing controller, and the matrix describing the uncertainty ellipsoid. Two spectral factorizations are performed to determine the numerator and denominator polynomials of the transfer function. The Nyquist robust stability margin can then be found as the Hoc norm of the transfer function. The method is applicable to both discrete and continuous systems. In Chapter 4, the construction of a controller robust with respect to ellipsoidal uncertainty is developed. The underlying method is due to Rantzer and Megretski (1994), and is applicable to continuous or discrete systems. In this chapter, the discrete case is considered, and further, the nominal controller for the system is assumed to be a predictive controller. Robust controllers that retain the nominal performance of the predictive controller are derived. The proposed method is extended so that the resulting robust controller exhibits integral action, assuring offset-free rejection of asymptotically constant disturbances. Chapter 5 considers a limiting-case robust control synthesis problem for ellipsoidal systems. The plant is assumed to be first-order and the controller considered is a static gain. This case is of interest because the analysis of Chapter 2 is not strictly applicable and also because it is possible to directly synthesize a maximally robust controller for the system. Chapter 6 discusses the future directions of this dissertation. Several possible extensions of this work are described, along with a discussion of their importance, and possible methods for solution of these problems are given.

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CHAPTER 2 STABILITY ANALYSIS FOR ELLIPSOIDAL SYSTEMS 2.1 Introduction This chapter discusses methods for stability analysis of ellipsoidally uncertain systems. For a continuous time system subject to ellipsoidal uncertainty, a necessary and sufficient condition for robust stability is derived. For discrete time systems, the necessary and sufficient condition of Guzzella et al. (1991), is discussed for comparison. In both the discrete and continuous time cases, the stability condition is that the magnitude of a realrational function of frequency be less than one. The convex nature of the uncertainty regions in the frequency domain allows an analytic form for this function to be derived. This function can have multiple local maxima and minima; a frequency sweep is required to find the global maxima. A necessary and sufficient condition for the robust stability of a continuous time system subject to ellipsoidal uncertainty is given in Section 2.2. The results of this section are continuous time counterparts of the results given in Guzzella et al. (1991), although an independent derivation is provided. Section 2.3 contains a brief review of the discrete time results of Guzzella et al. (1991), highlighting the differences between the continuous time and discrete time cases. Section 2.4 presents an example that uses the stability margin defined in the previous sections to compute a robust controller for a water heating system, and the performance of this controller is compared to controllers designed using a nominal performance measure. 7

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8 2.2 Stability Analysis Continuous Time A derivation of the robust stability analysis problem for ellipsoidally uncertain continuous time systems is given in this section. In subsection 1, the general ellipsoidal uncertainty structure is introduced. The closed-loop frequency domain uncertainty regions that arise due to the ellipsoidal uncertainty in the plant are discussed in subsection 2, along with the derivation of a graphical stability test for ellipsoidally uncertain systems. Subsection 3 contains the necessary and sufficient conditions for robust stability of the continuous time ellipsoidal uncertainty problem. Subsection 4 discusses the relationship between the parametric stability margin and the Nyquist robust stability margin. 2.2.1 Ellipsoidal Uncertainty The general plant considered in this analysis is the linear, strictly proper plant = fc„V' + ... + fc n _ BW s n + a n _\S n 1 + . . . + a 0 A(.s) which is represented by the coefficient vector P = K-1 ••• a 0 b n-\ b o] T e (2.1) (2.2) The values of the coefficients in this vector are uncertain, however, a nominal value of the vector is assumed known. The actual value of the parameter vector is represented by the nominal value plus an additive perturbation p = p°+5p (2.3) This perturbation vector is constrained to lie in an ellipsoid Tp in the parameter space. This ellipsoid is defined by a positive definite, symmetric matrix Q p such that % p = {s P E | 5p t q;'5p < 1 } (2.4) It is noted that the numerator and denominator polynomials of (2.1) can be expressed as a nominal polynomial and a perturbation polynomial as follows , bV) + AB(5) P{S) = — rr A u (s) + AAfO (2.5) where

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9 AA(s) = ' + . . . + 8 oq and AB(s) = $b n _[S n 1 + . . . + 8b 0 The coefficients of these perturbation polynomials are captured in the vector 8p 5p = [5a„_, ... 5a 0 8Vi S^ 0 ] T e ^ 2 " ( 2 . 6 ) An example of an ellipsoidal uncertainty region for a two parameter system is shown in Figure 2.1. Figure 2.1. Ellipsoidal uncertainty region Figure 2.2. General feedback interconnection Ellipsoidal uncertainty descriptions arise naturally in parameter identification techniques where the uncertainty ellipsoid is associated with the parameter-error covariance matrix. The covariance matrix can be used to define a matrix Q p that corresponds to a certain confidence level for the parameters. Ellipsoidal uncertainty descriptions are commonly encountered in chemical engineering applications where model parameters are found by fitting experimental data using linear or nonlinear regression techniques. Such uncertainty descriptions have been adopted in several studies, including Biernacki et al. (1987) where ellipsoidal domains are used for the analysis of systems characterized with weighted perturbation bounds. + v k Ellipsoidal parametric uncertainty models also appear in various other contexts. For example, the work of Agarwal and Bonvin (1989) on Kalman filtering provides a means for estimating the covariance matrix. The publications by Fogel and Huang (1982),

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10 Belfonte and Bona (1985), and Belfonte et al. (1990) discuss ellipsoidal outer-bounding algorithms which produce parameter estimates where the uncertainty is also constrained to lie inside ellipsoidal domains of the form (2.4). 2.2.2 Closed Loop Analysis Consider the problem of analyzing the robust stability of a feedback loop containing a fixed controller and a plant subject to ellipsoidal uncertainty. Figure 2.2 shows the general feedback structure adopted, where C(s ) is a proper controller given by C(s) = ^ m s n ' + ... + P 0 P(s) (2.7) s m +a m _ x s m 1 +... + a 0 a(s) and is represented by the coefficient vector c c=K,_i ... « 0 P„, ... M t s ‘* 2 ”‘ +1 < 2 8 > The plant P(s) is given by (2.1) and subject to the ellipsoidal uncertainty described by equations (2.3) and (2.4). It is assumed that the plant order is at least as large as the controller order, i.e., n>m. Furthermore, it is also assumed that the index k := n + m satisfies k >2. The case of k = 1, corresponding to a first order plant ( n = 1) and a constant controller (m = 0), is a limiting case that is discussed in detail in Chapter 5. The characteristic polynomial of the feedback loop G(5) = G°(5) + AG(5) (2.9) is the sum of nominal and perturbation polynomials, where G°(s) =
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11 s c = 1 0 a m-l 1 a 0 0 0 0 Pm 0 Pm-1 P 0 1 Po a m1 a 0 0 0 0 Pm Pm— 1 Po e 91 kxln and c = [oc m-\ ••• «o 0 ••• °] T e * k The nominal value of the characteristic vector is generated when the plant parameter vector assumes its nominal value, i.e.. g° = s cP ° + c = 8k 1 8k2 80 (2.13) The difference between the actual and nominal values of the characteristic vector is given by the perturbation term 5g:=g-g° =S c p + c-S c p°-c = S c (p-p°) = S c 5p (2.14) The elements of 8g are the coefficients of the polynomial AG(s) given in equation (2.1 1). It has been shown in Guzzella et al. (1991) that if 5p is constrained to lie in the set Tp, then 8g lies in the ellipsoidal uncertainty region where 5g T Qg'5g
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12 The image of the nominal polynomial G°(s) as s varies along the imaginary axis is G°0'©) = (sS g2 0)2 + •••) + j(g\® ~ S3® 3 +•••):= T*(e>) + 7X/(co) (2.17) Defining the vector it follows that where T(co) := Tfl(CQ) L X/((0) G W T(CO) = W T ((0)g° + t(C0) W T (co) = ... -to 2 0 l" wj(a>) ... 0 to 0 _wj((0)_ g * 2\k (2.18) The vector t(0)) is given by and t(co) = [(-l)* /2 co A t(to) = X o| g ^K 2 if k is even g if k is odd 0 The image of the polynomial AG(s) as s varies along the imaginary axis is given by AG(ja)) = ( 8 g 0 8 g 2 co 2 + ...) + ;( 8 g,co 8 g 3 co 3 + ...):= 8 x^( 0 ) ) + j 8 x,((o) (2. 1 9) and can also be expressed as the vector Sx((0) := SXfl(co) 81 ,( 0 )) = W T (co)8g Guzzella et al. (1991) show that as Sp takes on all possible values inside the parameter ellipsoid Ep, the region that AG(/'co) traces out, for co * 0, is an ellipse E^ in the complex plane. This ellipse is described as ? 2 where the matrix % w = |8x((o) g 8x T (oa)Q [0 1 8T(to) < lj Qco = W T (co)Q„W(to) g ST (2.20) ( 2 . 21 ) has full rank for all nonzero frequencies. At (0 = 0, the nominal polynomial is simply G°(jO) = go ( 2 . 22 )

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13 and the uncertainty region in the complex plane degenerates into a line segment defined by (2.23) where is the ( k,k ) element of the matrix Q g defined in (2.16) and k = n + m. Uncertainty regions for a non-zero frequency (0 and for to = 0 are shown in Figure 2.3. The frequency domain uncertainty set is called a value set, or template, and plays a major role in the stability analysis of ellipsoidal systems. This analysis, as well as the classical Nyquist stability criterion, is based on a result from complex variable theory called the principle of the argument. The principle of the argument states that as a complex-valued function F(s ) is evaluated along a closed, simple curve in the s-plane, along which F(s) has no zeros or poles, and interior to which F(s) is analytic, then the net change in angle is Aarg{F(s)} = 27t(n p -n z ) where n z and n p are the number of zeros and poles, respectively, of F(s) inside the contour. The principle of the argument can be used to test the stability of the closed loop of Figure 2.2 by applying the principle to the characteristic polynomial G(s). If the stability region is taken to be the open left half of the complex plane, and the contour is the classical Nyquist contour, traversed in the clockwise sense, then the polynomial G(s) has no zeros in the right half plane if and only if Aarg{G(s)} = 0 since a polynomial has no finite poles. The above angle change is the net change as the Nyquist contour is traversed, and is the sum due to the semi-circular part and the imaginary axis part. The change in angle due to the semicircular portion of the contour is -kn. This follows from the fact that on this part of the contour s = re jd where r — » and 0 varies from y to— y. Thus G(j) -> r k e jke

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14 and the net change in angle of G(s ) is -hi. Therefore, it follows that the change in angle of G(/co) as to varies from to °° must be kn for the closed loop to be stable. Since the coefficients of G(s) are real, the plot of G(/co) on (-<*>, 0] is symmetric to the plot on [0,-°°), fCK so the change in arg{G(/a>)} on [0,°°) must be — . This means that the plot of G(/co) £ must encircle the origin times in the counterclockwise direction. This important result is summarized in Lemma 2. 1 Lemma 2.1 A polynomial G(s) of degree k has all of its roots in the open left half k plane if and only if the plot of G(/co) encircles the origin of the complex plane — times in the counterclockwise direction as to varies from 0 to A very comprehensive discussion of the applications of the principle of the argument to stability testing, including a proof of the Routh-Hurwitz criteria, is given in Porter (1968). 2.2.3 Robust Stability Analysis A robust stability test can be constructed from the result of Lemma 2.1. The nominal characteristic polynomial G°(s) is assumed to be stable, and therefore the plot of G°(/co) has the correct number of encirclements for stability, as described by Lemma 2.1. Figure 2.3 shows the uncertainty ellipses 2^ at several frequencies. The band that is swept out by these ellipses is called a Nyquist envelope. Each point in a particular ellipse represents an allowable characteristic polynomial frequency response evaluated at that particular frequency, so the envelope represents the frequency response plots G(/'co) for all allowable characteristic polynomials. If the envelope does not include the origin, then all the frequency response plots have the same number of encirclements of the origin as the nominal polynomial, and thus all allowable characteristic polynomials are stable. However, if the Nyquist envelope contains or touches the origin, then at least one allowable characteristic polynomial has a different number of encirclements of the origin

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15 than the nominal polynomial and is unstable. Therefore, the system is robustly stable if and only if the value sets exclude the origin at all frequencies. The need for the value sets to exclude zero can also be explained in terms of the root locations of the polynomial G(s). The nominal polynomial G°(s) has all its roots in the left half-plane (LHP). As the parameter vector p assumes all possible values in Tp, the roots of G(s) move. In order for the roots to travel into the right half-plane (RHP), they must cross the imaginary axis. The possibility of the roots moving due to degree dropping has been ruled out by specifying that G(s) is monic. Continuity arguments can be used to establish that if any G(s) has roots in the RHP, then an allowable G(s) exists that has at least one root on the imaginary axis. However, this implies that G(/'co) = 0 for an allowable G(s) which in turn means that the origin must be included in the value set at that frequency. Therefore, all allowable characteristic polynomials have roots in the LHP if and only if the origin is excluded from every value set. Testing this zero exclusion condition can be accomplished using the critical direction method (Latchman et al. 1997), modified here for the analysis of polynomials. Consider the following quantities: i. The stability segment is the line segment joining the nominal curve G°(/(0) and the origin. Note that the length of the stability segment is simply G 0 (yco) . ii. The critical direction is defined as the direction of the stability segment, and is characterized by the unit vector G°0'a» d(;co) = q G°0'a>) iii. The critical perturbation radius p c (co) = maxja e 91 | G°(;to) + ad(y'co) e These quantities are illustrated in Figure 2.3. It is noted that the only part of the value set important for robust stability is that part that lies along the direction to the critical point. This is the fundamental premise of the critical direction method.

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16 Figure 2.3. Stability analysis quantities As stated before, the closed loop is unstable if any value set includes the origin. The value sets do not include the origin provided the critical perturbation radius is smaller than the length of the stability segment at every frequency. Therefore, a necessary and sufficient condition for robust stability is p c (co) < G O'co) V® g [0,°°) (2.24) Equation (2.24) is particularly useful when an analytic form for p c (CO) is available. As Lemma 2.2 shows, the ellipsoidal nature of the value sets allows such a form to be derived. Lemma 2.2 Suppose the nominal characteristic polynomial G°(/a>) given by equation (2.17) is subject to ellipsoidal uncertainty described by equations (2.20) and (2.23). Then the critical perturbation radius is ||x(co)|L p c «0)= , _ 2 . 0 (2.25a) a /t T ((0)Q co ‘t(CO) Pc(0) = V^U 0) = 0 (2.25b) Proof: The proof of Lemma 2.2 is given in the Appendix.

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17 By definition of X( 0 ), and from equation (2.22), it follows that G°(;'g>) =||x(co )|| 2 G°(yO) So to > 0 m = 0 (2.26a) (2.26b) The results of Lemma 2.2 and equation (2.26) can be used to write the necessary and sufficient conditions for robust stability given in (2.24) as t T (co) Qco'x(w) < ||)|U go w > 0 0 = 0 or they can be cast as x T (0)Q (1 ) 1 x(0) < 1 go < 1 0 > 0 0 = 0 The Nyquist robust stability margin introduced in Latchman et al. (1997) for rational transfer matrices can be reformulated for the case of polynomial systems. Define the frequency-dependent Nyquist robust stability margin as 1 %(m)'t T ( 0 )Q (O 1 x( 0 ) go 0 > 0 0 = 0 (2.27) and define the Nyquist robust stability margin as k N = sup %(m) oo>0 (2.28) Theorem 2.1 Suppose the ellipsoidally uncertain plant P(s) described by (2.1)-(2.4) and the controller C(s) described by (2.7) are joined in unity negative feedback as shown in

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18 Figure 2.2, with the nominal system stable. Then the system is robustly stable if and only if % < T Proof: By definition of both Nyquist robust stability margins and Lemma 2.2, it is immediately apparent that the necessary and sufficient condition given in (2.24) is satisfied Although equation (2.27) gives a closed-form expression that can be used to test robust stability, the frequency-dependent Nyquist robust stability margin is a rational function of frequency that may have local minima. This implies that the result of Theorem stability margin. While each point of the search requires little computational effort, it is more difficult to determine the range of frequencies on which to test robust stability. The results of Chapter 3 provide a method for testing robust stability that does not require a frequency sweep, and allows the value of t0 be calculated through a bisection search. 2.2.4 Computation of the Parametric Robust Stability Margin The parametric stability margin is defined as the minimum expansion (or contraction) of the uncertainty region in the parameter space required to bring the system to the edge of stability. For ellipsoidal uncertainty, the parameter space uncertainty region is contracted or expanded by multiplying the matrix by a scalar factor a. From equation (2.16), it follows that multiplying the matrix Q p by a scalar a results in Q g changing to aQ g . The new uncertainty region in the characteristic space is thus Using this definition of a scaled uncertainty set, the parametric stability margin for ellipsoidally uncertain systems can be expressed as if and only if % < 1. V 2. 1 must be tested by performing a frequency search to find the value of the Nyquist robust $ a min a(co) co

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19 where a(to) = min] a | G(jco,5g) = 0 aeSR L 5gea£^} The relationship between the parametric stability margin and the frequency-dependent Nyquist robust stability margin is detailed in the following lemma. Lemma 2.3 Suppose the ellipsoidally uncertain plant P(s) described by (2.1)-(2.4) and the controller C(s) described by (2.7) are joined in unity negative feedback as shown in Figure 2.2, with the nominal system stable. Then the parametric stability margin is * a Proof. Let the frequency that defines be denoted co*. Then % = %( (0 *) = t t (g)*)Q C 0 1 *t(cd*) Multiplying the matrix Q p by a scalar results in Q ? changing to aQ g and also Q m changing to aQo Thus, the system with the scaled uncertainty satisfies 1 Vot k N =k N (co*) = x T (a ,*{l WUo)*) ' '(“*) = fak N Since the factor a scales all the values of the frequency-dependent Nyquist robust stability margin, the maximum for the scaled system will occur at the same frequency as that for the original system. The parametric stability margin is defined as the minimum value of a that destabilizes the system, which corresponds to k N = 1. Therefore, % = 1 = VoT*% or a* = kjj 2 . V 2.3 Stability Analysis Discrete Time The stability analysis of discrete time ellipsoidally uncertain systems is very similar to that for continuous time systems. In this section, the few differences between the two cases are highlighted, as the complete derivation can be found in Guzzella et al. (1991).

