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Robust stability analysis and controller synthesis for systems with parametric uncertainties

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Robust stability analysis and controller synthesis for systems with parametric uncertainties
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Ji, Baowei
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English
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xvi, 109 leaves : ill. ; 29 cm.

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Algorithms ( jstor )
Conservatism ( jstor )
Critical points ( jstor )
Critical theory ( jstor )
Mathematical robustness ( jstor )
Polynomials ( jstor )
Term weighting ( jstor )
Transfer functions ( jstor )
Uncertain systems ( jstor )
Weighting functions ( jstor )
Control theory -- Mathematical models ( lcsh )
Dissertations, Academic -- Electrical and Computer Engineering -- UF
Electrical and Computer Engineering thesis, Ph.D
Robust control ( lcsh )
System analysis -- Mathematical models ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph.D)--University of Florida, 2004.
Bibliography:
Includes bibliographical references.
General Note:
Printout.
General Note:
Vita.
Statement of Responsibility:
by Baowei Ji.

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ROBUST STABILITY ANALYSIS AND CONTROLLER SYNTHESIS
FOR SYSTEMS WITH PARAMETRIC UNCERTAINTIES
















By

BAOWEI JI


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


2004

































Copyright 2004

by

Baowei Ji

















To my parents

and to

Songhong















ACKNOWLEDGMENTS


I wish I could find words to express my gratitude to my advisors (Dr. Haniph A. Latchman and Dr. Oscar D. Crisalle) for their guidance, support, and encouragement during my time at the University of Florida. Although I still have much to learn, I feel confident as I begin my career in systems control and other areas. This feeling certainly is primarily the result of the consistent inspiration and encouragement of my advisors. Many times I was frustrated by seemingly unsolvable problems. Dr. Latchman patiently listened to my messy reports and directed me to the right track. His energetic approach and responsible attitude encouraged me to seek the best in academic research and systems development. Even though I never took courses with Dr. Crisalle, we have had numerous research discussions, and I feel as if I have taken a series of courses on systems control engineering with him. I truly appreciate the availability of my advisors whenever I had questions. I am also grateful for their high standards on mathematical derivation and report organization, which will significantly benefit my career development.

I also wish to thank my other committee members (Dr. Norman G. Fitz-Coy and Dr. Tan Wong) for their willingness to serve on my Ph.D. committee. Dr. Fitz-Coy gave me tremendous help when I first came to the U.S. He motivated my research interest in systems and control engineering. Dr. Wong raised many insightful questions and gave very helpful guidance during my oral qualifying exam.

All of my friends and research team members deserve my sincere gratitude and appreciation. Regarding the preparation of this dissertation, I thank the help from Mr. Minkyu Lee, Mr. Yu-Ju Lin, Mr. Saleh Al-Shamali, Mr. Suman Srinivasan, Mr. Dave Tingling, Ms. Sheryl Latchman, Dr. Benjamin Harrison, and many others.


iv









As always, I thank my parents in China for their heartfelt love and support. For the past 5 years, my wife, Songhong has patiently and enthusiastically supported my study and research. Although she has even more work (with her J.D. program study at the Levin College of Law at the University of Florida), she always assists me with her dedicated love and timely thoughts. I give thanks for her love that makes my life much more colorful and meaningful.


v
















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS .................................... iv

LIST OF TABLES ....... ................................. viii

LIST O F FIG U RES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF NOTATIONS .............................. xi

ABSTRACT .................................. xiv

CHAPTER

1 INTRODUCTION ....... .............................. 1

2 CRITICAL DIRECTION THEORY .................... 6

2.1 Critical Direction Theory ...... ...................... 6
2.2 Calculating Critical Perturbation Radius ............ . 10
2.2.1 V alue Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Defining CPR for Nonconvex Value Sets .......... 15
2.2.3 Defining a New Perturbation Radius ................ 17
2.3 Conclusion ....... .............................. 21

3 REAL pu ANALYSIS ....... ............................ 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Transforming a Real Parametric Affine Problem into a Linear
Feasibility Problem . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 The Parametric Stability Margin (PSM) . . . . . . . . . . . . . . 27
3.4 Real-p Analysis based on Polynomial and Rank-One Matrix
A pp roach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Exact Calculation of the Robust Stability Margin and the Worst
Case Frequency Point . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 E xam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Simulation Reliability . . . . . . . . . . . . . . . . . . . . . . . . 45
3.8 C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 ROBUST CONTROL FOR PARAMETRIC PLANTS . . . . . . . . . . 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 48


vi










4.3 Preliminary: the Mixed-Sensitivity H, Robust Control . . . . . 51
4.4 Robust Control for Parametric Uncertain Plants . . . . . . . . . 53
4.5 MPR Weighting Approach and its Conservatism Analysis . . . . 56
4.6 ECPR Weighting Methodology . . . . . . . . . . . . . . . . . . . 59
4.7 C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 ROBUST CONTROL FOR PARAMETRIC PLANTS USING STATIC WEIGHTING APPROACHES . . . . . . . . . . . . . . . . . . . . . 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Parametric Robust Control using a Static Weighting Approach 65
5.3 Robust Stability Conditions and Stability Margin Calculation 67
5.4 E xam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5 Sim ulation Reliability . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.1 Solution with a Bisection Searching Procedure . . . . . . . 74
5.5.2 Solution with an Additive Searching Procedure . . . . . . 76
5.6 All-Solution Controllers . . . . . . . . . . . . . . . . . . . . . . . 80
5.7 C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 ROBUST CONTROL FOR PARAMETRIC PLANTS USING
DYNAMIC WEIGHTING APPROACHES . . . . . . . . . . . . . . . 86

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Sub-Robust Controller Synthesis Using a Dynamic MPR
W eighting Approach . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Robust Controller Synthesis: a Dynamic ECPR Weighting
Approach with Controller Tuning Algorithm . . . . . . . . . . 90
6.4 Example . . . . . . . . . . . . . . 93
6.5 Simulation Reliability . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6 C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . 98

R EFER EN C ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109


vii
















LIST OF TABLES

Table page

2.1 Comparison of the three definitions for the marginally unstable case 20

2.2 Comparison of the three definitions for the stable case . . . . . . . . . 20 5.1 Simulation with a bisection searching procedure . . . . . . . . . . . . . 76

5.2 Simulation with an additive searching procedure . . . . . . . . . . . . 80


viii
















LIST OF FIGURES

Figure page

2.1 Negative feedback loop of the uncertain system p(s) with a controller
c (s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Stability analysis for a uncertain system g(s) under unity negative
feed back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Critical direction and critical perturbation radius . . . . . . . . . . . 9

2.4 The value set of g(jw, q) at w = 0.7 for Example 2.1. . . . . . . . . . 13

2.5 Comparison of CPR definitions (2.22) and (2.4). . . . . . . . . . . . 16

2.6 The value set at the marginally stable condition: a = 1.8660, and
w = 4.7294. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 The value set at a = 1.8600, and w = 4.7294. . . . . . . . . . . . . . . 19

3.1 The true-or-false function f(a) vs. the scaling factor a. . . . . . . . . 29

3.2 The classical M - A structure used for analyzing robust stability . . 32

3.3 Comparison of the PSM with the complex parametric stability margin 40

3.4 Comparison of the real-p with the complex-p . . . . . . . . . . . . . . 41

3.5 Real-p by the optimization algorithm over one variable . . . . . . . . 42

3.6 Real-p by algorithm rank-one approach . . . . . . . . . . . . . . . . . 43

3.7 The uncertainty set with a* = 1.8489 at w* = 4.6389 . . . . . . . . . 44

3.8 Transformation between linear programming functions of Matlab and
Lab V IE W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 The negative feedback loop of the uncertain System p(s) with a
controller c(s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Stability analysis for an interval system g(s) under unity negative
feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 A standard H, synthesis problem in the mixed-sensitivity framework. 52

4.4 Standard Al - A loop for stability analysis. . . . . . . . . . . . . . . 54


ix










4.5 A system with parametric uncertainty in the standard M - A loop. . 54

4.6 The parametric controller synthesis problem that is recast into the
mixed-sensitivity H, synthesis framework. . . . . . . . . . . . . . . 56

4.7 M PR versus CPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.8 Complex maps between the original plant p(jw, E) and the transformed
system g(jOw, ) = c(jw)p(jw, c). . . . . . . . . . . . . . . . . . . . . 59

5.1 (a) Negative-feedback loop including the uncertain system p(s)
po(s) + 6(s) and a controller c(s); (b) unity-negative-feedback of
system g(s) = c(s)p(s); (c) mixed-sensitivity approach to the
uncertain feedback system . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Plots of pm and pe radii as functions of the uncertainty-size parameter C. 72 5.3 Plots of p, vs. e (dashed line) and pe vs. e (solid line). . . . . . . . . 73

5.4 Plots of the phases of the four controllers respectively. . . . . . . . . . 76

5.5 Plots of the magnitudes of the four controllers respectively. . . . . . . 77 5.6 Plots of the JT.(jw)J with the four controllers respectively. . . . . . . 78

5.7 Plots of the controller phases, controller magnitudes, and transfer gain
T,(jw)J with the three controllers respectively. . . . . . . . . . . . 84

6.1 Plot of the true-or-false function via the uncertainty level E. . . . . . . 89 6.2 The tuning process using a controller ct(s). . . . . . . . . . . . . . . . 92

6.3 Sub-robust controller design example using a dynamic MPR weighting
approach. ....... ................................ 94

6.4 Stability analysis for the parametric uncertain system (6.9) cascaded
with the designed robust controller(6.14). . . . . . . . . . . . . . . 96

7.1 Augmented feedback loop with performance weights. . . . . . . . . . 99

7.2 Nyquist plot for disturbance rejection . . . . . . . . . . . . . . . . . . 100

7.3 Augmented feedback loop with performance weights. . . . . . . . . . 102


x















LIST OF NOTATIONS


arg(z)

Iz
Re(z) Im(z)

.]
A AT
R n1 x< m



R+ RHQ

q

q
o-(A) o(A)

pA(A)



T (s) W1(s)

W2(s)


xi


Laplace variable Frequency, sometimes used as an input variable The argument of the complex z Magnitude

Real part of the complex z Imaginary part of the complex z Greatest-integer function Inverse of matrix A Transpose of matrix A Real-valued matrix with n rows and m columns elements Complex-valued matrix with n rows and m columns elements Set of nonnegative numbers Proper and stable analytical space The upper bound of a scalar q The lower bound of a scalar q The largest singular value of M The smallest singular value of M The structured singular value of a matrix A with uncertainties A H, norm of the transfer function T2,(s) The transfer function from the input w to the output z The sensitivity weighting functions The controller sensitivity weighting functions











W3(s) The complementary sensitivity weighting functions

pO(s) The nominal system function, i.e., without uncertainties

p(s) The transfer function representation of a plant p(jw) System p(s) at frequency w p(s, e) Similiar with p(s) except showing the effect
C(s) A controller

g(s ) g (s, () = C(s)p(s, C)
d,(jw) The critical direction of the value set of g(s, E)

dcP(jw) The effective direction of the value set of p(s, e)

kN(w) Nyquist robust stability margin

Pm(W) The maximum perturbation radius

Sm(W) The approximation to pm(w)

Pc(W) The critical perturbation radius

&c(W) The approximation to pc(w)
Pe(W) The effective critical perturbation radius

&e(W) The approximation to pe(w)

Pr (w) The perturbation radius

5r(W) The approximation to pr(W)

Q Parametric uncertainty region
,9 Q The boundary of Q

V(jw) The value set of system g(s)
a V(jw) The boundary of V(jw)

VC(jW) The critical value set of system g(s)

a Vc(jw) The boundary of V(jw)

Vc,,(jW) The effective value set of system p(s)
a V,,(jw) The boundary of V,,(jw)


xii












T

AP Bp Cp Dp



A B1 B2 C1 0 D12

C2 D21 0

inf max min sup CDT

CPR ECPR LF LFT LP MIMO

MPDA MPR MSM PSM SGT SISO


Numerical computation resolution State space representation of system p(s)






Augmented state space representation



Infinum

Maximum

Minimum

Supremum

The critical direction theory The critical perturbation radius The effective critical perturbation radius Linear feasibility Linear fractional transformation Linear programming A multi-input multi-output system Major principal direction alignment The maximum perturbation radius The multivariable stability margin The parametric stability margin The small gain theorem A single-input single-output system


xiii















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 STABILITY ANALYSIS AND CONTROLLER SYNTHESIS
FOR SYSTEMS WITH PARAMETRIC UNCERTAINTIES



By

Baowei Ji

May 2004

Chair: Haniph A. Latchman Cochair: Oscar D. Crisalle
Major Department: Electrical and Computer Engineering

This dissertation covers our study of both robust stability analysis and robust controller synthesis for systems with highly structured uncertainties. These problems have attracted much interest in advanced control engineering, but experienced limited development in the past two decades. In 1997, an important observation was introduced with the critical direction theory (CDT). According to the theory, on the Nyquist plane at any given frequency, only the uncertain plants located along a specified direction, namely the critical direction, are of relevance to the stability analysis; all other uncertain plants that do not lie in the critical direction can be ignored when dealing with stability analysis. This dissertation extended the theory to systems with highly structured uncertainties, and explored the CDT's advantages over other approaches.


xiv









Firstly, robust stability analysis was addressed in this dissertation for systems with real-valued uncertainties. For SISO systems with affine real-valued uncertainties, we transformed the problem into a linear feasibility (LF) problem, together with an algorithm for calculating the parametric stability margin that was defined as the maximuni scaling value of the uncertainties in maintaining the robust stability of the system. This algorithm used a bisection algorithm to achieve computational efficiency. Then we developed a linear programming (LP) and an iterative algorithm based on CDT for the same problem. All the three algorithms produced the same results for the examples simulated in this dissertation. Finally we related the parametric stability margin with the real p results reported recently in the literature, and found consistency in using the critical direction theory and p analysis. Our research on robust analysis provided an alternative real-p analysis with clear geometrical explanation. Furthermore, our research on robust analysis establishes a sound foundation for robust synthesis research reported as follows.

Then, we addressed robust controller synthesis for systems with highly structured uncertainties. We constructed a systematic controller synthesis methodology, combining the essence of CDT and Ho, design. An important attempt reported in the literature on robust controller design for systems with structured uncertainties was to rearrange the controller synthesis problem into the mixed-sensitivity synthesis framework, and then to use H, synthesis tools. The family of the uncertain plants was represented by one that overbounded all the uncertain systems. We explored the mechanism leading to conservatism of this overbounding operation, and then examined a static weighting based on the critical direction approach using a scalar to represent the uncertain systems for all frequencies. By applying this static weighting function, we dramatically reduced the conservatism incurred from the previous approach, and obtained stability margins that were 50% larger than those resulted from the previous approach. Finally, we developed two dynamic weighting approaches and


xv









obtained controllers that could accommodate even larger level of uncertainties than those obtained using the static weighting approach.

Our study shows that the critical direction theory is effective and promising for nonconservative stability analysis and controller synthesis for systems with highly structured uncertainties.

The critical direction theory provides necessary and sufficient condition for robust stability. However to address robust stability and robust sensitivity at the same time, one important entity was defined in this dissertation, together with analytical approach in calculating that parameter. Some ongoing research topics were mentioned at the end of this dissertation.


xvi















CHAPTER 1
INTRODUCTION


Systems and control has always played an indispensable role in electrical, mechanical, aerospace, chemical and other engineering design and industrial implementation. For the past century, control engineering has gone through significant development from classical control [13] [33] [56] (1920-1950), to modern control [38] [39] [55] (1960-1980), then to robust control [18] [64] [72] (1980 to present). Given a plant, the first question one may ask is whether the plant is stable. When using the term stability, generally, one means internal system stability (e.g., no blowout of circuit components), and/or externally bounded input bounded output stability (e.g., no loss of a prespecified trajectory) [37]. If the plant is stable, one may further ask whether any uncertainty is allowed inside the plant, and what is the maximum level of uncertainties that could be tolerated. If the plant is unstable, it may be necessary to design a controller (a kind of compensator and regulator) to stabilize the original plant while satisfying some predetermined specifications. Real life always carries uncertainties, such as manufactural imperfection, measurement noise, system modelling errors introduced by linearization and system identification, environment fluctuation, and physical aging and wearing, to name just a few.

To cope with system uncertainties, classical control defined gain margin and phase margin, which were measured separately. Unfortunately, uncertainties in real life usually involve variations of gain and phase simultaneously. A preclaimed infinity gain margin may easily be destroyed if there is a tiny change of the phase [6]. Modern control originated from and finds satisfying application in space and aircraft engineering. Nevertheless, only limited uncertainties are allowed in modern control


1







2

due to the requirements of precise state-space system representation and strict definition of cost functions. Robust control aims to achieve robustness and maintain system performance despite uncertainties. Unstructured uncertainties represented by disks, are well addressed by [ analysis and H, control. However, structured uncertainties (e.g., real-valued and phase-limited uncertainties) more closely resemble real-life uncertainties. An electrical board is an example of real-valued uncertainties with the expected variation of resistance values or capacitance values. This dissertation focuses on systems with highly structured uncertainties.

With an aim at analyzing robustness stability in an exact manner, the critical direction theory (CDT) was developed recently based on the Nyquist stability criterion [51]. Having functioned as the foundation of classical control, the Nyquist stability criterion deals with a fixed single-input-single-output (SISO) plant go(s) by plotting the complex values of go(jw) in the complex plane. The plot is called the Nyquist plot. The unity-negative-feedback-loop system is stable if and only if the number of counter-clockwise encirclements of the critical point in the Nyquist plot is equal to the number of open-loop system unstable poles. Now as uncertainties (say e) enter the plant, we have a family of plants (say g(s, e)) surrounding the nominal plant go(s). Imagining that all of the uncertain plants are plotted, we obtain tube-like areas composed of many curves surrounding the original plot in the complex plane. Then assuming that the number of open-loop unstable poles is the same for the family of plants g(s, e), it can be concluded that the whole family is stable if and only if the critical point is outside these tube-like areas. This is the critical point exclusion principle. To check system robustness stability, therefore, one must check whether the critical point is inside the plot of the plant family at each frequency. This can be done by checking the uncertain plants along only the critical direction, which is defined as the direction from the nominal point go(s) to the critical point. This brought about the basic idea of critical direction theory.







3


This dissertation extends the advantages of the critical direction theory to systems with more general structured uncertainties, and most importantly, develops methodologies of applying the theory for designing robust controllers in conjunction with H, tools.

We can roughly divide the dissertation into two parts: robust analysis and robust synthesis. Chapters 2 through 3 discuss robust analysis. Chapter 2 introduces the Nyquist stability criterion, gives insight into the critical direction theory, and describes the calculation of the critical perturbation radius based on the plot of value sets. A new definition of the perturbation radius is also presented for considering the tradeoff between system stability and sensitivity.

For SISO systems with affine real-parametric uncertainties, chapter 3 transforms the problem of finding the uncertainty stability margin into a linear feasibility (LF) problem, and presents a novel approach for solving from real-p. The approach recovers previous results with real-p analysis. A comprehensive example shows that all these approaches are equivalent in solving the specified affine problem. The results in this chapter demonstrates that the stability radii approach we proposed may be promising for solving more general real-p problems with more general structured uncertainties. In terms of computational complexity, the proposed algorithm for calculating the parametric stability margin (PSM), powered by a bisection algorithm, shows great advantage over that based on an additively searching process. A similar computation advantage is obtained with the parametric robust stability margin (PRSM) algorithm to find out the smallest stability margin and the corresponding worst-case frequency point. This research provides not only a new real-p analysis method, but also a solid foundation for controller synthesis in the second part of the research.

Chapters 4 through 6 focus on designing robust controllers that can stabilize as large an uncertainty as possible. Controller synthesis for systems with unstructured







4


uncertainties was well developed using H, tools; however, a similar problem for systems with parametric uncertainties has developed relatively slowly. A very important attempt in literature was to rearrange the problem into the mixed-sensitivity synthesis framework, and then to use well-developed H, synthesis tools where uncertain plants were represented by a scalar that measured the maximum radius of the plants deviated from the nominal point. In Chapter 4, we apply the critical direction theory to analyze the conservatism of the previous overbounding behavior in representing the whole family, thoroughly develop the parametric robust control methodology, and find out the exact weighting function that is defined as the effective critical perturbation radius.

Chapter 5 focuses on parametric robust control using a static weighting approach where the uncertain plants for all frequencies are represented by a scalar, namely the radius of a disk. We prove that there is only one kind of optimal controllers for static weighting even though the coefficients of the controllers may not be same for different design resolutions. Nevertheless, a much larger level of stabilizable uncertainties can be found using the new interpretation of the static weight, namely, the exact static weight associated with the critical direction theory. Finally, the relationship of allsolution controllers that satisfied the H, design criterion are discussed.

Chapter 6 makes an effort to find an even larger stabilizable uncertainty margin by designing a robust controller using dynamic weighting approaches, where the difference of the families of plants at different frequencies are taken into account. First we differentiate the overbounding function for each frequency and construct a sub-robust controller synthesis algorithm enhanced by the bisection algorithm. Then we establish another dynamic weighting approach, where only the necessary and sufficient uncertain plants at each frequency are considered in solving for the maximum stabilizable margin and a robust controller with the most powerful stabilizing capacity.







5

Finally some future research directions are discussed in Chapter 7 to continue our exciting and promising research work.















CHAPTER 2
CRITICAL DIRECTION THEORY


2.1 Critical Direction Theory

It is important to differentiate notations of uncertain systems p(s) and g(s), and notations of nominal systems po(s) and go(s) in the sequel. As shown in Figure 2.1 and Figure 2.2, there exists a relationship: g(s) = c(s)p(s). which indicates that a controller c(s) is included in g(s), similarly, go(s) = c(s)po(s). When discussing stability analysis (chapters 2-3), we always refer to Figure 2.2, even with a unity controller. All four notations are used in chapters 4-6 when we talk about controller synthesis.

Classical control primarily considers certain systems: for example, the nominal system go(s). Given a fixed system, the Nyquist stability criterion states that the system is closed-loop stable if and only if the map of the Nyquist contour of the open-loop system encircles the critical point (-1 + jO) in the anticlockwise direction a number of times equal to the number of unstable poles of the open-loop system. If the system is stable under unity-negative-feedback, then the critical point should be outside of the Nyquist contour of the open-loop system.

The similar argument can be extended from the nominal system go(s) to a family of uncertain systems g(s) = go(s) + 6_(s), where uncertain part cg(s) is bounded by A(s) . If we assume that g(s) and go(s) have the same number of open-loop unstable poles, then the whole family of uncertain systems under unity-negative-feedback is robustly stable if and only if the critical point -1 + jO is not in the value set of g(s) for all frequencies. This is generally called the critical point exclusion principle.


6









Latchman and Crisalle [49] proposed the critical direction theory (CDT) to elegantly measure this necessary and sufficient robust stability condition.

It is generally assumed that the nominal system go(s) is closed-loop stable under unity-negative-feedback, since for a nominally closed-loop unstable system po(s) that is stabilizable, a stabilizing controller can be easily found by using classical controller synthesis approaches and cascaded with po(s) to form go(s). Similar to other stability analysis approaches based on the Nyquist stability criterion, the critical direction theory assumes that g(s) and go(s) share the same number of open-loop unstable poles. The solid curve in Figure 2.3 represents the nominal system go(jw) at the prespecified frequency range; the shaded area represents the uncertainty family g(jwi) for a specific frequency point w, of that range. The critical direction theory uses the values shown in Figure 2.3


p(s)







e C(s) U
-- -- -- -- -- --- ---- -7





Figure 2.1: Negative feedback loop of the uncertain system p(s) with a controller c(s).


1. Critical line is the directed line that originates at the nominal point go(jw) and

passes through the Critical Point -1 + jO.

2. Critical direction
1 +~ go(jw)
d,(jw) := - = eiNdc (2.1)
11 + go(jW)I







8


g (s)


I 3(s)



r U







Figure 2.2: Stability analysis for a uncertain system g(s) under unity negative feedback.

where edAdc(w) is the angle of the critical direction. The critical direction may

be interpreted as the unit vector that defines the direction of the critical line.

3. Value set (also called the Image-Set in the literature)


V(w) := {g (jW) I g (jW) = go(jW) + 6g (jW), 6g (s) E A(s)} (2.2)

4. Critical value set

Vc(w) := {gc(jw) I gc(Jw) = go(jw)+6c(jw) = go(jw)+adc(jw), for some a c R+ (2.3)

5. Critical Perturbation Radius (CPR)


pc (w) := max{a I gc(jw) = go(jw) + ad,(jw) C V(w)} (2.4)
aE R+

6. The Nyquist stability margin

kN P:=W) (2.5)
|1+go(jw)|

Note that at every frequency w, the critical direction dc(jW) may be interpreted as a unit vector with its origin at go(jw) and pointing towards the point -1 + jO.







9


Im


d~TJ(c1))
________-lI-jO Re




go m + g (j q )dc (ja




CPR g0(o)O



Tg (O))
g0(j)


Figure 2.3: Critical direction and critical perturbation radius

The expression 6c(jw) = Qdc(jw) represents the set of perturbations with frequency response lying along a straight-line segment that joins the points go(jw) and -1 +j0. The critical perturbation radius (CPR) is defined as the distance from go(jw) to the intersection of the boundary of the critical value set with the straight-line segment that joins the points go(jw) and -1 + jO. The necessary and sufficient stability condition of the CPR is given in Theorem 2.1. Theorem 2.1 [49] Consider the uncertain system g(s) given in Figure (2.2) that is stable under unity feedback, and assume that g(s) and go(s) have the same number of open-loop unstable poles. Then, assuming that the critical template is convex, the uncertain system is stable under unity feedback if and only if kN < 1Proof: The proof is given in [49]. U







10


2.2 Calculating Critical Perturbation Radius

The substantial step in CDT is to calculate the CPR. For the case that there is only one intersection between the boundary of the value set of g(s) with the critical direction, [66] presents the following heuristic procedure for calculating the CPR. Algorithm for Calculating CPR

Step 1 Generate a fine grid of frequency points [i, W2,. . .. Wk, - , Wn] and set k 1. Step 2 Set W = Wk. End if k > n.

Step 3 Calculate go(jw) = xo + jyo and d(jw). Step 4 Find the complex point x*+ jy*, x*, y* E R, on the Nyquist plane where the

line go(jw) + ad(jw), a E R+, intersects with the value set boundary &V(W). Step 5 Set pc(w) = I(x* + jy*) - (xo + jyo)1. Set k = k + 1 and go to Step 2. End of algorithm.

For general non-convex value sets, there is no rigorous algorithm for searching for the intersections at present. One possible way is to partition each of the edges into small enough segments, each is assumed to have only one intersection with the critical direction. Then, the furthest intersection measured from go(jW) is taken as the effective point in calculation of the CPR. Even though it is less rigorous, this sub-division method is extensively used in studying systems subject to parametric uncertainties [12] [59].


2.2.1 Value Set

The calculation of CPR is based on the description of the value set boundary. The CPR's dependence upon the value set suggests that the results in this dissertation may be extended to general uncertain systems having multilinear, multiaffine and/or nonlinear perturbations. However, for simplicity and tractability, the major part of this dissertation is based on affine uncertain systems that have the following system transfer function:







11


q1N1(s) + q2N2(s) + ... + qN,(s)
g(s, q) =qiD(s) + q2D2(s) + . .. + qmDm(s) (2.6)

where Ni(s) and D,(s) are fixed polynomials, and q = (qj, q2,... qm) e Q is a vector of independent uncertainties.

Note that the affine uncertain system (2.6) can be equivalently expressed as


s'a + s-an_1 + ...+ a0 (2.7)
snb, + sn-1bn_1 + ... + bo where a, and bi are fixed linear combinations of the independent uncertainties q. Lemma 2.1 A cascaded system of an affine system with a structure-fixed controller is still an affine system.

Proof: As described in ( 2.6), an affine uncertain system can be expressed as


q) qN,(s) + q2N2(s) + . . .+ qmN(s) g(s) qiDi(s) + q2D2(s) + . .. + qmDm(s) (2.8)

where all qj are independent uncertainties.

By definition, a structure-fixed controller may be expressed as



c s) = (s) (2.9)

where a(s) and b(s) are constant coefficient polynomials of s.

Then, the cascaded system is



h(s, q) c(s)g(s, q)
q .a(s) Ni(s) + q2 - a(s) N2(s) + . . . + qm a(s) .Nm(s)
qi b(s) Di (s) + q2 b(s) D2(s) + . . . + q - b(s) Dm(s) qi N (s) + q2N(s) + . + qmN (s) iD1(s+ 2 s+.+ Mas







12


In comparison with (2.8), the cascaded system h(s) is also an affine uncertain system. U

The value sets of those affine uncertain systems are well-defined by the following mapping process from the edges of the parametric space Q to the boundary of the system space g(jw).

Lemma 2.2 (Affine Mapping Theorem) : Given a systems transfer function g(jw) and a closed affine interval perturbation defined as Q E R', the value set will be g(jw,q) for every q G 9Q where 9Q represents the edge of Q (hypercube of three dimensional perturbation). Then the boundary of the value set ag(jw, Q) is composed by g(jw, OQ) such that

1g (jw, Q) C g (jOw, 'Q) (2.11)

Proof: Refer to [12] for proof and related preliminary information. S

For affine uncertain systems, Fu [23] shows that the edges of value set of g(jw) are either circular arcs or straight-line segments. Accordingly, the intersections can be found analytically as explained in [5] and [66]. Example 2.1 Given a plant,


g(s, q) = c(s)p(s, q), (2.12)



c(s) = (0.3s + 1), (2.13)




2 + (4 + 0.4q, + 0.2q2)s + (20 + q - q3) s4 + (9.5 + 0.5q, - 0.5q2 + 0.5q3)s3 + (27 + 2q1 + q2)s2 + (22.5 - qi + q3)s + 0.1 (2.14)

where {ql, q2, q3} E Q are uncertainties, and satisfy -3 < q < 3, i = 1, 2, 3.