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20 In the discrete time formulation of the ellipsoidal stability analysis problem, the general linear plant considered is given by b„_, + be . B(z) Z n + a fl _iz n ~ l + ... + OQ ' A(z) (2.29) and the controller is C(z) = Pm *" 1 + • • • + Po P(^) (2.30) z m +a m _ lZ m ^+... + ocq a (z) The plant parameter vector p is exactly as defined in (2.2) and the uncertainty description is still given by (2.3) and (2.4). Also, the controller vector c is given by (2.8). The characteristic polynomial for the discrete time system is described by equations (2.9)(2.11) with the Laplace variable 5 replaced by the discrete time variable z. The characteristic polynomial degree remains k:=n + m. Furthermore, the uncertainty region “Eg is given by equation (2.15) and the uncertainty matrix Q g is given by (2.16). The primary difference between the two cases is the frequency domain analysis of the characteristic polynomial. The image of the nominal characteristic polynomial G°(z) as z varies along the unit circle is given by G V“) = (gS + . . . + cos[(* 1)0 )]gf_! + COSTCO]) +y(sin[a)]g 1 ° + . . . + sin [(* l)co + sin[£co]) (2.31) and the image of the perturbation polynomial is AG(e jC0 ) = (5g 0 + . . . + cos [(k ) + jfsinfcoiS^j + . . . + sin [(k l)co]5^ A: _ 1 ) (2.32) These polynomials can be expressed as the vectors T = Re G°(^ t0 )}' V Im < G 0 (e ;co )J x i_ = W T g°+t §x = Re {AG(e 7 '“)} "St/?" Im {AG(e y(0 )| 8x 7 = W T 5g and

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21 where t = [cos(k©) sin(kw)] T 6 9l 2 and W T := cos[(k-l)©] • • cos[©] f wj(©) sin[(k 1)©] • • sin[©] 0 _w}(©)_ e 9* 2 xk As 8p takes on all values inside "Ep, the uncertainty region AG( e JM ) that is generated around G°( at every frequency in the open interval (0,7t) is an ellipse = jfo e SR 2 5x T Q C0 1 8x < l} (2.33) where A(©) B(©) wJ(©)Q g W/j(©) w^(w)Q g w / (©) _B(©) D(w). w]'(©)Q g w /? (©) w]'(©)Q g w / (©) (2.34) The matrix Q w is rank deficient at the two frequencies 0 and K. At these points, the uncertainty regions are line segments 2b = {8 t*( 0) e SR | |5 t*( 0)| < ^wJ(0)Q g w*(0) } (2.35) £* = {&t*(tc) e SR | |5Tfl(7t)| < ^Jwl(K)Q g w R (K) J (2.36) The plot of all possible characteristic polynomials G(^) for co e [7t,27t] is symmetric about the real axis to the plot for co e [0,7i], thus only the range to e [0,7t] need be considered. The principle of the argument is used to deduce the root locations of the polynomial G°(z) by analyzing the number of encirclements of the origin of the plot of G°( as © varies between 0 and k. The only difference from the continuous time case is that now the stability region is the interior of the unit circle Izl = 1. The counterpart of Lemma 2.1 for discrete time systems is stated below.

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22 Lemma 2.4 [Guzzella et al. 1991] A polynomial G(z) of order k has all its roots inside the unit circle if and only if the number of counterclockwise encirclements of the The nominal characteristic polynomial is assumed to be a Schur polynomial, i.e., it has all its roots in the unit disc Izlcl. All allowable characteristic polynomials have the same degree as the nominal, so the plots of all allowable characteristic polynomials must have the same number of encirclements as the nominal polynomial for the loop to be robustly stable. Therefore, for robust stability, all the uncertainty templates must exclude the origin. As in the continuous time case, the highly structured nature of the uncertainty regions allows this zero exclusion requirement to be tested in a direct manner. Figure 2.4 shows the plot of the nominal polynomial as well as one ellipsoidal value set for the frequency point co*. The value sets at co=0 and co=7t are line segments. The stability segment at the frequency co is the line segment joining the nominal curve G 0 (V w *j and the origin. The length of the stability segment is simply IG°fe / ' a) jl. The critical perturbation radius p c (C0) is the distance from the center of the value set to its boundary along the stability segment. V“*) lm origin performed by the plot of G( e /c0 ) as co varies from 0 to 2k is equal to k. Figure 2.4. Nominal curve G°(ei®), value sets at frequencies co = 0, co*, n, the stability segment for co*, and the critical perturbation radius p c (co*)

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23 The robust stability condition for the discrete time case is based on the knowledge of the explicit form for the critical perturbation radius, which is given in Lemma 2.5 Lemma 2.5 Suppose the nominal characteristic polynomial G 0 ^ 03 ) given by equation (2.31) is subject to ellipsoidal uncertainty described by equations (2.33), (2.35) and (2.36). Then the critical perturbation radius is Pc(03) = H®)ll 2 go e (0,7t) (2.37a) Jt t (co)Q cd c((o) Pc(0) = tJ wJ(0)Q g w^(0) co = 0 (2.37b) PcW = ^wJ(7t)Q g W^(7t) © = n (2.37c) Proof: The proof of Lemma 2.5 is entirely analogous to the proof of Lemma 2.2 and is therefore omitted. The value sets % m do not include the origin provided the critical perturbation radius is smaller than the length of the stability segment at every frequency. Therefore, a necessary and sufficient condition for robust stability is P c (C0) < G°(e jU) ) V to e [0,7c] (2.38) Lemma 2.6 Suppose the conditions of Lemma 2.5 hold. Then the condition (2.38) is equivalent to the three conditions <1 to e (0,7t) (2.39a) x (co)Q a) x((o) w J?(0)Q„w/j(0) M<»l < 1 co = 0 (2.39b) Jwl(n)Q w R (n) ; — < 1 M*) to = n (2.39c)

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24 Proof: The form of p c (co) is given in Lemma 2.5. For 0 ) e (0,71), by definition. For the extreme points CD = 0 and co = n, T/(cd) = 0, and G°(e/C0 ) = |Tfl(co)|. Plugging these values into (2.38) yields (2.39) immediately. V Defining the frequency-dependent Nyquist robust stability margin as 1 *n(“) = x T (co)Q <0 1 x(ca) w «(0)QpW/j(0) M°)| |X^(7T)| co e (0,7t) CD = 0 CD = 7t (2.40) and the Nyquist robust stability margin as k N = sup & N (co) coe[0,7i] then the necessary and sufficient condition for robust stability of ellipsoidally uncertain systems is given in Theorem 2.2. Theorem 2.2 Suppose the ellipsoidally uncertain plant P(z ) described by (2.29), (2.2)-(2.4) and the controller C(z ) described by (2.30), are joined in unity negative feedback, with the nominal system stable. Then the system is robustly stable if and only if k N < 1 (2.41) Proof. Lemma 2.6 shows that the satisfaction of the three conditions (2.39) is the necessary and sufficient condition for robust stability. However, the quantities on the left hand side of (2.39) are identical to those on the right hand side of (2.40). It follows immediately that the necessary and sufficient condition for robust stability is that /: n (cd) < 1 for CD 6 [0,;t], or < 1, which is equation (2.41). V

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25 2.4 Design Example for Ellipsoidal Systems In this section, a simple water-heating control system with a discrete PI controller is used for illustrating the analysis and synthesis methods introduced in this chapter. The nominal plant parameters are used to tune three candidate PI controllers using standard techniques, and then the robustness of each controller is analyzed. It is verified that controllers cannot be robustified from knowledge of the nominal process alone. Finally, a robust control design based on the results of Section 3 is realized via numerical optimization. 2.4.1 Plant Model and Ellipsoidal Uncertainty The two-tank system with recycle shown in Figure 2.5 is used as a model for illustrating the robust analysis and synthesis techniques. The control objective is to maintain the temperature of the second tank (T 2 ) at a desired set point by manipulating the power (R) delivered by the heater located in the first tank. The actuation on the heater is performed through a zero order sample-and-hold element. The only available measurement is temperature T 2 . Fi F 2 1 r \ Tl Fi +F 2 t 2 Fi+F 2 F, Fi = 0.050 m^/min F 2 = 0.150 m-Vmin p = 1000 kg/m^ C p = 4286 J/kg-C V = 1 Figure 2.5. Two mixing tanks arranged in cascade with recycle stream. The actual plant model is generated by carrying out an energy balance in each tank. Assuming that the liquid volume V remains constant and equal in both tanks one arrives at the representation T 2 (s) _ kj R(s) x 2 s 2 +2xs + k 2 (2.42)

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where 26 r k, = = 0.07 — (F,+F 2 )pC p kW k 2 = Fl = 0.25 F,+F 2 V x = = 5 min Fj +F 2 The actuation on R is carried out by means of a sample-and-hold element with sampling period T=100 sec. The corresponding discrete transfer function representation, the actual plant, is then T 2 (z) = 0.003 lz + 0.0025 R(z) z 2 1.4932z + 0.5134 The identification of the nominal plant model is carried out using a standard leastsquares parameter identification method. The input-output data needed for identifying the nominal plant G(z; p°) and the uncertainty model “Ep is generated by applying a pseudorandom binary input to the heater and recording the temperature response. In the simulation it is assumed that the temperature data are acquired by a thermocouple which perturbs the measurements with an additive white Gaussian noise with variance g 2 =4.5 10 -3 C 2 . A total of 125 samples are gathered using a sampling period T=100 sec. The collected data is then regressed using the ARX model T 2 U)= b ' Z + b ° R(z) + e(z) z + ajz + a 0 where e(z) is assumed to be a white-noise sequence of unknown variance. Hence, the plant model and parameter vector for this representation are, respectively, P(z; P)= b ' Z + b Q — , n = 2 (2.44) z + aiz + a 0 and T P = h a 0 bi Z? 0 ] The least-squares procedure minimizes the functional J(p) = (y-Hp) T (y-Hp), where y is a vector of temperature measurements and H is the regression matrix (Draper and

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27 Smith 1981). The minimum is realized by the solution vector For the two-tank plant under study the solution is p° =[-1.1871 0.2087 0.0028 0.0038] 1 yielding the identified nominal plant model 0x 0.0028z + 0.0038 P(z] p ) = — (2.45) 1.1871z + 0.2087 An ellipsoidal uncertainty model is readily available as a by-product of the leastsquares identification technique. A standard result (Draper and Smith 1981; Crisalle and Bonvin 1991) states that a 100(l-a)% joint confidence region for the parameter estimate p° is given by the ellipsoidal domain given in (2.4) >2n with = {5peS\ 2 '' 5p T Q /? 1 5p
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28 Hence, using the above form for the matrix in (2.4) represents an ellipsoidal uncertainty model which can be interpreted as a region with a confidence level of 100(l-a)% depending on the value of the F-factor chosen. The fundamental uncertainty information is in fact contained in matrix £ whose eigenvectors determine the principal directions of the ellipsoid Tp. The remaining scalar factors adjust the volume of the ellipsoid without perturbing the principal directions. Setting the F factor equal to 2.37 leads to a 95% confidence region; however, it is also possible to choose the value of the F factor to encompass a family of uncertainties of larger or smaller scope. We adopt the value F=10 for our design, thus requiring that the controller be able to stabilize a family of plants belonging to a relatively large ellipsoidal region. 2.4.2 Controller Design The controller is a discrete proportional-integral (PI) compensator C(z;c)= PlZ + f° , c = [-1 p, P 0 ] T , m = 1 z 1 whose associated Sylvester matrix Sc and vector c are 0 1 -1 Pi Po 0 0 Pi Po. and c = [-1 0 0] T For the nominal plant and uncertainty description just described, three PI controllers are synthesized using a conventional tuning method. The designs proposed, denoted Cl, C2, and C3, are characterized by the tuning parameters P] and P 0 shown in Table 2.1. The control parameters for design Cl are found using the estimated model (2.45) and the tuning settings pi=Kc(l+0.5T/Ti) and Po=-Kc> where K c and Ti are the control gain and integralmode constants of a corresponding analog PI controller (Seborg et al. 1989). In order to find appropriate constants for the analog controller, the step-response of (2.42) is first

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29 approximated by a first-order lag and a pure delay equal to one half of the sampling period. Parameters K c and Tj are then determined from the correlations minimizing the ITAE (Seborg et al. 1989). Designs C2 and C3 are intended refinements to C 1 based on the observation that the [3 1 parameter for the latter is near a nominal stability limit. In fact, when Po=-98, a nominally stable loop is obtained for 98<(3i<149. Thus, the higher values of parameter Pi in designs C2 and C3 represent an attempt to move the controller parameters away from the nominal stability boundary. Clearly, the proposed tuning refinements are made based on only nominal stability considerations. Figure 2.6a shows that all three controllers produce nominally stable closed loops. Figure 2.6. Closed-loop responses of conventional controllers Cl, C2, and C3, and of the optimally robust controller C* to a 5 °C step change in setpoint. (a) Nominal response, (b) actual response. The stability robustness of all three candidate controllers is then analyzed using the critical direction technique. Values for the robustness parameter % are calculated by means

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30 of an exhaustive numerical search using a frequency increment Aco=0.01. The result obtained is given in the last column of Table 2.1. It is concluded that the loop involving C 1 is robustly stable (&n< 1)> but that neither controller C2 or C3 is robustly stable. In particular, controller C3 produces an unstable loop when it is used to control the actual plant, as shown in Figure 2.6b. Remarkably, in this example the intuitive idea of displacing parameter Pi away from the stability boundary leads to the loss of robustness in C2 and C3 because the tuning adjustments disregard the parametric uncertainties present. From Theorem 2.2 it follows that a controller may be considered optimally robust if it is capable of producing the smallest possible value of k nHence, the problem of robust synthesis reduces to the following optimization: c* = argmin &n(c) (2.48) ce§° where c* is the optimal controller, J° represents the set of all controllers which lead to stable loops with the nominal plant, and &n(c) is the robustness parameter (2.41) for a given controller c. The constrained optimization problem (2.48) is nonlinear and nondifferentiable. In this study the synthesis problems are solved by exhaustive numerical search over the control space. This approach is viable only because of the low dimensionality of the control space. For higher dimensions this approach would rapidly become computationally intractable due to the combinatorial explosion in the number of grid points. The exhaustive search is carried out by discretizing the space of control parameters with a grid of size A(3o=A(3 1=0.5, and then calculating the values of k^(c) at every grid point. Grid points corresponding to controllers that are not nominally stable are discarded, thereby gaining execution speed. The last row of Table 2.1 shows the numerically-determined optimal values for the control and robustness parameters for design C*. The optimal design is robust since it satisfies the constraint < 1, and it achieves the smallest value of &n (0.501). The

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31 optimal design stabilizes the nominal and actual plants, as is confirmed from the stepresponses shown in Figures 2.6a and 2.6b. Table 2.1. Control and robustness parameters for conventional controllers Cl, C2, and C3, and optimally robust controller C*. Controller Controller parameters Stability margin Design P. Po % Cl 101.0 -98.0 0.842 C2 115.0 -98.0 1.260 C3 127.0 -98.0 2.041 C* 31.0 -30.5 0.501 Figure 2.7 shows several contour plots of the functional &n(c) as a function of the controller parameters (3o and Pi. The contour &n(c) = 1 defines the boundary of robust stability. The optimal design is remarkably close to the nominal stability boundary, making it very sensitive to perturbations in the control parameters. For this reason, it may not be desirable to design a controller to optimize the stability margin alone. 2.4.3 Discussion The performance in servo-response tests is documented in Figures 2.6a and 2.6b for the various designs studied. Only the nominal and actual plants are considered. The figure shows that controller C* produces low overshoot levels and minor transient oscillations. However, caution must be exercised when attempting to generalize performance observations from this example because the optimally robust controller has been designed ignoring performance constraints.

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32 It is important to remark that functional &n(c) is not convex with respect to the control vector c. Profiles calculated using very fine grids also reveal that the functionals are non smooth. As a consequence, gradient-based optimization techniques are not suitable substitutes for the exhaustive search method. Figure 2.7. Contour plot on the (Po, Pi) plane of the functional k^{c). 2.5 Conclusions This chapter presents a necessary and sufficient condition for the robust stability of a continuous time system subject to ellipsoidal uncertainty. The condition is similar to that given in Guzzella et al. (1991), for discrete time systems. The test requires a frequency search to be performed. In order to avoid the problems associated with such a search, a new analysis method that does not require a frequency search is presented in Chapter 3.

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CHAPTER 3 STABILITY MARGIN CALCULATION FOR ELLIPSOIDAL SYSTEMS 3.1 Introduction The goal of this chapter is the development of a robust stability test for ellipsoidally uncertain systems that does not require a frequency search. It is shown that the necessary and sufficient condition of Theorem 2.1 can be transformed into a condition of the form r«L < i along with an auxiliary condition for co = 0. The function F(s ) is a stable, minimumphase, strictly proper transfer function. The infinity norm condition requires only the checking of the eigenvalues of a certain Hamiltonian matrix, and thus avoids a frequency sweep. Furthermore, computing the actual value of the norm can be done using a bisection search, thereby guaranteeing a prespecified level of accuracy. Section 3.2 describes the construction of the transfer function F(s). The new method of robust stability analysis using this transfer function is developed Section 3.3. The formulation of the appropriate transfer function F(z) for discrete systems is given in Section 3.4. An example is given in Section 3.5 to illustrate the proposed method. 3.2 Construction of E(s) To test the necessary and sufficient condition for robust stability established in Theorem 2.1, the function & N (oo) must be evaluated over the frequency range [0,°°). This frequency sweep is computationally expensive and is inherently imprecise since a finite number of frequency points must be used. An alternative is to construct a stable transfer 33

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34 function whose frequency response is equal to the function k N (to) for positive frequencies. Then the maximum of k N (to) for positive frequencies can be found as the infinity norm of this transfer function. Actually, as a first step in this construction, the square of the function & N (©) will be fitted to make the mathematical developments more tractable. From the resulting form, the appropriate F(s ) can be easily constructed. The function & N (w) is defined in Chapter 2 as 1 *n(®) = X T (0))Q co 1 X(03) a/ tfk .,1 So © > 0 © = 0 where x(©) = 'Re {G°0'©)}‘ 'X/j( ©)' Im {G 0 (;©)} X/(©) e W (3.1) (3.2) and A(©) B(©) wJ(©)Q g w^(©) wJ(©)Q g W/(©) B(w) D(©)_ w]’(w)Q g w^(©) w} (©)Q g w/(©) (3.3) The vectors w/?(©) and wX©) are defined in (2.18). Define /z(©) to be the square of the function that defines & N (©) for positive frequencies M©) = ~r ~\ x T «o)Q (0 1 t(©) (3.4) Even though Q' 1 is not defined at © = 0, a limiting value of /i(©) as © -> 0 is well defined, but this limiting value is not equal to & N (0) in general. Substituting equations (3.2) and (3.3) into (3.4) yields h((0) = A(©)D(©)-B 2 (©) A(©)t 2 (©) + D(©)x^(©) 2 B(©)x /? (©)t / (©) (3.5) The objective is to find a stable transfer function F(s ) that satisfies

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35 |F( 7 -co )| 2 = h( co) (3.6) for non-zero frequencies. To facilitate construction of such an F(s), Lemma 3.1 states useful properties of the function h( co). Lemma 3.1 The function h(c o) is a finite, non-negative, even, rational function of ( 0 , with numerator degree 4k-8 and denominator degree 4k-4. Proof: The proof of Lemma 3. 1 is given in the Appendix. V Lemma 3.1 is used in the proof of Theorem 3.1, which describes a factorization of h{ to) used to compute F(s). Theorem 3.1 The function //({ o) has a factorization of the form *») mv (3.7) where the polynomials m x X(s) = ^x r s r and Y(s) = ^ y r / r = 0 r = 0 have no zeros in the closed right half plane (RHP), have real coefficients, and mx = 2.k-4 and m Y = 2k-2. Proof: The proof is based on a result from spectral factorization, given by Rozanov (1967), which states that every non-negative, even, rational function of a scalar variable co has a factorization of the form (3.7) where X(s) and Y(s) have no zeros in the open RHP and have real coefficients. Lemma 3.1 shows that h{ co) satisfies the conditions of Rozanov, and thus has such a factorization. The degree of the numerator of h{ co) is 4&-8 and this is twice the degree of X(s). Similarly, the denominator degree of h( co) is 4k-4, and this is twice the degree of Y(s). Furthermore, since h( co) is finite and non-zero for all co, it follows that X(s) and Y(s) have no zeros in the closed RHP. V If the function F(s ) is constructed as

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36 then F(s) X(s) Y (s) (3.8) \F(jaf X(yto)X(-y‘(o) X(y'co) Y(yto)Y(-yto) YO'co) = h(c o) (3.9) and (3.6) is satisfied. It remains to calculate the coefficients of the polynomials X(s) and Y(s) so that (3.7) is satisfied. The first step is to express h( to) as a ratio of polynomials in yco so that the coefficients of X(/( o) and Y(/G)), and thus of X(s ) and Y(s), can be found. Theorem 3.2 The function /i(oo) defined in (3.5) is equivalent to h{ co) = — A(;co)DO'co)-B 2 0 co) A(y( 0 )T/(yco) + D(; 0 )x^(y(O) 2B(;o))x^(;co)x / (y'co) (3.10) where the coefficients of the polynomials A(y'(0), B(y'co), D(yc o), and X/0'(fl) depend only on the nominal coefficient vector g° and the matrix Q g . The polynomials XflO'G)) and X/Oco) are given by mi * rU ° ) ) Xs^Ca 0 ) 2 (=0 with m R = < -k k even , 2 (3.11a) ±(k1) k odd and W®) = Xg?2£-l)0'co) ( =1 with in | = -k A: eve n . 2 (3.11b) .^(fc + l) A: odd where g,° are the elements of the vector g° defined in (2.15), except for g® := 1. The polynomials A(;co), B(y'co), and D (joo) are given by and A(;(0) = X a 2^0'“) (=0 2( with n a = \k2 A: even I k 1 k odd (3.12a) and BO'co) = X b (2M)0'“) e=\ .v.^(2^-i) D(yto) = £ d 2 eU<°) e=\ 2( with n B = k 1 with n D = [ k 1 k even \k 2 A: odd (3.12b) (3.12c)

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37 Furthermore, the coefficients of the polynomials A(y'co), B(y'co), and 6(703) are given by a 2 ( • ^(2M) “ < X 9(k-2h),(k-2£+2h) h=0 m A X C l(k-2h),(k-2f+2h) h=Om A ) X C l(k-2h),(k-2^+2h+l) h=0 m 2 X C l(k-2h),(k-2^+2h+l) h=(£-m]) 0 < i < m A (m A + l)<^ = (2 (k1) m 2 = ±(k2) k even k odd k even k odd Proof: The proof of Theorem 3.2 is given in the Appendix. Theorem 3.2 gives an expression that can be used to compute the function h{ (0) as a rational function of 7(1). However, the coefficients of the polynomials X(s) and Y(s) are the desired quantities. It is not possible to extract a simple expression for the coefficients of

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38 X(s) and Y(s) in terms of the variable c, however. Therefore, the coefficients of these polynomials must be found numerically. The problem of computing the coefficients of the stable polynomials X(s) and Y(s) given the coefficients of the polynomials |X(y(D)| = X(j(0)X(-j(0) and |Y(yO))| = Y(yO))Y(-yO)) is a spectral factorization problem. Efficient algorithms for solving this problem, for discrete or continuous systems, are given in Jezek and Kucera (1985) and Kucera (1979). The algorithms presented in Jezek and Kucera (1985) are iterative, but are quadratically convergent, and can be used to compute the desired polynomial coefficients to a user-specified tolerance. 3.3 Robust Stability Testing Using F(s ) It follows from Section 3.2 that F(s) defined by equation (3.8) is strictly proper, stable and minimum phase. Now, since h( CD) is taken to be the square of the function that defines x(co), it follows that XU©) \F(M = YU©) for nonzero frequencies, implying that = X(;0))X( • /( ° ) = VW) = *(©) (3-14) i YU©)Y(-y©) sup *(©) = \\F(s)\\ o co>0 (3.15) Equations (3.15) and (3.1) for to = 0 imply that & N = max & N (co) = max co linolL . -Jtfk.k bo I (3.16) so that testing the condition of Theorem 2. 1 is equivalent to testing the two conditions \\F(s)\l < 1 (3.17a) So < 1 (3.17b)

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39 The function F(s ) is a strictly proper, stable transfer function, so the algorithm proposed by Boyd et al. (1989) can be used to compute ||i r (s)|| oo . This algorithm requires a state-space realization of F(s ) as input. The basic idea behind the algorithm is that if a strictly proper, stable, real-rational transfer function H(s) has the realization H(s) = C(sl-\y'B (3.18) then ItfWlL S Y if and only if the matrix M y := y 'BB 1 _-y C C -A' has at least one purely imaginary eigenvalue. This implies ||//(.s)|| oo < y if and only if M y has no purely imaginary eigenvalues. The function F(s) will have a state-space realization of the form given in equation (3.18) since it is strictly proper. F(s) is stable also, so the algorithm can be used to test the condition (3.17a) by checking if the matrix M := A BB 1 -C T C -A 1 (3.19) has any purely imaginary eigenvalues. The actual value of ||F(s)|L IS f° un d through a bisection search on y. Upper and lower bounds for ||F(s)|L in terms of the realization {A,B,C} are given in Boyd et al. (1989). A method for testing whether or not M y has any imaginary eigenvalues without actually computing the eigenvalues is also given. Such a method could be used to avoid any numerical problems computing eigenvalues. The robust stability test that has been developed is summarized below. Robust Stability Testing Algorithm 1) Obtain the nominal plant, the uncertainty matrix Q^, and a nominally stabilizing controller C(s). 2) Construct the vector g° and the matrix Q g .

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40 3) Compute the coefficients of the polynomials A(yco), B(y(o), D(yco), x^(y'co) and T/O'co) given in (3.11 )-(3. 1 3). 4) Construct the function h{ to) given in equation (3. 10). 5) Perform spectral factorizations to find Xfs) and Y(s) with roots in the left half of the complex plane. 6) Construct F(s) and obtain {A,B,C} such that F(s) = CfsI-Af'B. 7) Calculate iFfsJL using the algorithm of Boyd et al. ( 1989). 8) Robust Stability Test: Test the conditions given in (3.17a) and (3.17b). The advantage of this method for testing robust stability is that it avoids having to construct a frequency plot or perform a grid search. Testing condition (3.17a) requires only checking the eigenvalues of one Hamiltonian matrix constructed from the state-space matrices of F(s ). Calculating the coefficients of the polynomials used to find F(s) is straightforward. Furthermore, computing the actual value of the norm in (3.17a) can be done using a bisection search based on the algorithm mentioned above. Thus, the stability margin of the system can be obtained to any desired degree of accuracy. The necessary and sufficient condition for robust stability for the discrete time case is where p c (co) is given by equation (2.37). This condition is equivalent to the three conditions 3.4 Discrete Time Stability Analysis p c (to) < G°(e ju> ) V to e [0 ,tc] (3.20) < 1 (3.21a) < 1 M0)| (3.21b)

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41 M*) I < 1 (3.21c) As in the continuous time case, the endpoint conditions (3.21b) and (3.21c) will be left to test separately. Defining h( CO) as h( to) = — y ^ (o))Q (0 1 x(to) then it follows from the definitions in Chapter 2 that (3.22) h{ (0) = A(co)D(co)-B (co) A(co)i/ (co) + D(co)x|(co) 2B(co)x /? (co)x / (co) (3.23) The goal is to construct a stable transfer function F(z) that satisfies i2 \F(e j(a )\ = /i(co) (3.24) Invoking the discrete time version of Theorem 3.1 given in Rozanov (1967), it follows that /i(co) has a factorization of the form K co) = _ [X(^ M )| |Y(<>)| (3.25) where the polynomials m x IK'S X(z) = ^ and Y(z) = ^y r z r r = 0 r = 0 have all their zeros in the open unit circle. The problem is treated in a manner similar to that given in Section 3.2, except now the function h{ co) is rewritten as a function of c^“. Theorem 3.3 The function h{ co) defined in (3.22) is equivalent to A(e jti> )D(e j<0 ) B 2 (e j(i> ) h( co) = (3.26) A(e Jm )x] (e j(0 ) + D(e j(a )x 2 R (e j(0 ) 2B (e jl “ )x R (e jm )x , (e ja ) where the coefficients of the polynomials A(e yco ), B(e 7C0 ), D(e/t0 ), x R (e^) and x t {e^) depend only on the nominal coefficient vector g° and the matrix Q g . The polynomials x R (e^ m ) and x / (e' /t0 ) are given by

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42 M* yco ) = + *~ y£(0 ) 0.27a) £=0 and T/^ 05 ) = £g?(^“ e~ Jea ) (3.27b) ^=o where g° are the elements of the vector g° defined in (2.13), except for g® := 1. The polynomials A(e ;(0 ), B(e^), and D(e ya) ) are given by and and k 1 2k-2 = 4q u + 2Xq„ + !>,(<>“ + e'*") £=1 r=l 2k-2 B(e ja ) = X b r(^' r(0 e~ jm ) r=l k1 2*-2 D(^) = -2£q„ + Xd r (^“ + «->“) *=1 r=l (3.28a) (3.28b) (3.28c) The coefficients a r , b r , and d r are determined by the magnitude of r. Define the quantities a k X 2 92*-r-« ^=k-(m f -1 ) V k r = X 2c U r+( e=i (3.29) where mj = (r/2) if r is even and mi = (r1 )/2 if r is odd. Now, if 1 < r < k-l, then the coefficients are given as H | 9k-m 2 ,k-m 2 [ G + V +
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43 a r = b r = d r = L Qrri2,ni2 + X 2c k2k-r1^2 ,A72 2 m2 X 2c k2k-r-f U=1 r even r odd (3.31) where m 2 = k (r/2) if r is even and m 2 = k (r+l)/2 if r is odd. Proof: The proof of Theorem 3.3 is given in the Appendix. When h{ co) is computed according to equation (3.26), the result is h{ co) = 4*-4 X"< e=o e j(ae + e-W' e -j(4k-4)(0 < r 4k-4 J= 0 ' e j((+4k-4)m + g -;«-4A:+4)(oj| 4k-2 x^ £=0 'e j(oe + e~W\ e -j(4k-2)(o < 4k-2 Id, J= 0 ' e j((+4k-2)(0 + e -j((-4k+2)w j , where n ( and d^ are the coefficients that result from equations (3.27)-(3.31). This is equivalent to 8fc-8 (3.32) 8^-8 e ' m = e 2Ja -^~ 8k-4_ t=0 j(OH for appropriately calculated nq and d ( . As a result of the fact that many of the polynomial coefficients given by (3.28)-(3.31) are identical up to sign, many of the coefficients in (3.32) are exactly zero. The number of terms lost is the same for both the numerator and denominator, and is equal to 4(/c-l). Removing these terms gives a reduced form for h( co) (3.33) 4k-2 X" h( co) = -1=2— 4k-2_ Xd^ e (=0 jcof The polynomials X(e /C0 ) and Y(e /C0 ) are found by performing a spectral factorization using the results of Jezek and Kucera (1985) in a manner similar to the continuous time case. The final polynomials X(z) and Y (z) have degree 2k1, however, the first coefficient of XU) is identically zero.

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44 The transfer function that results in the discrete case is of comparable order to that obtained in the continuous case, and this order is roughly twice that of the nominal characteristic polynomial. In both cases, the transfer function is found by constructing a stable transfer function from the vector of coefficients of the nominal characteristic polynomial and the elements of the matrix describing the uncertainty ellipsoid. 3.5 Example In this section, an example is presented to illustrate the ideas introduced in this chapter. Consider the following nominal polynomial G°(s) = s 4 + 5.8600s 3 + 9.3954s 2 + 6.0126s + 5.3237 (3.34) represented by g°, and the associated uncertainty matrix Q g : "5.8600' "2.0425 2.3648 1.7252 1.2603" g°= 9.3954 2.3648 4.9454 3.5362 1.4123 6.0126 5.3237 Q g = 1.7252 1.2603 3.5362 1.4123 4.6202 2.2330 2.2330 3.6206 The polynomial (3.34) has roots at s = {-3.60, -2.00, -0. 13±0.85y}. Using the equations from Theorem 3.2 and the algorithm of Jezek and Kucera (1985), the spectral factors X(s) and Y(s) for this system are found to be X(s) = 2.12s 4 + 6.37s 3 + 9.58s 2 + 7.84s + 3.43 (3.35) and Y(s) = 1. 43s 6 + 6.92s 5 + 2 1. 7s 4 + 34.2s 3 + 34.9s 2 + 23.3s + 10.9 (3.36) The degree of the nominal polynomial in equation (3.34) is k = 4; it is easily verified that the degrees of X(s) and Y(s) match the specifications of Theorem 3.1. The roots of X(s) are s = {-1 ,02±0.60/, -0.48±0.96y}, and those of Y(s) are s = {-1 .30±2.22y, -0.99±0.63/, — 0. 13±0.90/} , showing that both polynomials are Hurwitz.

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45 Figures 3.1a and 3.1b show plots of the frequency-dependent Nyquist robust stability margin vs. frequency for the frequency range CD e [0, 10]. It is obvious that k^((0) is discontinuous at CD = 0 for this example, and k^(0) = 0.357420. Using equation (3.16), the value of &n is found to be k n = 0.971650, rounded to six digits, and since the condition (3.17) is satisfied, the polynomial is robustly stable. The plot in Figure 3.1a is constructed by evaluating (3.1) over the frequency range CD e [0, 10] using Acd = 0.01. The plot of the frequency response magnitude of X(s)/Y(s), evaluated at the same frequency points, is shown in Figure 3.1b. Figures 3.1c and 3. Id show a comparison of the results of the two methods. The maximum absolute difference between the two curves is less than 4xl0' 6 , while the maximum percentage difference is slightly greater than 0.0004%. The results of the frequency sweep method are shown in Table 3.1 for different values of Acd. The computed value of k n (rounded to six digits) as well as the number of floating point operations needed by a simple MATLAB routine are shown. It is interesting to note that the result for Acd = 0.3 is quite good, and in fact, much better than the result for Acd = 0.2. This is due to the fact that the maximum of &n(cd) occurs almost exactly at CD = 0.9. This point will be evaluated in the sweep using Acd = 0.3, yielding a much better estimate of k n than for the sweep using Acd = 0.2. Table 3.1. Floating point computations needed for a frequency search. Resolution Acd for the frequency sweep 0.3 0.2 0.1 0.01 0.971648 0.787774 0.971648 0.971648 flops 14,425 21,858 43,718 437,206

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abs. error % error 46 Figure 3.1. Plot of the Nyquist robust stability margin &n(0)) vs. frequency to. (a) Frequency sweep method, (b) Proposed method, (c) Percentage error between the two methods, (d) Absolute error between the two methods.

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47 Tables 3.2 and 3.3 show the results for the method proposed in this paper. In Table 3.2, the number of floating point computations needed to find both spectral factors is listed. These numbers represent the computations of a simple MATLAB program implementing the algorithm given in Jezek and Kucera (1985). The tolerance is taken to be the sum of squared errors between the generated coefficients of X(yco)X(-yco) or Y(yco)Y(-_/co) and the actual values of the coefficients. This tolerance must be satisfied by both polynomials X(s) and Y(s). The number of iterations of the algorithm in Jezek and Kucera (1985) required to attain this tolerance is also given in Table 3.2. For the computations presented in Table 3.3, the value IO -4 was used for the tolerance. Table 3.2. Floating point computations and iterations needed to compute X(s) and Y(s). Tolerance for computing spectral factors 10-2 io 4 io 6 io 9 iterations (X/Y) 4/6 4/6 5/7 5/7 flops 3682 3682 4337 4337 Table 3.3 shows the number of floating point computations needed to find IIF(.s)IL given X(s) and Y(s). The tolerance values listed are the relative tolerances required for termination of the algorithm used to compute IIF(5)IL, i.e., the estimated value of IIF(.y)IL is guaranteed to be within ±£llF(s)IL of the true value. It is very interesting to note that the number of computations required to find the spectral factors X(s) and Y(s) is small relative to the computation required to find IIF(5)IL. This is desirable, as the termination criterion for the spectral factorization algorithm can be stringent without seriously affecting the number of overall calculations needed to compute IIF(.s)IL. Thus, the computational burden of the proposed method is primarily dictated by the desired numerical accuracy of the Nyquist robust stability margin.

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48 Table 3.3. Floating point computations needed for computing IIF(s)IL. Relative tolerance £ for computing IIF(s)IL 0.1 0.01 0.001 0.00001 WF(s)\L 0.917423 0.973195 0.971801 0.971659 flops 181,669 268,629 358,405 573,145 For a tolerance as low as e = 0.001 used in computing the infinity norm, the computational burden for the proposed algorithm (358,405 + 3,682 = 362,087 flops) is lower than that for the frequency sweep using Aco = 0.01 (437,206 flops). If the relative tolerance level is increased to e = 0.00001, then the computational burden (573,145 + 3,682 = 576,827 flops) surpasses that of the frequency sweep using Aco = 0.01. However, the proposed algorithm affords a guaranteed level of accuracy in the final result that the frequency sweep cannot. 3.6 Conclusions In this chapter, a new robust stability test for ellipsoidal systems is presented. This test is based on construction of a strictly-proper, stable transfer function whose infinity norm is equivalent to the stability margin of the system. The proposed test avoids performing a frequency search, and can calculate the stability margin using a bisection search.