The value set of g(jw, q) at w = 0.7 is plotted in Figure 2.4 where one can see the twelve segments mapped from the twelve edges from the parametric set Q.







13


0







-0.5


E











-1.5
-1.5 -1 -0.5 0
Real
Figure 2.4: The value set of g(jw, q) at w = 0.7 for Example 2.1. The circle point inside the value set is the nominal point go(jw); the diamond point is the critical point -1 + JO; the critical line is also shown.

For more general cases such as multilinear perturbation and nonlinear perturbation, Theorem 2.2 needs to be revised to include some additional interior onedimensional manifold subset of Q in addition to 9Q. The interior one-dimensional manifold, which contributes to the boundary of the value set, can be identified using methods such as in [12] [24] [46]. Examples of multilinear interval systems are as follows [25] [12]:


800(1 + 0.1q1)(s + 2)
s(s + 4 + 0.2q2)(s + 6 + 0.3q3)(s + 10)'







14


(s-+ 2)(s2 + s + 1)(6.6s' + 13.582 + 15.5s + 20.4) (s + 1)(3 + a22 + 4s + ao)(s3 + b2S2 + 3.5s + 2.4)( Obviously, neither of them can be put into the affine parametric uncertain system form (2.6) introduced in Chapter 2. For multilinear perturbation and nonlinear perturbation, the following general mapping theory should be referred to include some additional interior one-dimensional manifold subset of Q in addition to the boundary of the uncertain space 0Q.

Theorem 2.2 (General Mapping Theorem) :Given a systems transfer function g(jw) and a closed general interval perturbation defined as Q E R',the value set will be g(jw, q) for every q E Q. There exists Te, a subset of Q, which is a set of one-dimensional manifolds that includes all the candidates that may contribute to the boundary of the value set 9g(jw, Q) such that


'0g(jw, Q) C g(jw, Tc) (2.17)

The interior one-dimensional manifold, which contributes to the boundary of the value set, needs to be identified by referring to [24] and [46].

A special kind of multilinear uncertain system g(s) = 2i is considered in [12] with the assumptions that

* Uncertainties q and r are independent;

" n(s, q) and d(s, r) are coprime over the whole uncertainty spaces H x A, where

q E H, r C A;

* d(s, r) $ 0 for all r C A and each w > 0.

The extremal set gE(s) is defined as




gE(s) (q E HK,r E AE) or (q E HE,r E AK)}, (2.18)
d(s, q) I







15


where HE(S), AE(s) and HK(s), AK(s) denote the corresponding manifolds and vertices in H and A. respectively.

The boundary of the value set is described by Theorem 2.3 based on the generalized Kharitonov polynomials.

Theorem 2.3 (page 475, [12])


&g(jW) C gE(jw), VW E [0, 00) (2.19)

where



gE(S) nK(S)J U nE(s)) (2.20)
dE (8 K (8)
and where the extremal polynomial manifolds nE(s) and dE(s) and the vertex sets nK(s) and dK(s) are defined as


nE (S) = {n(s, q) I q E H1E}, nK (S) = {n(s, q) I q E HK (2.21)

dE(s) = {d(s, r) I r E AE}, dK(s) = {d(s, r) I r E AK},

Proof: Refer to [12] for proof. U

These theorems suggest some methods to describe the value set of a multilinear uncertain system. A rule of thumb can be used to check the validity of these theorems since a value set at a specified frequency & can be roughly provided by plotting all points g(jw) for all possible combinations of uncertainty values, i.e., sweeping over the uncertainty space.


2.2.2 Defining CPR for Nonconvex Value Sets

The CPR definition (2.4) assumes that the critical value set, not necessarily the whole value set, is a convex set as shown in Figure 2.3. For general convex value sets, however, such a definition may incur conservatism as shown in Figure 2.5.

To cope with general cases, Baab et al. introduce another definition for the CPR [3]:







16


Im








-1+j 0 Re







- g0(jo)









Figure 2.5: The CPR (shown as p(w)) defined in (2.22) for a nonconvex value set is less conservative than that (shown as r) defined in (2.4) for a convex value set.




11 + go(j]w) - min{a I (w) = -1 + adc(jw) E 0 V(w)} if - 1 V(w)
pc~w) :=OE R
S+ go (I w) + +ni Ia (w) -1 + ad,(jw) E 0 V(w)} otherwise
aER
(2.22)

where a is a nonnegative scalar, OV(w) represents the boundary of the critical value set V(w), and finally, (w) is the closet distance from the critical point -1 + jA to the boundary of the critical value set.

Accordingly. this definition of the CPR is utilized to extend Theorem 2.1 for general non-convex value sets.







17


Theorem 2.4 Consider the uncertain system g(s) given in Figure (2.2) that is stable under unity feedback, and assume that g(s) and go(s) have the same number of openloop unstable poles. The uncertain system is stable under unity feedback if and only if


PC < 1 (2.23)
11+go(jW)
'where po(w) is given by (2.22).

Proof: The proof is given in [5].


2.2.3 Defining a New Perturbation Radius

So far two CPR definitions have been introduced, one for a convex critical value set, the other for nonconvex cases. Even though the later can represent the necessary and sufficient condition for robust stability, it can not tell the system sensitivity. In other words, both the definitions only check the uncertain plants along the critical direction, hence can not tell exactly when and how the family of the plants are approaching instability. We use Figures 2.6 and 2.7 to illustrate the situation.

We use the same plant equation g(s, q) in Example 2.1, but scale the uncertainties q, with a scalar a, i.e., -3a < qj < 3a, i = 1, 2, 3. It turns out that the worst case frequency point is w = 4.7294, where one obtains the smallest scaling factor o = 1.8660 for that the system is unstable. This smallest value is defined as the parametric stability margin in the following two chapters. The value set of the system g(s, aq) at the marginally stable conditions a = 1.8660, and w = 4.7294 is plotted in Figure 2.6. However, the case for a small variation of the scaling factor a is plotted in Figure 2.7 where a = 1.8600, w = 4.7294. Following either of the CPR definitions, the CPR is small and the Nyquist stability margin kN is far less than unity. Accordingly, one may predict that the real parametric stability margin is much larger than 1.8600, which contradicts the parametric stability margin a = 1.8660.









18


0.2


-0.2 1-


-0.4

E

_0.6k


-0.8 1-


-1


-1


-0.8


-0.6


-0.4


-0.2


Real

Figure 2.6: The value set at the marginally stable condition: a
4.7294.


0


0.2


= 1.8660, and w =


In recognizing this, we provide a more meaningful definition called a perturbation radius (PR):




1 + go(jw)| - min{a C(w) = -1 + aej' E 0 V(w), VO} if - 1 V(w) Pr (W) := j oj aj cER
'1 + go(j') + min{a I ((w) = -1 + aej9 C 0 V(w),VO} otherwise
aER
(2.24)

where a is a nonnegative scalar, where OV(w) represents the boundary of the value set V(w), and finally, where ((w) is the closet distance from the critical point -1 +jO to the boundary of the value set.

Different from the previous two CPR definitions in (2.4) and (2.22) that are associated with the critical value sets, the definition of PR in (2.24) checks the evolution


0 F







19
0.2


0


-0.2


-0.4

E
-0.6


-0.8


-1



-1 -0.8 -0.6 -0.4 -0.2 0 0.2
Real
Figure 2.7: The value set at a 1.8600, and w 4.7294.


of a whole value set around the critical point -1 + jO at each frequency, and reflects the shortest distance to the critical point among all the boundary points of the vale set. The computation algorithm and application of the definition can be found in [2].

Nevertheless, the newly defined perturbation margin still stipulates the necessary and sufficient condition for system stability as in the following theorem. Theorem 2.5 Consider the uncertain system g(s) given in Figure (2.2) that is stable under unity feedback, and assume that g(s) and go(s) have the same number of openloop unstable poles. The uncertain system is stable under unity feedback if and only if



1 Pr P) < 1 Vw, (2.25)
11 + gO0(j)|







20


where p(w) is given by (2.24).

Proof: The proof can be found in [2]. U Furthermore, it is straightforward to check the following equivalency among the three definitions (2.4), (2.22) and (2.24) in terms of system stability [2]:



Case: Stable Pr (() < 1 + go () -I > Pc(W) < 11 + go(jW)I, VW

Case: Marginally Unstable pr(w) =I + go(jw) - pc(w)) 1 + go(jw)1, Vw

Case: Unstable pr (w) > I1 + go(jW)| -- pc(w) > 1 + go(jO) , VW

Taking Figures 2.6 and 2.7 for example, we have the following two tables in measuring system stability. Table 2.1 shows the consistency of the three definitions for the case of marginal instability. Table 2.2 clearly shows one advantage of the PR definition (2.24) in reflecting the closeness of the condition of a = 1.8600 to the unstable condition of a = 1.8660 at w = 4.7294. More advantages of this new definition can be found in [2]. We will also apply the new definition in Chapter 7 for one kind of controller design problem.

Table 2.1: Comparison of the three definitions for the marginally unstable case I CPR (2.4) CPR (2.22) J PR (2.24)
radius value 4.553 4.553 4.553
radius value
i1+gO(jW)I
system stability unstable unstable unstable


Table 2.2: Comparison of the three definitions for the stable case I CPR (2.4) 1 CPR (2.22) PR (2.24) radius value 0.093 0.093 4.553
radius value 0.019 0.019 0.906
system stability stable stable almost unstable







21

2.3 Conclusion

The main advantage of the critical direction theory is that it provides necessary and sufficient conditions for robust stability in the presence of highly structured uncertainties with phase and directionality constraints. The critical direction technique opens up new avenues for robust analysis and could lead to novel approaches for robust control synthesis as demonstrated in the rest of this dissertation. We will first explore the advantages of CDT with robust analysis in the next chapter.















CHAPTER 3
REAL p ANALYSIS


3.1 Introduction

The problem of assessing robust stability and computing stability margins for SISO systems with parametric uncertainties has attracted considerable attention in the last decade. While more general parametric uncertainties such as multilinear dependence cannot be easily handled at present, it is now well known that affine parametric uncertainty problems result in a rank-one structured singular value (A) problem. Moreover, p for the rank-one problem is exactly equal to its upper bound that is obtained from a convex optimization problem even though the general robust stability margin computation problem is known to be NP-Hard and is computationally intractable. In last chapter, we had presented an explicit analytical expression for complex parametric uncertain systems. It is known that standard complex-pi analysis will likely incur conservatism when working with real parametric uncertainties. Many researchers have devoted their attentions to the exact calculation of real-p problem. This aspect of the problem is amply documented in the literature [14] [15] [30] [60]. Of particular interest are references [15] [67], which will be reviewed in Section 3.4.

While most of the aforementioned developments are based on the structured singular value, the approach using the stability radii is arguably less well-developed. This is mainly because of the nonconvexity or amorphous nature of the value sets. This chapter starts directly from the Nyquist stability theorem and works with system transfer functions to exactly calculate the stability radii for a special kind of parametric uncertainty, i.e., the so-called "affine parametric variation" problem.


22







23


The rest of the chapter is structured as follows. Following the introduction, Section 3.2 transforms the real affine parametric system into a linear feasibility problem to avoid the difficulty of working with nonconvex value sets. The stability radii, called the parametric stability margin (PSM), is defined in Section 3.3, where the exact calculation algorithm is also given. The previous representative results based on the structured singular value approach are reviewed and compared with our approach in Section 3.4. Section 3.5 presents a further exact calculation of the maximum real-p over a frequency range and the corresponding worst-case frequency point using a bisection algorithm. A comprehensive example is analyzed in Section 3.6. Simulation reliability is validated in Section 3.7. The final section concludes the chapter.


3.2 Transforming a Real Parametric Affine Problem into a Linear Feasibility Problem

Consider the nominal system go(s) and the family of uncertain systems g(s). The Nyquist stability criterion states that a system is closed-loop stable if and only if the map of the Nyquist contour of the open-loop system encircles the critical point in the anticlockwise direction a number of times equal to the number of unstable poles of the open-loop system. If we assume that the nominal system go(s) is stable under unity-negative-feedback, and that g(s) and go(s) have the same number of open-loop unstable poles, then the whole family of uncertain systems under unity-negativefeedback is robustly stable if and only if the critical point -1 + jO is not in the value set of g(s) for all frequencies s = jw. This is generally called "critical point exclusion", the extension of "the zero exclusion" principle applicable to uncertain polynomials. However, as shown in Chapter 2 (see page 18), the value set for an affine uncertain system is not necessarily convex, which imposes much difficulty in checking whether the critical point is in the value set or not. Fortunately, Baab et al. [5] demonstrate that the real parametric affine problem can be recast into a linear feasibility problem.

The standard affine problem is expressed as







24


no(s) + Z qjnj(s)
g (s, q) = (s)-, q C Q (3.1)
do(s) + E 1qidi(s)
where Q = {q E Rnr I < qj i, i 1,2,..., mr}, and q = [q1, q2, qm]T

Here, m, denotes the number of the uncertain entities.

The family of uncertain systems in (3.1) can be represented in the following vector-matrix form



noo01 .02 ' norm,

n 10 + n11 n 12 nj.'El, q2
[1s .- sm1 m j+


qmO - mL nrm2 . . mm, m

doo do1 d02 - - dqm


i . i ] dio + dil d12 -. - din, 2


n L1 dn2 ''' dnm qrm,
(3.2)

hence,



g(s, q) S (n + N, q) (3.3)
Sd T(do + Dm, q)
where sn and sd are vectors of length m+ 1 and n+ 1, respectively, containing powers of the Laplace variable s = u+jw; and n0 E R-+1, d0 E Rn+l, Nm, E R(m+1)x(m,) and Dm, E R(n+1)x(m,) are constants that represent the structure of the affine parametric uncertainty. The Nyquist image g(jw, q) is obtained by evaluating the vectors s, and Sd at the Hurwitz stability boundary s = jw to yield


sT = I - 2 -Js W4 ... (3.4)








25


Sd,w1T iw -2 - W3 4 ... I


(3.5)


Now the vectors s,, and Sd,w, which are constants for a particular frequency W, and n0, Nm,, d0 and Dm, can be separated into real and imaginary parts. Define


Sn,RT









Sd,R Ti


T


2 4


-W3 5


2 41 6




3 5 7


00 7201

n0.R n20 E REm/2]+1, Nm,,R - n21


n02 ' n22 -.-.-


n1l nil n12

nI n30 E RE~t1/ m n3l T32


nom, n2m,


' lm ,

-.- n3m,


c R( [m/2l+l)x(mr)



(3.10)


E RF(m+l)/21x(mr)



(3.11)


(3.6)


(3.7)


and


(3.8)


(3.9)







26


d01 d02
SR[n/21+1, DmR, = d21 d22


... dm, ... d2m,


C R( n/21+1)x(m,)


(3.12)


d12 ...

d32 ...


dim, d3m,


E Rx 1+/2 x(",)


(3.13)


where [-] represents the greatest-integer function.

The uncertain system becomes


Sl,R(nD,R + Nmr,R q) + sI(nI + Nm,,i ) - ) g (s, q) = n,~oR+NRq ,(3.14)
S ,R(doR + Dm,,R q) + j SI (doI + Dm,,, - q) To determine if the critical point -1 +j0 is a member of the value set of g(jw, q) for a particular frequency, it must be determined if there is a vector q E Q such that g(jw, q) = -1 + J0. Using the previous notation, the following equation is obtained




sn R(no,R + Nm,,R - q) + j sr 1(noj + Nm,,, - q)

[sern(do, + Dm, .R -q) + j sT (doj + Dmr,I q)] (-1 + J0)


Equating real and imaginary parts and rearranging terms yields


s,RNm,,R + S mSDmr,R -s,RO,R - sd1, sINm,,I + sD,,I -sI,no,j - s'1d0,1 I

Note that (3.15) has the form


(3.15)


do,R


d00 d20


d10 dr
doj d30 E RI(7+m ,2 Drn,,1 dai







27


A(w)q = b(w) (3.16)

where the real matrix A(w) E R2x"m, and the real vector b(w) C R2 depend on the uncertainty structure. This is a system of two linear equations in q. The solution represents all the points q in the parametric space that map under g(jw, q) to the critical point -1 + j0. Therefore, to determine whether the system is unstable, it suffices to determine if there exists a solution q E Q to (3.15). This is a standard linear feasibility (LF) problem of the form:


does there exist q such that

A(w)q = b(w)


3.3 The Parametric Stability Margin (PSM)

The parametric stability margin (PSM) is defined as follows:

Definition 3.1 Consider an interval plant g(s, q) with parametric uncertainties q C Q, where q < q, < q and cz(w) is an arbitrary positive scaling factor. Then the frequency-dependent parametric stability margin is the smallest scale factor a(w) such that g(jw,a(w)q) will satisfy


1 + g (jw, a(w) q) = 0, Vw (3.18)

The parametric stability margin is a nonnegative real scalar that can be interpreted as the minimal magnification (a(w) > 1) or contraction (a(w) < 1) of the parameter set Q that brings the closed-loop system to the limit of instability. Geometrically, the parametric stability margin represents the minimum tolerable blow-up factor. Note that the parametric stability margin is defined for each frequency.

Therefore, in order to calculate the stability margin, the decision of whether to increase or decrease the perturbation scaling factor a (a - iteration) is an essential







28


problem. For the LF problem given in Section 3.2, the decision of how to adjust a can be determined by a simple numerical comparison in the algorithm. Here, the true-or-false function of whether -1 E g(jw, q) is used as the testing index of the bisection method. Define the true-or-false function f(a) as:


f -1, (which means False) If - I g(jw, q) (3.19)
+1, (which means True) otherwise

If we can prove that the true-or-false function f(a) is monotonic with respect to a, then a bisection algorithm can be invoked and the search range over a can be dramatically reduced. To show the monotonicity of the true-or-false function, we should check the variation of the value set of g(jo, q) at a specified frequency point w with respect to the scaling factor a. The following lemma states an important property of the value set for interval systems that satisfies the monotonicity requirement. Lemma 3.1 Consider the interval plant g(s, aq) with a scaling factor a. The value set is monotonically non-decreasing as a increases.

Proof: The proof of this lemma is sketched as follows. Consider the uncertainties q E Q and scaling factors a, > 0, a2 > 0, such that a2 > a1. Then it follows that Q1 C Q2 where Qi = Qaiq and Q2 = QQ2q. Since the value sets g(jw,Q) and g(jw, Q2) are mapped from the hyperbox Qi and Q2 respectively, they certainly satisfy



g(jw, Qi) C g(jW, Q2) (3.20)

This concludes the proof. U

Following Lemma 3.1, it is straightforward to check that function f(a) increases monotonically while a increases from zero. As plotted in Figure 3.1, f(a) equals

-1 whenever a < a* which means the system is stable for these small levels of uncertainty and -1 + jO is outside the value set of g(jw, aq), and f(a) equals 1







29


whenever a > a* which means the system becomes unstable at a certain level of uncertainty and will continue to be unstable for even larger levels of uncertainty. a* represents that critical level of uncertainty, i.e., the PSM we are looking for at a given frequency point.

In conclusion, the true-or-false function satisfies the monotonicity requirement for the bisection method. This renders the following PSM algorithm possible.



f(a)
+1



a a

-1




Figure 3.1: The true-or-false function f(a) vs. the scaling factor a.


Algorithm PSM:(parametric stability margin solution) Step 1. Set up a fine grid of frequency points in the range (wIb, Wb). Step 2. Consider the next frequency point wk that has not yet been looked at. Set alb = 1; aub = B, where B is the largest possible value of a, up to oc. Step 3. a(k) = (alb + aub)/2. Check whether -1 E g(jwk, q).

* If -1 g(jwk, q), choose lower bound aeb = a(k). then iterate Step 3.

* Else, choose upper bound ac, = a(k). Now if (aub - alb)/alb < 7, where T

is small enough to approximate zero, exit this loop and go back to Step 2

to work on next frequency point. Else, iterate Step 3. End of algorithm.

Note that the algorithm with the a - iteration enhanced by the bisection algorithm achieves great computation efficiency compared with general a sweeping







30


method. Furthermore, in Section 3.5, an interleaved bisection algorithm is introduced to efficiently compute the worst-case stability margin and the corresponding worst-case frequency point. It is acknowledged that Bhattacharyya et al. [12] consider a similar stability analysis problem, but working with additive increase of the uncertainty, i.e., q = qO + Aq. By assigning zero to the characteristic polynomial of the family of uncertain systems, a Linear Programming (LP) problem is constructed to solve for the stability margin 3 where the variation of the uncertainty is bounded by 3. In this chapter, we are considering multiplicative increase of the uncertainty, i.e., q' = aq. Actually, it is in this multiplicative case that the parametric stability margin equals to the Multivariable Stability Margin, or the inverse of the well-known ft as shown in next section.

Motivated by the argument in [12], the PSM can be arranged as



inf a subject to
aqi A(w)q = b(w) and aqi < qi < ai for= .

Note that agq < qi < ai for i = 1, 2, . .., m, and the other constraint a > 0 can be easily expressed in matrix form:


1 0 0 ... 0

-1 0 0 ... q, 0

0 1 0 ... -42 0

0 -1 0 ... q 2 1 0
q
[ (3.21)



0 ... 0 1 -qM 0

0 ... 0 -1 q 0

0 ... 0 0 -1 0







31


namely,



Aconx < bcon (3.22)

where x := [q; o].

The combination of (3.21) and (3.22) is a standard linear programming (LP) problem. In Section 3.6, we show that a - iterationa algorithm with bisection algorithm yields the same results as with the standard LP problem solved using Matlab function fmincon.m.

In next section, some representative previous results of the same real parametric affine uncertainty problem are reviewed and compared.


3.4 Real-p Analysis based on Polynomial and Rank-One Matrix Approach

In general, stability problems can be conveniently studied in the framework of polynomials whose coefficients are affine functions of real or complex uncertainties. For polynomials with complex affine uncertainties, Latchman et al. [53] present explicit and exact stability conditions using critical direction theory and intuitive geometric arguments. Jie Chen et al. [14] [15] introduce a generalized notion of structured singular value and provide a necessary and sufficient condition concerning the robust D-stability of the polynomial with either complex unstructured or real parametric uncertainties alone. In order to facilitate comparing with the developments in this chapter, previous results on real parametric uncertainties are summarized in the following.

The characteristic polynomial of the uncertain system in (3.1) is: M,
p(jw, q) = n(jw, q) + d(jw, q) = po(jw) + pk (jw)qk(jw) (3.23)
k=1
As shown in [11], an affine real parametric uncertain system is equivalent to a rank-one problem.







32


W M N Zw
w~ Li z M
IU VI
r y r y

Figure 3.2: The classical M - A structure used for analyzing robust stability

Lemma 3.2 The unity negative feedback configuration of the SISO uncertain system (3.1) with affine uncertainty elements can be transformed into an equivalent MIMO M - A structure of the form shown in Figure 3.2 where the diagonal uncertainty matrix


A = diag{qi, q2, . ., q,} (3.24)

and the rank-one structural matrix M {Mik} with elements



Mik (s) - - P , ' = 1,2,.. ., , k = 1, 2,...,m (3.25)
po(s)
Furthermore, the nominal model matrix is V = , and the interconnecting
no(s)-ido(s)
matrices are N = {N,1}, where


ni(s)po(s) - pi(s)no(s)
(no(s) + do(s))do(s)
and U = {U1,k}, where


d0 (s)
U1,k do(s) k 1,2,..., m, (3.27)
no(s) + do(s)'







33


In [15], Jie Chen et al. provide a general formula for the generalized notion of y defined there. Furthermore, the real-p of the system with affine real parametric uncertainty reduces to the following convex optimization problem. Lemma 3.3 The structured singular value of the system with affine real parametric uncertainty as described in (3.25) is



A(s) = inf T Pk -S - + IM (3.28)
us)\k=1.s s
where the weighting constants (k) of the uncertainties are considered in the system matrix M, i.e.,


[pi(S) =pr [Y1 --[-mr] (3.29)
.Po (S) P0 (S)
Proof: Refer to [15] for the proof. U Of particular interest is the case q, = 1, which corresponds to the standard structured singular value. The analytical expression (3.28) involves solving a convex optimization problem in one real variable, i.e., x, and renders the real-p problem readily solvable.

In parallel with the real-p problem in the framework of polynomials, there is also a real-p analysis based on matrix operations. Referring to the left part of Figure 3.2, we can easily find the closed-loop transfer function between the input and the output, which is called the upper linear fractional transformation (LFT), denoted as FU(M A). The theoretical control insight is gained from this upper LFT representation:


y = FU(M - A)r = [V + UA(I - MA)-'N]r. (3.30)

Obviously, for the closed-loop to be well defined, the matrix I - MA must be invertible, i.e. det(I - MA) # 0. This leads to the definition of the multivariable stability margin (MSM) [18] [61]







34


MSM(w):= min{yldet(I - -yM(jw)A(jw)) = 0, ||A|.. < 1} (3.31)
-yCER+

By recalling the definition of the well-known P, the MSM is the inverse of p;


1
MSM:= (3.32)

Note that the MSM is precisely analogous to the definition of the parametric stability margin for a SISO system given in (3.18).

It is straightforward to denote matrix M in (3.25) as



M(s) u(s)v*

. pi(s) p2(s) pO (S) [1,1, ... ,1] (3.33)
Po(s)' po(s)' 'Po(s)]

Therefore, the real-p of the system with affine real parametric uncertainty reduces to a rank-one problem. For any bounded uncertainty A and a E R+,




PA (M) 1
min{# I det (I,, - #AM) , ||AJJl, < 1} = max{a I det In - M), 1A1<1} = max{a a = v*Au}

= max{a a = Y. okUk} (3.34)
k=1
As such, Young [67] indicates that the rank-one real-pi problem simply amounts to choosing 6k so that vectors {6kVkUk} for k = 1, - - - , m, add up to a positive real number, which is as large as possible. Certainly, one can search 6k from its lower bound to the upper bound while fixing all other 6 at either of their lower or upper bounds based on the well-known "Edge- Theorem". This is equivalent to a search over







35


2M segments. Young [67] proposes a much more efficient algorithm that is adjusted to work with purely real-p problem.

Algorithm Real-p: (Rank-one real-p solution) Step 1 Choose starting values for the real perturbations as 6k = sign(Re(vkUk)).

Then for all 6k we have Re(6kvkuk) > 0. Now compute S = sign (Im (E; 6kVkk)). Step 2 Rank all the components 6kvk'Uk by argument. Step 3 Consider the highest rank component that has not yet been looked at. Compute 6opt, which is the optimal value of this 6k, for 6opt E R unrestricted in sign or magnitude, and all the other real perturbations fixed at +1 or -1. Note: 60pt magnifies Re (Z_'= kvk Uk) while Im (Z o1 6kVkUk) < T where T is used to

approximate zero.

Step 4 If sign(60pt) = -sign(6k) and 1o6ptl > 1 and not all the components have been

looked at, then reassign 6k with max[-1, min[1, 6,pt]], and go back Step 3. Else, either find p with this 60pt or verify that no 6k exists to make the summation

add up to a real number.

End of algorithm.

Note that the algorithm requires at most a search over the real parameters, which grows linearly with m,, not as generally believed that the complexity in computing p grows exponentially with m, .

In Section 3.6, an example is provided to illustrate that the real-p over polynomial approach [15] and the real-p over rank-one matrix algorithm [67] are equal, and also equal to the inverse of the parametric stability margin introduced in Section 3.3.


3.5 Exact Calculation of the Robust Stability Margin and the Worst Case Frequency Point

Even though the MSM, the PSM, and the real-p are all defined for each frequency point, usually it is more important to find the minimal stability margin, or the maximal y over a frequency range from the perspective of robust analysis since







36


they denote the worst case stability margin. In this section we consider the problem of computing the worst case robustness stability margin for a SISO system with real parametric uncertainty. In the last few decades researchers have given considerable attention to the robustness margin problem for the case of unmodeled dynamics and parametric uncertainties. The most popular methods to date use Lyapunov methods and structured singular value p. For real parametric uncertainties, however, these methods could be very conservative. Furthermore, the degree of conservatism may be arbitrarily large.

Another method of computing the parametric stability margin, in principle, involves a domain splitting global search in both the frequency and parametric space. In [25], Gaston and Safonov developed a computational method for the exact robustness margin for a multilinear interval plant. In their approach the hypercube in the parametric space is divided into sub-cubes resulting in the union of these sub-convex hulls approximating the boundary of the value set.

The Parametric Robustness Stability Margin (PRSM) is defined as follows:

Definition 3.2 Consider an interval plant g(s, q) with parametric uncertainties q G Q., where q < qi < q and a is an arbitrary positive scaling factor. Then the PRSM is the smallest scaling factor satisfying



a* = inf {a 1 + g(jw, aq) = 0} (3.35)
W
In contrast with the parametric stability margin defined for each individual frequency, the parametric robustness stability margin (PRSM) is defined at the corresponding worst-case frequency point.

Generally, the procedure of finding the parametric robustness stability margin in the literature consists of two major steps. The first one is to find the upper and lower bounds of a for each frequency Wk where k = 0,1,... , m,. The second step is using numerical methods iteratively to obtain a(jwk). Finally, a plot of a(jwk) with







37


respect to Wk for k = 0,1, . . . , m, is created and the minimum of a(jwk) over the entire frequency range is the robustness margin.