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CHAPTER 4 ROBUST PREDICTIVE CONTROL DESIGN FOR ELLIPSOIDALLY UNCERTAIN SYSTEMS 4.1 Introduction The issue of designing controllers to robustly stabilize a system subject to real parametric uncertainty has received much attention in recent years. Since many physical uncertainties are most directly and effectively modeled by real parametric uncertainty, it is of great practical interest to include real uncertainty in the design of robust controllers. Unfortunately, the inclusion of real uncertainties greatly complicates both the robust control analysis and synthesis procedures. However, there are cases of interest where the uncertainty structure is such that the analysis and synthesis problems simplify considerably. This chapter considers such a case and establishes a procedure to design a robust predictive controller for a system subject to ellipsoidal parametric uncertainty. The standard method for analyzing the effect of uncertainty on a system is to construct a fictitious feedback loop where all the uncertainties are pulled out in a block-diagonal A matrix and all the known components, including the nominal plant, the controller, and any weights, are lumped into an M matrix. The robust control analysis problem is, given a fixed M and a set structure for A, find the smallest A that destabilizes the system. The structured singular value (Doyle 1982), p, is a stability margin defined by the size of the smallest destabilizing A as p(M) := {min a(A) I det(I AM) = 0} -1 where p(M) = 0 if no allowable A destabilizes the system. The robust control synthesis problem is, given a description of all possible As, design a controller that stabilizes the 49

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50 closed loop for any allowable A. It is the structure of the A matrix that determines the difficulty associated with solving the analysis and synthesis problems. The simplest uncertainty structure is where the matrix A is a full complex block, and only a norm bound on the whole matrix is known. This case is referred to as unstructured uncertainty, since no internal structure for the A matrix is known (or assumed). In this case, both the analysis and synthesis problems simplify considerably. The structured singular value for a full complex block A is simply p(M) = a(M). Furthermore, a controller is robustly stable if it generates an M matrix that satisfies HMf^ < flAf” 1 . Thus, Hoo design methods (Doyle et al. 1989), (Glover and Doyle 1988) can be used to find the optimal controller, or any sub-optimal controller that stabilizes the closed loop. In many cases of practical interest, the block-diagonal A matrix contains real and complex (possibly repeated) scalar blocks as well as norm bounded complex uncertainty blocks. If A contains real blocks as well as complex blocks, p(M) is referred to as mixed p. In general, the calculation of mixed p is NP hard (Braatz et al. 1994), (Braatz 1996), (Poljak and Rohn 1993). For both the purely complex and mixed cases, convex upper bounds for p exist (Doyle 1982), (Fan et al. 1991), and these are the basis for controller synthesis methods. The complex p-synthesis control design procedure attempts to find a controller minimizing p for the closed loop by iterating between computing an optimal scaling matrix D appearing in the upper bound and computing an Hoo controller K for this D, hence the name D-K iteration (Doyle 1985). The procedure is not jointly convex in D and K so there is no guarantee that the resulting controller is globally optimal, but it will be a locally optimal choice. The process of controller design for mixed p is similar to that for complex p, but now another scaling matrix is involved in the optimization, leading to the so-called D,G-K iteration method (Young 1996). This optimization is not jointly convex in D, G and K, so it may not yield the globally optimal controller. In the structured singular value framework, the individual elements of the A matrix are assumed to vary independently. Often it is more realistic to allow interactions between the

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51 uncertainties. One practical example is the ellipsoidal uncertainty description where the allowable coefficients of the system transfer function are restricted to lie inside an ellipsoid in the plant parameter vector space. This corresponds to a A matrix with real diagonal elements that is constrained by a weighted two-norm bound. Ellipsoidal uncertainty descriptions arise naturally in parameter identification techniques where the uncertainty ellipsoid is associated with the parameter-error covariance matrix. Ellipsoidal uncertainty descriptions are commonly encountered in chemical engineering applications where model parameters are found by fitting experimental data using linear or nonlinear regression techniques (Fogel and Huang 1982), (Belfonte and Bona 1985), (Belfonte et al. 1990). Methods for the computation of stability margins for ellipsoidal systems are given by several authors such as Guzzella et al. (1991), Biernacki et al. (1987), and Tsypkin and Polyak (1991). Biernacki et al. (1987) outline an iterative robust controller design procedure based on expanding the parameter space stability hyperellipsoid until it covers the allowable uncertainty region, but this procedure is not a convex optimization. The generalized structured singular value (Chen et al 1994a, Chen et al 1994b) is an extension of the structured singular value that takes into account interactions of the system uncertainties. The complex uncertainties, real uncertainties and complex block uncertainties are grouped together, and each group is norm bounded in an appropriate manner. Therefore, this is a more general framework than the mixed |i description. Recently, Braatz and Crisalle (1997) recast the ellipsoidal uncertainty problem in terms of the generalized structured singular value, allowing the straightforward inclusion of complex uncertainties in the stability margin calculation. However, there are no currently available controller design methods based on the generalized structured singular value. In some specialized cases, such as when the uncertain parameters appear in an affine manner in the numerator and denominator of a single input, single output system transfer function, the M matrix that results is rank one. Then the structured singular value is equal to its convex upper bound (Young 1994), (Young and Doyle 1996), and can be computed

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52 directly. The generalized structured singular value can also be computed in closed form (Chen et al. 1994b), (Braatz and Crisalle 1997). It is important to note that the restriction of the M matrix to be rank one does not depend on the number of uncertain parameters, but rather the manner in which they enter the system description. For the case of a rank one M matrix, Rantzer and Megretski (1994) derive a convex necessary and sufficient condition for robust stability that is used as the basis for design of a fixed-order Youla parameter whose coefficients are found using convex or quasi-convex optimization. The uncertainty description considered in this chapter yields a rank one M matrix, and the method of Rantzer and Megretski (1994) is used to compute an appropriate Youla parameter (Youla et al. 1976) that defines a robust predictive controller. Predictive control is a model-based control methodology that has found wide acceptance in industry. This is because predicitive control offers good performance, is easy to understand and formulate, and can accommodate input/output process constraints. The industrial success of the predictive control techniques is apparent by the variety of commercial predictive controllers that are available to the chemical processing industry through specialized vendors. Seborg (1994) reports that in oil refineries and petrochemical plants around the world, there are hundreds, perhaps thousands, of predictive controllers employed. It is possible to design predictive controllers using different plant representations, including finite impulse response (FIR) models, transfer function models and state space models. FIR based schemes include Dynamic Matrix Control (DMC) (Cutler and Ramaker 1980), Model Algorithmic Control (Mehra et al. 1979), and the quadratic DMC formulation of Garcia and Morshedi (1986). These methods are applicable only to stable plants, and often require large model orders, typically retaining 30 to 40 impulse response coefficients. Transfer function models are applicable to both stable and unstable plants, and lead to lower order representations. The well-known Generalized Predictive Control (GPC) technique (Clarke et al. 1987) and the MUSMAR approach (Greco et al. 1984) are

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53 examples of transfer function based predictive control formulations. Kwon and Pearson (1977) and Muske and Rawlings (1993) present state space formulations for predictive control. This large body of literature constitutes a rich source of knowledge to support the design and analysis of predictive controllers. There is currently a large amount of research focusing on the issue of stability of predictive control designs when the plant model is uncertain, such as the work of Zafiriou (1990) and Genceli and Nikolaou (1993). There is an interest in the research community to revisit the predictive control design techniques with the intention of including robustness features that guarantee stability or adequate performance when the plant model is uncertain. One interesting example is the robust quadratic DMC design including hard constraints studied by Zafiriou (1990). This work uses a contraction mapping first proposed by Economou (1985) to derive time-domain conditions for robust stability with respect to uncertainty in the impulse response coefficients of the nominal model, but this approach involves a very large numerical computation effort. Genceli and Nikolaou (1993) propose an analysis and synthesis method for predictive controllers based on FIR models, including constraints and using a linear cost functional. These authors use a parametric model uncertainty description that bounds the maximum deviations allowed for each pulseresponse coefficient, and obtain a sufficient condition for robust closed loop stability. The robustness of predictive controllers designed using transfer function representations is receiving increasing attention in the literature. Kouvaritakis et al. (1992) propose an alternative approach to GPC that employs a precompensator to stabilize the plant before the predictive design is carried out. The (^-parameterization procedure popularized by Youla (Youla et al. 1976) is employed in order to construct a final controller that is robust with respect to unstructured perturbations. The authors state rigorous necessary and sufficient conditions for robust stability; however, the approach proposed for synthesizing robust controllers is an approximate albeit practical scheme. The method consists of using polynomial or fixed-order transfer function approximations for

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54 the Youla parameter, and least-squares methods to identify the parameters of the robust design. Hrissagis et al. (1996) present a direct method for designing a predictive controller that is robust with respect to unstructured perturbations. In this approach, an appropriate Youla parameter is explicitly computed by solving a model-matching problem. This chapter presents a method for designing controllers that are robust with respect to real parametric uncertainty. The technique is based on the results of Rantzer and Megretski (1994), and therefore relies on using fixed-order approximations for the Youla parameter in the controller parameterization. The nominal predictive controller is designed using wellestablished methods and the nominal servo performance is retained by the robust controller. Furthermore, the design technique can be easily modified to incorporate integral action in the robust controller, allowing for the rejection of asymptotically constant disturbances. The chapter is organized in the following manner. The next section discusses the design of a predictive controller of the GPC type for a nominal transfer function plant model. The third section details the parameterization of all nominally stabilizing controllers through the use of the Youla parameter Q. The robust predictive control design based on the ellipsoidal uncertainty description is discussed in the fourth section. The fifth section contains the modifications of the design procedure required to incorporate integral action into the robust controller. A further modification to allow the inclusion of unstructured uncertainty is discussed in the sixth section. An example is given in the seventh section to illustrate the proposed design method. The design equations used to construct the nominal predictive controller are given in the final section. 4.2 Nominal Predictive Control Design Predictive control is a control methodology that is well documented in the literature. In particular, a wealth of knowledge is available to resolve crucial design issues such as nominal closed-loop stability, and parameter tuning (Lambert 1987; Mohtadi 1987).

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55 Predictive controllers are usually implemeted by executing at every sampling instant an algorithm that solves a quadratic optimization problem. For analysis purposes, it is desirable to represent the algorithmic controller in terms of transfer functions, allowing the utilization of classical z-domain tools for analyzing stability and performance. This section presents a brief review of the analysis technique discussed by Crisalle et al. (1989), which casts an algorithmic predictive control law of the GPC type into a form involving transfer function operators. The resulting nominal controller is used as the basis for the design of a robust controller. Consider the nominal process model A(z) y(z) = B(z) w(z) where y(z) and u(z ) are the process output and input, respectively, and A(z) and B(z) are the coprime polynomials A(z) = z n + a n -\Z n *+••• + 0o (4.2) B(z) = b m z' n + b m _\Z m ’+... + b 0 (4.3) of order n and m, respectively, where n>m. Predictive control involves the selection of future control moves that minimize the quadratic cost functional N y n u t) = £[r(t + i) y(t + i|t)f + X X[A«(t + i)] 2 (4.4) i=l i=0 where {r(t+/)} is the sequence of future values of the set point, {y(t+/lt)} is the sequence of predicted future values of the output, {Au(t+i)} is the sequence of future control increments, A is the move-suppression parameter used to penalize excessive control energy, and parameters N y and N u are the prediction and control horizons, respectively. The optimal control move, the w(t) that minimizes the functional J(t) for the prescribed set point sequence { r(t) } , is found by differentiating (4.4) with respect to the control moves, equating the result to zero, and solving for «(t). The predictive control law can also be cast in terms of transfer function operators as (Crisalle et al. 1989)

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56 R(z) / . tv w x S(z) n u(z) = T(z)r(z) y(z ) z z (4.5) where R(z) = z n + r n -\Z n ~ X +... + r 0 (4.6) S(z) = S n z n + s n _i z ,!_1 + ... + s 0 (4.7) x N v N v 1 T (z) = t N ' f z ' +t N y _jz • +... + qz (4.8) These operators satisfy ii o (4.9) and T(l) = S( 1) (4.10) The coefficients of the moving-average polynomial S(z), the regressor polynomial R(z), and the set-point advancement polynomial T(z) are functions of the tuning parameters N y , N u , and A, and of the model polynomials A(z) and B(z). Note that the predictive control law (4.5) includes an integrator if equation (4.9) is satisfied. A block-diagram representation of the predictive control structure is shown in Figure 4.1a. Specific equations for the polynomials (4.6) (4.8) are given in Section 4.8; further details of the derivation can be found in Crisalle et al. (1989). A formulation equivalent to (4.5) is also derived in McIntosh et al. (1991). Note that the transfer functions operating on u(z ) and y(z) in the nominal predictive controller (4.5) are biproper and of order n, the order of the nominal plant model. Note also that the set-point advancement polynomial T(z) is of degree equal to the prediction horizon N y . Since N y > n is a common tuning prescription (Clarke et al. 1987), the order of T(z) may exceed the order of R(z), making the control law noncausal with respect to the set-point signal. This noncausality is a natural consequence of the inclusion of future values of the set point in (4.4). Figure 4.1a shows that T(z) acts on the set point to

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57 produce the intermediate signal v(z)=T(z)r(z), which has the simple time-domain representation v(t) = t Ny r(t + N y ) + t N ^r(t + N y 1) + ... + hKt + 1) (4. 1 1) It is useful to remark that the nominal model (4.1) and the functional (4.4) are simpler versions of more elaborate formulations that improve the design performance at the expense of added complexity. Enhancements of the predicitve control law presented above, such as the inclusion of a lower prediction horizon parameter (Clarke et al. 1987), the addition of a weighted end-point term in (4.4) to guarantee stability for arbitrary parameter choices (Kwon and Byun 1989; Demircioglu and Clarke 1993), and the use of a filtered set point, can be accommodated within the framework proposed in this chapter through obvious modifications. Figure 4.1a illustrates the closed loop system established when the nominal predictive controller (4.5) is connected to the process (4.1). In addition to the set point signal r(t), the figure also shows an additive output disturbance signal d{ t). Note that the servo dynamics of the closed loop are fully characterized by the equations [ A(z)R(z) + B(z)S(z)] y(z ) = z"B(z)T(z) r(z) (4. 1 2) [A(z)R(z) + B(z)S(z)] u(z) = z" A(z)T(z) r(z) (4.13) Therefore, the stability of the closed loop for a given nominal predictive controller is contingent on the location of the roots of the characteristic polynomial A(z)R(z) + B(z)S(z). Furthermore, due to the presence of the integral action (4.9) in the controller and to the gain equality (4.10), the closed loop dynamics described by (4.12) are guaranteed to realize zero offset in the servo response. The integrator also guarantees perfect steady-state disturbance rejection for all disturbance signals that reach a constant steady-state. These desirable performance characteristics of the nominal controller will be preserved in the robust predictive controller designed in the following sections.

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58 Figure 4.1. (a) Structure of a nominal predictive controller, (b) Structure of the parameterized predictive controller including the Youla parameter Q(z). 4.3 Nominally Stabilizing Controller Parameterization In this section, all nominally stabilizing controllers are parameterized in terms of the nominal predictive controller (4.5) and a transfer function Q(z ) selected in the spirit of Wiener-Hopf design (Youla et al. 1976). However, a modification in the parameterization is introduced to achieve two important design requirements: (i) the parameterized controller must preserve the servo performance and the steady-state disturbance rejection properties of the nominal controller, and (ii) the parameterized controller must also be a predictive controller. Consider a nominal predictive controller (4.5) that stabilizes the closed loop system (4.12) (4.13). Provided the closed loop is stable, the nominal closed-loop characteristic polynomial A*(z) = A(z)R(z) + B(z)S(z) (4.14)

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59 is a Schur polynomial of degree 2n. As a first step in the parameterization of all nominally stabilizing controllers, a coprime fractional representation of the nominal plant model (4.1) is constructed p 0 (t) = B(z) = Nizl y A(z) M(z) (4.15) where N(z) and M(z) are proper and stable transfer functions that satisfy the Diophantine equation A(z)X(z)+M(z)F(z) = l (4.16) for some pair of stable and proper transfer functions X(z) and F(z). (Note the use of italicized capital letters for transfer functions, while polynomials are designated with plain capital letters.) A suitable (M(z), (V(z)) pair is readily derived from the nominal characteristic polynomial (4.14) as in Hrisaggis et al. (1996). First, the closed-loop characteristic polynomial is factored into the form A*(z) = Ai(z)A 2 (z), where both Ai(z) and A 2 (z) are Schur polynomials of degree n. If A*(z) contains complex poles then Ai(z) and A 2 (z) are constructed such that each complex-conjugate pair is contained in either Ai(z) or A 2 (z) to ensure that each polynomial factor has only real coefficients. Both sides of (4.14) are then divided by the factored characteristic polynomial to obtain A(z)R(z) | B(z)S(z) (4.17) Aj(z)A 2 (z) Aj(z)A 2 (z) Finally, stable and proper factorizations satisfying (4.16) are defined as M(z) := A(z) Aj(z) Â’ N(z):= B(z) Aj(z) (4.18) and nz y.= RU) (4.19) A 2 (z) A 2 (z) This result allows the nominal predictive control law (4.5) to be written in the equivalent form F(z) u{z) = Z(z) r(z) X(z) y(z) (4.20) where

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60 (4.21) The set of all solutions of (4.16) can be written in terms of the transfer functions (4.18) (4.19) and a stable, proper transfer-function Q(z ) through the well-known relations (Youla etal. 1976) Therefore, the set of all stabilizing controllers with the structure (4.20) is parameterized in the form to yield the control structure shown in Figure 4.1b. Setting Q(z )= 0 recovers the nominal predictive controller (4.20). In contrast to the standard Youla parameterization approach, the transfer function X(z) + M(z)Q{z) appears in the feedback path of Figure 4. lb, instead of appearing in the control block immediately preceding the plant. This approach, adopted from Hrissagis et al. (1996), in conjunction with factorizations (4.18) and (4.19) that make use of the nominal closed-loop polynomial, introduces the following highly desirable properties in the parameterized input-output maps. Property 4.1. The nominal control loop of Figure 4.1a and the parameterized control loop of Figure 4.1b have identical servo transfer functions y(z)/r(z) and u(z)/r(z). Proof. The proposition is proved by carrying out block-diagram algebra on each figure to derive in both cases the servo transfer functions y(z)/r(z) and u(z)/rfz). Property 4.2. Given that the nominal controller (4.5) is a predictive controller, then the parameterized controller (4.24) is also a predictive controller. X\z) = X(z) + M(z)Q(z) (4.22) nz)=Y(z)-N(zMz) (4.23) [Y(z) N(z)Q(z)] u(z) = Z(z) r(z) [X(z) + M(z)Q(z) ] y(z) (4.24) Proof. If (4.5) is a predictive controller, then by definition it yields a control sequence { w(t) } that minimizes the predictive performance index (4.4) for any prescribed set-point

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61 trajectory { r(t) } . For the given set point trajectory, it follows from Property 4.1 that the parameterized controller (4.24) will also produce the same control sequence due to the equality of the servo transfer function u(z)/r(z). It follows that the parameterized controller is also a predictive controller because it yields a control sequence that minimizes (4.4). Since any allowable parameter Q(z ) yields the same servo transfer functions y(z)/r(z) and u(z)/r(z), the parameterized controller has the intrinsic capability of preserving the nominal servo performance. Also note that although the terms containing Q(z) effectively cancel out in the servo transfer functions, the transfer function e(z)/v\(z) = M(z)[Y(z) N(z)Q(z)] in Figure 4.1b is affine in Q(z), as in the standard Youla parameterization method. 4.4 Robust Predictive Control Design In order to incorporate uncertainty into the plant description, the nominal plant transfer function shown in Figure 4.1b is now represented as the uncertain transfer function B(z) + AB(z) _ b m z' n + ... + b 0 +bb m z m + ... + bbo P{Z ) = (4.25) A(z) + AA(z) z n + a n -\Z n +... + #o +ba n _\Z n +... + 5ao where A(z) and B(z) are given in (4.2) and (4.3) and AB(z) = bb m z m + ... + bb Q (4.26) AA(z) = 8a n _ 1 z n_1 +... + 5fl 0 (4.27) The values of the coefficients of AA(z) and AB(z) are not known explicitly, however, the perturbation vector 8p = [5a„_ 1 ... da Q bb m ... 8/? 0 ] T e cr' ! +" !+1 (4.28) composed of the coefficients of AA(z) and AB(z), is constrained to lie in an ellipsoid Up in the parameter space. This ellipsoid is defined by a positive definite, symmetric matrix Q p such that

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62 E p = j8p e 9T +m+1 | 5p T Q” 1 8p < l} (4.29) The objective of the robust control design is to select a suitable Youla parameter Q(z) such that the resulting controller stabilizes the family of plants described by Ep. The work of Rantzer and Megretski (1994) considers robust controller design for any system where the uncertainties can be extracted from the closed loop as shown in Figure 4.2, where it is noted that the signal vv is scalar. This formulation immediately restricts this technique to rank one M-A problems. The affine uncertainty description in equation (4.25) results in a rank one M-A problem, so this formulation can be accommodated by the technique proposed by Rantzer and Megretski (1994). It is straightforward to transform the system shown in Figure 4.1b, now with an uncertain plant, into the form of Figure 4.2. This is done by first extracting the uncertainty vector 8p out of the plant block, and then constructing the appropriate closed loop transfer function T zw from the uncertainty block output w to the uncertainty block input z. The first step in this process is achieved by first rewriting the plant block as shown in Figure 4.3. w Figure 4.2. Feedback structure from Rantzer and Megretski. w 5p T Figure 4.3. Augmented plant with uncertainty vector 8p. For the model given in equation (4.25), Figure 4.4 shows explicitly the internal structure of the augmented plant given in Figure 4.3. The closed loop shown in Figure 4.3 is given by z 'Gil U) Gi 2 (z)" w _y_ _G 2 i(z) G 2 2(z)_ u (4.30)

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63 From Figure 4.4, it is straightforward to show that both Gn(z) and G\ 2 (z ) are transfer function vectors with n+m+1 rows while G 2 i(z) and G 22 U) are scalar transfer functions. The exact form for these transfer functions is given below. G n (z) A (z) ~1 A(z) 0 0 Gji(z) = -1 G\2(z) z /, ~ 1 B(z) A 2 (z) B(z) A 2 (z) z m A (z) G 2 2(z) 1 _ ~A(z) B(z) A(z) (4.31a) (4.31b) Figure 4.4. Augmented plant in detail. When the plant block in Figure 4.1b is replaced by that shown in Figure 4.3, the resulting closed loop is transformed to the form shown in Figure 4.5. The overall closed loop from w to z becomes T zw , where the transfer function vector T zw is given by

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64 T IW = 7j + T 2 Q = {G„-G, 2 MXG 21 } + {-G, 2 M 2 G 2 ,}e (4.32) The expressions in (4.32) for the transfer function vectors T\(z ) and 72(z) simplify to z" _1 R(z) ' A 1 (z)A 2 (z) A?(z) R (z) B (z) A,(z)A 2 (z) and T 2 (z) = A?(z) z m S(z) z m A(z) Aj(z)A 2 (z) A\ (z) s'(z) A (z) A i (z)A 2 (z)_ A f(z) _ Note that all elements of both 7i(z) and T 2 (z) are stable, strictly proper transfer functions, and that both Ti(z) and Tjiz) have n+m+1 rows. Figure 4.5. RST configuration with the augmented plant. w z Figure 4.6. Closed loop from w to z.

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65 In the controller design method of Rantzer and Megretski (1994), the approach is to express the Youla parameter Q(z) as a ratio of stable proper transfer functions (3(z) and a(z) in the form Q{z) = P(z)/a(z) (4.34) The main result of Rantzer and Megretski (1994) is that the closed loop in Figure 4.6 is stable for perturbations 5p satisfying I8pl < 1 if a(z) and (3(z) satisfy |Re{7j (e j0} )a(e J(0 ) + T 2 (e ;co )P(e 7 “ )}|^ < Re{a( 1 , then |Re{r 1 (^“)a(^' w ) + r 2 (e 7 ' C0 )p(^“)}| J < Re{a(e 7 '“)} for |5p| < y (4.37) Therefore, if a (z) and |3(z) are selected to solve the optimization Re{7](e 7CO )a(e y “) + r 2 (e 7 ' (D )p(e 7 ' (0 )}| < min max <( cc.peRH^, ooe[0,7t] Re {a(0 Vco e [0, 7t] (4.39) then the resulting closed loop will be robustly stable for the largest possible value of I8pl.

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66 Since a(z) and (3(z) can be of any order (such that the resulting Q(z) is proper), this is an infinite dimensional optimization. This optimization is converted to a finite dimensional problem by restricting a(z) and (3(z) to be of the form N 2N+1 oc(x,z) = 1 + %x k a k (z) and (3(x,z) = X'TtMz) (4.40) *=i k= N+l where x = [^ ••• jc 2 n+i] T e ^ 2N+I Quiz) Then Q(z) is a rational transfer function of order N 2N+1 := (4.41) i + k = i The functions CL k (z) and Pj.(z) should be chosen such that the limiting (? N (z) lim 0 N (z) = lim {(3(x,z)/a(x,z)} N— » N— (4.42) can approximate any stable transfer function Q(z ) arbitrarily well. Here, the basis functions a k (z) and P^(z) are chosen as a k(z) = P*(z) ( -az + 1 ") k (4.43) V z-a y where a is taken to be a real scalar that satisfies Id < 1. This choice constrains the basis functions to be all-pass functions. An interesting choice for a is simply a = 0, reducing the basis functions to the form a kiz) = Mz) = z (4.44) Obviously, a ratio of polynomials in z _1 can recover any rational transfer function as N grows without bound. Choosing basis functions of the form (4.43), the functions a(x,z) and (3(x,z) become N a(x,z) = 1 + Yj x k k = 1 (~az + 1 \ k \ z-a y 2N+1 and P(x,z) = X x k ( -az + 1 lt-N+1 (4.45) *=n+i v z-a y Then, the closed loop in Figure 4.6 is stable for all allowable perturbations 8p (that is, all 8p satisfying I8pl < 1) if a(x,z) and P(x,z) satisfy

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67 Re{r 1 (^“)a(x,^ w )+r 2 (^“)p(x,e yw )}|' / < Re{a(x,e ya) )} (4.46) The constraint in equation (4.29) is Sp T Q“’8p < 1. A new vector 8 can be defined 5 := Qp 1/2 Sp (4.47) i j j where the symmetric positive definite matrix Q p satisfies Q, = Qp 2 Qp' 2 (4.48) Then the vector 8 satisfies ||8||., < 1 for all allowable 8p. The block diagram shown in Figure 4.3 can be adjusted to use the new vector 8 as the perturbation vector, but a factor Qj/ 2 must be included in the plant matrix G(z), as shown in Figure 4.7. The new vector z is related to z simply by z = Q p z. Therefore, the transfer vectors G\\ and G \2 are changed to Qj, /2 Gn and Qj, /2 Gi 2 , respectively. Plugging these new expressions into equation (4.31) yields T iw = Q 'i\ w = 2 7j Q^' 2 72 e (4.49) Note that all the elements of the matrix Q^, /2 are real, and that the dual norm to the Euclidean 2-norm is the Euclidean 2-norm itself. Therefore, equation (4.35) becomes |q|/ 2 Re|r 1 (e 7 “)a(x,e 7W )-ir 2 (e/<0 )|3(x,e 7 “)||| 2 < Re{a(x,r*“)} Va)e[0,7t] (4.50) Figure 4.7. Augmented plant with new uncertainty vector 8. Alternatively, the norm used to measure 8p can be defined as |6p| := q: i,2 s p and then the dual norm in equation (4.35) is Ixl* :=' q!/ 2 x (4.51) 2 (4.52)

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68 which yields exactly the result shown in equation (4.50). Equation (4.50) cannot be satisfied if Reja(x,e yt0 )j < 0 for any frequency. Therefore, provided that min Re{a(x,e 7a) )} > 0 (4.53) coe[0,7t] *• J then the closed loop is robustly stable (for a specific value of x) if max (flg[0,7t] Qj, /2 Re{7i(e^)a(x,^'®) + T 2 (e j( °Mx,e j<0 )} Re|a(x,e 7<0 )| < 1 (4.54) As outlined above, then the optimally robust controller (actually the optimal vector x) can be found by solving the optimization program x opt = arg min max coe[0,7t] q;/ 2 r e{r,(Oa(xy w ) + r 2 (^)P(x,^)}|| Re {a(x,e 7C0 )} (4.55) subject to min Re{a(x,eyC0 )} > 0 CO€[0,7t] ^ J (4.56) for larger and larger values of N. In practice, a value of N will be found such that the value Qp 2 Re{r 1 (e ; “)a(x,e ; “) + r 2 (e/0) )P(x,e/0) )}| max (O6[0,7t] Re joc(x,e ;to )j (4.57) does not decrease appreciably as N is increased further. The optimal vector x°P l for this value of N is considered to describe the optimal controller. Alternatively, the value of N can be chosen to enforce a maximum degree condition on the resulting controller. The final form of the robust predictive controller is C N (z) X(z) + M(z)<2 n (z) Y(z)N(z)Qti(z) (4.58)

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69 which will be biproper and of degree 2n+N if there are no common factors that cancel in forming Cn(z). Thus, a specific value for N can be chosen, and the "optimal" Q N (z) found for that value of N. The use of basis functions of the form (4.45) results in the following desirable properties of the optimization (4.55) (4.56) as outlined below. Lemma 4.1. If the basis functions a(x,z) and (3(x,z) are given by equation (4.45), then the objective function £(x) := max coe[0,7t] Qp 2 Re{7j (e j(0 )a(x, e j ® ) + T 2 (e j( ° )P(x, e j(0 )} Reja(x,e 7C0 )j (4.59) is a quasi-convex function of x, and the constraint function 0(x) := min Re a)e[0,7t] {a(x,
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70 d [ Qp 2 {A(co*)x + b(a>*)} dx 1 + c T (co*)x *(x) (Boyd and Barratt, 1991). Similarly, if the value to =£0* that satisfies min Reja(x,e ;w )) = 1 + c T (co*)x roe[0,7t] *• J (4.63) (4.64) is found, then an appropriate subgradient ^(x) for the constraint function (4.60) is *00 = ~(l + c T ( 0 )*)x) (4.65) To compute the subgradients for the constraint and objective functions, the extreme frequencies that satisfy equations (4.62) and (4.64) must be found. The details of this process are discussed in the following subsection. 4.4.1 Constraint Testing and Objective Function Value Computation The optimization (4.55)-(4.56) can be performed using standard methods for constrained quasi-convex optimization, such as the ellipsoidal method outlined in Boyd and Barratt (1991). This method requires the value of the objective function and its subgradient if the current point is feasible (satisfies the constraint (4.56)), and the value of the constraint and its subgradient if the constraint is violated. As commented above, computation of either subgradient requires finding the frequency that satisfies (4.62) or (4.64). The case of the objective function is considered first, since the methods used are very similar to the methods introduced in Chapter 3. If the constraint (4.56) is satisfied, then the function |Qj/ 2 Re{r 1 (c^)a(x,c^) + r 2 (c^)P(x,e Jto )}|| 2 UX> := Re{a(x.O} < 4 66 > is non-negative for all (0 e [0,7t]. Therefore, it is possible to find a stable rational transfer function whose magnitude matches the value of ^(x) for all co e [0, 7c] , using the methods of Chapter 3. Then the objective function value is found as the Hoo norm of this transfer

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71 function, and it is also possible to compute the maximizing frequency so that the subgradient can be computed. There are several additional points that must be addressed before the method of Chapter 3 is used to find the transfer function fit to (4.66). The first is that only the real part of T[(e/ “)a(x,e/a) ) + r 2 (e/ “)P(x,e/0) ) is retained. Since the quantity 7’ 1 (e' /CO )oc(x,e/a) ) + ^(e-^Plx,^ 03 ) is a transfer function vector, this is equivalent to computing the real part of each element. Given any general polynomial in z f(z) = f np z" P ++ fo++ f-n n z'"" (4.67) then the real part of this polynomial, evaluated along the unit circle, is Rejfte^)! = f n cos(n p co) + ...+ f 0 + .,. + f_„ n cos(n n co) (4.68) and if it assumed that n n >n p , then Re jf(e/0) )J = fo + (f_, +f 1 )cos(o)) + ... + (f np + f_ np )cos(n p co)f ( _ n l)Cos((n p + l)co) + ... + f_ nn cos(n n co) (4.69) This is equivalent to the frequency response magnitude of the polynomial fr(z):=-f_ nn z nn +... + -f( _ i)Z p +-(f_„ +f n )z p +... + -(f_i +fi)z + f 0 + -(f_l+fl)z 1 +... + |(f_n +fn )z np +|f(-n -\)Z " P 1 +... + ^-f_ nn Z " n (4.70) where it is noted that the frequency response of (4.70) is purely real by definition. Therefore, it is possible to compute the coefficients of a polynomial whose frequency response magnitude is equivalent to the real part of any individual element of the vector T\(ej (i> )a(x,e-i (i) ) + 7’2(e/W )P(x,e/0) ), and all of these elements can be stacked to forma matrix of coefficients. The effect of the matrix multiplication by Q^ /2 is accounted for simply by multiplying the matrix of coefficients on the left by Q|/ 2 . Each polynomial is then multiplied with itself to achieve the effect of the squaring involved in the two-norm, all of the coefficients corresponding to like powers are summed, and finally a spectral factor is

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72 found for the numerator to account for taking the square root. Before this is done, a common denominator for the two-norm part and Re|a(x,e/<0 )| is found and extracted. A spectral factor for Reja(x,c/C0 )j is also found, but it is multiplied by itself since there is no need to take the square root of Re|a(x,c j,0) )| . The constraint (4.56) can be tested using the KalmanYakubovich-Popov lemma. The form of this lemma given in Rantzer (1993) is reproduced here: Lemma 4.2.(Kalman-Yakubovich-Popov) Given G(s) = C(sI-A) _1 B + D with A stable, the following statements are equivalent: (i) G(s ) + G*(s ) < 0 forallRes>0 (ii) There exists a positive definite, symmetric matrix P such that p o' A B A t C T 1 p O' 0 I C D + b t d t J 0 i < 0 (4.71) For a scalar G(s), condition (i) becomes Re(G(s)) < 0. For the discrete time case, the transfer function cx(x,z) is transformed into continuous time by means of a bilinear transformation. Define 1 + 5 ^ a(x,z) = a x, 1 -s) := a'(x,s) (4.72) and note that a i-r n X, V 1 + Z -1 = a(x,z) If Re{a'(x,y(o)} > 0 for to > 0 implies Re a x, l-e’*' 1 + e -ye since 7 (4.73) > 0 for 0e[O,7t], then. Re [ ( i-,-y 0 YI a x, 1 + e -ye = Re M^ e )} (4.74) Lemma 4.2 can be used to test Re{a'(x,yco)} > 0 instead of Re ja(x,e yC0 j| . Now \-e 1 -h 2 / sin 0 — 1 / sin0 = J 1 + e -ye 2 + 2cos0 1 + COS0 (4.75)

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73 so if co:= ( sin0 , then as 0 varies from 0 to n, to varies from 0 to Therefore, with Vl + cos0 this definition of co, Reja'(xjco)} > 0 for to > 0 implies Re^, va ^ 1 + e ^ j 0 e [0,Tt]. The transfer function a'(x,s ) is a scalar, but we wish to test Re(cc'(x,s)) > 0 so a x, > > 0 for we take a'Cx,^ = -G(s). A state space description of a'(x,s) can be found easily in controller canonical form, and the state matrix A will be stable by definition of the basis functions chosen. Given the state space description of a\x,s ) as [A,B,C,D], Re( a'(x,s)) > 0 for all Re s > 0 becomes equivalent to the existence of a positive definite, symmetric matrix P such that p o' A B A T -C T 1 p o' 0 I -C -D + b t -D t J 0 i < 0 (4.76) This condition is a linear matrix inequality (LMI) feasibility problem in the matrix variable P, and (4.76) can be checked using the LMI toolbox in Matlab. This formulation of the constraint is very useful in the continuous time version of this controller design method. 4.5 Robust Control Design With Steady State Disturbance Rejection Offset-free regulation in the presence of asymptotically constant disturbances will not be attained in general from the robust controllers designed using the methods of the previous section. In this section, a simple modification derived in Hrissagis et al. (1996) is discussed that ensures that the robust controller obtained provides integral action. The nominal predictive controller (4.5) leads to the nominal regulation transfer function y(z) = A(z)R(z) d(z ) A(z)R(z) + B(z)S(z) 1 ' from which it follows that limy(f) = 0 for step disturbances as well as for other /— >00 disturbances with a constant steady state because R( 1) = 0. On the other hand, the nominal regulation transfer function for the parameterized robust controller (4.24) is y(z) _ A(z)R(z) A(z)B(z) d(z) A(z)R(z) + B(z)S(z) A?(z) Q(z) (4.78)