Here, we propose a new approach to compute the worst case robustness margin directly from Nyquist stability criterion. A key step in the new approach is the use of the bisection algorithm. The robust stability test for a given uncertainty size can be accomplished by a simple numerical comparison. This has immediate implication for numerical efficiency, since a well defined mechanism is now available to measure the effect of increasing the uncertainty size without computing the entire value set. Moreover, instead of searching for robustness margin at each sampled frequency, this new approach increases the scale of uncertainty size a for a range of the frequencies until the first a violates the robust stability criterion at some frequency. This process simultaneously identifies a restricted range for both a and w and an efficient bisection search identifies the robustness margin. Note that the specified frequency tolerance may affect the searching of the global minimum a*. Special care should be taken for all degenerate cases.

Algorithm PRSM (Parametric Robust Stability Margin) Step 1 Start with initial scale ai = 0, initial frequency range F, a increasing step

Aa, and set performance criterion T.

Step 2 Check whether -1 E g(ji, q), where wi E F

" If -1 V g(j'w, q), Vwj, choose a = ao + Aa, then iterate Step 2.

* If at any wi, -1 E g(jwi, q), choose upper bound as, = a and lower bound

alw=0 - .A.z

Step 3 Set the frequency range F equal to F' = [wow, Wp], where

0 W*0W = w1 such that -1 V g(jwk,q),VWk < WI and -1 E g(jwi,q). If

W10W = oWP, {number of points in F'} =2*{number of points in F'}, then
iterate Step 3; Else, go to Step 4.







38


* w,, = W, such that -1 E g(jwu_1, q) and -1 g(jWk, q), VWk > w., then

go to Step 4.

Step 4 Bisect the scale such that a = (alow + aup)/2. Step 5 Check whether -1 E g(jwk, q)where Wk E F'

* If -I g(jWk, q), Vw E F', then a,= a; go back to Step 4.

* If at any wi, -1 E g(jwi, q), check whether the program should end:

- If (PUP - WiO0)/wiOW > 7 or (aGU - a01W)/alow > r, Go back to Step 3

for further reduction of the frequency range F'.

- If (O, - wiO0)/wiO < T and (aup - alow)/aow < T, go to Step 6.

Step 6 a* = (ajo,, + aou)/2 is the robustness margin, w* (wow + WUP)/2 is the

corresponding first unstable frequency. End of algorithm.


3.6 Example

The following example illustrates the use of the PSM algorithm and compares our results with those based on other approaches.

Consider the uncertainty system g(s, q)


g (s, q) =(s, q) (3.36)
d(s, q)


n(s, q) = s2 + (4 + 0.4q, + 0.2q2)s + (20 + q - q3) (3.37) d(s, q) = s4 + (9.5+0.5q1 - 0.5q2+0.5q3)s3 + (27+ 2qi + q2)s2+ (22.5 - qi + q3)s +0.1 (3.38)

where {qi,q2,q3} E Q, and -3
System (3.36) is a modified version of a model investigated by Fu [23]. Note that coefficients of the transfer function depend affinely on the parameters q E Q. The nominal plant go(s) = g(s, qO) is obtained with qO = (0, 0, 0).







39


Its characteristic polynomial is:



p(s, q) = n(s, q) + d(s, q) (3.39)


p(s, q) s4 + (9.5 + 0.5qi - 0.5q2 + 0.5qq3)s3 + (28 + 2q, (3.40)
+q2)8(2 + (26.5 - 0.6qi + 0.2q2 + q3)s + (20.1 + qi - q3)

Referring to (3.23), we have the following entities


po(s) = s4 + 9.5s3 + 28s2 + 26.5s + 20.1 p1(s) = 0.5s3 + 2s2 - 0.6s + 1 (3.41)

p2(s) = -0.5s3 + s2 + 0.2s

p3(s) = 0.5s3 + s - 1
Given s = jw, the nominal point of p(jw, q) is found to be



po(jW) w4 - 9.5jw3 - 28w2 + 26.5jw + 20.1 (3.42)

From Lemma 3.2, M (Mik), where Mik(s) =1- , i= ,2, 3; k 1, 2, 3
PO (S)
and A = diag{qi, q2, q2}.

Result 1: (Complex p Solution)

When the uncertainties are purely complex, it was shown in [53] that an exact analysis formula can be obtained, which is the same as that given in [15]. Moreover, the result is precisely the same as the value of p calculated with Matlab function ssv. m.

Result 2: (Real-p Solution with Algorithm PSM and the standard LP algorithm)

With Matlab function linprog.m, and T = 0.01, the parametric stability margin is calculated by Algorithm PSM and shown in Figure 3.3 together with the MSM, i.e. the reciprocal of complex-p obtained in Result 1. Obviously, standard complex-p solution introduces much conservatism when dealing with real structured uncertainty.








40


This also can be seen from Figure 3.4 where both real-p and complex-p are shown. With Matlab function fmincon.m, the solution of the standard LP problem is obtained, which is exactly same with that solved from Algorithm PSM.


(Stability Radii) Algorithm PRSM 100


9080

- PRSM
70- MSM-Complex Assumption


6050 0


U40-


30


20-


10-


0
101 100 101
Frequency w in rad/sec


Figure 3.3: Comparison of the PSM with the complex parametric stability margin



Result 3: (Jie Chen et al. [15] Optimization Solution) Refer to (3.29), the weighting constants are Yk = 3, k = 1, 2, 3.








41


(Stability Radii) Algorithm PRSM


10
Frequency w in rad/sec


Figure 3.4: Comparison of the real-p with the complex--p


Pi Pmr(s) T
po(s) pO(s) )

3p(S) 3 P1s) -3Pas
IpjFO() pc (s) PO (5)
-3p2() -3p(s) -3P2()

-3(s) -3p3(s) -3p3(s)
POWs POWs POWs


F 3 ( Pk(s)
Re '7k
.k=1 PO(s)


+x Im 7k Pk(S)I
( PO (s


0.7


0.6 1


0.5k


=5
E


U) U)


0.4


0.3


0.2


0.1


(1'


Complex Mu Real Mu
F -U


10~1


101


M(s)


p(s)=inf
xER


(3.43)








42

With the Matlab optimization toolbox, this real-p is calculated and shown in Figure 3.5.


0.7


JieChen 1994, ssv with Polynomial approach















1 0


0.6


0.5


0.41


0.31


0.21


0.1


10- 10* 0
Frequency w in rad/sec


Figure 3.5: Real-p by the optimization algorithm over one variable Result 4: (Young [67] Algorithm Rank-One Solution)


101


M(s) = u(s)v*


(3.44)


where


u(s) = Pi(s) P2(8) P3(8) T
PO(s)' pO(s)' P0(s)


(3.45)


and


v = [3, 3, 3]T


E


(3.46)








43


Set T = 0.01, the real-p can be worked out by using the rank-one algorithm. The result is shown in Figure 3.6.

By comparing figures 3.4, 3.5 and 3.6, the three methods are equivalent in solving the real-p problem with affine parametric uncertainty. It is worthy to indicate again that Jie Chen et al. [14] [15] study uncertain polynomials, while Young [67] solves the problem specified in the rank-one case from the definition of p. This chapter recovers the previous results from the stability radii perspective by searching in the value set. Moreover the Algorithm with Bisection will get the exact stability margin at the exact worst frequency point as shown in Result 5.


Young 1994, ssv with Rank One Problem Algorithm
0.7 1 I I I


0.6 F


0.51


0.41


E


0.31


0.21


0.1


10-


100
Frequency w in rad/sec


Figure 3.6: Real-p by algorithm rank-one approach


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


101







44


Result 5: The Worst Case Stability Margin and the Corresponding Frequency

For the interval plant given in this example, the exact robustness margin found by the stability radii approach is a* = 1.8489 at w* = 4.6389. As shown in Figure 3.7, the critical point -1 + JO is right on the boundary of the value set at w* while a = a*. Note that the minimal stability margin obtained by using complex-p approach is 1.1786 - only about 64 percent of the minimal stability margin resulted from real-p analysis. The comparison reveals that real-p analysis is critical to eliminate the conservatism incurred by using standard complex-p analysis for real-parametric problems.


The Uncertain Set at alpha=1.8489 and w= 4.6389
0.1


0.05


09



-0.05


-0.1


-0.15


-0.2


-0.25
-1 -0.8 -0.6 -0.4 -0.2 0
Re


Figure 3.7: The uncertainty set with * = 1.8489 at w* = 4.6389







45


Matlab LabVIEW
z = x --x
XAex b max Cz
Aex = be -Aex > -be z (3.47)
Acx < b, - Acx > -b, M z > b
-X>-XU z>0
Cz= -f(x)


Figure 3.8: Transformation between linear programming functions of Matlab and LabVIEW

3.7 Simulation Reliability

To check the reliability of the simulation programs, we construct two kinds of programs using linear programming functions from different commercial software: Matlab and LabVIEW. As listed in Figure 3.8, those two functions use different expressions of the variables and their conditions. Note that the equality in Matlab program can be expressed as inequalities in LabVIEW program. We obtained the same results from these two different programs. This further verifies the simulation besides the confirmation between the results in this dissertation and these using other approaches reported in [15] [671.


3.8 Conclusion

By transforming the affine parametric uncertain system problem into a linear feasibility problem, this chapter presents a novel simple approach based on Nyquist stability analysis for the problem of finding robust stability margins for SISO systems with real parametric uncertainties. The approach recovers previous results based on real-p analysis for affine parametric uncertainty problem. The approach is promising and can be considered as a significant step towards the solution to the general robust stability analysis problem. Also, in terms of computation complexity, the PSM algorithm with bisection method to solve the frequency-dependent stability margin shows great advantage over general a - iteration. A similar computation advantage







46

is obtained with the PRSM algorithm to solve for the smallest stability margin and the corresponding worst-case frequency point by reducing the searching time with ever-reduced frequency range and alpha-searching range.















CHAPTER 4
ROBUST CONTROL FOR PARAMETRIC PLANTS


4.1 Introduction

Results on stability analysis for systems with parametric uncertainties have emerged steadily over the years, including outstanding results such as the edge theorem, the mapping theorem, the generalized Kharitonov theorem [12], the critical directional theory (CDT) [4], and many others. However, nonconservative robust controller synthesis for interval plants remains a relatively unsolved problem for which only few results are available. An early attempt to synthesize a robust controller was given in [40], where interval plants are overbounded by a constant uncertainty disk in the frequency domain. By assigning this constant bound as the weighting function, standard results from the H, synthesis theory are then applied to solve for a robustly stabilizing controller. This approach shows great promise because it is quite simple and makes effective use of the well-developed H, synthesis methodology. To facilitate comparison with the alternative results developed in the following chapters, we refer to the method in [40] as the maximum perturbation radius (MPR) approach. Unfortunately, the MPR weighting strategy may introduce conservatism since the overbounding operation guarantees only sufficient conditions for robust stability. Other researchers have tried to attack the synthesis problem using fixed-structure controllers such as PID, or other low-order controllers [16] [28] [31]. Yet, it can be argued that these approaches suffer from inherent shortcomings by restricting attention to only fixed-structure controllers.

As discussed in Chapter 2, the critical direction theory [4] [51] gives necessary and sufficient stability conditions that involve a critical perturbation radius (CPR).


47







48


Motivated by this fact, the conservatism of the MPR weighting is explored in detail in the following chapters. Furthermore, the critical direction theory is applied to introduce an exact weighting strategy that results in an even larger stability margin than that based on the MPR weighting. The effective critical perturbation radius (ECPR) weighting strategy proposed in our work recovers the simplicity of the MPR weighting while reducing the conservatism induced by uncertainty over-bounding.

The rest of the chapter is organized as follows. Sections 4.2- 4.4 set up the problem of robust controller design for systems with parametric uncertainties, present necessary preliminaries of mixed-sensitivity Ho, synthesis, and finally recast the parametric problem into the mixed-sensitivity framework. The MPR weighting and its conservatism are discussed in Section 4.5 where an example is also given. The main results are introduced in Section 4.6, where we define the ECPR weighting function, and provide a dynamic robust controller synthesis approach.


4.2 Problem Formulation

Consider a SISO linear uncertain plant represented by a rational transfer function, p(s, e), in cascade with a controller c(s), as shown in Figure 4.1. The family of uncertain plants can be represented by jp(s, e) - po(s)I < 16(s, e)I where po(s) is a known nominal SISO system, and 6(s, e) is a perturbation characterized by a set of real parameters, namely, the coefficients of the numerator and denominator polynomials. Each parameter in turn is an element of a real interval of width 2e centered about the nominal value of the corresponding parameter of the nominal plant po(s). Therefore, the uncertainty description for the problem constitutes a real hypercube A(e) with sides of width 2e. The value e is a non-negative scalar that represents the level of uncertainty. In general, system pa(s) is neither open-loop stable nor unitynegative-feedback system stable. A stabilizing controller c(s) needs to be designed so that the closed-loop system of Figure 4.1 is robustly stable, i.e., stable over all uncertain plants p(s, e).







49


As is common in all Nyquist-based robustness studies, it is assumed that the set of allowable perturbations A(c) is such that systems p(s, E) and po(s) have the same number of open-loop unstable poles.

This assumption can be checked by the following steps:

Step 1. Use the Routh-Hurwitz criterion for the denominator Kharitonov polynomials of p(s, c) to find an uncertainty level el such that the number of unstable roots of the family of denominator-polynomials of p(s, ) does not change for

all c < el [40].

Step 2. Use the edge theorem to find another uncertainty level 62 such that there is

no unstable pole-zero cancellation in the family of p(s, 6) for all c < 62 [17].

Step 3. Define the upper bound of the uncertainty as c, = min{61, 62} and require

that < e,,.

After a candidate controller c(s) is designed, define g(s, 6) = p(s, c)c(s) as shown in Figure 4.2. Note that the nominal system is go(s) := po(s)c(s), and the uncertainty is 6g (s, c) := 6(s, c)c(s). For robust stability of system g(s, 6), the following necessary conditions should be satisfied:

(Bi) The nominal system go(s) = po(s)c(s) is stable under unity-negative-feedback.

(B2) The family of uncertain systems g(s, c) and its nominal system go(s) have the

same number of open-loop unstable poles.

Therefore, besides the assumption made for system p(s, c), one further condition needs to be checked after a candidate controller is designed. If a candidate controller c(s) has unstable zeros, one must ensure that there is no unstable pole-zero cancellation between the zeros of controller c(s) and the poles of the family of p(s, 6) for all 6 < e. A similar analysis should be carried out for the possible unstable pole-zero cancellation between the unstable poles of controller c(s) and the unstable zeros of the original systems p(s, c). A candidate controller should be rejected if any forbidden pole-zero cancellation occurs.







50


There are primarily two problems to be resolved. The first is to determine the parametric stability margin max, which is defined in [40] as the maximum C for which the entire family of interval plants p(s, c) is robustly stabilizable. The second problem is to synthesize a stabilizing controller that can robustly stabilize the family of plants for all E < Emax





r+ + (S) I







Figure 4.1: The negative feedback loop of the uncertain System p(s) with a controller c(s). This figure is usually used when dealing with a controller design problem.


g(s)



,(Ss)





r U+








Figure 4.2: Stability analysis for an interval system g(s) under unity negative feedback. This figure is usually used when dealing with a stability analysis problem.







51


4.3 Preliminary: the Mixed-Sensitivity H, Robust Control

The mixed-sensitivity minimization problem is usually considered as the most important one for H, synthesis problems and receives a lot of attention in the H" synthesis literature because the problem can be considered as the combination of the robust control problem and the performance optimization problem. The mixedsensitivity minimization problem is an extension of the weighted sensitivity minimization problem introduced by Zames [70].

Consider the classical feedback structure with weighting strategy shown in Figure 4.3 where Po(s) is the nominal transfer function of a given plant and W1, W2 and W3 are design specification weighting transfer functions. Note that, Po(s) instead of po(s) is used here to denote that the systems are not restricted to SISO cases. The same meaning applies to other capital notations of Figure 4.3. The design goal is to control the gain of transfer functions from the reference r to output vector Yi = [y11, Y12, Y13]T to be less than a certain level -y.
It is straightforward to check that


yu(s) = Wi(s)(I + L(s))-'r(s) Y12(S) = W2(s)c(s)(I + L (s))-1r(s) (4.1)

Y13(S) = W3(s)L(s)(I + L(s))- r(s) where L(s) = Po(s)c(s).

Now by defining


S(s) := (I + L(s))1

R(s) := c(s)(I + L(s))-' (4.2)

T(s) := L(s)(I + L(s))-1 = I - S(s), the mixed-sensitivity synthesis problem is to find a stabilizing controller c(s) such that y is minimized and







52


W S(s)

1lTyi0(s)1c W2R(s) < Y < 1. (4.3)

W3T(s)

Note that, generally -y is required to be less than one for robust stability of the whole system where the structure and size of uncertainties is absorbed into the plant Po(s), and weighting functions W1(s), W2 (s) and W3 (s).

Traditionally, L(s) and T(s) are called the open-loop transfer function and the closed-loop transfer function respectively. I + L(s) is the return difference transfer function and the two matrices S(s) and T(s) are known as the sensitivity function and the complementary sensitivity function, respectively. Similarly, WiS, W2R and W3T are called the weighted sensitivity function, the weighted control sensitivity function, and the weighted complementary sensitivity function respectively. IITY1'(s) I I is called the mixed-sensitivity cost function because it penalizes all the three sensitivity functions simultaneously [6]. Minimizing the infinity norm of the mixed sensitivity function corresponds to the optimization of the robustness of the controller. Thus the choice of the weights W1, W2 and W3 is an important issue and this choice typically depends on the kind of application as well as engineering insights into a physical problem.

Wi y11
W1
Y12
W2

r Y13
-b C P W


Y


Figure 4.3: A standard H,,, synthesis problem in the mixed-sensitivity framework.







53


4.4 Robust Control for Parametric Uncertain Plants

The following theorem is proposed to help illustrate the transformation of the controller synthesis problem posed in Figure 4.1 into the mixed-sensitivity framework. Theorem 4.1 Suppose the system in Figure 4.1 is stable when 6(s) is zero. Then the size of the largest stable 6(s) for which the system remains stable is


1
1 1(s)11- < (4.4)
||R(s)11.
where R(s) := c(s)(I + L(s))1 as defined in (4.2).

Proof: This theorem can be proved invoking the Small Gain Theory. Based on the Sandberg-Zames' Small Gain Theory [72], the M - A system in Figure 4.4 is internally stable for any stable A(s) satisfying


1
||AWs)|| < (4.5) ||M(s)||o
where both M(s) and A(s) are assumed stable.

Considering the system in Figure 4.1, we have


w(s) = c(s)r (s) - c(s)[z(s) + po(s)w(s)]


w(s) = 1+ (s) r(s) - 1+c(so(s) z(s) (4.6)


w(s) = R(s)r(s) - R(s)z(s) where R(s) := c(s) as defined in (4.2).
1+C(S)po(S)
In order to use the Small Gain Theory, the system in Figure 4.1 can be rearranged as shown in Figure 4.5 imitating the M - A structure in Figure 4.4. According to the Small Gain Theory, the closed-loop system in Figure 4.1 is stable for any stable 6(s) satisfying the following condition:







54


6(s)I. < 1(4.7)
1R(s)JI.
This finishes the proof.


W1 el

+ A




+ W2
M
e 2 +


Figure 4.4: Standard M - A loop for stability analysis.




6(s)




F +
-R(s) + (-r)



Figure 4.5: A system with parametric uncertainty in the standard M - A loop.


For unstructured uncertainties A(s), the Small Gain Theory gives a necessary and sufficient stability condition. But for general structured uncertainties including the parametric case, the small gain condition is only sufficient to guarantee internal stability of the M-A loop. In Section 4.6, however, we will apply the critical direction theory and derive a necessary and sufficient condition for the internal stability of the closed-loop in Figure 4.1.









As a consequence of Theorem 4.1, it is common to specify the stability margins of control system via singular value inequalities such as


|W2(s)11. < I I Rs)JI. or,
(4.8)

1VW2(s)R(s)jj. < 1

where |W2(jw)II, = sup o(W2(jw)) is the size of the largest anticipated and "effective" plant perturbations.

By rearranging the system in Figure 4.1 into the M - A form, a stabilizing controller is required to satisfy (4.8) for closed-loop stability. Comparing (4.8) with (4.3), the closed-loop system configuration in Figure 4.1 is actually a special case of the standard Ho, synthesis problem, i.e., the mixed-sensitivity robust synthesis problem with W1(s) [ W3(s) [ (see Figure 4.6). The design criterion is



stabilizing c(s){fT~2() K1 Ty2 lcsp~)} ~.

(4.9)
mi P, W2(s)R(s)| o < 1
6lubilmg C(S)
The remaining problem is how to choose the weighting function W2(s) to represent the "effective" part of the uncertainty 6(s). In general, the more closely the approximate weighting functions are to the exact sufficient and necessary stability conditions, the less conservatism the resulting controllers will incur. We argue that weighting functions associated with sufficient robust stability conditions, e.g., weighting functions associated with the size of the largest plant perturbations (defined as the MPR in Section 4.5), usually yield conservatism in controller design. Hence we seek a weighting function W2(s) related to an exact measure of robust stability by referring to the critical direction theory.







56


r e
r- C(S)


W2(s)





u y
Y1


Figure 4.6: The parametric controller synthesis problem that is recast into the mixedsensitivity H,, synthesis framework.

4.5 MPR Weighting Approach and its Conservatism Analysis

Firstly, consider the weighting method by overbounding the value set of p(jW, f) at each frequency w. The largest perturbation radius pm(w, c) for the system p(jOw, C) at each frequency w is defined as (see Figure 4.7):


(4.10)


Pm(W,e)= max 6(jw)j= max Ip(jW,) -po(jW)
TA(g) oE s)
The largest pmP, c) over all frequencies will be denoted by


p,(c) := max {6(jw)I}
W, JEA(Ed)


(4.11)


For the linear perturbation case, at each frequency the pm(w, 6) in the family of transfer functions p(jw, c), will be associated with a point on one of the extreme segments, which is defined by the Kharitonov segments. Searching for the pm(w, E) for the case of linear perturbation is very simple, since there are only at most 32 extreme segments to be considered [40]. For affine parametric uncertain system, according to the mapping theorem introduced in Chapter 2 (see page 12), the MPR for each







57


Im


-1+ 0 Re











~MPR
POCo ))







Figure 4.7: MPR versus CPR

frequency point can be found out by searching over the boundaries of the value set mapped from the edges of the parametric space.

For SISO systems, we have



10Li, 0)1 = p(j ) - Po (jW)| I Pm(W, f) I Pm (). (4.12)

Referring to Theorem 4.1, either pm(s, c) or the constant Pm(E) can be used as the weighting function W2(s) in the H, design to synthesize a stabilizing controller for the parametric uncertain system of Figure 4.1, and equivalently of Figure 4.6. This represents one of the first attempts to attack controller synthesis for systems with parametric uncertainties [12] [40].

The MPR weighting scheme shows much promise because the approach is simple and directly applies the well-developed H, synthesis framework. Unfortunately, as shown in the following, the MPR weighting approach may incur conservatism since







58


the overbounding of the value set of p(jw, e) provides only sufficient conditions for robust stability.

To analyze the conservatism that may be introduced by taking the constant uncertain bound based on the MPR as the weighting function W2(8), we now compare the value sets of system p(s, E) and g(s, c).

As indicated in Section 4.2, system g(s, c) corresponds to c(s)p(s, c). After multiplication by the controller c(jw), the value set of p(jw, E) may be shifted, rotated, and/or magnified or contracted to produce the value set of g(jw, 6). Figure 4.8 illustrates the behavior of the corresponding variation caused by the controller c(jow) between the value set of p(jwj, c) and that of g(jwj, c) at a frequency wi.

As illustrated in Figure 4.8: g,(jwi, c) represents the intersection of the critical direction dc(jwi) and the boundary of the value set of g(jws, e); and, p,(jw, e) = g* "',,) namely, the original plant point p.(jUw, E) in the value set of p(jw, c) that is mapped to g.( Oi, e) in the value set of g(jw, ).

From the CPR definition (2.4) on page 8, it follows that


Pc(W, ) = g:(jW, f) - go (W)I
= c(jW)p,(jW, () - c(jW)po(jw)| (4.13) = c(jOW) Ip(jW, 6) - POOW)
As we noted in Section 4.4, an effective weighting function W2(s) should appropriately represent the uncertain plants. According to the critical direction theory, together with Equation (4.13), one should use 1p,(Jw, c) - po(jw)I as the weighting function in designing the controller c(s). Note that, generally, pm(w, f) > |p(jW, C) po(jw)l. Therefore, conservatism may be incurred when using W2(jw) = Pm(W, e). Even more conservatism may be resulted in using weighting function W2(jw) = Pm(C), as done with the constant MPR weighting approach.







59


t lin


Pm ( w ( ,
P.(j e) J


Re






p0(jO)


(a)


"IM
-1+ j0




g'06
Re


\d (jw,)











(b)


Figure 4.8: Complex maps between the original plant p(jOw, () and the transformed system g(jw, c) = c(jw)p(jw, e). The darkened areas represent the uncertainty value sets. The point g,(jwi, c) in (b) defines a disk of radius pc(wi, c), whereas its image p,(jwi, c) in (a) defines a disk of radius p,(wi, E). The overbounding disk of radius pm(wi) in (a) circumscribes the disk of radius pe(wi, E), and leads to more conservative robust-stability estimates.

4.6 ECPR Weighting Methodology

We revisit Figure 4.8 to seek a better weighting function. The family of critical uncertain plants V(jow, E) of g(jw2, 6) can be represented by g(j'W, ) = g(j'W, E) ejld Pi), where Od.(wi) is the angle of the critical direction defined in (2.1). Consider a certain controller c(jwi) = jc(jw)je'c(wi), where Oc(wi) is the angle of the controller. Divided by controller c(Ijwi), the family of critical uncertain plants g(jwj, 6) is mapped back to a specific family of original plants p(jwj, e) expressed as:


(4.14)


p(jwj, 6) gpjwi, 6))ejedwi)
PO'W 0 = IC U ji) Iei' (wi) Z- p(j~j,' E) Ie jld,,p(Wi)


where Od,,(Wi) = Od (Wi) - 0c(wi).

Likewise, the critical point (-1+ jO) of the value set of g(jw, E) is mapped back to a specific point of -. In other words, the specific line originating from pO(jwi) to cULjwj







60


in the value set of p(j'w, c) is mapped to the critical line originating from go(jwi) to (-1 + jO) in the value set of g(jwi, 6). Hence, we name the direction originating from po(jwi) to - to be the effective direction d,,(jw) of the value set of p(j'w, E).

Parallel to the critical direction theory, we could obtain the following observation. Given the controller c(s), only the uncertainty plants along the effective direction de,p(s), not over the whole value set of p(s, e), need to be considered in solving for necessary and sufficient conditions for the robust stability of the system of Figure 4.1.

Formally similar to the definition of the CPR (2.4) (see page 8) for the value set of g(jw, E), the effective critical perturbation radius (ECPR) for the value set of p(jw, E) is defined as


Pe(w, c) max{a po(jw) + ade,,(jw) E VC,,(w, E)} (4.15)
aeR+
where Vc,p(w, 6) is the uncertain plant p(jw, E) along the effective direction de,,(jW). For simplicity, only convex Vc,p(w, 6) is considered in the following. The results can be easily extended to non-convex cases.

Lemma 4.1 There is a mapping relationship between the ECPR of p(jw, 6) and the CPR of g(jw, E) as follows:



Pc(W,6) = Pe(W, 1)1c(jW). (4.16)

Proof: Without loss of generality, the equality (4.16) can be proved using the items in Figure 4.8.

By the ECPR definition (4.15), p,(w, c) =|p.(jw, c) - po(jw)1.

By the CPR definition (2.4), pc(w, E) g,(jw, e) - go (jw).

Notice the mapping relationship: g,(jw, e) = c(jw)p,(jw, E) and go(jw) = c(jw)po(jw). Therefore,







61


Pc(w,e) c(jw)p(jOw, ) - c(jOW)po(jw)

c(jOw) p(jOw, ) - po(jO)1 (4.17)

SC(jw)pe (w, E).
This finishes the proof.

When the controller equals to zero at a frequency, the point ) is at infinite distance off the nominal point po(jw). In this case, the effective direction of the value set of p(s, c) is not specified. However, since the upper bound of pe(w, e) can be infinitely large, we can replace pe(w, c) with pm(w, e) for simplicity. Theorem 4.2 Assume that the closed-loop system of Figure 4.1 is nominally stable; systems p(s, e) and po(s) have the same number of open-loop unstable poles; and there is no unstable pole-zero cancellation between a controller c(s) and the family of systems p(s,e). Then, the closed-loop system of Figure 4.1 is robustly stable if and only if

Pe (W,) < 1. (4.18)
1 + c(jOw)po(jOW)
Proof: From the critical direction theory stated in Theorem 2.1 of page 9, the closed-loop system of Fig. 4.1 is stable under the given assumptions if and only if kN < 1. Using the definition (2.3) in page 9, it follows that


kN <1 > sup PC(W) <1
W 1+go(jw)

Finally, from Lemma 4.1, the following equivalent necessary and sufficient stability conditions are obtained


kN < 1 S P (W) c(jw) 1
1 + c(jw)po(jw) 1


0







62


There is a transfer relationship from the input w(s) of the uncertain plant 6(s, e) to its output z(s) in Figure 4.1:


z() = C(S) w(s) = TaW(s)w(s). (4.19)
1 + c(s)po(s)

Hence, Equation (4.18) of Theorem 4.2 can be expressed as


Pe(W, )Tzw(jW)||o < 1. (4.20)

Comparing the with the well-known result from the Small Gain Theorem, we have obtained a similar stability analysis result, but for systems with parametrically structured uncertainties.