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74 Because the synthesis procedures described in the previous section do not necessarily yield a Youla parameter satisfying Q{\) = 0, the robust predictive controller may display unacceptable nominal regulation performance at the steady state, unless the nominal plant has an integrator (that is, A(1)=0) or is a self-regulating process (B(l) = 0). Clearly, the robust predictive controller will attain perfect steady state disturbance rejection for all the plants belonging to the uncertain family (4.25) only if the Youla parameter has a zero gain, that is, 2(1) = 0. This gain constraint can be introduced in the robust predictive controller design through a simple modification of the factorizations (4.18) (4.19). First, the integrator is extracted from the nominal predictive controller by writing R(z) = (z-l)R'(z), and then (4.17) is rewritten in the form A(z)(z-l)R'(z) , B(z)S(z) Aj(z)A 2 (z) ' Aj(z)A 2 (z) 1 Introducing the modified coprime factorization M (2):= Ozi)A« ^BU) and X(z):= zAj(z) S(z) Y(z):= A(z) zR'(z) (4.79) (4.80) (4.81) A 2 (z) A 2 (z) A A A A leads to operators that satisfy the Diophantine equation N(z)X(z)+M(z)Y(z)= 1. Note that this modified factorization is equivalent to augmenting the nominal plant with an integrator. z Thus the modified plant output is y := y so that (4.30) becomes z — 1 Gii(z) G\ 2 (z) w y\ [g 2 \(z) G 22 (z)] l u_ (4.82) where G 2 \(.z) = — ^—G 21 (z) z — 1 (4.83) G 22 (z) = G 22 {z) z — 1 (4.84) These modified transfer functions are then substituted into (4.32) resulting in

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75 71 = G| | — G l2 MXG 2l = G]]—G] T 2 — — G^Af^G^l — — G\ 1 '7 12 (iziW_s ' z J Vz-17 7 12 V Z z-\) G 2 i G 21 = 71 (4.85) fz-lV \ Z ) (4.86) It is now possible to solve for a parameter £> N (z) using the design procedure detailed in the previous section. After a solution is found, the Youla parameter £?n(z) used in the parameterized predictive control structure of Figure 4. lb is constructed by re-associating the augmented-plant integrator with the controller to obtain (4.87) 2n,i(z) = ^^<2n(z) The final robust predictive controller design for this case is obtained by substituting the Youla parameter (4.87) and the factorizations (4.18) (4.19) into the structure (4.24). The resulting controller includes an integrator since (4.87) satisfies the zero-gain condition 0n(1) = O. 4.6 Inclusion of Unstructured Uncertainty High-frequency dynamics that are often neglected in system modeling can be represented in most cases as unstructured uncertainty. In this section, unstructured uncertainty is incorporated into the uncertainty description and a modified algorithm for controller design is presented. The structural constraint on the uncertainty description in Figure 4.2 is that the signal w must be a scalar. Therefore it is possible to include an element A(z) along with the vector 8p in the uncertainty block, provided w remains a scalar. Figure 4.8 shows one possible combination. This structure is similar to the additive uncertainty model that appears in the literature in that if all the elements of 8p are zero, then the plant block is equivalent to the transfer function P(z) = B(z) A(z) + A(z) (4.88)

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76 or P(z) = P°(z ) + A(z) (4.89) Figure 4.8 Plant block with unstructured element A(z) included. This is the standard additive uncertainty model provided ||A(z)|| oo < 1. Obviously, an appropriate stable, proper weighting function can be extracted from any stable A(z) as A(z) = W(z)A(z) so that ||A(z)|| oo < 1. When the elements of 8p are non-zero, then the general transfer function from u to y is P(z) B(z) + AB(z) + A (z) A(z) + AA(z) A(z) + AA(z) (4.90) and a suitable weighting function can be extracted such that P(z) B(z) + AB(z) + A (z) + AA(z) A(z) A(z) + AA(z) W(z)A(z) (4.91) which satisfies ||A(z)| <1. The inclusion of the unstructured element A(z) changes the stability condition (4.35) to the following form (Rantzer and Megretski, 1994)

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77 Qp 2 Re{7i 5 (e/(D )a(x,e/W ) + 7^ (^ /C °)P(x,e/C0 )|j^ + r 1 A (e ;l0 )a(x,^ l0 ) + r2 A (e ; ' 0) )(3(x,^ 0) )| < RejaCx,^ 10 )} (4.92) £ ? for to g [0, 7t] . The transfer function vectors T°(z) and T 2 (z) are exactly the Ti(z) and Ti(z) given in (4.33), while 7j A (z) and T 2 A (z) are simply 7| a (z) = -W(z)M(z)X(z) and T 2 A (z) = -W(z)M 2 (z) (4.93) 4.7 Examples 4.7.1 Example 1 Consider the stable nominal plant model p (z) = ~— — (4.94) Z .4 a m with the corresponding nominal plant parameter vector p =[-.4 5] . The matrix Q p defining the plant parameter uncertainty ellipsoid is -0.09 1.0 (4.95) The allowable uncertainty region described by this ellipse is shown in Figure 4.9. It is noted that the range a^ g (-1,1) defines all stable plant models. A nominal predictive controller is designed for the system using the tuning parameters N y = 1, N u = 1, and X = 0, resulting in the following controller and prefilter polynomials R(z) = z-1 (4.96) S(z) = 0.28z-0.08 (4.97) T(z) = 0.20z (4.98)

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78 Figure 4.9 Allowable uncertainty region for the example. It is readily verified that these polynomials satisfy (4. 10) (4. 1 1). The nominal closed loop polynomial is A *(z) = z 2 (4.99) and the polynomials Aj(z) and A 2 (z) are taken as Ai(z) = z and A 2 (z) = z. For this example, the pole of the basis function is taken to be a = 0. It is straightforward to show that the perturbation 8p = [0.5 0.25 ] t (4.100) yields 5p 8p = 0 .9978 and is therefore allowable. The plant model resulting from this perturbation is 5 25 P(z)=—~ (4.101) z +.1 The closed loop polynomial resulting from this plant model and the nominal controller is G(z) = z 2 + 0.57z 0.52 (4.102)

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79 which has a root outside the unit circle, showing that the nominal controller is not robustly stable. Robust controllers with and without integral action are designed for the choice N = 1. The optimal vector x for the robust control design without integral action is x opt = [-0.0008 -0.2002 0.0338] 7 (4.103) The objective function value (4.59) for this vector is C(x opt ) = 0.4683 The Youla parameter corresponding to x°P l is -0.20022 + 0.0338 Qi (z) := 0.0008 (4.104) (4.105) The final form (equation 4.58) of the robust predictive controller for this Youla parameter is Q(z) = 0.0798^+0.03372 -0.0135 2 2 + 0.00022 -0.1682 (4.106) The modified design procedure, using equations (4.85) and (4.86) in the optimization (4.55) (4.56), yields the optimal vector x opt = [0.2448 -0.0959 -0.0332] 1 (4.107) which corresponds to the following Youla parameter — 1"\ —0.09592 0.0332 Qli(z) = \ z J 2 + 0.2448 (4.108) The final form (equation 4.58) of the robust predictive controller for this Youla parameter is c u(z) = 0.184l2 3 +0.08962 2 -0.01152-0-0133 2 3 0.27572 2 0.55832 0. 1660 (4.109) The servo performance of the nominal predictive controller, along with both robust controllers, is shown in Figure 4.10. No uncertainty is assumed present in the plant model, i.e., 8p=0. A unit step change in the set point occurs at t = 0 and t = 50 sec. A step disturbance of magnitude 0.2 occurs at t = 12 sec. By design, both the nominal

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80 40 60 Time, t 100 Figure 4.10 Performance comparison of the three different control designs: the nominal controlled:), robust controller without integral action (-.) and with integral action D C O o < o c o o 0.25 0.2 0.15 0.1 0.05 0 -0.05 0.1 -0.15 0 20 40 60 80 100 Time, t Figure 4.11 Control actions produced by the three different control designs: the nominal controlled:), robust controller without integral action (-.) and with integral action (--).

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81 Figure 4.12 Performance comparison of the different control designs for the perturbed plant: the nominal controlled:), robust controller without integral action (-.) and with integral action Figure 4.13 Control actions produced by the different control designs for the perturbed plant: the nominal controlled:), robust controller without integral action (-.) and with integral action (— ).

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82 predictive controller and the robust controller with integral action exhibit no steady-state offset, while the robust controller designed without integral action does produce offset. The disturbance rejection properties of the three designs are quite similar. Figure 4.1 1 shows the control actions produced by the three control designs. Again, all three controllers show similar behavior, and it is noted that all three designs require only modest control actions in regulation and disturbance rejection for the nominal plant. Figure 4.12 shows the response of the three predictive controllers when the plant is given by the model (4.101). As noted above, the nominal controller produces an unstable closed loop for this specific plant model, and the vertical range has been restricted to highlight the responses of the two stable designs. The robust controllers both produce stable closed loops with the perturbed plant, however, there are more noticeable transients after the set point changes. The disturbance rejection properties of the two robust designs are still quite good. The steady-state offset of the original robust control design is still very apparent. Figure 4.13 shows the control actions produced by the three control designs in controlling the perturbed plant. The nominal controller requires actions of larger magnitude than shown on this plot; the vertical range is truncated to highlight the control actions of the robust controllers. It is of interest to find out if it is possible to design a predictive controller for the nominal plant that is robustly stable. Table 4.1 shows the results of designing predictive controllers for different values of N y and N u . The value of % is given for the choices of N y and N u shown, and as discussed in Chapter 2, values less than one correspond to a robustly stable controller. For all the entries of Table 4.1, the move-suppresion parameter A. is set equal to zero. For all the entries where N u > 1 the predictive controller given in equations (4.96)-(4.98) is recovered, which is not robustly stable.

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83 Table 4.1 Nyquist robust stability margin for nominal predictive controllers designed using various N y and N u values. Ny \ NU 1 2 3 4 5 2 0.9165 1.0954 3 0.8595 1.0954 1.0954 4 0.8332 1.0954 1.0954 1.0954 5 0.8187 1.0954 1.0954 1.0954 1.0954 4.7.2 Example 2 The second example is the stable nominal plant model 0.30z + 0.42 1 z 1 + 1.20z +0.54 (4.110) with the corresponding nominal plant parameter vector p° =[1.20 0.54 0.30 0.42]'. The matrix Q p for this example is 0.1000 0.0490 0.0140 0.0105 0.0490 0.0700 0.0275 0.0130 0.0140 0.0275 0.1200 -0.0208 0.0105 0.0130 -0.0208 0.0600 (4.111) A nominal predictive controller is designed for the system using the tuning parameters N y = 4, N u = 2, and X = 0, resulting in the following controller and prefilter polynomials R(z) = z 2 0.5402z 0.4598 (4.112) S(z) = 0.2197z 2 + 1.5860z + 0.591 1 (4. 1 13) T(z) = 1.3037z 4 0.88 14z 3 + 0.5 101z 2 + 1 .4643z (4. 1 14) The nominal closed loop polynomial is A*(z) = z 4 + 0.7257z 3 (4.115)

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84 and the polynomials Ai(z) and A 2 (z) are taken as Aj(z) = z 2 +0.7257z and A 2 (z) = z 2 . Again, the pole of the basis function is taken to be a = 0. It is straightforward to show that the perturbation 5p = [0.07 -0.10 -0.27 0.10] T (4.116) yields 8p T Q~'Sp = 0.9627 and is therefore allowable. The plant model resulting from this perturbation is P(z) = 0.03Z + 0.52 z 2 + 1.27z + 0.44 (4.117) The closed loop polynomial resulting from this plant model and the nominal controller is G(z) = z 4 + 0.7364z 3 0.5440z 2 + 0.0208z + 0. 105 1 (4. 1 18) which has a root outside the unit circle, thus the nominal controller is not robustly stable. Robust controllers with and without integral action are designed for the choice N = 2. The optimal vector x for the robust control design without integral action is x opt =[0.6397 0.3024 -0.4932 -2.6443 -1.7309] 1 (4.119) The objective function value (4.57) for this vector is C(x opt ) = 0.8793 The Youla parameter corresponding to x°P' is Q 2 (z) -0.4932z 2 -2.6443z1.7309 z 2 +0.6397z + 0.3024 (4.120) (4.121) The final form (equation 4.56) of the robust predictive controller for this Youla parameter is C 2 (Z) -0.2735z 5 1.3502z-2.2453z 3 1.4338z 2 -0.1335z +0.1297 z 5 + 0.9732z 4 + 0.5697z 3 + 0.8074z 2 + 0.2559z 0. 1009 (4.122) The modified design procedure, using equations (4.72) and (4.73) in the optimization (4.53) (4.54), yields the optimal vector x opt =[0.5843 0.3764 0.0660 -1.0629 -0.8583] 1 (4.123) which corresponds to the following Youla parameter

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85 fz-l') 0.0660z 2 1 .0629 z 0.8583 (4.124) { z J z 2 + 0.5843z + 0.3764 The final form (equation 4.56) of the robust predictive controller for this Youla parameter is Note that the degree of this controller is only 5, since two pole-zero cancellations at the origin occur when the controller is constructed. The servo performance of the nominal predictive controller, along with both robust controllers, is shown in Figure 4.14. No uncertainty is assumed present in the plant model, i.e., 8p=0. A unit step change in the set point occurs at t = 0 and t = 50 sec. A step disturbance of magnitude 0.2 occurs at t = 12 sec. By design, both the nominal predictive controller and the robust controller with integral action exhibit no steady-state offset, while the robust controller designed without integral action does produce offset. The disturbance rejection properties of the three designs are quite similar. Figure 4.15 shows the control actions produced by the three control designs. Again, all three controllers show similar behavior, and it is noted that for this example all three designs require more substantial control actions in regulation and disturbance rejection for the nominal plant. Figure 4.16 shows the response of the three predictive controllers when the plant is given by the model (4.105). The nominal controller produces an unstable closed loop for this specific plant model, so the vertical range has been restricted to highlight the responses of the two stable designs. The robust controllers both produce stable closed loops with the perturbed plant, however, there are very noticeable oscillations after the set point changes. The steady-state offset of the original robust control design is still very apparent. 0.2857z 5 + 0.8241z 4 + 1.7302z 3 + 2.5980z 2 + 2.0468z + 0.6249 z 5 + 0.7500z 4 0.056 lz 3 0.3488z 2 0.8590z 0.4861 (4.125)

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86 1.4 QD o 1.2 1 0.8 0.6 0.4 0.2 IMA . I I It I \ A / II II u » 0 0.2 0 20 40 60 Time, t 80 100 Figure 4.14 Performance comparison of the three different control designs: the nominal controlled:), robust controller without integral action (-.) and with integral action (— ). Figure 4.15 Control actions produced by the three different control designs: the nominal controlled :), robust controller without integral action (-.) and with integral action (— ).

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87 Figure 4.16 Performance comparison of the different control designs for the perturbed plant: the nominal controller(:), robust controller without integral action (-.) and with integral action Figure 4.17 Control actions produced by the different control designs for the perturbed plant: the nominal controller(:), robust controller without integral action (-.) and with integral action (— ).

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88 Figure 4.17 shows the control actions produced by the three control designs in controlling the perturbed plant. The nominal controller requires actions of larger magnitude than shown on this plot; the vertical range is truncated to highlight the control actions of the robust controllers. The robust design with integral action requires smaller initial control actions than the standard robust design. Both designs require similar control efforts after the onset of the disturbance. Table 4.2 shows the values of the Nyquist robust stability margin for several different predicitive controllers designed using the values of N y and N u shown. For all the designs, X = 0. It is interesting to note that as with Example 1, only the designs with N u = 1 are robustly stable. However, the Nyquist robust stability margin does not continually decrease with increasing N u , as is seen from the first column of Table 4.2. The entries marked with an X represent N y and N u pairs that result in a controller that is not nominally stable. Table 4.2 Nyquist robust stability margin for nominal predictive controllers designed using various N y and N u values. Ny ^ NU 1 2 3 4 5 2 0.8767 X 3 0.8372 3.2163 X 4 0.8701 2.1186 1.7620 X 5 0.8690 2.0701 1.7132 1.8914 X 4.8 Design Equations for Nominal Predictive Control This section provides specific design equations used to synthesize a nominal predictive controller following the approach of Crisalle et al. (1989). An equivalent formulation is

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89 given by McIntosh et al. (1991). The final design equations for the polynomials (4.6) (4.8) that appear in the predictive control law (4.5) are: R(z) = z n ,-i Sr i+^'Xwr 1 ) i'=l d-z" 1 ) S(z) = z n ' N y , r=l N, T (z)=lk iZ l r=i (4.126) (4.127) (4.128) where the design operators F ,-(z _1 ) and T^z -1 ) and the coefficients k L , i= 1,2, ..., N y are determined from the process model according the following procedure. First, rewrite the nominal plant model (4.1) (4.3) in the equivalent form A_j(z _1 ) y(z) = z _1 B_j(z -1 ) u(z) (4.129) involving inverse powers of z, where A_j(z _l ) and B_j(z -1 ) are related to (4.2) and (4.3) in an obvious manner and are of the form A_i(z 1 ) = l + r3_]jZ z 2+... + a_i n z n (4.130) B_!(z 1 ) = b_ lj0 +Z?_iiz -1 +... + b_ lnh z~ nb (4.131) To obtain the design operators F^z -1 ), which are polynomials of degree n (the order of the plant), solve the set of Diophantine equations E i (z" 1 )A(z _1 )A_ 1 (z" 1 ) + z" , F i (z _1 ) = l, i= 1, 2, .... N y (4.132) which also yields the intermediate polynomials E ( (z _1 ) of degree M. The second design operators, the polynomials T,(z _1 ) of degree are obtained by decomposing the product E t (z -1 )B_ 1 (z -1 ) in the form E,(z -1 )B_ 1 (z -1 ) = G,(z -1 ) + z -, r,(z _1 ) (4.133)

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90 where polynomials G /(z -1 ) of degree j'-l are known as the dynamic polynomials, and are characterized by the fact that their coefficients are the sampled values of the step-response of the plant (2.6.10). Note that the polynomials T^z -1 ) are identically zero if n\> equals 0. In turn, the coefficients of the dynamic polynomials are used to define the nonzero elements of the Toeplitz matrix G^ u known as the truncated dynamic matrix, which contains only N u columns. Finally, the coefficients /q , i=\, 2, ..., N y are obtained as the components of the gain vector A: T =[k j k 2 ... k^ y ], calculated from the expression A design technique to synthesize robust predictive controllers for systems subject to ellipsoidal uncertainty has been presented. The technique uses a constrained quasi-convex optimization to determine the coefficients of a fixed order Youla parameter to robustly stabilize the system. The robust controller retains the nominal servo performance of the original predictive controller designed for the nominal system. A straightforward modification is given to allow the incorporation of integral action into the robust controller design. Furthermore, the design can accommodate unstructured uncertainty, as long as the problem remains equivalent to a rank one M-A problem. (4.134) 4.9 Conclusions