Similarly, as mentioned in Section 4.4, there is a transfer relationship from the system input r(s) to the controller output u(s) and further to the system output Y12(s) in Figure 4.6:


Y12(s) = TYl2(s)r(s) = W2(s)Tu,(s)r(s), (4.21)

and

Tu,(s) C(S) r (s). (4.22)
1 + c(s)po(s)
Using the ECPR pe(w, c) as the weighting function W2(jw), Equation (4.18) of Theorem 4.2 can be expressed as


IIPe(W, ()Tur(jW) 11 < 1 (4.23)

alternatively,

ITyi2r(jW)jj. < 1. (4.24)

One can easily identify Equation (4.24) with the general expression of a mixedsensitivity problem as stated in Section 4.4. Corollary 4.1 The radius pe(w, e) is the exact weighting function in designing controller c(s).







63


Proof: This property can be easily derived from necessary and sufficient condition given in Theorem 4.2.

As we argued in last section, when the MPR is used as the weighting function W2(jW), jTyir(jW)j = |pm(E)Turo(jw) < 1 is merely sufficient for system stability, but no longer necessary. Here again, one can see how the MPR weighting approach incurs conservatism.


4.7 Conclusion

In this chapter, we take advantages of the critical direction theory in defining necessary and sufficient stability conditions, and develop a systematic method of choosing the exact weighting function, namely, the ECPR that allows for parametric uncertain systems to be robustly controlled using the H" synthesis tools. The following two chapters will continue this topic and focus on controller design using static weighting functions and dynamic weighting functions respectively.















CHAPTER 5
ROBUST CONTROL FOR PARAMETRIC PLANTS USING STATIC WEIGHTING APPROACHES


5.1 Introduction

This chapter continues the discussion in the last chapter and focuses on using only static weighting functions. The plant is given as one SISO system p(s, c) including a parametric uncertain part 6(s, e), together with the same assumptions as listed in Section 4.2. We will address two issues: the first is to find the maximum stabilizable margin cmax; the second is to design a controller that can stabilize the closed-loop system of Figure 5.1(a) with uncertainties up to the maximum level 6max Recall that the problem is recast into the mixed-sensitivity form and the well-developed H"' design tools can be used to cope with the parametric uncertain plants.

As discussed in last chapter, an first attempt in robust controller design for parametric uncertain system used a constant overbounding method [40]. Its conservatism is analyzed in detail in the last chapter. This chapter argues that a much larger parametric stability margin could be obtained using a static weight based on the critical direction theory, namely, the uncertain parts 6(s, E) for all frequencies are represented by a constant ECPR, instead of a constant MPR. Moreover, we revisit H, robust control, and discuss in detail all the stabilizing controllers that satisfy the design criterion, which are called all-solution controllers. The next chapter will develop a dynamic weighting method to further reduce conservatism in determining the maximum stabilizable uncertainties and in designing a robust controller.

The rest of the chapter is organized as follows. Section 5.2 explores the characteristics of static weighting. Then an upper bound of the static weight is found


64







65


0 ((s)
C(S) & P(S)


(a)


5g (s)





(b)

z 1
-+W2(s)

c(S p0(s)



(c)

Figure 5.1: (a) Negative-feedback loop including the uncertain system p(s) P +(s)+ 6(s) and a controller c(s); (b) unity-negative-feedback of system g(s) = c(s)p(s); (c) mixed-sensitivity approach to the uncertain feedback system. and used for finding a stability margin in Section 5.3. Two illustrative examples are presented in Section 5.4. Simulation reliability is discussed in detail in Section 5.5. Some important derivation and simulation on all-solution controllers are studied in Section 5.6. Conclusive remarks are given in the final section.

5.2 Parametric Robust Control using a Static Weighting Approach

Lemma 5.1 When using static weighting approach to synthesize a controller for systems of Figure 5.1(c), a controller that minimizes ||Tzw(s)|| also minimizes I|Tu (s)|I.







66


Proof: We denote a controller c(s) that minimizes |Tzw(s)ca as


c0,(s) = arg min flTz2(s)flO. (5.1)
stabilizing c(s)

As discussed earlier, Tz.(s) = W2(s)Ts.(s), hence,


cop(s) = arg min IW2(s)T..(s)) 2. (5.2)
stabilizing c(s)

Given W2(s) = r, r C R+,


c0p(s) arg min flrTw(s)|o
stabilizing c(s)
= arg min rIIT.w(s) I (5.3)
stabilizing c(s)
= arg min T.-(s) I .
stabdlizing C(S)
This finishes the proof. Theorem 5.1 For the static weighting approach, the controller design criterion

min sup |Tuw(jw)| is equivalent to
stabilizing c(s) w

1
max inf +po(jW) . (5.4)
stabilizing c(s) W C(jw) Proof: Following the proof in Lemma 5.1,

cop(s) = arg min flT-W(s)IIK
stabilizing c(s)
c(jW)
=arg inZncs sup cjw
stabilizg C(S) W I1+c(jw)po(jw) = arg max s s co) (5.5)
stblzn ) Wp I+~u)P~w = arg max inf 1+c(jw)po(jwj)
stabilizing c(s) w cjw) = arg max inf Ij+ po(jw)
stabilizing c(s) w C



Expression (5.4) exposes an alternative interpretation of the controller design procedure, namely, a search over all stabilizing controllers to find one that maximizes







67


the minimal distance between po(jw) and the point -'. Note that both of the design criteria of Theorem 5.1 involve only the nominal plant po(s); hence, the problem is independent of the uncertain plant 6(s, e). This is a natural consequence of using a static weight.

The alternative static weight proposed in this chapter adopts r = pe(c), so that


W2(s) = Pe((), (5.6)

where

Pe(E) max pe(w, e), (5.7)

The next section discusses the attributes and advantages of using this static weight.


5.3 Robust Stability Conditions and Stability Margin Calculation

From Figure 4.8 of page 59, it is obvious that robust stability is lost when the effective radius pe(wi, e) is equal to the distance between the points - ) and po(jo) at some frequency wi. In such a case, a point on the boundary of the plant value set is mapped by the controller to the critical point -1 + JO in the Nyquist plane. Consider an optimal controller c0p(s) designed using the static weighting approach, and define pu := inf C( I) + po(jw) . We obtain the following theorem. Theorem 5.2 Let cop(s) be an optimal controller designed using the static weighting approach, and let pe(c) be the corresponding constant ECPR for an uncertainty A. The closed-loop of Figure 5.1(a) is robustly stable if pe(C) < pe.

Proof: Condition pe(c) < p' implies

1
pe(c) < inf + po(jO)
Scop(jw)

which is equivalent to

Pe(C) cop(jw)
Pe ( Icop jW) < 1,Vw.
1 OPU(j)pOU(j )







68


A further condition is obtained by considering the definition of the constant ECPR weight (5.7),

Pe(W, ) 1 + ) < 1, Vw. (5.8)
I + c"0(j)p0(jW)
According to Theorem 4.2 of page 61, condition (5.8) implies the stability of Figure 5.1(a). This finishes the proof. U

The sufficient stability condition in Theorem 5.2 validates the utilization of the constant ECPR as the static weighting function, as expressed in (5.6).

The method proposed in [40] to compute the stability margin, denoted as (mar, used the constant MPR weighting, i.e., W2(s) = pm, and then, e and hence pm is increased until pm = p' = p'. However, as shown in the examples in Section 5.4, much larger stability margin can be found based on the static ECPR weighting, i.e., calculating cma, from a plot of Pe(c) vs. e, instead of pm(E) vs. c.

Alternatively, we can find Emax(w) at each frequency by iteratively increasing e from zero until -1 is in the value set of p(jw, e), then the stability margin fmax is the smallest value of emaz(w) among all frequencies. A more efficient solution can be constructed by referring to the linear programming algorithm posed in [12] [50]. Thus, by interpreting the static weight W2 as a constant ECPR weight (W2(s) = pe()), instead of a constant MPR weight, a larger parametric stability margin may be found.

From Theorem 5.1, one can obtain an alternative meaningful expression of p',

1
p m T.(8) (5.9)
stabilizing c(s)

Further connections with the classical H. static weighting design approach can be established by considering a system in which the weight is given by W2(s) = r, and the controller in question c0,(s) attains the value min |IT.(s)H|o as defined stabilizing c(s)
in (5.1). Now let xrna denote the value of r such that TrnazT "(S) I = 1. Then it







69


follows that rmax min I1r.(S)||.. Hence
stabilizing c(s)

r""" = pe (5.10)


In next section, two examples are given to illustrate the substantial improvement in stability margin assessment using constant ECPR weighting.

5.4 Examples

Example 5.1 Consider the interval plant discussed in [12] [17] [28] [40],


p~s~e) 5s + q1
P(s ) = 2s+q (5.11)
s2+ q2s + q3

with intervals


qi E [4 - c, 4 + c], q2 E [2 - E, 2 + E], q3 E [--15 - c, -15 + ], (5.12) where e represents the level of uncertainty. The parameter space A(e) is characterized by a hyperbox whose edges vary independently with a length 2e each. Note that to satisfy the assumption of the family of systems p(s, e) made in Section 4.2, eu =

7.5 [17] [40].

As indicated earlier, a robust controller needs to be synthesized, and the maximum stability margin in terms of static weighting needs to be found. In the following, we list in detail the steps in solving the problems.

Step 1: Analyze the problem

We have the closed-loop system of Figure 5.1(a) where po(s,e) is the plant p(s,e) with q1 = 4, q2 = 2, and q3 = -15, and where 6(s, e) = p(s, e) - po(s, e). Note that plant po(s, e) can be represented in state-space equations as:


4o(t) = APOxPO(t) + BpOu(t) (5.13)
yp0(t) =C~0xp0(t) + D~u(t)


where







70


Ap [ -2 15 Bp = ,C0 =[ 0 0 ], DpO = 0. (5.14)
1 0 0

Step 2: Recast the problem into the mixed-sensitivity form

We transfer the closed-loop system of Figure 5.1(a) to that of Figure 5.1(c) and have W1(s) =Wa(s) = 0, and the uncertain part 6(s, ) is represented by W2(s).

Step 3: Express the mixed-sensitivity problem using the Lower-LFT form

Ab 6hown in Figure 5.1(c) toyether with W'(s) = r, r E R+, it is straightforward to obtain the following state space representation of the augmented plants:


xPO(t) Ap0xp, (t) + Ow(t) + Bpu(t)

z(t) = 0xP"(t) + Ow(t) + ru(t) (5.15)

y(t) -Cp0xp.(t) + w(t) - D0u (t)

If expressing it in G - K form, we have


APO 0 BP0

G 0 0 r (5.16)

-Cp, 1 -Dp,

Note that by assigning r = 1, we get the state space representation for the transfer function Tuu(s).

Step 4: Design the robust controller via static weighting method
Using -v-iteration, the controller that minimizes ||TUW(s)| = 1+po(s)c(s) found to be


3603.7935s + 18018.9673
s 2 + 1434.5016s - 2312.4499 (5.17)

It is straightforward to check that there is no unstable point cancellation between the unstable pole of c(s) (5.17) and the zeros of p(s,e) for e up to eu.







71


Step 5: Find the maximum stabilizable margin using the static ECPR weighting

According to Theorem 5.1, the upper bound of the exact static weight is pu=

0.3950. The ECPR pe(e) is calculated and plotted versus e shown as the solid line in Figure 5.2. It is observed that (,a = 5.2511 when pe(() = 0.3950.

Step 6: Compare with the cnax found using the static MPR weighting As shown by the dashed line that reflects the variation of pm(e) vs. c, one gets e = 2.8300 when pm(e) = 0.3950 (the limiting value of W2). The value e = 2.8300 was proposed in [40] as the parametric stability margin in the case of the constant weighting method using the MPR approach. Therefore, the parametric stability margin calculated by the MPR weighting approach (e = 2.8300) is only about 54 percent of that obtained by the ECPR weighting approach (e = 5.2511).









1-2


1 II 1


0.9


0.8 --0.7
PM

0.6


' 0.5r "'= 0.r3950

0.4


0.3- Pe


0.2


0 .1 -0
0 1 2 3 4 5 6
E



Figure 5.2: Plots of pm and pe radii as functions of the uncertainty-size parameter C. The value 3 = 0.3950 is the limiting value of the static weight W2(s) found via a standard unstructured H,, approach. Example 5.2 Consider the interval plant given in /40], 30s + qi
p(s,ce) = sq1(5.18)
s3 + q2s2 + q3S + q4

with intervals




qi E [10 -- c, 10 + E], q2 E [-3 - e,-3 + e,(5.19) q3 E [-4 -c, -4 + e], q4 E [12 -, 12 + e. (5.20)


Note that to satisfy the assumption of the family of systems p(s, c) made in


Section 4.2, cu = 6.0.









73


Using -y-iteration, the controller that minimizes ||Tu,(s)||K C(S )s 1+po(SWcSIO

found to be



30s + 10 (5.21)
s3 - 3s2 - 4s + 12' and W1(s) = W (s) = 0, WV2(s) r, where r is a constant. It turns out that at W2 = 0.4023, p(XY00) approximates unity and the following controller is found




3617.6s2 + 4562.3s - 5345.9 s3 + 1468.3s2 + 18620.7s + 6605.8


1.


Q.


1.6


1.4


1.2-


0) CL


0.8 0.6


0.4 0.2


01
0


0.5


1.5


2


2.5


E


Figure 5.3: Plots of Pm vs. E (dashed line) and pe vs. e (solid line)


The stability margin was reported in [40] to be 0.8500, which actually corresponds to pu = 0.4023, as shown by the dashed line in Figure 5.3. However, by interpreting







-





-

%=0.402
-


I







74


the value 0.4023 as the pe, and the stability margin is c = 1.2800 where pe = 0.4023. Again, the conservatism introduced by previous approach can be reduced by taking the constant weight W2 as the constant effective bound, instead of the constant overbounding uncertainty.


5.5 Simulation Reliability

As discussed in this chapter, the optimal controller designed by static weighting should be unique that minimizes |Tu,(s)lo over all stabilizing controllers. However, one may find that different controllers are reported in [40] [12] [36] and here. These different controllers are calculated with the same algorithm and similar programs, but with different design resolutions that specified the degree of closeness to the minimal value. In our simulation, we developed two kinds of programs and showed that the results are consistent. The first program uses commercial Matlab software hinfopt.m with a bisection algorithm called -y-iteration, and finds the optimal controller by minimizing Tu(s)11, over all stabilizing controllers. Similar to the solving procedure given in [40], the second one finds the optimal controller by minimizing ||Tz,(s)||o over all stabilizing controllers by increasing the weighting scalar r step by step, called an additive searching procedure.


5.5.1 Solution with a Bisection Searching Procedure

To work with HTu (s).. , the weighting function r is fixed as unity so that the augmented plant G is fixed. The controller design resolution T is the only factor that affects the designed controller when using a bisection algorithm to minimize JjT,(s) |o over all stabilizing controllers. For illustration purpose, we list four controllers computed by a same program based on Matlab software hinfopt.m but with different resolutions T.

For simplicity, the controllers are expressed in matrix form:







75


Cnum s S -T 1]
c(s) = (5.23)
Cden S S 1]T

where the nominator vector cnum equals [ n ... n ], and the denominator vector Cden equals [ d- . . d, do ]. Controller 1 with _ = 10-6

Cnum = 108 * [ 0 0.37748739677966 1.88743698863214 1

Cden = 107 * [ 0.00000010000000 1.49422174558615 -2.43793793780942 1;
(5.24)

Controller 2 with T = 10-2

Cnum = 105 * [ 0 1.47459789570386 7.37298947851135 ]

Cden = 104 * [ 0.00010000000000 5.83775000382779 -9.52194474311628 ];
(5.25)

Controller 3 with T = 10-2

Cnum = 104 * [ 0 0.92197910324932 4.6098955162450

Cden = 103 * [ 0.00100000000000 3.65750061702856 -5.93944837515235 ];
(5.26)

Controller 4 with T = 3.1 x 10-3

Cnum = 103 * [ 0 0.57981456953645 2.89907284768224

Cden = 102 * 0.01000000000000 2.37509933774846 -3.59463576158953 1;
(5.27)

As shown in Figures 5.4 and 5.5, all these four controllers actually have similar values of phase and magnitude, and as shown in Figure 5.6, they produce the similar closed-loop gain ITs,,(jw)l as expected by the design procedure. Nevertheless, the small difference between the controllers reflect the difference of pre-specified design









76


200
controller at c=0.000001 controller at T=0.0001
- controller at t=0.01 150- - - controller at t=0.031



100

-0

50


0
0
Cd

L -50-100



-150 I
0 100 200 300 400 500 600 700 800 900 1000
frequency w Figure 5.4: Plots of the phases of the four controllers respectively. Note that the plots of the case of T = 10-6 is almost overlapped with the case of T 10-'.


resolution T. One can check the Table 6.1 and concludes that the values of ||Ts,(s)l|m are made smaller and smaller as the resolution goes finer and finer.

Table 5.1: Simulation with a bisection searching procedure

controller 1 controller 2 controller 3 controller 4 11T.()| II... 2.5263163 2.5264456 2.5283951 2.5600000
resolution T 10-6 10-4 102 3.1 x i0-


5.5.2 Solution with an Additive Searching Procedure

Different from the first algorithm, now we work with a transfer function Tzw(s) that has a plant G with a varying element D12 r. The optimal controller is found by additively increasing r till no stabilizing controller exists. Then the maximum static









77


0 100 200 300 400 500
frequency o


600 700 800 900 1000


Figure 5.5: Plots of the magnitudes of the four controllers respectively. Note that the plots of the case of T = 10-6 is overlapped with the case of T = 10-4.


3


-controller at %=0.000001 .
controller at r=0.0001- - controller at T=0.01
- -controller at r=0.031


2.5


2


1


0.5









78


2.


.:1.


0


0 100 200 300 400 500
frequency o


600 700 800 900 1000


Figure 5.6: Plots of the Ts.(jw)| with the four controllers the plots of the case of T = 10-6 is overlapped with the case


respectively. Note that of T = 10-4.


5




2




5







controller at c=0.000001
controller at t=0.0001 .5 - controller at T=0.01
- - controller at c=0.031



A







79


weight is found together with a stabilizing controller that is taken as the optimal controller. In this case the increasing step is the design resolution. In our simulation, we used the commercial Matlab function hinf.m and have found the corresponding maximum weights in designing the controllers reported in [40] [12]. Consider the Controller in [40]

With r = 0.3911755, we found the following controller:

cnum = 10 * [ 0 0.64788818856446 3.23944094282226 ]

Cden = 102 * [ 0.01000000000000 2.64455741306764 -4.03427788447897 ];
(5.28)

which is close to that reported in [40]:


647.9(s + 5)
647s)s = (5.29)
C(s) s2 + 264.456s - 403.411 Consider the Controller in [12]

With r = 0.3957261755, we found the following controller:

cnum = 105 * [ 0 0.27999876755765 1.39999383778789 1

cden = 104 * [ 0.00010000000000 1.10912845491568 -1.80682537381042 ];
(5.30)

which is close to that reported in [12]:


28000(s + 5)
c (S) = 280(+)(5.31) s2 + 11090s - 18070(
In Matlab function hinf.m, the a is set as unity [72] [62]. Therefore. the closer the gain flTzes||o approaches to unity, the more powerful the corresponding controller is in terms of closed-loop stability. One can concludes from Table 6.2 that the values of ||T,.s||K more closely approach unity as the weighting function r approaches the maximum weight whose value is affected by the step size in additive searching algorithm.







80


Table 5.2: Simulation with an additive searching procedure

controller (5.28) controller (5.30)
11T.zW)| I1 0.99992994752899 0.99999996334698
static weight r 0.3911755 0.3957261755


5.6 All-Solution Controllers


As well-known in standard robust control [72] [64], given


A B1 B2

G= C1 0 D12

C2 D21 0

which satisfies some proper assumptions, H, design has all-solution controllers K(s) = FL (F(s), Q(s)) (


where


A"
F(s)= Fo

-C2


-ZOLo Z B2

0 I

I 0


AO := A + -y-2BB*Xo + B2F. + ZccL. C2

F, := -B*Xo, Lo, := -YoC2*, Z := (I - X.)~

X,,, := Rz'c(H,,), YO, := Ri'c(J,,)

Q (s) E RH., I|Q(s)ll. < .


Now for Q(s)


AQ BQ ,one has the following state space representaCQ DQ


tions:


5.32)


5.33)


(5.34)


(5.35) (5.36) (5.37) (5.38)







81


(F A,,XF u FCXF +

U2 -C2XF +


Z.Lxy + Z.B2Y2

0 + Y2 y + 0


(Q = AQxQ + BQU2 Y2 = CQxQ + DQU2


(A. - Z.B2DQC2)xF +

-BQC2)XF +

(FQ - DQC2)XF +


ZB2CQXQ

AQXQ


+


CQXQ +


(ZOL. - Zc B2DQ)y

BQy DQy
(5.41)


The all-solution controller K(s) is:



AK BK

CK DK


A. - Z.B2DQC2

-BQC2


FQ - DQC2


Z.B2CQ

AQ


CQ


(5.42)


-ZL, + ZOB2DQ

BQ

DQ


If Q(s) = 0, one gets a central controller



Kcentral(s) = A
F 0

We point out two observations from expressions (5.42-5.43):


and,


(5.39)


Therefore,



XF



U2


(5.40)


(5.43)







82


" All solution controller K(s) is strictly proper if and only if Q(s) is strictly

proper;

* The central controller KcentaI(s) is always strictly proper.

Regarding the controller design, given different complementary controllers Q(s), one will get different optimal controllers c(s) that minimize the transfer gain ITs.(s) | even though a same high resolution value T is used. However, as shown in the following example, all those controllers have values close to each other for any frequency since they all should minimize ||T.(s)||2. Example 5.3 We continue the discussion of Example 5.1 but with a focus on allsolution controllers. We select the resolutionT = 10-6, and compute three controllers with different complementary controllers Q(s). Using the matrix expression in Equation (5.23), the three controllers are listed below and compared in Figure 5.7.

Controller 1:



Q, (S) = 0 (5.44)

Cnum = 10 * [ 0 0.37748739677966 1.88743698863214

Cden = 107 * [ 0.00000010000000 1.49422174558615 -2.43793793780942 ];
(5.45)

This is exactly the same controller of Equation (5.24).

Controller 2:


s + 0.7
Q2(8) = (5.46)
s + 53







83


Cnum 1010 * [ 0.00000000252632 0.00000001440000 1.97425909399972

9.87129542578808 ]

Cden = 109 * [ 0.00000000100000 0.00000005136842 0.78147797179945

-1.27504154133096 1;
(5.47)

Controller 3:



Q3(s) = + 2 +0.01 (5.48)
s2 +41s+ 100



Cnum = 103 * [ 0.00000000025263 0.00000000133895 0.15363737087079

1.14563650049922 1.88724824019991 1 Cden = 102 * [ 0.00000000100000 0.00000003936842 0.60814834996409

0.50183158281841 -2.43769414374154 ]; (5.49)

As shown in Figure 5.7, all three controllers have almost the same values of controller phases and magnitudes, and hence produce almost the same transfer gains jTuw(s)f|x. Actually, there is no difference among the values of |Tm(jw)| at any frequency up to the eighth digit after the decimal point.

5.7 Conclusion

This chapter focuses on the H, methods using constant weighting strategies for systems with parametric uncertainties. The conservatism of earlier work in the literature is exhibited and a much larger parametric stability margin is obtained.

Even though this chapter shows that the ECPR weighting strategy leads to a much larger parametric stability margin than the MPR weighting method does, only constant weighting methods are considered in this chapter. Referring to (4.13) on









84















200- - controller with Q (s)
V1
controller with Q2(s)
e 100 - _ controller with Q3(s)

0 0

o -1000 100 200 300 400 500 600 700 800 900 1000
CL
frequency o

a) 8 - - controller with Q, (S)
controller with Q2(s)
controller with Q3(s)
CD

0)
C 2
0) 2
E 0 100 200 300 400 500 600 700 800 900 1000
frequency O

2.5263- -controller with Q(S)
2.5263- controller with Q2(s)
2.5263 - controller with Q2(s)
2.5263-- cnroerwh 3(S)
2.5263-
2.52632.5263 - I I I
0 100 200 300 400 500 600 700 800 900 1000
frequency o


Figure 5.7: Plots of the controller phases, controller magnitudes, and transfer gain T.(jw)| with the three controllers respectively. Note that all the three plots at each sub-figure are overlapped. Actually, there is no difference among the values at any frequency up to the eighth digit after the decimal point.




Full Text

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ROBUST STABILITY ANALYSIS AND CONTROLLER SYNTHESIS FOR SYSTEMS WITH PARAMETRIC UNCERTAINTIES By BAOWEI JI 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 2004

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Copyright 2004 by Baowei Ji

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To my parents and to Songhong

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ACKNOWLEDGMENTS I wish I could find words to express my gratitude to my advisors (Dr. Haniph A. Latchman and Dr. Oscar D. Crisalle) for their guidance, support, and encouragement during my time at the University of Florida. Although I still have much to learn, I feel confident as I begin my career in systems control and other areas. This feeling certainly is primarily the result of the consistent inspiration and encouragement of my advisors. Many times I was frustrated by seemingly unsolvable problems. Dr. Latchman patiently listened to my messy reports and directed me to the right track. His energetic approach and responsible attitude encouraged me to seek the best in academic research and systems development. Even though I never took courses with Dr. Crisalle, we have had numerous research discussions, and I feel as if I have taken a series of courses on systems control engineering with him. I truly appreciate the availability of my advisors whenever I had questions. I am also grateful for their high standards on mathematical derivation and report organization, which will significantly benefit my career development. I also wish to thank my other committee members (Dr. Norman G. Fitz-Coy and Dr. Tan Wong) for their willingness to serve on my Ph.D. committee. Dr. Fitz-Coy gave me tremendous help when I first came to the U.S. He motivated my research interest in systems and control engineering. Dr. Wong raised many insightful questions and gave very helpful guidance during my oral qualifying exam. All of my friends and research team members deserve my sincere gratitude and appreciation. Regarding the preparation of this dissertation, I thank the help from Mr. Minkyu Lee, Mr. Yu-Ju Lin, Mr. Saleh Al-Shamali, Mr. Suman Srinivasan, Mr. Dave Tingling, Ms. Sheryl Latchman, Dr. Benjamin Harrison, and many others. IV

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As always, I thank my parents in China for their heartfelt love and support. For the past 5 years, my wife, Songhong has patiently and enthusiastically supported my study and research. Although she has even more work (with her J.D. program study at the Levin College of Law at the University of Florida), she always assists me with her dedicated love and timely thoughts. I give thanks for her love that makes my life much more colorful and meaningful. v

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS iv LIST OF TABLES viii LIST OF FIGURES ix LIST OF NOTATIONS xi ABSTRACT xiv CHAPTER 1 INTRODUCTION 1 2 CRITICAL DIRECTION THEORY 6 2.1 Critical Direction Theory 6 2.2 Calculating Critical Perturbation Radius 10 2.2.1 Value Set 10 2.2.2 Defining CPR for Nonconvex Value Sets 15 2.2.3 Defining a New Perturbation Radius 17 2.3 Conclusion 21 3 REAL ii ANALYSIS 22 3.1 Introduction 22 3.2 Transforming a Real Parametric Affine Problem into a Linear Feasibility Problem 23 3.3 The Parametric Stability Margin (PSM) 27 3.4 Real-p, Analysis based on Polynomial and Rank-One Matrix Approach 31 3.5 Exact Calculation of the Robust Stability Margin and the Worst Case Frequency Point 35 3.6 Example 38 3.7 Simulation Reliability 45 3.8 Conclusion 45 4 ROBUST CONTROL FOR PARAMETRIC PLANTS 47 4.1 Introduction 47 4.2 Problem Formulation 48 vi

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4.3 Preliminary: the Mixed-Sensitivity H ^ Robust Control 51 4.4 Robust Control for Parametric Uncertain Plants 53 4.5 MPR Weighting Approach and its Conservatism Analysis .... 56 4.6 ECPR Weighting Methodology 59 4.7 Conclusion 63 5 ROBUST CONTROL FOR PARAMETRIC PLANTS USING STATIC WEIGHTING APPROACHES 64 5.1 Introduction 64 5.2 Parametric Robust Control using a Static Weighting Approach . 65 5.3 Robust Stability Conditions and Stability Margin Calculation . . 67 5.4 Examples 69 5.5 Simulation Reliability 74 5.5.1 Solution with a Bisection Searching Procedure 74 5.5.2 Solution with an Additive Searching Procedure 76 5.6 All-Solution Controllers 80 5.7 Conclusion 83 6 ROBUST CONTROL FOR PARAMETRIC PLANTS USING DYNAMIC WEIGHTING APPROACHES 86 6.1 Introduction 86 6.2 Sub-Robust Controller Synthesis Using a Dynamic MPR Weighting Approach 87 6.3 Robust Controller Synthesis: a Dynamic ECPR Weighting Approach with Controller Tuning Algorithm 90 6.4 Example 93 6.5 Simulation Reliability 96 6.6 Conclusion 97 7 CONCLUSIONS AND FUTURE WORK 98 REFERENCES 103 BIOGRAPHICAL SKETCH 109 vii

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LIST OF TABLES Table page 2.1 Comparison of the three definitions for the marginally unstable case . 20 2.2 Comparison of the three definitions for the stable case 20 5.1 Simulation with a bisection searching procedure 76 5.2 Simulation with an additive searching procedure 80 viii

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LIST OF FIGURES Figure page 2.1 Negative feedback loop of the uncertain system p(s) with a controller c(s) 7 2.2 Stability analysis for a uncertain system g(s) under unity negative feedback 8 2.3 Critical direction and critical perturbation radius 9 2.4 The value set of g(jcv, q) at lo = 0.7 for Example 2.1 13 2.5 Comparison of CPR definitions (2.22) and (2.4) 16 2.6 The value set at the marginally stable condition: a = 1.8660, and tv = 4.7294 18 2.7 The value set at a — 1.8600, and u — 4.7294 19 3.1 The true-or-false function f(a) vs. the scaling factor a 29 3.2 The classical M — A structure used for analyzing robust stability . . 32 3.3 Comparison of the PSM with the complex parametric stability margin 40 3.4 Comparison of the real-// with the complex-// 41 3.5 Real-// by the optimization algorithm over one variable 42 3.6 Real-// by algorithm rank-one approach 43 3.7 The uncertainty set with a* = 1.8489 at uj* = 4.6389 44 3.8 Transformation between linear programming functions of Matlab and LabVIEW 45 4.1 The negative feedback loop of the uncertain System p(s ) with a controller c(s) 50 4.2 Stability analysis for an interval system g(s) under unity negative feedback 50 4.3 A standard H ^ synthesis problem in the mixed-sensitivity framework. 52 4.4 Standard M — A loop for stability analysis 54 IX

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4.5 A system with parametric uncertainty in the standard M — A loop. . 54 4.6 The parametric controller synthesis problem that is recast into the mixed-sensitivity H ^ synthesis framework 56 4.7 MPR versus CPR 57 4.8 Complex maps between the original plant p(ju, e) and the transformed system g(ju, e) = c(jw)p(jw, e) 59 5.1 (a) Negative-feedback loop including the uncertain system p(s) = Po(s) + S(s) and a controller c(s); (b) unity-negativefeedback of system g(s) — c(s)p(s ); (c) mixed-sensitivity approach to the uncertain feedback system 65 5.2 Plots of p m and p e radii as functions of the uncertainty-size parameter e. 72 5.3 Plots of p m vs. e (dashed line) and p e vs. e (solid line) 73 5.4 Plots of the phases of the four controllers respectively 76 5.5 Plots of the magnitudes of the four controllers respectively 77 5.6 Plots of the \T uw (ju>)\ with the four controllers respectively 78 5.7 Plots of the controller phases, controller magnitudes, and transfer gain \T U w{ju)\ with the three controllers respectively 84 6.1 Plot of the true-or-false function via the uncertainty level e 89 6.2 The tuning process using a controller c t (s) 92 6.3 Sub-robust controller design example using a dynamic MPR weighting approach 94 6.4 Stability analysis for the parametric uncertain system (6.9) cascaded with the designed robust controller(6.14) 96 7.1 Augmented feedback loop with performance weights 99 7.2 Nyquist plot for disturbance rejection 100 7.3 Augmented feedback loop with performance weights 102 x

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LIST OF NOTATIONS s w arg(z) M Re(z) Im(z ) [•1 A " 1 A T j^nxrn Qnxm R + RH oc q q a(A) a(A) Ma (A) ||^(s)||oo T zw (s ) Wl(s) W 2 (s) Laplace variable Frequency, sometimes used as an input variable The argument of the complex z Magnitude Real part of the complex z Imaginary part of the complex z Greatest-integer function Inverse of matrix A Transpose of matrix A Real-valued matrix with n rows and m columns elements Complex-valued matrix with n rows and m columns elements Set of nonnegative numbers Proper and stable analytical space The upper bound of a scalar q The lower bound of a scalar q The largest singular value of M The smallest singular value of M The structured singular value of a matrix A with uncertainties A Hoo norm of the transfer function T zw (s) The transfer function from the input w to the output z The sensitivity weighting functions The controller sensitivity weighting functions xi

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W 3 (s) The complementary sensitivity weighting functions Po(s) The nominal system function, be., without uncertainties p(s) The transfer function representation of a plant p{ju) System p(s) at frequency u P{s,e ) Similiar with p(s) except showing the effect c(s) A controller g{s,c) g{s,e) = c(s)p{s,f) d c {ju) The critical direction of the value set of g(s, e) d c ,p{jw) The effective direction of the value set of p(s, e) k N (u) Nyquist robust stability margin Pm (^) The maximum perturbation radius Pm (^) The approximation to p m (uj) Pc(w) The critical perturbation radius Pc(w) The approximation to p c (o>) PeM The effective critical perturbation radius PeM The approximation to p e (uj) PrM The perturbation radius p r (u) The approximation to p r (co) Q Parametric uncertainty region dQ The boundary of Q V(M The value set of system g(s) d V(ju) The boundary of V (ju>) Vc(juj) The critical value set of system g(s) d V c (juj) The boundary of V c (ju>) ^c,p (i w ) The effective value set of system p(s) 9 VcO'w) The boundary of V CtP (ju) Xll

PAGE 13

r Numerical computation resolution A p B p State space representation of system p(s c p Dp A A A c x 0 D\2 Augmented state space representation c 2 An 0 inf Infinum max Maximum min Minimum sup Supremum CDT The critical direction theory CPR The critical perturbation radius ECPR The effective critical perturbation radius LF Linear feasibility LFT Linear fractional transformation LP Linear programming MIMO A multi-input multi-output system MPDA Major principal direction alignment MPR The maximum perturbation radius MSM The multivariable stability margin PSM The parametric stability margin SGT The small gain theorem SISO A single-input single-output system

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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 STABILITY ANALYSIS AND CONTROLLER SYNTHESIS FOR SYSTEMS WITH PARAMETRIC UNCERTAINTIES By Baowei Ji May 2004 Chair: Haniph A. Latchman Cochair: Oscar D. Crisalle Major Department: Electrical and Computer Engineering This dissertation covers our study of both robust stability analysis and robust controller synthesis for systems with highly structured uncertainties. These problems have attracted much interest in advanced control engineering, but experienced limited development in the past two decades. In 1997, an important observation was introduced with the critical direction theory (CDT). According to the theory, on the Nyquist plane at any given frequency, only the uncertain plants located along a specified direction, namely the critical direction, are of relevance to the stability analysis; all other uncertain plants that do not lie in the critical direction can be ignored when dealing with stability analysis. This dissertation extended the theory to systems with highly structured uncertainties, and explored the CDTÂ’s advantages over other approaches. xiv

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Firstly, robust stability analysis was addressed in this dissertation for systems with real-valued uncertainties. For SISO systems with affine real-valued uncertainties, we transformed the problem into a linear feasibility (LF) problem, together with an algorithm for calculating the parametric stability margin that was defined as the maximum scaling value of the uncertainties in maintaining the robust stability of the system. This algorithm used a bisection algorithm to achieve computational efficiency. Then we developed a linear programming (LP) and an iterative algorithm based on CDT for the same problem. All the three algorithms produced the same results for the examples simulated in this dissertation. Finally we related the parametric stability margin with the real fi results reported recently in the literature, and found consistency in using the critical direction theory and // analysis. Our research on robust analysis provided an alternative real-/z analysis with clear geometrical explanation. Furthermore, our research on robust analysis establishes a sound foundation for robust synthesis research reported as follows. Then, we addressed robust controller synthesis for systems with highly structured uncertainties. We constructed a systematic controller synthesis methodology, combining the essence of CDT and H <*, design. An important attempt reported in the literature on robust controller design for systems with structured uncertainties was to rearrange the controller synthesis problem into the mixed-sensitivity synthesis framework, and then to use H 0 0 synthesis tools. The family of the uncertain plants was represented by one that overbounded all the uncertain systems. We explored the mechanism leading to conservatism of this overbounding operation, and then examined a static weighting based on the critical direction approach using a scalar to represent the uncertain systems for all frequencies. By applying this static weighting function, we dramatically reduced the conservatism incurred from the previous approach, and obtained stability margins that were 50% larger than those resulted from the previous approach. Finally, we developed two dynamic weighting approaches and xv

PAGE 16

obtained controllers that could accommodate even larger level of uncertainties than those obtained using the static weighting approach. Our study shows that the critical direction theory is effective and promising for nonconservative stability analysis and controller synthesis for systems with highly structured uncertainties. The critical direction theory provides necessary and sufficient condition for robust stability. However to address robust stability and robust sensitivity at the same time, one important entity was defined in this dissertation, together with analytical approach in calculating that parameter. Some ongoing research topics were mentioned at the end of this dissertation. xvi

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CHAPTER 1 INTRODUCTION Systems and control has always played an indispensable role in electrical, mechanical, aerospace, chemical and other engineering design and industrial implementation. For the past century, control engineering has gone through significant development from classical control [13] [33] [56] (1920-1950), to modern control [38] [39] [55] (1960-1980), then to robust control [18] [64] [72] (1980 to present). Given a plant, the first question one may ask is whether the plant is stable. When using the term stability, generally, one means internal system stability ( e.g ., no blowout of circuit components), and/or externally bounded input bounded output stability (e.g., no loss of a prespecified trajectory) [37]. If the plant is stable, one may further ask whether any uncertainty is allowed inside the plant, and what is the maximum level of uncertainties that could be tolerated. If the plant is unstable, it may be necessary to design a controller (a kind of compensator and regulator) to stabilize the original plant while satisfying some predetermined specifications. Real life always carries uncertainties, such as manufactural imperfection, measurement noise, system modelling errors introduced by linearization and system identification, environment fluctuation, and physical aging and wearing, to name just a few. To cope with system uncertainties, classical control defined gain margin and phase margin, which were measured separately. Unfortunately, uncertainties in real life usually involve variations of gain and phase simultaneously. A preclaimed infinity gain margin may easily be destroyed if there is a tiny change of the phase [6]. Modern control originated from and finds satisfying application in space and aircraft engineering. Nevertheless, only limited uncertainties are allowed in modern control 1

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2 due to the requirements of precise state-space system representation and strict definition of cost functions. Robust control aims to achieve robustness and maintain system performance despite uncertainties. Unstructured uncertainties represented by disks, are well addressed by p analysis and control. However, structured uncertainties (e.g., realvalued and phase-limited uncertainties) more closely resemble real-life uncertainties. An electrical board is an example of real-valued uncertainties with the expected variation of resistance values or capacitance values. This dissertation focuses on systems with highly structured uncertainties. With an aim at analyzing robustness stability in an exact manner, the critical direction theory (CDT) was developed recently based on the Nyquist stability criterion [51]. Having functioned as the foundation of classical control, the Nyquist stability criterion deals with a fixed single-input-single-output (SISO) plant g 0 {s) by plotting the complex values of go{ju)) in the complex plane. The plot is called the Nyquist plot. The unity-negative-feedback-loop system is stable if and only if the number of counter-clockwise encirclements of the critical point in the Nyquist plot is equal to the number of open-loop system unstable poles. Now as uncertainties (say e) enter the plant, we have a family of plants (say g(s,e)) surrounding the nominal plant go{s). Imagining that all of the uncertain plants are plotted, we obtain tube-like areas composed of many curves surrounding the original plot in the complex plane. Then assuming that the number of open-loop unstable poles is the same for the family of plants g(s,e), it can be concluded that the whole family is stable if and only if the critical point is outside these tube-like areas. This is the critical point exclusion principle. To check system robustness stability, therefore, one must check whether the critical point is inside the plot of the plant family at each frequency. This can be done by checking the uncertain plants along only the critical direction, which is defined as the direction from the nominal point g 0 (s) to the critical point. This brought about the basic idea of critical direction theory.

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3 This dissertation extends the advantages of the critical direction theory to systems with more general structured uncertainties, and most importantly, develops methodologies of applying the theory for designing robust controllers in conjunction with Z/qo tools. We can roughly divide the dissertation into two parts: robust analysis and robust synthesis. Chapters 2 through 3 discuss robust analysis. Chapter 2 introduces the Nyquist stability criterion, gives insight into the critical direction theory, and describes the calculation of the critical perturbation radius based on the plot of value sets. A new definition of the perturbation radius is also presented for considering the tradeoff between system stability and sensitivity. For SISO systems with affine real-parametric uncertainties, chapter 3 transforms the problem of finding the uncertainty stability margin into a linear feasibility (LF) problem, and presents a novel approach for solving from real-/i. The approach recovers previous results with real-p analysis. A comprehensive example shows that all these approaches are equivalent in solving the specified affine problem. The results in this chapter demonstrates that the stability radii approach we proposed may be promising for solving more general real-p problems with more general structured uncertainties. In terms of computational complexity, the proposed algorithm for calculating the parametric stability margin (PSM), powered by a bisection algorithm, shows great advantage over that based on an additively searching process. A similar computation advantage is obtained with the parametric robust stability margin (PRSM) algorithm to find out the smallest stability margin and the corresponding worst-case frequency point. This research provides not only a new real-/r analysis method, but also a solid foundation for controller synthesis in the second part of the research. Chapters 4 through 6 focus on designing robust controllers that can stabilize as large an uncertainty as possible. Controller synthesis for systems with unstructured

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4 uncertainties was well developed using H tools; however, a similar problem for systems with parametric uncertainties has developed relatively slowly. A very important attempt in literature was to rearrange the problem into the mixed-sensitivity synthesis framework, and then to use well-developed H ^ synthesis tools where uncertain plants were represented by a scalar that measured the maximum radius of the plants deviated from the nominal point. In Chapter 4, we apply the critical direction theory to analyze the conservatism of the previous overbounding behavior in representing the whole family, thoroughly develop the parametric robust control methodology, and find out the exact weighting function that is defined as the effective critical perturbation radius. Chapter 5 focuses on parametric robust control using a static weighting approach where the uncertain plants for all frequencies are represented by a scalar, namely the radius of a disk. We prove that there is only one kind of optimal controllers for static weighting even though the coefficients of the controllers may not be same for different design resolutions. Nevertheless, a much larger level of stabilizable uncertainties can be found using the new interpretation of the static weight, namely, the exact static weight associated with the critical direction theory. Finally, the relationship of allsolution controllers that satisfied the H ^ design criterion are discussed. Chapter 6 makes an effort to find an even larger stabilizable uncertainty margin by designing a robust controller using dynamic weighting approaches, where the difference of the families of plants at different frequencies are taken into account. First we differentiate the over bounding function for each frequency and construct a sub-robust controller synthesis algorithm enhanced by the bisection algorithm. Then we establish another dynamic weighting approach, where only the necessary and sufficient uncertain plants at each frequency are considered in solving for the maximum stabilizable margin and a robust controller with the most powerful stabilizing capacity.

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5 Finally some future research directions are discussed in Chapter 7 to continue our exciting and promising research work.

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CHAPTER 2 CRITICAL DIRECTION THEORY 2.1 Critical Direction Theory It is important to differentiate notations of uncertain systems p(s) and g(s), and notations of nominal systems p 0 (s) and g 0 (s) in the sequel. As shown in Figure 2.1 and Figure 2.2. there exists a relationship: g(s) = c(s)p(s), which indicates that a controller c(s) is included in g(s), similarly, g 0 (s) = c(s)p 0 (s). When discussing stability analysis (chapters 2-3), we always refer to Figure 2.2, even with a unity controller. All four notations are used in chapters 4-6 when we talk about controller synthesis. Classical control primarily considers certain systems: for example, the nominal system go(s). Given a fixed system, the Nyquist stability criterion states that the system is closed-loop stable if and only if the map of the Nyquist contour of the open-loop system encircles the critical point (— 1 + jO) in the anticlockwise direction a number of times equal to the number of unstable poles of the open-loop system. If the system is stable under unity-negative-feedback, then the critical point should be outside of the Nyquist contour of the open-loop system. The similar argument can be extended from the nominal system g 0 (s) to a family of uncertain systems g(s) = go(s) + 6 g (s), where uncertain part 5 g (s) is bounded by A(s) . If we assume that g(s) and go(s) have the same number of open-loop unstable poles, then the whole family of uncertain systems under unity-negative-feedback is robustly stable if and only if the critical point — 1 + jO is not in the value set of g(s) for all frequencies. This is generally called the critical point exclusion principle. 6

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7 Latchman and Crisalle [49] proposed the critical direction theory (CDT) to elegantly measure this necessary and sufficient robust stability condition. It is generally assumed that the nominal system go(s) is closed-loop stable under unity-negative-feedback, since for a nominally closed-loop unstable system p 0 (s) that is stabilizable, a stabilizing controller can be easily found by using classical controller synthesis approaches and cascaded with p 0 (s) to form g 0 (s). Similar to other stability analysis approaches based on the Nyquist stability criterion, the critical direction theory assumes that g(s) and g 0 (s) share the same number of open-loop unstable poles. The solid curve in Figure 2.3 represents the nominal system g 0 (jui) at the prespecified frequency range; the shaded area represents the uncertainty family g(juji) for a specific frequency point of that range. The critical direction theory uses the values shown in Figure 2.3 p(s) Figure 2.1: Negative feedback loop of the uncertain system p(s ) with a controller c(s). 1. Critical line is the directed line that originates at the nominal point go(ju) and passes through the Critical Point —1 + jO. 2. Critical direction 1 + 9o(jw) l +9o(ju)\ = e d c (Ju) := J S d c (u) ( 2 . 1 )

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8 gO) r *<£>u -4 y Figure 2.2: Stability analysis for a uncertain system g(s) under unity negative feedback. where e j6d ^ is the angle of the critical direction. The critical direction may be interpreted as the unit vector that defines the direction of the critical line. 3. Value set (also called the Image-Set in the literature) V(u) {9{ju) | g(juj) = g Q (ju) + 6g(jtd),6g(s) G A(s)} (2.2) 4. Critical value set V c (ui) := {9c(ju) | 9cU u ) = So ljw)+&c(jv) = 9o(ju)+ad c (ju), for some a E F(, } (2.3) 5. Critical Perturbation Radius (CPR) p c (ui) := max {a \ g c (ju) = g 0 {ju)) + ad c (ju ) G V c (u>)} (2.4) a&R+ 6. The Nyquist stability margin kN — Pc(w) l+9o(ju)\ OO (2.5) Note that at every frequency u, the critical direction d c (ju) may be interpreted as a unit vector with its origin at go(jui) and pointing towards the point —1 + jO.

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9 Figure 2.3: Critical direction and critical perturbation radius The expression 6 c (ju>) = ad c (ju) represents the set of perturbations with frequency response lying along a straight-line segment that joins the points g Q (jui) and —1 + jO. The critical perturbation radius (CPR) is defined as the distance from go(ju) to the intersection of the boundary of the critical value set with the straight-line segment that joins the points go (jcu) and —1 + j 0. The necessary and sufficient stability condition of the CPR is given in Theorem 2.1. Theorem 2.1 [49] Consider the uncertain system g(s) given in Figure (2.2) that is stable under unity feedback, and assume that g(s ) and go{s) have the same number of open-loop unstable poles. Then, assuming that the critical template is convex, the uncertain system is stable under unity feedback if and only if kpf < 1. Proof: The proof is given in [49].

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10 2.2 Calculating Critical Perturbation Radius The substantial step in CDT is to calculate the CPR. For the case that there is only one intersection between the boundary of the value set of g(s) with the critical direction, [66] presents the following heuristic procedure for calculating the CPR. Algorithm for Calculating CPR Step 1 Generate a fine grid of frequency points [uq, u; 2 , . . . , ujk, , uj n \ and set k = 1. Step 2 Set u = uq. End if k > n. Step 3 Calculate go(jui) = x 0 + jy 0 and d(ju>). Step 4 Find the complex point x* + jy*, x* ,y* € R , on the Nyquist plane where the line go(juj) + ad(jui),a € R + , intersects with the value set boundary dV(co). Step 5 Set p c (a>) = |(x* + jy*) — ( x 0 + jyo ) |Set k = k + 1 and go to Step 2. End of algorithm. For general non-convex value sets, there is no rigorous algorithm for searching for the intersections at present. One possible way is to partition each of the edges into small enough segments, each is assumed to have only one intersection with the critical direction. Then, the furthest intersection measured from go(jui) is taken as the effective point in calculation of the CPR. Even though it is less rigorous, this sub-division method is extensively used in studying systems subject to parametric uncertainties [12] [59]. 2.2.1 Value Set The calculation of CPR is based on the description of the value set boundary. The CPR’s dependence upon the value set suggests that the results in this dissertation may be extended to general uncertain systems having multilinear, multiaffine and/or nonlinear perturbations. However, for simplicity and tractability, the major part of this dissertation is based on affine uncertain systems that have the following system transfer function:

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11 r \ _
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12 In comparison with (2.8), the cascaded system h(s) is also an affine uncertain system. The value sets of those affine uncertain systems are well-defined by the following mapping process from the edges of the parametric space Q to the boundary of the system space g(ju>). Lemma 2.2 (Affine Mapping Theorem) : Given a systems transfer function g(jw) and a closed affine interval perturbation defined as Q G R m , the value set will be g(ju>, q) for every q G dQ where dQ represents the edge of Q (hypercube of three dimensional perturbation) . Then the boundary of the value set dg(jui, Q ) is composed by g(jtu,dQ) such that dg(ju, Q) C g(ju, dQ) (2.11) Proof: Refer to [12] for proof and related preliminary information. For affine uncertain systems, Fu [23] shows that the edges of value set of g(jui) are either circular arcs or straight-line segments. Accordingly, the intersections can be found analytically as explained in [5] and [66]. Example 2.1 Given a plant, g(s, q) = c(s)p(s, q ), ( 2 . 12 ) c(s) = (0.3s + 1), (2.13) / \ s 2 + (4 + 0.4g! + 0.2g 2 )s + (20 + gi q 3 ) s'* + (9.5 + 0.5, q) at 00 = 0.7 is plotted in Figure 2.4 where one can see the twelve segments mapped from the twelve edges from the parametric set Q.

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Imag 13 o 0.5 -1 1.5 -1 Figure 2.4: The value set of g(ju>, q) at w = 0.7 for Example 2.1. The circle point inside the value set is the nominal point go{ju)-, the diamond point is the critical point —1 + jO; the critical line is also shown. For more general cases such as multilinear perturbation and nonlinear perturbation, Theorem 2.2 needs to be revised to include some additional interior onedimensional manifold subset of Q in addition to dQ. The interior one-dimensional manifold, which contributes to the boundary of the value set, can be identified using methods such as in [12] [24] [46]. Examples of multilinear interval systems are as follows [25] [12]: -1 0.5 Real 800(1 + 0.1gi)(s + 2) Sl S ’^ s(s + 4 + 0.2g 2 )(s + 6 + 0.3g 3 )(s + 10) ’ (2.15)

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14 0 2 (s,a,b) = (5 + 2)(s 2 + 5 + l)( 6 . 6 s 3 + 13.5s 2 + 15.5g + 20.4) (s + l)(s 3 + a 2 s 2 + 4 s + a 0 )(s 3 + 6 2 s 2 + 3.5s + 2.4) Obviously, neither of them can be put into the affine parametric uncertain system form (2.6) introduced in Chapter 2. For multilinear perturbation and nonlinear perturbation, the following general mapping theory should be referred to include some additional interior one-dimensional manifold subset of Q in addition to the boundary of the uncertain space dQ. Theorem 2.2 (General Mapping Theorem) -.Given a systems transfer function g(jco) and a closed general interval perturbation defined as Q E R m ,the value set will be g(ju,q) for every q £ Q. There exists T c , a subset of Q, which is a set of onedimensional manifolds that includes all the candidates that may contribute to the boundary of the value set dg(ju>, Q ) such that The interior one-dimensional manifold, which contributes to the boundary of the value set, needs to be identified by referring to [24] and [46]. with the assumptions that • Uncertainties q and r are independent; • n(s, q) and d(s, r) are coprime over the whole uncertainty spaces n x A, where q G II, r E A; • d(s, r) 7^ 0 for all r G A and each cu > 0. The extremal set ^(s) is defined as Sg(ju, Q ) c g(ju, T 0 ) (2.17) A special kind of multilinear uncertain system g(s) = is considered in [12] (2.18)

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15 where n#(s), A E (s) and n^s), A E {s) denote the corresponding manifolds and vertices in II and A, respectively. The boundary of the value set is described by Theorem 2.3 based on the generalized Kharitonov polynomials. Theorem 2.3 (page 475, [12]) dg{ju) C g E (jca), Vuj € [0, oo) (2.19) where 9e{s) fnjM) .. ( n E {s) \ \d E (s)J \d K (s)J ’ ( 2 . 20 ) and where the extremal polynomial manifolds n E (s) and d E (s) and the vertex sets n K (s ) and d K (s ) are defined as n E {s) = {n{s,q) | qeU E }, n K (s) = {n(s, q) \ q G 11#} d E {s) = {d(s, r) | re A E }, d K (s) = {d(s, r) | reA x }, Proof: Refer to [12] for proof. These theorems suggest some methods to describe the value set of a multilinear uncertain system. A rule of thumb can be used to check the validity of these theorems since a value set at a specified frequency u can be roughly provided by plotting all points g(ju) for all possible combinations of uncertainty values, i.e., sweeping over the uncertainty space. 2.2.2 Defining CPR for Nonconvex Value Sets The CPR definition (2.4) assumes that the critical value set, not necessarily the whole value set, is a convex set as shown in Figure 2.3. For general convex value sets, however, such a definition may incur conservatism as shown in Figure 2.5. To cope with general cases, Baab et al. introduce another definition for the ( 2 , 21 ) CPR [3]:

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16 Figure 2.5: The CPR (shown as p(u;)) defined in (2.22) for a nonconvex value set is less conservative than that (shown as r) defined in (2.4) for a convex value set. Pc{u) = |l + 0o(jw)| min{\a | f(w) = -1 + ad c (juj) ) G d K(^)} otherwise ( 2 . 22 ) where a is a nonnegative scalar, dV c (co) represents the boundary of the critical value set V c (u), and finally, £(u;) is the closet distance from the critical point —1 + jO to the boundary of the critical value set. Accordingly, this definition of the CPR is utilized to extend Theorem 2.1 for general non-convex value sets.

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17 Theorem 2.4 Consider the uncertain system g(s) given in Figure (2.2) that is stable under unity feedback, and assume that g(s) and go(s) have the same number of openloop unstable poles. The uncertain system is stable under unity feedback if and only if 1 + 9o(ju)\ 00 < 1 where p c {aj) is given by (2.22). Proof: The proof is given in [5]. (2.23) 2.2.3 Defining a New Perturbation Radius So far two CPR definitions have been introduced, one for a convex critical value set, the other for nonconvex cases. Even though the later can represent the necessary and sufficient condition for robust stability, it can not tell the system sensitivity. In other words, both the definitions only check the uncertain plants along the critical direction, hence can not tell exactly when and how the family of the plants are approaching instability. We use Figures 2.6 and 2.7 to illustrate the situation. We use the same plant equation g(s , q) in Example 2.1, but scale the uncertainties qi with a scalar a, i.e., —3ct < qi < 3a, i = 1,2,3. It turns out that the worst case frequency point is u — 4.7294, where one obtains the smallest scaling factor a = 1.8660 for that the system is unstable. This smallest value is defined as the parametric stability margin in the following two chapters. The value set of the system g(s, aq) at the marginally stable conditions a = 1.8660, and u> = 4.7294 is plotted in Figure 2.6. However, the case for a small variation of the scaling factor a is plotted in Figure 2.7 where a = 1.8600, u = 4.7294. Following either of the CPR definitions, the CPR is small and the Nyquist stability margin k ^ is far less than unity. Accordingly, one may predict that the real parametric stability margin is much larger than 1.8600, which contradicts the parametric stability margin a = 1.8660.

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Imag 18 Real Figure 2.6: The value set at the marginally stable condition: a — 1.8660, and co = 4.7294. In recognizing this, we provide a more meaningful definition called a perturbation radius (PR): i \l + go(ju)\ min{a | ((w) = -1 + ae je £ d V(u)y0} if-l^F(u;) Pr((*>) = < ^ 1 |1 + 9o{ju)\ + rnin{a \ £( oj ) = — 1 + ae J0 G d V(u>), V0} otherwise V. (2.24) where a is a nonnegative scalar, where dV (u>) represents the boundary of the value set V(lj), and finally, where C(w) is the closet distance from the critical point -1 +j0 to the boundary of the value set. Different from the previous two CPR definitions in (2.4) and (2.22) that are associated with the critical value sets, the definition of PR in (2.24) checks the evolution

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Imag 19 -1 0.8 0.6 0.4 0.2 0 0.2 Real Figure 2.7: The value set at a = 1.8600, and t o = 4.7294. of a whole value set around the critical point -1 + jO at each frequency, and reflects the shortest distance to the critical point among all the boundary points of the vale set. The computation algorithm and application of the definition can be found in [2], Nevertheless, the newly defined perturbation margin still stipulates the necessary and sufficient condition for system stability as in the following theorem. Theorem 2.5 Consider the uncertain system g(s) given in Figure (2.2) that is stable under unity feedback, and assume that g(s) and go{s) have the same number of openloop unstable poles. The uncertain system is stable under unity feedback if and only if Pr(u) 1 +9o(J u )\ < 1 Vo (2.25)

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20 where p(cu) is given by (2.21). Proof: The proof can be found in [2]. Furthermore, it is straightforward to check the following equivalency among the three definitions (2.4), (2.22) and (2.24) in terms of system stability [2]: Case: Stable p r (w) < |1 + go{joj)\ <=> p c {w) < |1 + go (ju)\, Vw Case: Marginally Unstable p r (ui) = |1 + go(ju)\ 4=^ p c { uj) = |1 + go(ju)\, Vuj Case: Unstable p r (u>) > |1 + go{ju)\ <*==> p c (oj) > |1 + go(ju)\, Vu; Taking Figures 2.6 and 2.7 for example, we have the following two tables in measuring system stability. Table 2.1 shows the consistency of the three definitions for the case of marginal instability. Table 2.2 clearly shows one advantage of the PR definition (2.24) in reflecting the closeness of the condition of a = 1.8600 to the unstable condition of a 1.8660 at uj = 4.7294. More advantages of this new definition can be found in [2]. We will also apply the new definition in Chapter 7 for one kind of controller design problem. Table 2.1: Comparison of the three definitions for the marginally unstable case CPR (2.4) CPR (2.22) PR (2.24) radius value 4.553 4.553 4.553 radius value |l+9oO'w)l 1 1 1 system stability unstable unstable unstable Table 2.2: Comparison of the three definitions for the stable case CPR (2.4) CPR (2.22) PR (2.24) radius value 0.093 0.093 4.553 radius value |i+go(FUI 0.019 0.019 0.906 system stability stable stable almost unstable

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21 2.3 Conclusion The main advantage of the critical direction theory is that it provides necessary and sufficient conditions for robust stability in the presence of highly structured uncertainties with phase and directionality constraints. The critical direction technique opens up new avenues for robust analysis and could lead to novel approaches for robust control synthesis as demonstrated in the rest of this dissertation. We will first explore the advantages of CDT with robust analysis in the next chapter.