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CHAPTER 5 ANALYTIC SOLUTION TO A LIMITING SYNTHESIS PROBLEM 5.1 Introduction This chapter presents an analytic solution to the limiting case of the robust control design problem discussed in Chapter 2. The analysis method developed in Chapter 2 is not strictly applicable to the limiting case of a first order plant and constant controller. This is the case where the characteristic polynomial is of degree 1, i.e., k := n + tn = 1. For this case, an analytic solution to the robust synthesis problem is available. This chapter is organized in the following manner. Section 2 presents the limiting case discrete time problem definition. Section 3 presents the results of the robustness analysis for the special case. These results are used to produce the solution to the robust synthesis problem, which is presented in Section 4. The continuous time limiting case robustness design is presented in Section 5. An example is presented in Section 6 that illustrates the different possible results of the robust synthesis procedure. Concluding remarks are presented in Section 7. 5.2 Problem Statement Consider the closed loop system shown in Figure 5.1, where the compensator is a static-gain controller k 6 9?, with k * 0, and the plant is the discrete time system Pfc p >-r^r < s -‘> which is characterized by the parameter vector p = [a (3] T . 91

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92 k P(z) i L Figure 5.1. Feedback loop with plant P(z) and controller k. The plant parameter vector is modeled as p = p° + 5p (5.2) where p° = [a 0 (3°] T , (3° * 0, is a vector containing the known nominal values of the plant parameters, and the uncertainty 8p is an element of the set E p := {5p e 9* 2 | Sp^-'Sp < l} (5.3) 9*2x2 where q 9il 912 L 912 922 j is a symmetric positive-definite matrix. With the plant model (5.1) placed in the closed-loop of Figure 5.1, the characteristic polynomial is simply G(z) = z + g where g = a + k(3. It is obvious that the closed loop will be robustly stable if and only if all possible values of g satisfy Igl < 1. Therefore, it must be established how the ellipsoidal uncertainty (5.3) affects the characteristic polynomial coefficient. The relationship between the plant parameter vector p and the characteristic polynomial coefficient g is given by the linear map g = Sp where S = [1 k]. The nominal values of the plant parameters map to a nominal characteristic coefficient g° = Sp° or, explicitly, g° = a 0 + k(3°. It follows that the uncertainty in the coefficient of the characteristic polynomial 8g = g g° is of the form Sg = S8p (5.4) where Sg is a real scalar. As 8p takes on all values in Ep, (5.4) generates a family of characteristic polynomials g = { G(z) I g = g° + Sg, Sg = SSp, Sp g Ep } (5.5)

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93 It is not difficult to show that when 8p belongs to the set Ep defined in (5.3), then 5g given by (5.4) lies in a continuous, real interval Eg that is symmetric with respect to zero. That is, Eg is of the form E g = {Sg e 91 | |8g| < Sg*, Sg*e 91, Sg*> 0} (5.6) The general theory of Guzzella et al. (1991) is not applicable to the limiting plant/controller pair considered in this chapter because the line-segment (5.6) is in fact a degenerate ellipse. Since the condition for robust stability is Igl < 1, the endpoint 8g* of the characteristic interval Eg is of particular significance to the analysis of robustness. The explicit form for Sg* is given in Theorem 5.1 Theorem 5.1 Consider the closed-loop system of Figure 5.1, with k e 9? and the family of plants P(z) described by (5.1)-(5.3). Then the uncertainty 8g in the coefficient of the characteristic polynomial lies in the interval £ g = j Sg e 91 | |8g| < ^|sQS f] | (5.7) Proof. Writing the endpoint of the interval Eg in (5.6) as 8g* = S8p* where 8p* some element of Ep, it follows by linearity that 8p* must lie on the boundary of the ellipse Ef, i.e., Sp* satisfies 8p* T Q“'Sp* = 1 (5.8) The value of 8g* is found by solving the optimization problem Sg* = max S8p (5.9) 8pe £ p where Ep is defined in (5.3). This constrained optimization problem is solved using the Lagrange multiplier method. Let J(8p) = SSp ^(Sp T Q -1 8p l) be the Lagrangian, where X > 0 is the unknown multiplier. Equating to zero the derivative of J(Sp) with respect to 8p and solving for the optimal value of Sp yields Sp* = r‘QS T (5.10)

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94 The Lagrange multiplier is readily obtained by substituting (5.10) into the constraint (5.8), yielding \ 2 = SQS T . Substitution of (5.10) into 5g* = S8p* yields 8g* = r'SQS T (5.11) from which it follows that 5g* = (SQS T ) 1/2 ( 5 12 ) The quadratic form SQS T = qn + 2qi2k + q 22 ^ 2 is a positive scalar for all values of k, since S is never equal to the null vector and Q is a positive-definite matrix. Therefore, the uncertainty Sg satisfies |8g| < VSQS T (5.13) and the uncertainty region 2^ given in (5.6) can be rewritten in the form 2 g = { 5g e 9t I |8g| < t/sQS 7 } V It is of interest to note that if the continuous-time limiting case is considered, the resulting closed loop characteristic polynomial is exactly identical to the discrete time case except that the discrete variable z is replaced by the Laplace variable s. Therefore, the analysis of the previous section holds without modification except in this case the robust stability criterion is that g be positive for all values of the uncertainty 5g. The continuous time case will be considered further in Section 5. 5.3 Robustness Analysis This section presents a criterion for testing whether a given controller k places the roots of all of the members of the characteristic polynomial family £ inside the unit circle. Theorem 5.2 Consider the closed-loop system of Figure 5.1, with k e 9^ and the family of plants P(z) described by (5.1)-(5.3). Furthermore, let q = SQS T . Then the closed loop system is robustly stable if and only if lg°l + Vq < 1 (5.14)

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95 Proof. Again, the closed loop is robustly stable if and only if Igl < 1. Since g = g° + 8g, if lg° + 5gl < 1, then the theorem is proved. From the triangle inequality and (5.13) |g| = |g° + &g By the definition of q, it follows that < g ^[sQS 1 (5.15) |g| * It is not hard to verify that equality will hold for some member of the uncertainty set (5.3). Therefore, the condition Igl < 1 is equivalent to g Vq < 1 Remark 1 . It is noted that the inequality lg°l + Vq < 1 is a more restrictive condition than lg°l < 1, so a controller must be nominally stabilizing to be robust. Remark 2 . The condition lg°l + Vq < 1 is equivalent to checking the Schur stability of the extreme polynomials of g, namely Gj(z) = z + g° + Vq and G 2 (z) = z + g° Vq . This is the method suggested in Barmish (1994) for analyzing the robust stability of low-order interval polynomials. Theorem 5.2 provides a necessary and sufficient condition for testing the robustness of a controller k. However, a slightly modified version of this condition is used in the synthesis discussion to follow. Therefore, before the robust synthesis problem is addressed, a robustness parameter r(k) is defined and the condition presented in Theorem 5.2 is restated in terms of this new parameter. Let r(k) := 1 lg°l Vq (5.16) where r(k) G 91 can be interpreted as the difference between the magnitude of the maximal root of Q and the critical magnitude Igl = 1 . Corollary 5.1. The feedback loop of Figure 5. 1 is robustly stable if and only if r(k) > 0.

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96 5.4 Robust Synthesis This section considers the problem of finding a controller k that yields a robustly stable closed loop. A robust controller must yield a nominal characteristic polynomial that is stable. Let 7C = {k e 91 : lg°l < 1} be the set of all controllers that lead to a stable nominal closed loop. It is straightforward to show ac= jk e and X= jk € Using the result of Corollary 5.1, define the (possibly empty) set of robustly stabilizing controllers Tfo = (k e 91 : r(k) >0). The goal of the robust synthesis procedure is to find the optimal controller k = arg max r(k) (5.18) ke9C -(1 + a ) 1-a P 0 ’ o0 P L if P° > 0 (5.17a) ^l-a° -(1 + aV Q 0 ’ r.0 if (3° < 0 (5.17b) The controller k is called optimally robust if r(k) >0. To highlight the dependence on the controller parameter k, the function r(k) can be written in the form r(k) = 1 a 0 + k(3° ^q n + 2q, 2 k + q 22 k 2 (5.19) Although (5.19) is a simple function, due to the presence of the absolute-value operator the derivative with respect to k does not exist at the point ryO ko=-j^ (5.20) Consequently, the maximum of r(k) cannot be computed simply by setting the first derivative of r(k) with respect to k equal to zero and solving for an optimal k. This problem is avoided using an alternate definition of r(k). Let r(k) = min { n(k) , r 2 (k) } (5.21) where and r i (k) := 1 g° Vq (5.22)

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97 r 2 (k) := 1 + g° Vq (5.23) It follows that r(k) = ri(k) when g° > 0, and r(k) = r 2 (k) when g° < 0. Furthermore, ri(ko) = r 2 (ko) since g°(ko) = 0, where ko is defined in (5.20). Note that rj(k) and r 2 (k) are continuously differentiable functions of k. The problem of finding the maximum of r(k) is solved by analyzing individually the maxima of rj(k) and r 2 (k). An analytical solution is possible because functions rj(k) and r 2 (k) are concave. Denote as k, = (3° 91.922 9l2 0x2 9 22 (P°) dr i the solution to ^ = 0, and denote as k 2 = 922 922 922 P |9ll922 ~ 9 1 2 (5.24) 0x2 922 \ 922 (P ) (5.25) dr? the solution to ^7 = 0. If k] and k 2 are real, they are the global maximizers of ri(k) and r 2 (k), respectively. The numerator of the fraction inside the radical in (5.24) and (5.25) is positive because q 1 i9 22 -q 12 = det(Q) > 0. If q 22 -(P 0 ) 2 < 0, then both k\ and k 2 are complex, and neither of the branch functions ri(k) and r 2 (k) has a real maximum. Theorem 5.3 presents a comprehensive solution to the robust synthesis problem in terms of the three controllers ko, k), and k 2 . Theorem 5.3. Consider the closed-loop system of Figure 5.1, with k e 91 and the family of plants P(z) described by (5.1)-(5.3). Also define ko, k) and k 2 as in (5.20), (5.24), and (5.25), respectively. Then the optimal controller is given by k =i ki k 2 l k 0 if a 0 + k 1 [5 0 > 0 and kj e 91 if a 0 + k 2 P 0 < 0 and k 2 e 91 otherwise Furthermore, if r(k) > 0, then the optimal controller is robust. (5.26a) (5.26b) (5.26c) Proof. The proof of Theorem 5.3 is given in the Appendix.

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98 Theorem 5.3 presents the complete analytic solution to the limiting case robust synthesis problem of a first order plant subject to ellipsoidal uncertainty and constant controller. The optimal controller is found simply by checking the sign of g° at the controllers kj and k 2 If neither of these two controllers satisfies the conditions (5.26a) and (5.26b), then the optimal controller is koIt is remarked that the algorithm in Theorem 5.3 results in a unique choice of controller. 5.5 Continuous Time Limiting Case In the limiting case for continuous systems, the plant (5. 1) is Kr-v'-jh (5 ' 27) The uncertainty in the coefficients is still given by (5.2)-(5.3), and the controller is still a static gain k. The resulting characteristic polynomial is G(s) = s + g where g = g° + 8g, and the uncertainty 5g lies in the interval £ g = { 5g e 51 | |8g| < VSQS T } (5.28) For the continuous time case, the necessary and sufficient condition for robust stability is that the characteristic polynomial coefficient g be positive. Since |Sg| < VSQS T (5.29) it follows that if g° is less than or equal to zero, the closed loop is not robustly stable since the allowable perturbation 8g = 0 yields an unstable closed loop. Therefore, it is assumed that g° > 0, so that the nominal closed loop is stable. In this case, the only way that the coefficient g can become negative is by passing through the origin, implying that g = 0 for an allowable 8g. This implies that if the nominal system is stable, then robust stability is guaranteed if and only if g * 0 for all allowable 8g. This condition is equivalent to g° > VSQS T (5.30) For the continuous time case, the robustness parameter r(k) is defined as

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99 r(k) = g° Jioi 7 (5.31) Corollary 5.2. The continuous time limiting case feedback loop is robustly stable if and only if r(k) > 0. Computing the range of values of k that lead to a stable nominal closed loop is straightforward. Let 3C={ke SK : g° > 0 } be the set of all controllers that lead to a stable nominal closed loop. It is straightforward to show Using the result of Corollary 5.2, define the possibly empty set of robustly stabilizing controllers = {k e !?Cl r(k) >0}. The goal of the robust synthesis procedure is to find the optimal controller This controller is optimal only if two conditions are met: 1) a 0 + ki(3° > 0 and 2) k\ is real. If (P 0 ) 2 > q22 then ki is imaginary, and furthermore r(k) — > «= as |k| — » °° {where the sign of k is chosen so that g° > 0}. In this case there is no finite controller that maximizes the robustness parameter r(k), and an appropriate robustly stabilizing controller should be chosen using additional criteria. This situation is not as simple as the discrete time case where the optimum controller can be found directly. However, in this case a (5.32a) and (5.32b) k opt = arg max r(k) (5.33) Assuming that g° > 0 then (5.34) (5.35)

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100 robustly stabilizing controller is guaranteed to exist, even if a large (magnitude) value of k is required to stabilize the loop. If (P 0 ) 2 < q 22 then k[ is real. This implies that if k\ satisfies a 0 + kiP° > 0, then it is the unique solution to (5.33), i.e., ki = k opt . If a 0 + ki (3° < 0, then there is no maximum. 5.6 Example The theory discussed in this chapter is illustrated with a simple example. Consider the nominal plant P(z; p°) = where P° = 0.08 and a 0 will take different values for this example. Let the parametric uncertainty description £^be described by (5.3) through the matrix [ 0.02 0.01 ‘ Q = L 0.01 0.02 . When a 0 = 0.2, condition (5.26a) of Theorem 5.3 is satisfied because kj=-0.094e 01 and a°+k]P° = 0.192 > 0; hence, the global maximizer of r(k) is k = -0.094. The branches rj(k) and r 2 (k) are plotted along with the function r(k) in Figure 5.2a. Note that in this case, both branch maxima kj and k 2 lie to the right of the intersection point k 0 . When a 0 = -0.25, condition (5.26b) of Theorem 5.3 is satisfied because k 2 =1.094e 01 and a°+k 2 P° = -0.162 < 0; hence, the global maximizer of r(k) is k = 1.094. The functions r(k), rj(k), and r 2 (k) are plotted in Figure 5.2b, where in this case both branch maxima lie to the right of k 0 . When a 0 = -0.04, condition (5.26c) of Theorem 5.3 is satisfied because o^+kjP 0 = -0.048 < 0 and a°+k 2 P° = 0.048 > 0; hence, the global maximizer of r(k) is k = Icq = 0.50. The results of this case are presented in Figure 5.2c, where the maximum occurs at the intersection point k = Icq.

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101 (a) (b) (c) Figure 5.2. Plots of the branch functions r^k), r 2 (k), and r(k) for the three cases of the example. The global maximizers are: (a) k = k] = -0.094, (b) k = k 2 = 1.094, and (c) k = k 0 = 0.50. 5.7 Conclusions The limiting case design problem considered in this chapter is indeed short, simple, and straightforward. The payoff is that the result for testing robustness is simply checking the positivity of a function containing the nominal characteristic coefficient and the elements of the matrix describing the uncertainty in the plant model parameters. Also, the procedure for finding the maximally robust controller (for the discrete time case) is a short if-then-else sequence, readily adaptable to computer implementation. The case where maximization leads to the controller kj or k 2 (for the discrete time case k] for the continuous time case) in fact appears to be rather rare. To see why this is so,

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102 notice that to satisfy the condition leading to this case, the variance of the estimate of the process gain must be greater than the square of the estimate itself. The examples presented consider a plant with a small gain, where the variance of the gain estimate is a large percentage of the nominal value. Although the above discussion appears to imply that the results presented are of limited usefulness, this is not true. It so happens that the robust stability criteria that are derived in this paper are the same criteria that occur at the singular frequencies 0 and n in the general discrete time case. The scrutiny provided in this chapter thus provides insight into the more general case.

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CHAPTER 6 FUTURE DIRECTIONS 6.1 Ellipsoidal Systems with Delays Many systems of practical interest are subject to delay. In the continuous time case, the presence of a delay term transforms the characteristic polynomial into a so-called quasipolynomial. The results of Chapter 2 are not directly applicable to this delay case. In the discrete time case, if the delay is an integer multiple of the sampling time, then the resulting system is still a ratio of polynomials and the results presented in this work are applicable. It is of interest to see if stability analysis results for delay ellipsoidal systems can be derived. As a first step, a constant delay should be considered, and if analysis results are obtained, the case of variable delay could be analyzed. 6.2 Performance Constraints Most controllers are not designed solely on the basis of stability robustness. Performance requirements are usually present, although it may not be possible to directly incorporate these requirements into the controller design. Usually these constraints are posed as time-domain constraints, such as settling time limits, but they may also be constraints such as input saturation. If a convex optimization can be constructed to maximize the performance of the system, then the stability constraint that is optimized in the controller design methodology of Chapter 4 can be taken simply as a constraint in this new convex optimization. The controller design procedure then becomes a robust performance problem. 103

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104 6.3 LMI-based Robust Control Design It has been proposed (Rantzer 1993) that the finite dimensional optimization for constructing robust controllers can be reformulated as an LMI problem, or more specifically, as a generalized eigenvalue problem subject to an LMI constraint. Such a formulation would be able to take advantage of the powerful methods for solving LMIs that have been developed recently. The matrices involved in the LMI are constructed from a state space realization of the transfer function vectors T\ and Tj along with the basis functions tXk and [3kIf it is possible to derive a method for constructing these state space matrices given the nominal plant, controller, and basis functions, then this optimization method would be far more efficient than employing standard quasi-convex optimization routines.