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CHAPTER 3 REAL // ANALYSIS 3.1 Introduction The problem of assessing robust stability and computing stability margins for SISO systems with parametric uncertainties has attracted considerable attention in the last decade. While more general parametric uncertainties such as multilinear dependence cannot be easily handled at present, it is now well known that affine parametric uncertainty problems result in a rank-one structured singular value (//) problem. Moreover, // for the rank-one problem is exactly equal to its upper bound that is obtained from a convex optimization problem even though the general robust stability margin computation problem is known to be NP-Hard and is computationally intractable. In last chapter, we had presented an explicit analytical expression for complex parametric uncertain systems. It is known that standard complex-// analysis will likely incur conservatism when working with real parametric uncertainties. Many researchers have devoted their attentions to the exact calculation of real-// problem. This aspect of the problem is amply documented in the literature [14] [15] [30] [60]. Of particular interest are references [15] [67], which will be reviewed in Section 3.4. While most of the aforementioned developments are based on the structured singular value, the approach using the stability radii is arguably less well-developed. This is mainly because of the nonconvexity or amorphous nature of the value sets. This chapter starts directly from the Nyquist stability theorem and works with system transfer functions to exactly calculate the stability radii for a special kind of parametric uncertainty, i.e., the so-called “affine parametric variation” problem. 22

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23 The rest of the chapter is structured as follows. Following the introduction, Section 3.2 transforms the real affine parametric system into a linear feasibility problem to avoid the difficulty of working with nonconvex value sets. The stability radii, called the parametric stability margin (PSM), is defined in Section 3.3, where the exact calculation algorithm is also given. The previous representative results based on the structured singular value approach are reviewed and compared with our approach in Section 3.4. Section 3.5 presents a further exact calculation of the maximum real-// over a frequency range and the corresponding worst-case frequency point using a bisection algorithm. A comprehensive example is analyzed in Section 3.6. Simulation reliability is validated in Section 3.7. The final section concludes the chapter. 3.2 Transforming a Real Parametric Affine Problem into a Linear Feasibility Problem Consider the nominal system go(s) and the family of uncertain systems g(s). The Nyquist stability criterion states that a system is closed-loop stable if and only if the map of the Nyquist contour of the open-loop system encircles the critical point in the anticlockwise direction a number of times equal to the number of unstable poles of the open-loop system. If we assume that the nominal system go(s) is stable under unity-negative-feedback, and that g(s) and go(s) have the same number of open-loop unstable poles, then the whole family of uncertain systems under unity-negativefeedback is robustly stable if and only if the critical point — 1 + jO is not in the value set of g(s) for all frequencies s — jui. This is generally called “critical point exclusion”, the extension of “the zero exclusion” principle applicable to uncertain polynomials. However, as shown in Chapter 2 (see page 18), the value set for an affine uncertain system is not necessarily convex, which imposes much difficulty in checking whether the critical point is in the value set or not. Fortunately, Baab et al. [5] demonstrate that the real parametric affine problem can be recast into a linear feasibility problem. The standard affine problem is expressed as

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24 »(»,<) = + (3.1) do(s) + E£i Qidi( s ) where Q = {q G R mr | ft < ft < ft, * = 1, 2, , m r }, and q = [ft, g 2 , ... , q mT ] T Here, m r denotes the number of the uncertain entities. The family of uncertain systems in (3.1) can be represented in the following vector-matrix form 1 s q) = hence, s m— 1 s m / \ n 0 o n 0 1 no2 nom r ft nw + nn nn n\m T 92 V fT'mO nm2 nmm T 1 1 / 1 s s n ~l s n / 1 Ro o doi do2 • i s o ft d\o + d a dn d\m T 92 V 1 a3 o i dn\ d n 2 dnm r 9m r \ (3.2) 9{s, q) = s n T (no + N mr • q) (3.3) Sd T (d 0 T Dm r q) where s n and Sj are vectors of length m + 1 and n + 1, respectively, containing powers of the Laplace variable s = a+jcu ; and n 0 G R m+1 , d 0 G R n+1 , N mr G _R( m + 1 )x(^r) anc j D mr G J R("+bx(m r ) are constants that represent the structure of the affine parametric uncertainty. The Nyquist image g(ju>, q) is obtained by evaluating the vectors s n and Sd at the Hurwitz stability boundary s = ju to yield 3 n,o; 1 ju -u 2 ju 3 u 4 (3.4)

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25 Sd,J = ( 3 . 5 ) 1 ju) —UJ 2 —juj 3 UJ 4 Now the vectors s niW and Sd )a; , which are constants for a particular frequency u>, and n 0 ,N mr ,do and D mr can be separated into real and imaginary parts. Define s„,r 1 -UJ 2 UJ 4 -a; 6 ( 3 . 6 ) T S n ,I = UJ —UJ 3 u 5 —uo 7 ( 3 . 7 ) Sd.R 7 — 1 -UJ 2 UJ 4 -UJ 6 ( 3 . 8 ) and T Sd,I = UJ —UJ 3 UJ 5 —UJ 7 ( 3 . 9 ) ™00 noi n 02 • ^0 m T no,H — 1 O CN ... 1 eR ,m/2 1+1 , N m ,,„ = n 21 ™22 ’ ^2 m r £ ft(\m/2]+l)x(m r ) ( 3 . 10 ) nio 1 3 3 to ^1 m r n 0,7 = n 30 G c>i(m+l)/2] tvj _ n 31 ^32 ‘ ^3 m r ^f(m+l)/2] x(m r ( 3 , 11 )

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26 1 o o doi do2 dom r do,// — <^20 s R'" /2 ' + \ D m „„ = ^21 d,22 • d2m r g J^(r«/2l+l)x(m r ) (3.12) dio dn d\2 • dlm r do,/ = 1 o CO ... € Dm „,= ^31 ^32 d^m r G i?d n+1 )/ 2 l x ( m r) (3.13) where [•] represents the greatest-integer function. The uncertain system becomes S n,fl( n 0,fl + N m r ,R • q ) +j s n,/(n 0 ,/ + N mr>/ • q) p(s,q) (3.14) s d,fi( d o,fl + D mri /e • q) + j sJ J (do,/ + D mrj/ • q) To determine if the critical point — 1 -f jO is a member of the value set of g(ju> , q) for a particular frequency, it must be determined if there is a vector q e Q such that g{ju, q) = -1 + j 0. Using the previous notation, the following equation is obtained s r,/i( n o,fi + N mrtR • q) + j s^ / (n 0 ,/ + N mri/ • q) = [ S d,fl(do,fl + Dm r ,fi • q) + j s dy(do,/ + D m T ,I ' q)] (~ 1 + j0) Equating real and imaginary parts and rearranging terms yields S n,R^m r ,R + s d,« D m r ,fi S n,I^rn r ,I + Note that (3.15) has the form ~ S n,R n 0,R S n,/ n 0,/ s di/do,/ (3.15)

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27 A(uj)q = b(uj) (3.16) where the real matrix A{u) G R 2xirir and the real vector b(u) G R 2 depend on the uncertainty structure. This is a system of two linear equations in q. The solution represents all the points q in the parametric space that map under g(ju), q) to the critical point —1 + jO. Therefore, to determine whether the system is unstable, it suffices to determine if there exists a solution q G Q to (3.15). This is a standard linear feasibility (LF) problem of the form: does there exist q such that A(u)q = b(io) (3.17) 3.3 The Parametric Stability Margin (PSM) The parametric stability margin (PSM) is defined as follows: Definition 3.1 Consider an interval plant g(s, q ) with parametric uncertainties q G Q, where q < qi < q and a(u j) is an arbitrary positive scaling factor. Then the frequencydependent parametric stability margin is the smallest scale factor a(u) such that g(juj,a(u)q) will satisfy 1 + g(ju,a(w)q) = 0, Vcu (3.18) The parametric stability margin is a nonnegative real scalar that can be interpreted as the minimal magnification (a(u) > 1) or contraction (a(u;) < 1) of the parameter set Q that brings the closed-loop system to the limit of instability. Geometrically, the parametric stability margin represents the minimum tolerable blow-up factor. Note that the parametric stability margin is defined for each frequency. Therefore, in order to calculate the stability margin, the decision of whether to increase or decrease the perturbation scaling factor a (a — iteration) is an essential

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28 problem. For the LF problem given in Section 3.2, the decision of how to adjust a can be determined by a simple numerical comparison in the algorithm. Here, the true-or-false function of whether -1 e g(j co,q) is used as the testing index of the bisection method. Define the true-or-false function f(a) as: { —1, (which means False) If — 1 4 a(ju), q) (3.19) +1, (which means True) otherwise If we can prove that the true-or-false function f(a) is monotonic with respect to a, then a bisection algorithm can be invoked and the search range over a can be dramatically reduced. To show the monotonicity of the true-or-false function, we should check the variation of the value set of g(ju), q) at a specified frequency point u> with respect to the scaling factor a. The following lemma states an important property of the value set for interval systems that satisfies the monotonicity requirement. Lemma 3.1 Consider the interval plant g(s, aq ) with a scaling factor a. The value set is monotonically nondecreasing as a increases. Proof: The proof of this lemma is sketched as follows. Consider the uncertainties q E Q and scaling factors ax > 0, a 2 > 0, such that a 2 > a iThen it follows that Q\ C Q 2 where Q x = Q aiq and Q 2 = Q a2 q Since the value sets g(jco,Q 1 ) and g{ju,Q 2 ) are mapped from the hyperbox Q x and Q 2 respectively, they certainly satisfy g{ju,Q x ) C g(ju,Q 2 ) (3.20) This concludes the proof. Following Lemma 3.1, it is straightforward to check that function f(a) increases monotonically while a increases from zero. As plotted in Figure 3.1, f(a) equals -1 whenever a < a* which means the system is stable for these small levels of uncertainty and -1 +j 0 is outside the value set of g(jw,a q), and f(a) equals 1

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29 whenever a > a* which means the system becomes unstable at a certain level of uncertainty and will continue to be unstable for even larger levels of uncertainty, a * represents that critical level of uncertainty, i.e., the PSM we are looking for at a given frequency point. In conclusion, the true-or-false function satisfies the monotonicity requirement for the bisection method. This renders the following PSM algorithm possible. Figure 3.1: The true-or-false function f(a) vs. the scaling factor a. Algorithm PSM: (parametric stability margin solution) Step 1. Set up a fine grid of frequency points in the range (ujib, oo u b). Step 2. Consider the next frequency point u>k that has not yet been looked at. Set aib = 1; a ub = B , where B is the largest possible value of a, up to oo. Step 3. a(k) = (a tb + a u j)/ 2. Check whether —1 e g(juik,q). • If — 1 ^ g{juk, q), choose lower bound = a(k). then iterate Step 3. • Else, choose upper bound a up = a(k). Now if ( a u b — ctib)/aib < r, where r is small enough to approximate zero, exit this loop and go back to Step 2 to work on next frequency point. Else, iterate Step 3. End of algorithm. Note that the algorithm with the a — iteration enhanced by the bisection algorithm achieves great computation efficiency compared with general a sweeping

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30 method. Furthermore, in Section 3.5, an interleaved bisection algorithm is introduced to efficiently compute the worst-case stability margin and the corresponding worst-case frequency point. It is acknowledged that Bhattacharyya et al. [12] consider a similar stability analysis problem, but working with additive increase of the uncertainty, i.e., q = q 0 + A q . By assigning zero to the characteristic polynomial of the family of uncertain systems, a Linear Programming (LP) problem is constructed to solve for the stability margin f3 where the variation of the uncertainty is bounded by /3. In this chapter, we are considering multiplicative increase of the uncertainty, i.e., q' — aq. Actually, it is in this multiplicative case that the parametric stability margin equals to the Multivariable Stability Margin, or the inverse of the well-known /r as shown in next section. Motivated by the argument in [12], the PSM can be arranged as inf a subject to a 0 can be easily expressed in matrix form: 1 0 0 . . • -
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31 namely, Arnr)X br. (3.22) where x := [q; a]. The combination of (3.21) and (3.22) is a standard linear programming (LP) problem. In Section 3.6, we show that a — iterationa algorithm with bisection algorithm yields the same results as with the standard LP problem solved using Matlab function fmincon.m. In next section, some representative previous results of the same real parametric affine uncertainty problem are reviewed and compared. 3.4 Real-/i Analysis based on Polynomial and Rank-One Matrix Approach In general, stability problems can be conveniently studied in the framework of polynomials whose coefficients are affine functions of real or complex uncertainties. For polynomials with complex affine uncertainties, Latchman et al. [53] present explicit and exact stability conditions using critical direction theory and intuitive geometric arguments. Jie Chen et al. [14] [15] introduce a generalized notion of structured singular value and provide a necessary and sufficient condition concerning the robust D-stability of the polynomial with either complex unstructured or real parametric uncertainties alone. In order to facilitate comparing with the developments in this chapter, previous results on real parametric uncertainties are summarized in the following. The characteristic polynomial of the uncertain system in (3.1) is: TTlr P(M q) = n(ju, q) + d(ju, q) = p 0 (juj ) + ^ Pk{ju)q k {jcu) (3.23) k = 1 As shown in [11], an affine real parametric uncertain system is equivalent to a rank-one problem.

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32 Figure 3.2: The classical M — A structure used for analyzing robust stability Lemma 3.2 The unity negative feedback configuration of the SISO uncertain system (3.1) with affine uncertainty elements can be transformed into an equivalent MI MO M — A structure of the form shown in Figure 3.2 where the diagonal uncertainty matrix A = diag{q u q 2 , . . . ,q mr } (3.24) and the rank-one structural matrix M = {M ik } with elements Mik (s) Pi(s) Po{s) modi matrices are N = {Ni t i}, where , i = 1 , 2 , . . . , m r , k = l,2,...,m r (3.25) Furthermore, the nominal model matrix is V — , and the interconnecting AT _ n l {s)p 0 (s) Pi(s)n 0 {s) ,_ 10 _ *1,1 ( _ f \ , J / _\\ j / '"T (n 0 (s) + d 0 (s))d 0 (s) (3.26) and U = {U\ t k}, where U lJt do(s) n 0 (s) + d 0 (s) , A: = 1, 2, . . . , m r (3.27)

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33 In [15], Jie Chen et al. provide a general formula for the generalized notion of H defined there. Furthermore, the real-// of the system with affine real parametric uncertainty reduces to the following convex optimization problem. Lemma 3.3 The structured singular value of the system with affine real parametric uncertainty as described m (3.25) is ffis) = inf x&R E Lfc=i Re 7 fc Pk{s) Z Po{s) + x Im 7 fc Pk{s) : Po(s) Qr 1 /Qr (3.28) where the weighting constants ffik) of the uncertainties are considered in the system matrix M , i.e., M Pi{s) Pm r {s ) n T [7l-"7m r ](3.29) _Po(s) po(s) Proof: Refer to [15] for the proof. Of particular interest is the case q r — 1, which corresponds to the standard structured singular value. The analytical expression (3.28) involves solving a convex optimization problem in one real variable, i.e., x, and renders the real-/i problem readily solvable. In parallel with the real-/x problem in the framework of polynomials, there is also a real-/r analysis based on matrix operations. Referring to the left part of Figure 3.2, we can easily find the closed-loop transfer function between the input and the output, which is called the upper linear fractional transformation (LFT), denoted as F U (M — A). The theoretical control insight is gained from this upper LFT representation: y = F U (M A )r = [V + UA{I MA)~ 1 N]r. (3.30) Obviously, for the closed-loop to be well defined, the matrix I — M A must be invertible, i.e. det(I — M A) ^ 0. This leads to the definition of the multivariable stability margin (MSM) [18] [61]

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34 MSM(u ) := mw{ 7 |det(/ 'yM(ju)A(ju))) = 0, IIAIloo < 1} (3.31) J6R+ By recalling the definition of the well-known fi, the MSM is the inverse of MSM := — (3.32) Ha (AT) Note that the MSM is precisely analogous to the definition of the parametric stability margin for a SISO system given in (3.18). It is straightforward to denote matrix M in (3.25) as M(s) u(s)v* Pl{s) Ms) . po(s)’ M s Y Pm r (a) Po(s) . T [ 1 , 1 , ... , 1 ] (3.33) Therefore, the real-// of the system with affine real parametric uncertainty reduces to a rank-one problem. For any bounded uncertainty A and a e R + , HA (M) 1 min{P | det (/„ — /3AM) , ||A||oo < 1} AM\ |. . , . I Halloo < 1} max{oi | det I n a max{a | a = v*Au} m T max {a \ a = S k v k u k } k = 1 (3.34) As such, Young [67] indicates that the rank-one real-// problem simply amounts to choosing 5 k so that vectors {S k v k u k } for k = 1 , • • • , m r add up to a positive real number, which is as large as possible. Certainly, one can search 6 k from its lower bound to the upper bound while fixing all other 5 at either of their lower or upper bounds based on the well-known “Edge-Theorem”. This is equivalent to a search over

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35 2 mr segments. Young [67] proposes a much more efficient algorithm that is adjusted to work with purely real-// problem. Algorithm Real -g: (Rank-one real-p solution) Step 1 Choose starting values for the real perturbations as S k = sign(Re(v k u k )). Then for all S k we have Re(S k v k u k ) > 0. Now compute S = sign ( Im hvkU k )) . Step 2 Rank all the components S k v k u k by argument. Step 3 Consider the highest rank component that has not yet been looked at. Compute 5 opt, which is the optimal value of this S k , for 6 opt € R unrestricted in sign or magnitude, and all the other real perturbations fixed at +1 or —1. Note: 5 opt magnifies Re (XX=i ^k v k^ k ) while Im (£X=i S k v k u k ) < r where r is used to approximate zero. Step 4 If sign(5 opt ) = —sign(5 k ) and \S opt \ > 1 and not all the components have been looked at, then reassign 6 k with max[1, min[l,6 opt \], and go back Step 3. Else, either find g with this S opt or verify that no 6 k exists to make the summation add up to a real number. End of algorithm. Note that the algorithm requires at most a search over the real parameters, which grows linearly with m r , not as generally believed that the complexity in computing g grows exponentially with m r . In Section 3.6, an example is provided to illustrate that the real-/x over polynomial approach [15] and the real-// over rank-one matrix algorithm [67] are equal, and also equal to the inverse of the parametric stability margin introduced in Section 3.3. 3.5 Exact Calculation of the Robust Stability Margin and the Worst Case Frequency Point Even though the MSM, the PSM, and the vea\-g are all defined for each frequency point, usually it is more important to find the minimal stability margin, or the maximal g over a frequency range from the perspective of robust analysis since

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36 they denote the worst case stability margin. In this section we consider the problem of computing the worst case robustness stability margin for a SISO system with real parametric uncertainty. In the last few decades researchers have given considerable attention to the robustness margin problem for the case of unmodeled dynamics and parametric uncertainties. The most popular methods to date use Lyapunov methods and structured singular value p. For real parametric uncertainties, however, these methods could be very conservative. Furthermore, the degree of conservatism may be arbitrarily large. Another method of computing the parametric stability margin, in principle, involves a domain splitting global search in both the frequency and parametric space. In [25], Gaston and Safonov developed a computational method for the exact robustness margin for a multilinear interval plant. In their approach the hypercube in the parametric space is divided into sub-cubes resulting in the union of these sub-convex hulls approximating the boundary of the value set. The Parametric Robustness Stability Margin (PRSM) is defined as follows: Definition 3.2 Consider an interval plant g(s, q ) with parametric uncertainties q € Q, where q
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37 respect to u k for k = 0,1,..., m r is created and the minimum of a(joj k ) over the entire frequency range is the robustness margin. Here, we propose a new approach to compute the worst case robustness margin directly from Nyquist stability criterion. A key step in the new approach is the use of the bisection algorithm. The robust stability test for a given uncertainty size can be accomplished by a simple numerical comparison. This has immediate implication for numerical efficiency, since a well defined mechanism is now available to measure the effect of increasing the uncertainty size without computing the entire value set. Moreover, instead of searching for robustness margin at each sampled frequency, this new approach increases the scale of uncertainty size a for a range of the frequencies until the first a violates the robust stability criterion at some frequency. This process simultaneously identifies a restricted range for both a and lo and an efficient bisection search identifies the robustness margin. Note that the specified frequency tolerance may affect the searching of the global minimum a * . Special care should be taken for all degenerate cases. Algorithm PRSM (Parametric Robust Stability Margin) Step 1 Start with initial scale cq = 0, initial frequency range F, a. increasing step A a, and set performance criterion r. Step 2 Check whether —1 G g(juii, q), where uji G F : • If — 1 ^ g(juJi , q), Vwi, choose a = a 0 + Aa, then iterate Step 2. • If at any a;*, —1 G g{jcOi, q), choose upper bound a up = a and lower bound Otlow ^ A Oc. Step 3 Set the frequency range F equal to F' = [ui ow , uj up \, where • ui ow ui such that -1 ^ g{ju k , q), and -1 G g(ju> i+i,q). If oJiow = u up , {number of points in F'} =2*{number of points in F'}, then iterate Step 3; Else, go to Step 4.

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38 • U) up = L 0 U such that -1 E c/(jw u _i,q) and -1 £ g{ju k , q), > u u , then go to Step 4. Step 4 Bisect the scale such that a = ( ai ow 4 a up )/2. Step 5 Check whether —1 E g(jco kl q)where a i k E F' • If — 1 ^ g{jco k , q), Vw G F', then ai ow = a ; go back to Step 4. • If at any coi, —1 G g(jcoi, q), check whether the program should end: If ( co up uJi ow )/u)i ow > t or ( a up ai ow )/cn ow > r, Go back to Step 3 for further reduction of the frequency range F' . If (u4 p ~ u^iow) / '^iow — and (cr^p &iow') / &iow ^ j go to Step 6. Step 6 a* = (&iow + a„p)/ 2 is the robustness margin, u>* = ( uji ow 4o; up )/2 is the corresponding first unstable frequency. End of algorithm. 3.6 Example The following example illustrates the use of the PSM algorithm and compares our results with those based on other approaches. Consider the uncertainty system g(s, q) g{s, q) n(s.q) d(s, q) (3.36) 7i(s, q) — 4 (4 + 0.4(7! + 0.2q 2 )s 4 (20 + 91 — Qs) (3.37) d(s, q) = s 4 4 (9.5 4 0.5^i — 0.5q 2 4 0.593)5 ^ 4 (27 4 2<7i 4 < 72)^ 4 (22.5 — q\ 4 <73)5 4 0.1 (3.38) where {qi, q 2 , 93} e Q, and -3 < <7* < 3, i = 1, 2, 3. System (3.36) is a modified version of a model investigated by Fu [23]. Note that coefficients of the transfer function depend affinely on the parameters q E Q. The nominal plant g 0 (s ) =
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39 Its characteristic polynomial is: p(s,q) = n(s, q) + d(s, q) I s 4 + (9.5 + 0.5Q! 0.5q 2 + 0.5 q 3 )s 3 + (28 + 2q x P(s, q) = < -\-q2)s^ + (26.5 — 0.6
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Stability Margin 40 This also can be seen from Figure 3.4 where both real-// and complex-/^ are shown. With Matlab function fmincon.m, the solution of the standard LP problem is obtained, which is exactly same with that solved from Algorithm PSM. (Stability Radii) Algorithm PRSM Figure 3.3: Comparison of the PSM with the complex parametric stability margin Result 3: (Jie Chen et al. [15] Optimization Solution) Refer to (3.29), the weighting constants are 7 *, = 3, k — 1, 2, 3.

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41 (Stability Radii) Algorithm PRSM Figure 3.4: Comparison of the real-/i with the complex-/r M{s) Pl(«) Pm r ( s ) 1 • [71. • Po(s)' ’ Po(s) . QPi(s) °Po(s) qPi(s) °Po(s) 3 Pi(s) po(s) 3 P 2 (s) Po(s) _qP 2 {£l °Po(s) 3 P 2 M Po(s) oP 3 (s) °Po(s) qP 3 (s) °Po(«) 3 P 3 l£l °Po« J p{s) = inf xER E .jt=i &|7 ^) + *M 7 ‘Sm. (3.43)

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real mu 42 With the Matlab optimization toolbox, this real-// is calculated and shown in Figure 3.5. JieChen 1994, ssv with Polynomial approach Figure 3.5: Real-// by the optimization algorithm over one variable Result 4: (Young [67] Algorithm Rank-One Solution) M(s) = u(s)v* where and «(s) Pi(s) P2(s) . Po(s)’ Po(«)’ « = [3, 3, 3] t (3.44) (3.45) (3.46)

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real mu 43 Set r = 0.01, the real-// can be worked out by using the rank-one algorithm. The result is shown in Figure 3.6. By comparing figures 3.4, 3.5 and 3.6, the three methods are equivalent in solving the real-// problem with affine parametric uncertainty. It is worthy to indicate again that Jie Chen et al. [14] [15] study uncertain polynomials, while Young [67] solves the problem specified in the rank-one case from the definition of /i. This chapter recovers the previous results from the stability radii perspective by searching in the value set. Moreover the Algorithm with Bisection will get the exact stability margin at the exact worst frequency point as shown in Result 5. Young 1 994, ssv with Rank One Problem Algorithm Figure 3.6: Real -/i by algorithm rank-one approach

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44 Result 5: The Worst Case Stability Margin and the Corresponding Frequency For the interval plant given in this example, the exact robustness margin found by the stability radii approach is a* = 1.8489 at ui* = 4.6389. As shown in Figure 3.7, the critical point —1 + jO is right on the boundary of the value set at co* while a = a*. Note that the minimal stability margin obtained by using complex-// approach is 1.1786 — only about 64 percent of the minimal stability margin resulted from real-// analysis. The comparison reveals that real-// analysis is critical to eliminate the conservatism incurred by using standard complex-// analysis for real-parametric problems. The Uncertain Set at alpha=l .8489 and w= 4.6389 Figure 3.7: The uncertainty set with a* — 1.8489 at ui* = 4.6389

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45 Matlab ' min f(x) 'j , A e x — b e l I A c x < b c [ l Xi < X b e —A e x > —b e —A c x > —b c —x > —x u , Cz = —f{x) Lab VIEW { max Cz Mz > b z> 0 (3.47) Figure 3.8: Transformation between linear programming functions of Matlab and LabVIEW 3.7 Simulation Reliability To check the reliability of the simulation programs, we construct two kinds of programs using linear programming functions from different commercial software: Matlab and LabVIEW. As listed in Figure 3.8, those two functions use different expressions of the variables and their conditions. Note that the equality in Matlab program can be expressed as inequalities in LabVIEW program. We obtained the same results from these two different programs. This further verifies the simulation besides the confirmation between the results in this dissertation and these using other approaches reported in [15] [67]. 3.8 Conclusion By transforming the affine parametric uncertain system problem into a linear feasibility problem, this chapter presents a novel simple approach based on Nyquist stability analysis for the problem of finding robust stability margins for SISO systems with real parametric uncertainties. The approach recovers previous results based on real-/r analysis for affine parametric uncertainty problem. The approach is promising and can be considered as a significant step towards the solution to the general robust stability analysis problem. Also, in terms of computation complexity, the PSM algorithm with bisection method to solve the frequency-dependent stability margin shows great advantage over general a — iteration. A similar computation advantage

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46 is obtained with the PRSM algorithm to solve for the smallest stability margin and the corresponding worst-case frequency point by reducing the searching time with ever-reduced frequency range and alpha-searching range.

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CHAPTER 4 ROBUST CONTROL FOR PARAMETRIC PLANTS 4.1 Introduction Results on stability analysis for systems with parametric uncertainties have emerged steadily over the years, including outstanding results such as the edge theorem, the mapping theorem, the generalized Kharitonov theorem [12], the critical directional theory (CDT) [4], and many others. However, nonconservative robust controller synthesis for interval plants remains a relatively unsolved problem for which only few results are available. An early attempt to synthesize a robust controller was given in [40], where interval plants are overbounded by a constant uncertainty disk in the frequency domain. By assigning this constant bound as the weighting function, standard results from the H ^ synthesis theory are then applied to solve for a robustly stabilizing controller. This approach shows great promise because it is quite simple and makes effective use of the well-developed H ^ synthesis methodology. To facilitate comparison with the alternative results developed in the following chapters, we refer to the method in [40] as the maximum perturbation radius (MPR) approach. Unfortunately, the MPR weighting strategy may introduce conservatism since the overbounding operation guarantees only sufficient conditions for robust stability. Other researchers have tried to attack the synthesis problem using fixed-structure controllers such as PID, or other low-order controllers [16] [28] [31]. Yet, it can be argued that these approaches suffer from inherent shortcomings by restricting attention to only fixed-structure controllers. As discussed in Chapter 2, the critical direction theory [4] [51] gives necessary and sufficient stability conditions that involve a critical perturbation radius (CPR). 47

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48 Motivated by this fact, the conservatism of the MPR weighting is explored in detail in the following chapters. Furthermore, the critical direction theory is applied to introduce an exact weighting strategy that results in an even larger stability margin than that based on the MPR weighting. The effective critical perturbation radius (ECPR) weighting strategy proposed in our work recovers the simplicity of the MPR weighting while reducing the conservatism induced by uncertainty over-bounding. The rest of the chapter is organized as follows. Sections 4.24.4 set up the problem of robust controller design for systems with parametric uncertainties, present necessary preliminaries of mixed-sensitivity synthesis, and finally recast the parametric problem into the mixed-sensitivity framework. The MPR weighting and its conservatism are discussed in Section 4.5 where an example is also given. The main results are introduced in Section 4.6, where we define the ECPR weighting function, and provide a dynamic robust controller synthesis approach. 4.2 Problem Formulation Consider a SISO linear uncertain plant represented by a rational transfer function, p(s,e), in cascade with a controller c(s), as shown in Figure 4.1. The family of uncertain plants can be represented by |p(s,e) — Po(s)\ < |5(s, e)| where po(s) is a known nominal SISO system, and 6(s,e) is a perturbation characterized by a set of real parameters, namely, the coefficients of the numerator and denominator polynomials. Each parameter in turn is an element of a real interval of width 2e centered about the nominal value of the corresponding parameter of the nominal plant po(s). Therefore, the uncertainty description for the problem constitutes a real hypercube A(e) with sides of width 2e. The value e is a non-negative scalar that represents the level of uncertainty. In general, system po(s) is neither open-loop stable nor unitynegative-feedback system stable. A stabilizing controller c(s) needs to be designed so that the closed-loop system of Figure 4.1 is robustly stable, i.e., stable over all uncertain plants p(s,e).

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49 As is common in all Nyquist-based robustness studies, it is assumed that the set of allowable perturbations A(e) is such that systems p(s,e) and po(s) have the same number of open-loop unstable poles. This assumption can be checked by the following steps: Step 1. Use the Routh-Hurwitz criterion for the denominator Kharitonov polynomials of p(s, e) to find an uncertainty level e\ such that the number of unstable roots of the family of denominator-polynomials of p(s, e) does not change for all e < ci [40]. Step 2. Use the edge theorem to find another uncertainty level e 2 such that there is no unstable pole-zero cancellation in the family of p(s, e) for all e < e 2 [17]. Step 3. Define the upper bound of the uncertainty as e u — min{e i,e 2 } and require that e < e u . After a candidate controller c(s) is designed, define g(s, e) — p(s, e)c(s) as shown in Figure 4.2. Note that the nominal system is g 0 (s) := p 0 (s)c(s), and the uncertainty is 6 g (s, e) := 5(s, e)c(s). For robust stability of system g(s, e), the following necessary conditions should be satisfied: (Bl) The nominal system go(s) = p 0 (s)c(s ) is stable under unity-negative-feedback. (B2) The family of uncertain systems g(s, e) and its nominal system go(s) have the same number of open-loop unstable poles. Therefore, besides the assumption made for system p(s,e), one further condition needs to be checked after a candidate controller is designed. If a candidate controller c(s) has unstable zeros, one must ensure that there is no unstable pole-zero cancellation between the zeros of controller c(s) and the poles of the family of p(s, e) for all e < e u . A similar analysis should be carried out for the possible unstable pole-zero cancellation between the unstable poles of controller c(s) and the unstable zeros of the original systems p(s, e). A candidate controller should be rejected if any forbidden pole-zero cancellation occurs.

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50 There are primarily two problems to be resolved. The first is to determine the parametric stability margin e max , which is defined in [40] as the maximum e for which the entire family of interval plants p(s, e) is robustly stabilizable. The second problem is to synthesize a stabilizing controller that can robustly stabilize the family of plants for all £ (-maxFigure 4.1: The negative feedback loop of the uncertain System p(s) with a controller c(s). This figure is usually used when dealing with a controller design problem. Figure 4.2: Stability analysis for an interval system g(s) under unity negative feedback. This figure is usually used when dealing with a stability analysis problem.

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51 4.3 Preliminary: the Mixed-Sensitivity H ^ Robust Control The mixed-sensitivity minimization problem is usually considered as the most important one for synthesis problems and receives a lot of attention in the H ^ synthesis literature because the problem can be considered as the combination of the robust control problem and the performance optimization problem. The mixedsensitivity minimization problem is an extension of the weighted sensitivity minimization problem introduced by Zames [70]. Consider the classical feedback structure with weighting strategy shown in Figure 4.3 where Pq(s) is the nominal transfer function of a given plant and W\, W 2 and are design specification weighting transfer functions. Note that, P 0 (s ) instead of po{s) is used here to denote that the systems are not restricted to SISO cases. The same meaning applies to other capital notations of Figure 4.3. The design goal is to control the gain of transfer functions from the reference r to output vector U\ = [2/1 1 ) 2/12 ? 2/13] 7 " t° be less than a certain level 7. It is straightforward to check that 2/11(5) = Wi(s)(I + L(s))~ l r(s) 2/i2(s) = W 2 (s)c(s)(I + L(s))~ 1 r(s) (4.1) 2/13(5) = W 3 (s)L{s)(I + L(s))~ 1 r(s) where L(s ) = Po(s)c(s). Now by defining S(s) := (I + Hs))1 R{s) := c(s)(/ + L(s))1 (4.2) T(s) := L(s)(I + L(s))~ 1 = I S(s), the mixed-sensitivity synthesis problem is to find a stabilizing controller c(s) such that 7 is minimized and

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52 ||2~t/ir( s ) || oo Wrfis) W 2 R{s) W 3 T(s ) < 7 < 1. (4.3) OO Note that, generally 7 is required to be less than one for robust stability of the whole system where the structure and size of uncertainties is absorbed into the plant Po{s), and weighting functions VFi(s), W 2 {s) and VF 3 (s) . Traditionally, L(s) and T(s) are called the open-loop transfer function and the closed-loop transfer function respectively. I + L(s) is the return difference transfer function and the two matrices S(s) and T(s ) are known as the sensitivity function and the complementary sensitivity function, respectively. Similarly, WiS, W 2 R and W 3 T are called the weighted sensitivity function, the weighted control sensitivity function, and the weighted complementary sensitivity function respectively. ||T J/ir (s)|| 00 is called the mixed-sensitivity cost function because it penalizes all the three sensitivity functions simultaneously [6]. Minimizing the infinity norm of the mixed sensitivity function corresponds to the optimization of the robustness of the controller. Thus the choice of the weights Wi,W 2 and W 3 is an important issue and this choice typically depends on the kind of application as well as engineering insights into a physical problem. Figure 4.3: A standard H ^ synthesis problem in the mixed-sensitivity framework.

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53 4.4 Robust Control for Parametric Uncertain Plants The following theorem is proposed to help illustrate the transformation of the controller synthesis problem posed in Figure 4.1 into the mixed-sensitivity framework. Theorem 4.1 Suppose the system in Figure 4-1 is stable when S(s) is zero. Then the size of the largest stable 6(s) for which the system remains stable is ll«Wll< iisTit < 44 > where R{s) c(s)(I + L(s ))~ 1 as defined in (4-2). Proof: This theorem can be proved invoking the Small Gain Theory. Based on the Sandberg-Zames’ Small Gain Theory [72], the M — A system in Figure 4.4 is internally stable for any stable A(s) satisfying l|A(s)ll “ < HMMIL (4 ' 5) where both M(s) and A(s) are assumed stable. Considering the system in Figure 4.1, we have w(s) = c(s)r(s) c(s)[z(s) +p 0 (s)w(s)] w ( s ) = (46) w(s) = R(s)r(s) — R(s)z(s) where R(s) := 1+c (f)p 0 ( J ,) as defined in (4.2). In order to use the Small Gain Theory, the system in Figure 4.1 can be rearranged as shown in Figure 4.5 imitating the M — A structure in Figure 4.4. According to the Small Gain Theory, the closed-loop system in Figure 4.1 is stable for any stable 5(s) satisfying the following condition:

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54 This finishes the proof. Halloo < 1 TOIU (4.7) Figure 4.4: Standard M — A loop for stability analysis. Figure 4.5: A system with parametric uncertainty in the standard M — A loop. For unstructured uncertainties A(s), the Small Gain Theory gives a necessary and sufficient stability condition. But for general structured uncertainties including the parametric case, the small gain condition is only sufficient to guarantee internal stability of the M — A loop. In Section 4.6, however, we will apply the critical direction theory and derive a necessary and sufficient condition for the internal stability of the closed-loop in Figure 4.1.

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55 As a consequence of Theorem 4.1, it is common to specify the stability margins of control system via singular value inequalities such as ||M / 2 (s)|| (X) < or, (4.8) || W ' 2 (*) fl(*)|| 00 <1 where ||W / 2 (ja;)|| 00 = supa(W 2 (jco)) is the size of the largest anticipated and “effecCJ tive” plant perturbations. By rearranging the system in Figure 4.1 into the M — A form, a stabilizing controller is required to satisfy (4.8) for closed-loop stability. Comparing (4.8) with (4.3), the closed-loop system configuration in Figure 4.1 is actually a special case of the standard Hoc synthesis problem, i.e., the mixed-sensitivity robust synthesis problem with Wi(s) = [ ]> W 3 (s) = [ ] (see Figure 4.6). The design criterion is min {||r rW2 (s) stabilizing c(s) <11 T, W 2 (s)c{s) l+c(s)po(s) }, i.e.. (4.9) min || W 2 (sJRtsJtloo < 1 stabilizing c(s ) The remaining problem is how to choose the weighting function W 2 (s) to represent the “effective” part of the uncertainty 5(s). In general, the more closely the approximate weighting functions are to the exact sufficient and necessary stability conditions, the less conservatism the resulting controllers will incur. We argue that weighting functions associated with sufficient robust stability conditions, e.g., weighting functions associated with the size of the largest plant perturbations (defined as the MPR in Section 4.5), usually yield conservatism in controller design. Hence we seek a weighting function W 2 (s) related to an exact measure of robust stability by referring to the critical direction theory.

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56 Figure 4.6: The parametric controller synthesis problem that is recast into the mixedsensitivity .f/oo synthesis framework. 4.5 MPR Weighting Approach and its Conservatism Analysis Firstly, consider the weighting method by overbounding the value set of p(ju),e) at each frequency to. The largest perturbation radius p m (u>,e ) for the system p(jui,e) at each frequency uj is defined as (see Figure 4.7): p m (u;,e) = max |<5(jo;)| = max \p(ju, e) Po(ju)\ (4.10) <5,e) in the family of transfer functions p(juj,e), will be associated with a point on one of the extreme segments, which is defined by the Kharitonov segments. Searching for the p m (w, e) for the case of linear perturbation is very simple, since there are only at most 32 extreme segments to be considered [40]. For affine parametric uncertain system, according to the mapping theorem introduced in Chapter 2 (see page 12), the MPR for each

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57 Figure 4.7: MPR versus CPR frequency point can be found out by searching over the boundaries of the value set mapped from the edges of the parametric space. For SISO systems, we have |tf(M £ )l = I P0’ w > c ) -PoM| < p m (uj,e ) | < p m (e). (4.12) Referring to Theorem 4.1, either p m (s,e) or the constant p m (e) can be used as the weighting function W^s) in the H ^ design to synthesize a stabilizing controller for the parametric uncertain system of Figure 4.1, and equivalently of Figure 4.6. This represents one of the first attempts to attack controller synthesis for systems with parametric uncertainties [12] [40]. The MPR weighting scheme shows much promise because the approach is simple and directly applies the well-developed synthesis framework. Unfortunately, as shown in the following, the MPR weighting approach may incur conservatism since

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58 the overbounding of the value set of p(ju , e) provides only sufficient conditions for robust stability. To analyze the conservatism that may be introduced by taking the constant uncertain bound based on the MPR as the weighting function ^(s), we now compare the value sets of system p(s, e) and g(s, e). As indicated in Section 4.2, system g(s,e) corresponds to c(s)p(s,e). After multiplication by the controller c(ju>), the value set of p(juu,e) may be shifted, rotated, and/or magnified or contracted to produce the value set of g(jco,e). Figure 4.8 illustrates the behavior of the corresponding variation caused by the controller c(jcoi) between the value set of p(jcji,e) and that of g(juji,e) at a frequency cjj. As illustrated in Figure 4.8: g t (juji,e) represents the intersection of the critical direction d c (ju>i) and the boundary of the value set of g(juji,e)\ and, p*(juJi,e) = , namely, the original plant point t) in the value set of p(jui,e) that is mapped to g*(jui,e) in the value set of g(jco,c). From the CPR definition (2.4) on page 8, it follows that Pc{ w,e) = \g*{juj,e) g 0 (juj)\ = \c{ju)p t {juj,e) c{ju)po{ju)\ (4-13) = |c(jw)|b*0'w,e) -p 0 (ju})\ As we noted in Section 4.4, an effective weighting function W 2 (s) should appropriately represent the uncertain plants. According to the critical direction theory, together with Equation (4.13), one should use | p*(jw,e) — Po(ju)\ as the weighting function in designing the controller c(s). Note that, generally, p m (u), e) > |p*(jw, e) — Po(ju)\. Therefore, conservatism may be incurred when using = p m (ui,e). Even more conservatism may be resulted in using weighting function W 2 (jui) = p m (e), as done with the constant MPR weighting approach.

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59 Figure 4.8: Complex maps between the original plant p(jco, e) and the transformed system g(juj,e) = c(ju)p(juj,e). The darkened areas represent the uncertainty value sets. The point ) in (b) defines a disk of radius p c (u>i,e), whereas its image p*(ju>i,e) in (a) defines a disk of radius p e (u;,,e). The overbounding disk of radius p m (uji ) in (a) circumscribes the disk of radius p e (uji, e), and leads to more conservative robust-stability estimates. 4.6 ECPR Weighting Methodology We revisit Figure 4.8 to seek a better weighting function. The family of critical uncertain plants V c (juji, e) of g(juji, e) can be represented by g(juJi, e) — | g(ju>i, t)\e^ 6dc ( Ui \ where 9 dc (uji ) is the angle of the critical direction defined in (2.1). Consider a certain controller c(juii) = \c(ju>i) \e^ ec ^'\ where 9 c {uJi) is the angle of the controller. Divided by controller c(jcoi), the family of critical uncertain plants g(juji,e) is mapped back to a specific family of original plants p{ju>i , e) expressed as: P(jui, e) \g(juj i ,e)\e J0 ‘ i c( Ui) where O dcv {ui) = 9 dc (ui) 9 c (coi). (4.14) Likewise, the critical point (—1 +j 0) of the value set of g(joJi, e) is mapped back to a specific point of In other words, the specific line originating from Po(juji) to

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60 in the value set of p(juii, e) is mapped to the critical line originating from go{j^i) to (—1 + jO) in the value set of g(jui,e). Hence, we name the direction originating from Po(ju)i) to to be the effective direction d CtP (juOi) of the value set of p(jw i: e). Parallel to the critical direction theory, we could obtain the following observation. Given the controller c(s), only the uncertainty plants along the effective direction d c>p (s), not over the whole value set of p(s,e), need to be considered in solving for necessary and sufficient conditions for the robust stability of the system of Figure 4.1. Formally similar to the definition of the CPR (2.4) (see page 8) for the value set of g(ju,e), the effective critical perturbation radius (ECPR) for the value set of p(ju>, e) is defined as p e (w,e) := max{a \ p 0 (Ju) + ad CtP (ju) € V c


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61 p c (uj,e) = \c{ju)p*(ju,e) c(juj)p 0 {juj)\ c(ju)\\p*{ju,e) -Po(ju)\ c(ju)\p e {u,e). (4.17) This finishes the proof. When the controller equals to zero at a frequency, the point is at infinite distance off the nominal point p 0 (ju>). In this case, the effective direction of the value set of p(s, e) is not specified. However, since the upper bound of p e (uj, e) can be infinitely large, we can replace p e (uj,e) with p m (uj,e) for simplicity. Theorem 4.2 Assume that the closed-loop system of Figure f.l is nominally stable; systems p(s,e ) and p 0 (s ) have the same number of open-loop unstable poles; and there is no unstable pole-zero cancellation between a controller c(s) and the family of systems p(s,e). Then, the closed-loop system of Figure 4.1 is robustly stable if and Proof: From the critical direction theory stated in Theorem 2.1 of page 9, the closed-loop system of Fig. 4.1 is stable under the given assumptions if and only if kx < 1. Using the definition (2.3) in page 9, it follows that Finally, from Lemma 4.1, the following equivalent necessary and sufficient stability conditions are obtained only if (4.18)

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62 There is a transfer relationship from the input w(s) of the uncertain plant S(s, e) to its output z(s) in Figure 4.1: *00 = i , j*) n r\ w ( s ) = T zw (s)w(s). (4.19) 1 + c[s)po(s) Hence, Equation (4.18) of Theorem 4.2 can be expressed as \\Pe{^>^)T Z w{j^>)\\oo < 1(4-20) Comparing the with the well-known result from the Small Gain Theorem, we have obtained a similar stability analysis result, but for systems with parametrically structured uncertainties. Similarly, as mentioned in Section 4.4, there is a transfer relationship from the system input r(s) to the controller output u(s) and further to the system output 2 / 12 ( 3 ) in Figure 4.6: ?/i 2 (s) = Ty ur (s)r{s) = W 2 (s)T ur (s)r(s), (4.21) and T ur {s) = c{s) -r(s). (4.22) 1 + c(s)p 0 (s) Using the ECPR p e (u>,e) as the weighting function W 2 (ju), Equation (4.18) of Theorem 4.2 can be expressed as \\pe{^i ^)T ur {jco) 1 1 oo < 1 (4.23) alternatively, ll^yi2r(j , ' w )||oo < 1(4.24) One can easily identify Equation (4.24) with the general expression of a mixedsensitivity problem as stated in Section 4.4. Corollary 4.1 The radius p e (uj,e) is the exact weighting function in designing controller c(s).

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63 Proof: This property can be easily derived from necessary and sufficient condition given in Theorem 4.2. As we argued in last section, when the MPR is used as the weighting function W 2 (ju), ||T yi2r (jfa/)||oo = \\Pm(t)T ur (jw)||oo < 1 is merely sufficient for system stability, but no longer necessary. Here again, one can see how the MPR weighting approach incurs conservatism. 4.7 Conclusion In this chapter, we take advantages of the critical direction theory in defining necessary and sufficient stability conditions, and develop a systematic method of choosing the exact weighting function, namely, the ECPR that allows for parametric uncertain systems to be robustly controlled using the synthesis tools. The following two chapters will continue this topic and focus on controller design using static weighting functions and dynamic weighting functions respectively.

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CHAPTER 5 ROBUST CONTROL FOR PARAMETRIC PLANTS USING STATIC WEIGHTING APPROACHES 5.1 Introduction This chapter continues the discussion in the last chapter and focuses on using only static weighting functions. The plant is given as one SISO system p(s, e) including a parametric uncertain part S(s,e), together with the same assumptions as listed in Section 4.2. We will address two issues: the first is to find the maximum stabilizable margin e max ; the second is to design a controller that can stabilize the closed-loop system of Figure 5.1(a) with uncertainties up to the maximum level e max . Recall that the problem is recast into the mixed-sensitivity form and the well-developed H ^ design tools can be used to cope with the parametric uncertain plants. As discussed in last chapter, an first attempt in robust controller design for parametric uncertain system used a constant overbounding method [40]. Its conservatism is analyzed in detail in the last chapter. This chapter argues that a much larger parametric stability margin could be obtained using a static weight based on the critical direction theory, namely, the uncertain parts <5(s,e) for all frequencies are represented by a constant ECPR, instead of a constant MPR. Moreover, we revisit Hoc robust control, and discuss in detail all the stabilizing controllers that satisfy the design criterion, which are called all-solution controllers. The next chapter will develop a dynamic weighting method to further reduce conservatism in determining the maximum stabilizable uncertainties and in designing a robust controller. The rest of the chapter is organized as follows. Section 5.2 explores the characteristics of static weighting. Then an upper bound of the static weight is found 64

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65 (a) S g (s) So(*)->0 — 1 (b) (c) Figure 5.1: (a) Negative-feedback loop including the uncertain system p(s) — p 0 (s) + 8(s) and a controller c(s); (b) unity-negative-feedback of system g(s) = c(s)p(s); (c) mixed-sensitivity approach to the uncertain feedback system. and used for finding a stability margin in Section 5.3. Two illustrative examples are presented in Section 5.4. Simulation reliability is discussed in detail in Section 5.5. Some important derivation and simulation on all-solution controllers are studied in Section 5.6. Conclusive remarks are given in the final section. 5.2 Parametric Robust Control using a Static Weighting Approach Lemma 5.1 When using static weighting approach to synthesize a controller for systems of Figure 5.1(c), a controller that minimizes ||T ztl) (s)|| 00 also minimizes ||T u11 ,(s)|| 00 .

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66 Proof: We denote a controller c(s) that minimizes ||r zU) (s)||oo as c op {s) = arg min ||T, ll ,(s)|| 0 o. stabilizing c(s) As discussed earlier, T zw (s) = VF 2 (s)T uu ,(s), hence, c op {s) = arg min ||W / 2 (s)T uu ,(s)|| 00 . stabilizing c(s) Given VF 2 (s) = r, r € R + , ( 5 1 ) (5,2) c™(») = a rg min J|rT„(s)||; stabilizing c(s) = arg min r||T ul0 (s)|| c stabilizing c(s) = arg min ||T uw (s)|| 00 stabilizing c(s ) (5.3) This finishes the proof. Theorem 5.1 For the static weighting approach, the controller design criterion min sup \ T uw (ju)\ is equivalent to stabilizing c(s) u max inf stabilizing c(s) w 1 c(jw) + Po(J u ) Proof: Following the proof in Lemma 5.1, Cop(^) arg min II T uw { stabilizing c(s) arg min sup stabilizing c(s) u / arg max ( su. stabilizing c(s) V 0, arg max inf stabilizing c(s) LJ arg max inf stabilizing c(s) U c((M l+c(juj)po(jui) c(ju) l+c(juj)po(jui) l+c(jui)p 0 (jgj) c(ju) -1 at) (5.4) (5.5) Expression (5.4) exposes an alternative interpretation of the controller design procedure, namely, a search over all stabilizing controllers to find one that maximizes

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67 the minimal distance between po(juj) and the point — ^jy • Note that both of the design criteria of Theorem 5.1 involve only the nominal plant Po( s ); hence, the problem is independent of the uncertain plant 6(s,e). This is a natural consequence of using a static weight. The alternative static weight proposed in this chapter adopts r — p e (e), so that W 2 (s) = p e (e), (5.6) where p e {e) \= max p e {bo,e), (5.7) LJ The next section discusses the attributes and advantages of using this static weight. 5.3 Robust Stability Conditions and Stability Margin Calculation From Figure 4.8 of page 59, it is obvious that robust stability is lost when the effective radius p e {wu e) is equal to the distance between the points — and Po{ju)i) at some frequency In such a case, a point on the boundary of the plant value set is mapped by the controller to the critical point —1 + jO in the Nyquist plane. Consider an optimal controller c op (s ) designed using the static weighting approach, and define p“ := inf + Po(ju) . We obtain the following theorem. Theorem 5.2 Let c op (s) be an optimal controller designed using the static weighting approach, and let p e (e) be the corresponding constant ECPR for an uncertainty A. The closed-loop of Figure 5.1(a) is robustly stable if p e (e) < p“. Proof: Condition p e (e) < p“ implies 1 Pe{t) < inf Cop {j LO ) + PoO'w) which is equivalent to Pe(t) Cop(ju) 1 + c op (ju))p 0 (juj ) < 1 ,Vw.

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68 A further condition is obtained by considering the definition of the constant ECPR weight (5.7), Pe(w,e)Cop(j u ) < 1 ,Vcd. (5.8) 1 + c op (ju)p 0 {juj) According to Theorem 4.2 of page 61, condition (5.8) implies the stability of Figure 5.1(a). This finishes the proof. The sufficient stability condition in Theorem 5.2 validates the utilization of the constant ECPR as the static weighting function, as expressed in (5.6). The method proposed in [40] to compute the stability margin, denoted as e max , used the constant MPR weighting, i.e., 1T 2 (s) = p m , and then, t and hence p m is increased until p m = p 1 ^ = p“. However, as shown in the examples in Section 5.4, much larger stability margin can be found based on the static ECPR weighting, i.e., calculating e max from a plot of p e (e) vs. e, instead of p m (e) vs. e. Alternatively, we can find e max (uj ) at each frequency by iteratively increasing e from zero until Cop(ju) is in the value set of p(juJ, e), then the stability margin e max is the smallest value of e max (uj) among all frequencies. A more efficient solution can be constructed by referring to the linear programming algorithm posed in [12] [50]. Thus, by interpreting the static weight W 2 as a constant ECPR weight (IP^s) = p e (e)), instead of a constant MPR weight, a larger parametric stability margin may be found. From Theorem 5.1, one can obtain an alternative meaningful expression of p“, 1 Pe mm stabilizing c(s) I T uw (s) (5.9) Further connections with the classical static weighting design approach can be established by considering a system in which the weight is given by lp 2 (s) = r, and the controller in question c op (s) attains the value min || T uw OOlloo as defined stabilizing c(s) in (5.1). Now let r max denote the value of r such that ||r mai T tU0 (s)|| oo = 1. Then it

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69 follows that r max = — 7 — 1 n T 77 m Hence 771271 ||-2 uw (.5 )||oo stabilizing c(s) r max = p u e . (5.10) In next section, two examples are given to illustrate the substantial improvement in stability margin assessment using constant ECPR weighting. 5.4 Examples Example 5.1 Consider the interval plant discussed in [12] [17] [28] [40], with intervals P(s,e) 5s + q\ s 2 + q 2 s +
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70 -2 15 1 O i *5 o II o II 1 0 1 o 1 ,C P0 = [ 0 0 ],D P0 = 0. (5.14) Step 2: Recast the problem into the mixedsensitivity form We transfer the closed-loop system of Figure 5.1(a) to that of Figure 5.1(c) and have W^i(s) = W^s) = 0, and the uncertain part S(s,e) is represented by W 2 (s). Step 3: Express the mixedsensitivity problem using the Lower-LFT form As shown in Figure 5.1(c) together with W 3 (s) = r, r G R + , it is straightforward to obtain the following state space representation of the augmented plants: Zpo W — ^po x Po ( t ) + 0 w(t) + B Po u(t) z(t) = 0x po W + 0 w(t) + ru(t ) y(t) = -C po X po {t) + w{t) — D Po u(t) G — K form, we have A Po 0 B P0 G = 0 0 r -Cpo 1 ~~D Po (5.15) (5.16) Note that by assigning r = 1 , we get the state space representation for the transfer function T uw (s). Step 4 : Design the robust controller via static weighting method Using r y-iteration, the controller that minimizes ||Ti iUJ (s) ||oo = found to be l+po(s)c(s) IS 3603.7935s + 18018.9673 , ^ ~ s 2 + 1434.5016s 2312.4499' ^ ' ’ It is straightforward to check that there is no unstable point cancellation between the unstable pole of c(s) (5.17) and the zeros ofp(s,e) for e up to e u .

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71 Step 5: Find the maximum stabilizable margin using the static ECPR weighting According to Theorem 5.1, the upper bound of the exact static weight is = 0.3950. The ECPR p e (c ) is calculated and plotted versus e shown as the solid line in Figure 5.2. It is observed that t max = 5.2511 when p e (e) = 0.3950. Step 6: Compare with the t max found using the static MPR weighting As shown by the dashed line that reflects the variation of p m (e) vs. e, one gets e = 2.8300 when p m (e) — 0.3950 (the limiting value of W 2 ). The value e — 2.8300 was proposed in [f.0] as the parametric stability margin in the case of the constant weighting method using the MPR approach. Therefore, the parametric stability margin calculated by the MPR weighting approach (e = 2.8300J is only about 54 percent of that obtained by the ECPR weighting approach (e — 5.2511,).

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72 Figure 5.2: Plots of p m and p e radii as functions of the uncertainty-size parameter e. The value /3 = 0.3950 is the limiting value of the static weight W 2 (s) found via a standard unstructured H 0 0 approach. Example 5.2 Consider the interval plant given in [40], 30s + qi p(s,e) = 5 s 3 + q 2 s 2 -Iq 3 s + q 4 with intervals (5,18) q\ G [10 — e, 10 + e], q 2 G [—3 — e, —3 + e, (5.19) q 3 G [—4 — e, —4 -(c], q\ G [12 — e, 12 + e. (5.20) Note that to satisfy the assumption of the family of systems p(s, e) made in Section 4-2, e u — 6.0.