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APPENDIX 1. Proof of Lemma 2.2 The case of co = 0 is proved first. From Figure 2.3 it follows that p c (0) is half of the length of the line segment EoFrom (2.23) it follows that P c(°) = V^7 which is equation (2.25b). For to > 0, the function p c (to) corresponds to the distance from the center of the ellipse Eco to its boundary, and can therefore be considered the length of a vector lying on the boundary of E^. Let this vector be represented by St*((D) which satisfies (8t * (co)) T Q” 1 (8 t * (co)) = 1 (A 1.1) and by definition ||St * (o))|| 2 = p c (co) Considering equation (2.19), Sx*(co) also represents a perturbation polynomial AG(/C0*) that can be written in Cartesian and polar forms as AGO'co*) = 8 tr(o)) + ;8xi(co) = p c (o))e 70 (A 1.2) for some angle 0. Using equations (2.18) and (2.21), the positive definite, symmetric matrix Q u can be written as Qco = ’A(to) B(co)" _B(co) D(to)_ wJ(co)Q g w^(co) wJ(a))Q g w^(o)) wJ(co)Q g W/(co) wJ(C0)Q g W 7 (G)) (A1.3) It is noted that the elements A(oo), B(co), and D(co) are real polynomials in to. Equation (A 1.3) and the definition of 8x can be used to rewrite (Al.l) as 105

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106 D(8t^) 2 + A(dx*j) 2 2B(8t^)(8t/) = det(Q w ) (A 1.4) where the dependence on co has been dropped from all terms. Now, from (A 1.2) it follows that S'tfl = p c (O))cos(0) and 8i/ = p c (co)sin(0) Substituting these expressions into (A 1.4) yields p 2 (co)[d cos 2 (0) + A sin 2 (0) 2B sin(0)cos(0)] = det(Q u ) or Pc(®) = *\2 Vd et(Q m ) Vdcos 2 (0) + Asin 2 (0) 2Bsin(0)cos(0) (A1.5) All that remains is to determine 0. Write G°(/'co) in polar form as G(;‘co) = r z e where from (2.19) it follows that r t = (A 1.6) (A1.7) G u O’(o) = ||t(cd)|| 2 The vectors given by (A 1.2) and (A 1.6) are colinear, however, they point in opposite directions. This implies cos(a) = -cos(0) and sin(a) = -sin(0). The real and imaginary parts of G°(j(Q) can be expressed in terms of the quantities in equation (A 1.6) as x R = r x cos(a) and T/ = r T sin(a). These relationships can be combined to give and cos(0) = sin(0) = r x Substituting these expressions into (A 1.5) and simplifying yields r x ydetlQJ Pc(“) = d(t|) + a(t5)-2B(t w t / ) Finally, using equations (A 1.3) and (A 1.7) results in

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107 P r (CO) = I ^X T (03)Q“ 1 X((0) which is equation (2.25a). This completes the proof. V 2. Proof of Lemma 3.1 The proof of this Lemma is broken down into three separate parts, each covering a portion of the claims in the Lemma. Property 1 . h{ co) is a real-rational function of co 2 , and thus an even function of 0). Furthermore, the numerator degree of this function is 4/c-8 and the denominator degree is 4*-4. Proof: First, it is shown that all elements of the right-hand side of (3.5) are polynomials in CO. Consider the elements of the matrix Q w given in (3.3). From the form of the vector wr it is not hard to see that A is a polynomial in even powers of co with a constant term, and the degree of A is 2k-4 if k is even and 2k-2 if k is odd. Similarly, D is also a polynomial in even powers of co, but with no constant term. The degree of D is 2k-2 if k is even and 2k-4 if k is odd. B is a polynomial in odd powers of co, and it has no constant term either. The degree of B is 2k-3. Since the elements of Qg are real, it follows that all these polynomials have real coefficients. Thus the numerator of h( co) is a polynomial in even powers of co with lowest term co 2 . The leading degree of this polynomial is 2k-6. The leading coefficient is a minor of the positive definite matrix Q g so it is always positive. Looking at the definitions of Xr and X/ , it is straightforward to see that Xr is a polynomial in even powers of co with a non-zero constant term. The degree of this polynomial is k if k is even, and k 1 if k is odd. Also, it can be seen that X/is a polynomial in odd powers of co with no constant term, with degree k 1 if k is even, and k if k is odd. Thus X£ and Xj are polynomials in even powers of co, but x^ has a constant term while x {

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108 does not. Also, x^X/ is a polynomial in odd powers of co with no constant term. From this it follows that all three elements of the denominator of h{ co) are polynomials (with real coefficients) in even powers of 0) with lowest term to 2 . It is straightforward to show that the degree of this polynomial is Ak-2. However, both the numerator and denominator have lowest order terms to 2 . Therefore, a factor of co 2 can be cancelled from each, lowering the numerator degree to Ak-8 and the denominator degree to Ak-A. Property 2. /z(c o) is a finite, non-negative function for co > 0. Proof: The matrix Q u is positive definite for co > 0. Furthermore, X is never equal to the null vector. Therefore, x T Q w *x > 0 for co > 0. From (3.4) it follows that h~ x { co) = x T Q u 'x. Therefore, h{ co) < °° for co > 0. All of the elements of qJ are bounded for finite co, so x T Q w 'x < °° for 0 < co < °°. Note that /z(co) -> 0 as co -> °°, which follows immediately from the degree conditions of Property 1 . Therefore, 0 < h( co) < « for co > 0 Property 3 . h( 0) is a finite, positive quantity. Proof: In the proof of Property 1, it is noted that both numerator and denominator polynomials of h( co) have lowest order terms co 2 . Thus h( 0) = 0/0. However, once the factor of co 2 is cancelled from both polynomials the value of h( 0) can be extracted. It is a straightforward but tedious exercise to show that this process yields H 0 ) = 4k,k(go) 2 2 4k.k4k-l.k-l ~ 4k,k-l + 4k-i ,k-i (S? ) 2 ' 24k.k-i(go)(g?) (A2.1) where q* j is the (i,j) element of the matrix Q g . Since the numerator is a minor of a positive definite matrix, it is a strictly positive, finite quantity. Thus V4k,k4k-l,k-l > 4k,k-l (A2.2)

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109 It remains to show that the denominator of (A2.1) is also strictly positive and finite. The elements qk,k and qk1 ,k-i are diagonal elements of a positive definite matrix, so they are positive. Since g® and gg are real numbers, it follows that the first two terms in the denominator are positive. It is assumed that both g° and gg have the same sign, as is necessary if the nominal closed loop is to be stable. Thus if q^k-i ^ 0, the denominator of (A2.1) is positive. Now consider the case where qk,k-i > 0Using (A2.2), it follows that q k ,k(gS) 2 + qk-i, k -i(g?) 2 2 qk,k-i(go g?) q k ,k(go ) 2 + qk-l.k-ltg ?) 2 2 Vqk,kqk-l,k-l (go g^) 0x2 0 „0x ^k.k (go) ~ V q k-l,k-l (g?) > 0 Since this proves the denominator is also strictly positive and finite, the final result is 0 < h( 0) < Taken together, the three properties prove all the claims of Lemma 3.1. 3. Proof of Theorem 3.2 In order to prove the Theorem, the relationships between A(co), B(co), D(co), and T/(co) and A(j(o), B(jco) , D(jco), T/^jco), an d ^/(j®) are derived and then it is shown that when equations (3.1 1) and (3.12) are substituted into equation (3.10), equation (3.5) is recovered. The first step is to write the various polynomials from (3.5) as polynomials in jco. In order to do so, the vectors w R and w /, introduced in equation (2.18), are rewritten as and Wp = 0 (j( 0 ) 4 0 (jco) 4 0 (j(o) . xO T W, = or 1 (j ®) 5 o (jo ,) 4 o (jco ) 1 o • \3 (A3.1) (A3. 2) It follows from the definition of T(co) that

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110 if k is even, and if k is odd. Defining **(®) = g8+." + g*-2(j®)* 2 +(j“) A M®) = go + -. + g*-i(j “)* _1 nic **(j®) = X§2^(j«) 2/ 1=0 where m R = -k k even 2 .^(fc-1) ko dd and g k := 1 , it follows that M®) = * r { j®) Equations (A3.3)-(A3.4) are simply (3.1 la). Similar to the above case, */(®) = 0' 1 )(g?(j®) 1 +... + gJ-i(jto)* _1 ) if k is even, and = Ci‘ 1 )(gf(j«) 1 +-+g°-i(jto)*" 1 ) + (j^Xj®)* if k is odd. Defining m. 0 where and g k := 1, it follows that */(j®) = £g2*-i(j®) (=0 \k /ceven m, = < , 2 \Uk + 1) k odd */(®) = (T^/G®) or */0®) = jx/(a>) (A3. 3) (A3. 4) (A3. 5) (A3. 6) (A3 .7) (A3. 8) Equations (A3.6)-(A3.7) are just (3.1 lb).

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Ill A proof of all equations (3. 1 2)-(3. 1 3) is not given, rather, one polynomial is discussed. The polynomial A(co) is defined in equation (2.34) as A(g>) = wJ(co)Q g w^((0) We assume k is odd, the case where k is even is similar. Then wj (a>)Q g w*(a>) = [(jco)* 1 ••• (jco) 2 0 (jco) 0 ]^, ••• q k ] k 1 (jco) (jto) 0 . (j©)° . where qj represents column i of the matrix Q g . Thus A(co) = [(jco)*' 1 ••• Gco)° ][qyt Gco)° + q^_ 2 (j a) ) 2 +••• + Qi Geo)*' 1 = (jco)°{q w (jco) 0 +q w _ 2 (jco) 2 +... + q /U (jco)* '} + ••• + Cjco)* -1 {q u (jco) 0 + %k -2 O®) 2 + • • • + qu Geo)* -1 } = Qk,k + { ( ik,k-2 )(jCO) 2 + ...+ (q3,l -+-qi, 3 )Gco) 2A:_4 +q U iaco) 2 * -2 Several lengthy calculations allow this to be written as 1* A A(co) = A(jco) = £a 2 ^(jco) 2£ (A3. 9) where nA = k1, and if m A is defined as n A /2, then i X 9(k-2h),(k-2C+2h) 0 < £ < m A a 2r * h=0 I 9(k-2h),(k-2M-2h) (m A + l)<^
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112 However, the formula (A3. 10) still holds. The proofs of equations (5.12b,c) and (5. 13b,c) are entirely similar and are not included. However, it is noted that B(jco) = jB(co) (A3. 11) and D(jco) = -D(to) (A3. 12) where B(co) and D(oo) are given in (2.34) and B(jco) and D(jco) are given by (3.12b) and (3.12c). Then, substitution of (A3. 5), (A3. 8), (A3. 9), (A3. 11), and (A3. 12) into (3.10) yields equation (3.5) V 4. Proof of Theorem 3.3 In order to prove the Theorem, the relationships between A(co), B(co), D(co), T/?(co), and T/(co) and A(e 7
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113 then *,(«*) = ls° l (e i ' m +
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114 A(co) = \t( ej2Qt '° '° + 2 + e-M k -^)q u ( =1 k 1 k + l£ ^L;(2k-f-p)co + e Kp-()(a + e -Xp-i)a + e -A2k-^-p)(o *=1 p =£+l In the above expression, it is noted that 2(k-£) > 0, 2k-£-p > 0, and p-l > 0. The expression also shows that if a r is the coefficient of ^ the coefficient of e' jm is also a r for r > 0. Therefore, only the nonnegative powers will be considered. First consider the coefficients of J®. Only the first sum contributes, and this contribution is seen to be l 4 *-l Yfrw +
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115 If r is even, but k
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116 5. Proof of Lemma 4.1 The constraint function is <|)(x) := min Re{a(x,e/ “)} we[0,7t] 1 1 The function RejaCx,^®)} with the choice of a (x,z) given by (4.43) is equivalent to Reja(x,e , ' a> )j = l + c T (co)x where c(to) = [c](co) ••• c N (co) 0 ••• 0] T e9I 2N+1 The functions Cj(co) are given by (-ae J * + lV Cj(o)) = Re P J® _ a V e a j for i = 1,...,N Since (A5.1) (A5.2) (A5.3) Rejocf^x 1 +(1A.)x 2 ,e 7 “)| = 1 + c t (o))[A,x 1 +(1-?i)x 2 = X + Xc T (co)[x' j + (1 X) + (1 X)c T (( 0 )[x 2 ] = X Re{a(x 1 ,
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117 -max CO -Re|a(x,e 7 “)|| = min Reja(x,e ;W )j and the negative of a convex function is concave, it follows that <{)(x) = min Re|a(x,e/a) )j is a concave function of x. This result also means that all values of x satisfying y is a convex set for all real y. The objective function is £(x) = max ^(x,co) CO£[0,7l] First it is shown that the function (A5.4) C(x,m) Qp 2 Re{7j ( e ja )a(x, e ja ) + T 2 (e ja )[3(x, e ja )} Re{a(x,e 710 )} 2 (A5.5) is quasi-convex, which means that it satisfies £(X,x' +(l-^)x 2 ,co) < max|^(x 1 ,(o),^(x 2 ,co)| for any X s [0, 1] . This constraint is less restrictive than convexity, but it implies that the set of all x for which £(x,co)82 is assumed without loss of generality. Since the set of all x that satisfy (|)(x) > 0 is convex, if it assumed that x 1 and x 2 are elements of this set, all convex combinations of x 1 and x 2 are also inside this set, and are thus feasible points for the objective function. Now, using the affine nature of the constraint C(Ax' +(l-X)x 2 ,co) QfRej +(1 X)x 2 ,e j(0 ) + T 2 (e Ja )$(Xx l + (1 X)x 2 ,e j(a )] I2 Re a(Xx' + (l-X)x 2 ,e j(0 )

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118 Q;/ 2 ARe{r 1 (^“)a(x\^) + r 2 (^“)P(x I ,^ w )} Re{a(Ax' + (1 A)x 2 ,e 7(0 )} + Q;, /2 (1-A)R e{rK^“)a(x 2 ,e^) + r 2 (^ M )(3(x 2 ,^ C0 )}|| o ^Q|/ 2 Re [ 7j (*> )a(x‘ , e ja ) + T 2 (e 7 “ )p( x 1 , ) i ARe{a(xV“) + (1 A) Re{a(x 2 ,e/0) ) (1 A)|qJ/ 2 Re | [7](^“)a(x 2 ,^“) + 7' 2 (^ (0 )(3(x 2 ARej [a(x 1 ,e'“)] | + (1 A)Reja(x 2 ,e 7 “)j 1 A5, Re| a(xV®)] > + (1 — A)5 2 Re | [a(x 2 ,e ; “)| ARejo^x 1 ,^®) | + (1 A)Re{a(x 2 ,e 7 ®)J i S^ARejafx 1 ,^®) } + (l-A)Re{a(xV®)}) ARe{ a(xV®)} + (l-A)Re{a(xV®)} This implies = 5 l ^(Ax 1 +(1A)x 2 ,(o) < max(^(x 1 ,a)),^(x 2 ,co)) = 8] and therefore, the function £( x,co ) is quasi-convex. Again, the maximum of a family of quasi-con vex functionals is also quasi-convex, so the objective function (A5.4) is therefore quasi-convex. V 6. Proof of Theorem 5.3 Proof. Since from definition (5.21) r(k) is the minimum of two concave functions, it follows that r(k) must have a unique global maximum (Rockafellar 1970). Hence, the conditions for selecting an optimizing controller in (5.26a)-(5.26c) must yield a unique result. The structure of the proof is as follows. First, the conditions in (5.26a) are assumed, and are shown to lead to the choice kj for the optimal controller. Second, the

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119 conditions of (5.26b) are assumed, and are shown to lead to the choice of k 2 for the optimal controller. Next, it is shown that the conditions in (5.26a) and (5.26b) are incompatible, i.e., they cannot be simultaneously satisfied. Finally, the remaining compatible combinations of signs of a 0 + k i (3° and a 0 + k^P 0 are considered, along with the case where both k\ and k 2 are complex, and these two possibilities are shown to lead to the choice of ko for the optimal controller, demonstrating (5.26c) and completing the proof. Let Ij denote the range of values of k where a 0 + kp° > 0, and let I 2 denote the range of values of k where a 0 + kp° < 0, such that Ij u {ko} u I 2 = (—<»,oo). The result (5.26a) is shown by demonstrating that r(k) < ri(ki) on the subintervals Ii and I 2 , and at the point koAssume that k] e and a 0 + kiP° > 0. It is obvious that r(k) < ri(ki) for all k e I), because r(k) is defined by the branch rj(k) on Ij, and k] is the maximum of n(k). The function r(k) = r 2 (k) on I 2 , but from definitions (5.22) and (5.23) it follows that r 2 (k) < rj(k) on I 2 , and thus r(k) < ri(k[) for all k e I 2 because rj (k j ) is the maximum value of rj(k). Finally, since r(ko) = ri(ko), it follows immediately that r(ko) < ri(kj). This proves (5.26a). The proof for (5.26b) is analogous. Next it is shown that the conditions a 0 + k 1 P° > 0 and a 0 + k 2 P° < 0 cannot be simultaneously satisfied. The definitions of k\ and k 2 given in (5.24) and (5.25) can be rewritten as k\ = Ci C 2 P 0 and k 2 = ci + C 2 P 0 , where cj, C 2 e 9?, and C 2 > 0 when k ( and k 2 are real. Thus if a 0 + k)P° = a 0 + cip° C 2 (P 0 ) 2 > 0, then a 0 + cip° + C 2 (P 0 ) 2 >0. Recognizing that a 0 + k 2 P° = a 0 + C 1 P 0 + c 2 ( P °) 2 , it follows that if g° = a 0 + kp° > 0 at kj, then g° > 0 at k 2 In a similar manner, it can be shown that if g° < 0 at k 2 , then g° < 0 at ki also. Therefore, the only possibility not covered by the conditions on the sign of g° in (5.26a) and (5.26b) is the case where a 0 + kiP° < 0 and a 0 + k 2 p° > 0. To complete the proof it is shown that this case leads to the choice k = koThis pair of inequalities implies that r(kj ) = r 2 (kj) and that r(k 2 ) = n(k 2 ). Therefore, the branches rj (k) and r 2 (k) do not define the curve r(k) at their maximum points. Since the function rj (k) has a

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120 unique maximum, this implies that the function r(k) does not attain its maximum on Ij, since ^ * 0 everywhere on Ij. Similarly, r(k) does not attain its maximum on I 2 . Therefore, the only possible maximum is the point ko. This implies that k = ko. In the case that a 0 + ki (3° = 0, this implies that the maximum of n(k) occurs at the point kj = ko. It can be shown that if g° = 0 at ki, then g° > 0 at k 2 , and thus the maximum occurs at k] = koSimilarly, if k 2 = ko, it can be shown that the maximum occurs at k 2 = koIn the case where kj and k 2 are complex, neither rj(k) or r 2 (k) has a real maximum. Again, r(k) does not acheive its maximum on Ij or I 2 , therefore this condition implies that k = ko. This proves (5.26c). Finally, from Corollary 5.1, it is concluded that the optimal controller is robust if r(k) <0. V

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BIOGRAPHICAL SKETCH Harry Michael Mahon was bom in Jacksonville, Florida on April 28, 1970. He graduated with a B.S. in chemical engineering from the California Institute of Technology in June, 1992. Mr. Mahon joined the graduate program at the University of Florida in August of 1992. He graduated with a Ph.D. in chemical engineering and an M.S. in electrical engineering in May 1998. 126

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Oscar D. Crisalle, Chairman Associate Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. A . Spyfos A. Svoronos Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard B. Dickinson Assistant Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. CL*** Thomas E. Bullock Professor of Electrical and Computer Engineering

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1998 Winfred M. Phillips Dean. College of Engineering Karen A. Holbrook Dean, Graduate School