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73 Using 7 iteration , the controller that minimizes ||T uu ,(s)||oo found to be c ( s ) l+Po(s)c(s) is Po{s ) = 30s + 10 (5.21) s 3 3s 2 -4s + 12Â’ and Wi(s) = W 3 (s) = 0, ^(s) = r, where r is a constant. It turns out that at W 2 = 0.4023, p(X 00 F 00 ) approximates unity and the following controller is found c(s) 3617.6s 2 + 4562.3s 5345.9 s 3 + 1468.3s 2 + 18620.7s + 6605.8' (5.22) Figure 5.3: Plots of p m vs. e (dashed line) and p e vs. e (solid line) The stability margin was reported in [40] to be 0.8500, which actually corresponds to = 0.4023, as shown by the dashed line in Figure 5.3. However, by interpreting

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74 the value 0.4023 as the p e , and the stability margin is e = 1.2800 where p e = 0.4023. Again, the conservatism introduced by previous approach can be reduced by taking the constant weight as the constant effective bound, instead of the constant overbounding uncertainty. 5.5 Simulation Reliability As discussed in this chapter, the optimal controller designed by static weighting should be unique that minimizes ||T UJiU (s )|| 00 over all stabilizing controllers. However, one may find that different controllers are reported in [40] [12] [36] and here. These different controllers are calculated with the same algorithm and similar programs, but with different design resolutions that specified the degree of closeness to the minimal value. In our simulation, we developed two kinds of programs and showed that the results are consistent. The first program uses commercial Matlab software hinfopt.m with a bisection algorithm called 7 -iteration, and finds the optimal controller by minimizing ||7^^(s) Hoc over all stabilizing controllers. Similar to the solving procedure given in [40], the second one finds the optimal controller by minimizing ||T zu; (s )|| 00 over all stabilizing controllers by increasing the weighting scalar r step by step, called an additive searching procedure. 5.5.1 Solution with a Bisection Searching Procedure To work with ||T uw (s)||oo, the weighting function r is fixed as unity so that the augmented plant G is fixed. The controller design resolution r is the only factor that affects the designed controller when using a bisection algorithm to minimize ||TÂ’ uu ,(s)|| 0O over all stabilizing controllers. For illustration purpose, we list four controllers computed by a same program based on Matlab software hinfopt.m but with different resolutions r. For simplicity, the controllers are expressed in matrix form:

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75 ^num[ S l • • • s if c(s) (5.23) Cden [ • • • 5 If where the nominator vector c num equals [ ni • • • Tli Tl 0 ] , and the denominator vector c den equals [ d t d x d 0 ] Controller 1 with r = 10~ 6 Cnum = 10 8 * [ 0 0.37748739677966 1.88743698863214 ] Cden = 10 7 * [ 0.00000010000000 1.49422174558615 -2.43793793780942 ]; (5.24) Controller 2 with r = 10~ 2 Cnum = 10 5 * [ 0 1.47459789570386 7.37298947851135 ] Cden = 10 4 * [ 0.00010000000000 5.83775000382779 -9.52194474311628 ]; (5.25) Controller 3 with r = 10~ 2 Cnum = 10 4 * [ 0 0.92197910324932 4.6098955162450 ] c d en = 10 3 * [ 0.00100000000000 3.65750061702856 -5.93944837515235 ]; (5.26) Controller 4 with r = 3.1 x 10 -3 Cnum = 10 3 * [ o 0.57981456953645 2.89907284768224 ] Cden = 10 2 *[ 0.01000000000000 2.37509933774846 -3.59463576158953 ]; (5.27) As shown in Figures 5.4 and 5.5, all these four controllers actually have similar values of phase and magnitude, and as shown in Figure 5.6, they produce the similar closed-loop gain \T uw (ju)\ as expected by the design procedure. Nevertheless, the small difference between the controllers reflect the difference of pre-specified design

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phase of controllers (degree) 76 Figure 5.4: Plots of the phases of the four controllers respectively. Note that the plots of the case of t = 10 -6 is almost overlapped with the case of r = 10 -4 . resolution r. One can check the Table 6.1 and concludes that the values of ||T ulu (s)|| 00 are made smaller and smaller as the resolution goes finer and finer. Table 5.1: Simulation with a bisection searching procedure controller 1 controller 2 controller 3 controller 4 ||T UU ;(JW) 00 2.5263163 2.5264456 2.5283951 2.5600000 resolution r IQ -6 1(T 4 10“ 2 3.1 x 10~ 3 5.5.2 Solution with an Additive Searching Procedure Different from the first algorithm, now we work with a transfer function T zw (s) that has a plant G with a varying element D \2 = r. The optimal controller is found by additively increasing r till no stabilizing controller exists. Then the maximum static

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77 3 r 2.5 =»1.5 0.5 \ \ \ controller at x=0.000001 controller at x=0.0001 controller at x=0.01 controller at x=0.031 0 100 200 300 400 500 600 700 800 900 1000 frequency m Figure 5.5: Plots of the magnitudes of the four controllers respectively. Note that the plots of the case of r = 10 -6 is overlapped with the case of r = 10~ 4 .

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78 2.5 "si .5 0.5 \ \ \ controller at x=0. 000001 controller at x=0.0001 controller at x=0.01 controller at x=0.031 100 200 300 400 500 600 700 frequency to 800 900 1000 Figure 5.6: Plots of the \T uw (ju)\ with the four controllers respectively. Note that the plots of the case of r = 10 -6 is overlapped with the case of r = 10~ 4 .

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79 weight is found together with a stabilizing controller that is taken as the optimal controller. In this case the increasing step is the design resolution. In our simulation, we used the commercial Matlab function hinf.m and have found the corresponding maximum weights in designing the controllers reported in [40] [12]. Consider the Controller in [40] With r = 0.3911755, we found the following controller: Cnum = 10 3 * [ o 0.64788818856446 3.23944094282226 ] Cden = 10 2 *[ 0.01000000000000 2.64455741306764 -4.03427788447897 ]; (5.28) which is close to that reported in [40]: 647. 9(s + 5) _ s 2 + 264.456s 403.411 ' Consider the Controller in [12] With r = 0.3957261755, we found the following controller: (5.29) Cnum = 10 5 * [ 0 0.27999876755765 1.39999383778789] Cden = 10 4 *[ 0.00010000000000 1.10912845491568 -1.80682537381042 ]; (5.30) which is close to that reported in [12]: 28000(s + 5) < S i “ s 2 + 11090s 18070' (5.31) In Matlab function hinf.m , the 7 is set as unity [72] [62], Therefore, the closer the gain Hr^slloo approaches to unity, the more powerful the corresponding controller is in terms of closed-loop stability. One can concludes from Table 6.2 that the values of HT^sHoo more closely approach unity as the weighting function r approaches the maximum weight whose value is affected by the step size in additive searching algorithm.

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80 Table 5.2: Simulation with an additive searching procedure controller (5.28) controller (5.30) \\T zw (j“) 00 0.99992994752899 0.99999996334698 static weight r 0.3911755 0.3957261755 5.6 All-Solution Controllers As well-known in standard robust control [72] [64], given A Bi b 2 G = Ci 0 D\ 2 c 2 D 2 \ 0 which satisfies some proper assumptions, H ^ design has all-solution controllers K{8) = F L (F(8),Q{*)) (5.33) where Aoo __ 7 T ^00-^00 Z oc B 2 F{s) = Foe 0 I -c 2 / 0 Aqo A + 7 2 B\B\X ) (5.37) Q(s) e RHoo, ||£J(s)||oo < 7 . (5.38) Now for Q(s) = a q Bq Cq Dq one has the following state space representations:

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81 and, Therefore, Xp u «2 AqqXf F x F + —C2XF + Zoo Too y + Z 00 B 2 y 2 O + 2/2 2 / + 0 ^ i<5 = 21 qXq + BqU2 ^ V 2/2 CqXq + DqU 2 J \ / (5.39) (5.40) / , xf — (+00 ~ Z 00 B 2 DqC2)x f + -Zoo-E^Cq^q — (ZooZ/qo — Z oa B2Do)y \ Xq = BqC^Xf + AqXq + ^ U 2 = (Zq — DqC2)xf + CqXq + The all-solution controller K(s) is: Bqy Dqy (5.41) K(s) = Ak B k Ck D k Aqo Z 00 B 2 -BqQ Aq Fq — DqC 2 Cq If Q(s) = 0, one gets a central controller K central {$) Z oqLqq + ZooB2Dq Bq Dr Aqo 1 8 N 8 1 Foo 0 1 We point out two observations from expressions (5.42-5.43): (5.42) (5.43)

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82 • All solution controller K(s) is strictly proper if and only if Q(s) is strictly proper; • The central controller K centra i(s) is always strictly proper. Regarding the controller design, given different complementary controllers Q(s), one will get different optimal controllers c(s) that minimize the transfer gain ||T UU) (s)|| 00 even though a same high resolution value r is used. However, as shown in the following example, all those controllers have values close to each other for any frequency since they all should minimize (s) ||oo. Example 5.3 We continue the discussion of Example 5.1 but with a focus on allsolution controllers. We select the resolution r = 10 -6 , and compute three controllers with different complementary controllers Q(s). Using the matrix expression in Equation (5.23), the three controllers are listed below and compared in Figure 5.7. Controller 1: Qi(s) = 0 ( 5 . 44 ) Cnum = 10 8 * [ o 0.37748739677966 1.88743698863214 ] Cden = 10 7 *[ 0.00000010000000 1.49422174558615 -2.43793793780942]; (5.45) This is exactly the same controller of Equation (5.2f). Controller 2: Qiis) s + 0.7 7+53 (5.46)

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83 Cnum Cden 10 10 * [ 0.00000000252632 0.00000001440000 1.97425909399972 9.87129542578808 ] 10 9 * [ 0.00000000100000 0.00000005136842 0.78147797179945 -1.27504154133096 ]; (5.47) Controller 3: , s 2 + 0.3s + 0.01 ys(S ' s 2 + 41s + 100 (5.48) Oitim C-den 10 3 * 10 2 * [ 0.00000000025263 0.00000000133895 1.14563650049922 1.88724824019991 [ 0.00000000100000 0.00000003936842 0.50183158281841 -2.43769414374154 0.15363737087079 ] 0.60814834996409 ]; (5.49) As shown in Figure 5.7, all three controllers have almost the same values of controller phases and magnitudes, and hence produce almost the same transfer gains ||7uiu(s)||ooActually, there is no difference among the values of \T uw (jui)\ at any frequency up to the eighth digit after the decimal point. 5.7 Conclusion This chapter focuses on the H <*> methods using constant weighting strategies for systems with parametric uncertainties. The conservatism of earlier work in the literature is exhibited and a much larger parametric stability margin is obtained. Even though this chapter shows that the ECPR weighting strategy leads to a much larger parametric stability margin than the MPR weighting method does, only constant weighting methods are considered in this chapter. Referring to (4.13) on

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84 0 0 D) 0 *o _0 O c o o 0 0 0 -C Q. 0 0 "5 C O O 'o 0 D Z3 C O) 0 E 200 100 0 -100 2.5263 _ 2.5263 S 2.5263 5 t 2.5263 2.5263 2.5263 controller with Q^s) controller with Q 2 (s) controller with Q 3 (s) 100 200 300 400 500 frequency ro 600 700 800 900 1000 100 200 300 400 500 frequency co 600 700 800 900 1000 — controller with Q (s) controller with Q 2 (s) _ controller with Q 3 (s) 100 200 300 400 500 frequency to 600 700 800 900 1000 Figure 5.7: Plots of the controller phases, controller magnitudes, and transfer gain \T uw (ju>)\ with the three controllers respectively. Note that all the three plots at each sub-figure are overlapped. Actually, there is no difference among the values at any frequency up to the eighth digit after the decimal point.

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85 page 58, the exact weighting function should be p*(s) —po(s), which means a dynamic weighting function should be used if one wants to further reduce conservatism in calculating the parametric stability margin. This will be considered in next chapter.

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CHAPTER 6 ROBUST CONTROL FOR PARAMETRIC PLANTS USING DYNAMIC WEIGHTING APPROACHES 6.1 Introduction This chapter continues the discussion of the last two chapters and focuses on using dynamic weighting functions. We are working on one SISO plant p(s, e) including a parametric uncertain part 5(s,e), together with the same assumptions as listed in Chapter 4. We restated our mission here. Two issues need to be addressed: the first is to find the maximum stabilizable margin e max ; the second is to design a controller that can stabilize the closed-loop system of Figure 4.1 (on page 50) with uncertainties up to the maximum level e mai . As discussed in the last chapter, a much larger parametric stability margin is obtained using a static weight based on the critical direction theory rather than a worst case overbounding approach. As pointed out in the end of the last chapter, a dynamic weighting method needs to be developed in order to further reduce conservatism in determining the maximum stabilizable uncertainties and in designing a robust controller. The rest of the chapter is organized as follows. Section 6.2 exploits a dynamic maximum perturbation radius (MPR) weighting approach and constructs an e-iteration procedure to synthesize a sub-robust controller in terms of stability capacity. Then, a tuning procedure is established in Section 6.3 to apply the dynamic effective critical perturbation radius (ECPR) weighting function. As usual as with other chapters, an comprehensive example, and simulation reliability are presented in Sections 6.4 and 6.5. Conclusion is given in Section 6.6. 86

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87 6.2 Sub-Robust Controller Synthesis Using a Dynamic MPR Weighting Approach Given the MPR p m (w,e), one needs a further transformation to use the design tools that requires that weighting functions take the forms of stable and real rational functions, because p m {u, e) is just a set of non-negative values over a specified frequency range. This situation is reminiscent of the well-known D — K iteration in p synthesis in which the frequency-dependent scalings D(u) are approximated by a transfer function D(s) [6]. Similarly, using a numerical inversion technique, a stable, minimum-phase, and real rational transfer-function approximation p m (s) to p m (u;,e) can be found in the sense that |Pm(jw,e)| = pm{w,e), Vw. (6.1) Now consider the problem posed in Chapter 4 to use H 0 0 methods to design a robust controller that can stabilize as large an uncertainty as possible. As described there, the controller synthesis problem for parametric uncertain systems can be recast into the mixed-sensitivity framework, and the maximum perturbation radius (MPR) may be used as a weighting function in the synthesis tools. Due to the overbounding effect of the MPR around the value set for a certain size of uncertainty, it is arguably true that the controller synthesized by using the MPR weighting function is one of the robust controllers. However, this is not equal to saying that the larger the MPR is, the more robust the synthesized controller is. The hope of utilizing larger weighting functions to get a more robust controller may fail by merely blowing up the p m (s,e). The reason is similar to the derivation in Equation (5.3). Referring to Figure 4.6 in page 56, and let W^s) = a p m (s,e), a G R + . Under the TCo synthesis criterion that minimizes ||T’y 12 r( s )||oo an d stabilizes the closed-loop system, the optimal controller is fixed due to the equivalency shown in the following:

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88 Cop{s) arg min ||T yiar (s)||oo stabilizing c(s) arg min ||T ur (s) • W 2 (s)||oo stabilizing c(s ) ( 6 . 2 ) = arg min Q.||T ur (s) stabilizing c(s) = arg min ||T ur (s) • p m (s, e)||oostabilizing c(s ) Nevertheless, it makes sense to increase the size of uncertainty e and utilize the correspondingly expanded p m (s,e) as a more conservative weighting function based on the following lemma. Lemma 6.1 Consider the interval plant p(s,e) with uncertainty e. Then the value set is monotonically nondecreasing when the uncertainty e increases. Proof: The proof of this lemma is sketched as follows. Denote by A(e) the coefficient space of the interval system. Consider uncertainty levels ei,e 2 , such that D < e 2 . Then it follows that Ai C A 2 where Ai = A(ei) and A 2 = A(e 2 ). Since the value sets p(ju, Ai) and p(ju, A 2 ) are mapped from the their coefficient spaces Ai and A 2 respectively, they certainly satisfy p(ju, Ai) C p(juj, A 2 ) (6.3) This concludes the proof. According to the lemma, the corresponding MPR is also monotonically nondecreasing when the uncertainty level e increases. We define e sub max as the largest value of e such that the system p(s, e) in Figure 4.1 is stabilizable using only the MPR weighting approach. The e sub max can be found by increasing e and testing the stability of controllers synthesized using the corresponding dynamic MPR weighting functions. Note that we call this margin a sub-maximum stability margin because of MPR’s conservatism nature that will

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89 be clear in next section. A computationally efficient algorithm for e sub ma x can be designed using a bisection algorithm. Define the true-or-false function /(e) as: /(e) 1 , + 1 ) ^ If the interval system is stabilizable by the MPR weighting approach otherwise / (6.4) Figure 6.1: Plot of the true-or-false function via the uncertainty level e. Following Lemma 6.1, it is straightforward to check that function /(e) is monotonically non-decreasing while e increases from zero (see Figure 6.1). Hence, the true-or-false function satisfies the monotonicity requirement by any bisection method. This renders the following algorithm possible. Algorithm for Sub-robust Controller Synthesis Step 1 Set the initial bounds of e as ei b — 0, e ub = e u , where e u is obtained based on the discussion in Section 4.2 (on page 48); and, assign a typical value between 10 -2 and 10 -4 to r as the tolerance for the stopping criterion. Step 2 Let e = ( t ib + e ub )/2. Check whether a robust controller can be designed merely using the MPR weighting approach.

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90 • If the interval system is stabilizable, set lower bound = e, and iterate Step 2. • Else, set upper bound e up = e. Now if (e u b — eib)/tib < t, set e S ub-max — (f/6 + e U (,)/ 2 and end the iteration. Else, iterate Step 2. End of Algorithm. Having decided a stability margin, we can design a robust controller using the Hoo synthesis techniques in conjunction with the MPR weighting function at t S ub-maxIts advantages over static weighting approaches is shown in an example of Section 6.4. However, we call this controller a sub-robust controller because there may exist a robust controller that can stabilize the system for e = e max > e S ub-max even though the MPR weighting function fails in designing an admissible controller for e max because of the conservatism introduced by the overbounding effect. The next section extends the approach to the use of a dynamic weighting strategy based on the critical direction theory. 6.3 Robust Controller Synthesis: a Dynamic ECPR Weighting Approach with Controller Tuning Algorithm Recall the derivation in Chapter 4, the CPR definition (2.4) on page 8 implies that p c {uj,e) = \g*{ju,e) g 0 (ju)\ = \c{ju)p*{ju,e) c(juj)p 0 (ju)\ (6-5) = \c{ju)\\p*(ju,e) -p 0 (ju)\ It is clear that conservatism may be introduced using the over-bounding weight p m (s) instead of p e (s ) = | p*(jco) p 0 (ju>)\. Therefore, even though the interval system is unstabilizable by the MPR weighting approach at e su bmax , there may still exist other weighting functions that will result in an admissible controller. A non-conservative robust controller should be designed using the necessary and sufficient weighting

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91 function p e (s). Unfortunately, the ECPR is defined after the robust controller has been specified, instead of before the design of the controller. To work out this seemingly contradictory situation, we now propose a tuning method to find the maximum stability margin e max and an associated robust controller c rc (s ). Given a controller c(s), the following notation is used: Pe( u ) = Pc{w) I c(juj)\' ( 6 . 6 ) Note that p e [u) is equivalent to the ECPR defined in (4.15) on page 60, but here c(s) may contains some tuning parameters. Hence, different ECPR weights exist for different controllers c(s). Now recall that from the sub-robust controller synthesis algorithm given in the last section, the sub-maximal stabilizable uncertainty c su bma x was found. Choose a certain uncertainty level e f > e su b-max , a non-admissible controller c su ft(s) can be obtained using the dynamic MPR approach. Our design takes c su( ,(s) as a starting point. A simple but effective tuning approach is implemented by cascading c su b{s) with the following tuning controller c t (s ) that has two tuning parameters k and z c t (s) = k s + z s + p' (6.7) To reduce the computation complexity, the parameter p is taken as a constant chosen arbitrarily in the range: 1 < p < 20. In the sequel we use p = 10. Inserting Cj(s) into the loop, a new ECPR called the tuned ECPR is calculated as n (u) PcO _ Pc(u) ' Ct(ju)c sub (jw) ct{ju)c su b(ju) (0 g) = \Pt(jw) Po{jw)\ where Pt(ju) is a certain boundary point in the neighborhood of p*(s) (see Figure 4.8 (on page 59) where p*(s) is plotted). Note that now g(s) = c t (s)c sub (s)p(s).

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92 Figure 6.2: The tuning process using a controller c t (s). Hence, a new weighting function is obtained by tuning c t (s). For each p e ,t{s) resulting from varying the tuning parameters k and z, a new controller c(s) can be designed using the synthesis tool. For each c(s), the Nyquist robust stability margin k N of the closed-loop with p(s) and c(s) is calculated and checked. Whenever /cjv < 1, it means that the uncertainty level e t is stabilizable using the ECPR weighting. Finally, the desired parameters k and z are those for which the corresponding synthesized controller c(s) results in the minimum k^ of the closed-loop system. The following lists the steps of the tuning process in searching for a robust controller. Algorithm for Robust Controller Synthesis Step 1 Following the e-iteration algorithm, let c su b(s ) be the controller synthesized using the MPR approach at a certain value of e t > e s „h_ mai . Define c t (s) = Step 2 Fix 2 at 10, and vary k within a suitable range around unity, then calculate p e ,t{s) of the system g(s) — c t (s)c SU b(s)p(s) . Synthesize a controller to replace Csub(s) using the H 0 0 synthesis with p e ,t( s )Assign c su6 (s) to c(s) in Figure 4.1, and calculate k^. Let k rc equal the value of k that corresponds to the minimal value of k^ and go to Step 3. Step 3 Fix k at k rc , and vary factor z within the vicinity of 10, then calculate p e ,t ( s ) of the system g(s) — c t (s)c su b(s)p(s). Synthesize a controller to replace c su b(s) using the H ^ synthesis with p e ,t{ s )Assign c su b(s) to c(s) in Figure 4.1, and

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93 calculate k^. Let z rc equal the value of z that corresponds to the minimal value of kiy. If the minimal value of k n is less than unity, a robust controller c rc (s ) is found with the tuning parameters k rc and z rc , or the algorithm should be restarted with a proper value e, satisfying e su bmax < e < e t . End of algorithm. The above tuning procedure involves a step-by-step process, searching alternatively over parameter k and then z. A global search over the parameters k and z is certainly also possible, but is much more computationally demanding. Our preliminary experience suggests that the simple heuristic algorithm proposed above works quite well, as illustrated by the following example. 6.4 Example Consider the interval plant given in [17] [28] [40]. with intervals p(s,e) 5s + q i s 2 + q 2 s + q 3 (6,9) qi £ [4 — e, 4 + e], q 2 £ [2 — e, 2 + e], q 3 £ [— 15 — e, — 15 + e]. (6.10) The parameter space is characterized by a hyperbox A(e) whose edges vary independently with a length 2e each. Note that to satisfy the assumption of the family of systems p(s,e) in Section 4.2 (on page 48), e u = 7.5 (refer to [17] [40]). Result 1: Design of a sub-robust controller The simulation result of the true-orfalse function /(e) in the sub-robust controller synthesis process is shown in Figure 6.3. Similarly, The sub-maximum stability margin can also be found by checking whether the Nyquist robust stability margin

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94 Figure 6.3: Sub-robust controller design example using a dynamic MPR weighting approach. The dashed-line represents the true-or-false function /(e) vs. e; the solid line represents the change of k n vs. e. is less than unity or not, instead of checking the true-or-false function. Both approaches show that: Csub-max = 6.2499. (6.11) Result 2: Design of a robust controller At e > 6.2499, the interval system becomes unstabilizable by the MPR weighting. However, we cannot assert that e su bmax is the upper bound of the robust stability margin because the MPR weighting may incur conservatism. The controller synthesized by the MPR weighting at e = 7 is

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95 ( 6 . 12 ) where n sub {s ) = 16981.5925s 4 + 229086.4782s 3 + 1106292.6284s 3 +2118935. 9323s + 959678.4155 (6.13) d sub (s ) = s 5 + 3377.3566s 4 732.4843s 3 162599.2866s 3 -565429.0321s 296531.9683. After the tuning process, k = 0.4, and z — 11.6 are found to result in a robust controller called c rc (s), n rc (s) = 13487.7972s 4 + 127457.5715s 3 + 372751.7843s 2 +374990. 3134s + 58480.1182 (6.15) d rc {s) = s 5 + 581.2072s 4 29230.1744s 3 131929.4094s 2 -143703.8491s 24276.0372 Note that it is straightforward to check that there is no unstable pole-zero cancellation between the unstable pole of c rc (s) in (6.14) and the zeros of p(s) with uncertainty level e up to e u . With the controller c rc (s) the system in Figure 4.1 with e = 7 has k n = 0.9593. As shown in Figure 6.4, the stability margin of the system in Figure 4.1 with controller c rc (s) is found to be e max = 7.1796, which is very close to the upper bound of the uncertainty level 7.5. Compared with the stability margin value of 4.0 obtained in (6.14) where

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Stability Margin at Each Frequency 45 96 Stability Margin of the System Cascaded with the Robust Controller Figure 6.4: Stability analysis for the parametric uncertain system (6.9) cascaded with the designed robust controller(6.14). the last chapter, the robust controller synthesis strategy proposed here is much more 6.5 Simulation Reliability The simulation reliability discussed in Section 5.5 in the last chapter is also applicable here because the same programs are used here except that dynamic weighting functions instead of scalar values are used for this chapter.

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97 6.6 Conclusion The main contribution of this chapter is to introduce necessary and sufficient weighting functions and to use the H ^ method in designing a non-conservative robust controller for systems with parametric uncertainties. A sub-robust controller synthesis algorithm enhanced by a bisection algorithm is constructed. Then a tuning procedure is established to find the maximum stabilizable uncertainty and to design a robust controller to stabilize that level of uncertainties. The simulation results, using a widely referred example in the literature, demonstrate that this chapter proposes a very promising strategy in synthesizing robust controllers for systems with parametric uncertainties.

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CHAPTER 7 CONCLUSIONS AND FUTURE WORK Chapters 4-6 have discussed robust controller design using the critical direction theory and the H 0 0 synthesis knowledge. We are given a nominal unstable plant p 0 (s) and are asked to find a controller that can robustly stabilize the family of uncertain plants p(s,e). We focus on internal stability of the closed-loop system without discussing system performance. We try to point out some research directions for future work in this promising area. Before the discussion, we firstly point out that the parametric robust control methodology needs some adjustment when dealing with a nominally closed-loop stable interval plant. Otherwise the optimal controller is always zero according to the synthesis criterion, namely, c op (s) := arg min ||T 2/12r (s)|| 0O = 0. (7.1) stabilizing c(s) To make sense in controller synthesis for a nominally closed-loop stable system, one solution is to assign a certain function to fUi(s), i.e. , including the tradeoff between the weighted sensitivity function and the weighted control sensitivity function while synthesizing a non-conservative controller for the whole plant. Recall that the new definition of perturbation radius given in Chapter 2 involves robustness sensitivity. Therefore, one might try to bridge the sensitivity weighting function Wi(s) with the perturbation radius, and then utilize both weighting functions for parametric robust control. More formally, the sensitivity function W\ ( s ) may be introduced by referring to the performance circles as defined in [64]. Performance circles are defined around 98

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99 the critical point -1 + j 0, and are utilized to analyze the nominal performance and robustness performance. For example, the nominal performance is analyzed as follows for the input-disturbance rejection problem shown in Figure 7.1. The nominal performance is defined as follows. Figure 7.1: Augmented feedback loop with performance weights. Definition 7.1 (page 45, [64]) Nominal performance of the feedback loop of Figure 7 . 1 is achieved if and only if the weighted output remains bounded by unity, that is, ||VF !/ (s)?/(s)||2 < 1 , for all disturbances in the set {d 6 £ 2 , IMII 2 < 1}> and for all other external inputs to the system equal to zero. Then, the following theorem is given on page 46 of [64]. Theorem 7.1 (page 46, [64]) The feedback system of Figure 7.1 achieves nominal performance, as defined in Definition 7.1, if and only if ||W tf (a)5(s)W' d (s)|| 00 < 1, (7-2) where S(s) is the input sensitivity matrix. Proof: Refer to [64] for the proof. Accordingly, the optimal controller, in the sense of providing optimal disturbance rejection, can be found by solving an optimal control problem, in this case of the form

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100 where min ||VF(s)S(s)||oo, stabilizing c(s) (7.3) W(s) := W,(s)W d (s). Figure 7.2 interprets graphically the nominal performance condition by means of a Nyquist plot. Recall that for a SISO system, S(s) = l/(l+Po{jw)c(ju))). Define g 0 (s) = p 0 (s)c(s), then (7.2) is equivalent to \W{jw)\<\l + g 0 {jw)\, Vcu. (7.4) Nominal performance is achieved if and only if the Nyquist plot of the nominal loop g 0 (ju) is outside of the disk \W d (juj)\. More results on nominal performance for the racking problem, nominal performance for the measurement noise attenuation, and robustness performance are addressed in [64]. Figure 7.2: Nyquist plot for disturbance rejection.

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101 Notice that the perturbation weighting functions Wd(s) and W y (s) are primarily defined for unstructured cases in [64]. One can apply the critical direction theory and measure the nominal performance only along the critical direction for structured cases. In terms of robust stability, we have obtained an exact criterion in Chapter 4 (see page 61). Theorem 7.2 Assume that the closed-loop system of Figures f.l is nominally stable; systems p(s, e) and po(s) have the same number of open-loop unstable poles; and there is no unstable pole zero cancellation between a controller c(s) and the family of systems p(s,e). Then, the closed-loop system of Figures f.l is robustly stable if and only if ||p e (a;,e)i?(s)|| 00 < 1, (7.5) where R(s) = 1+c ^ oW y Finally robust performance is defined as: Definition 7.2 (page 45, [64]) Robust performance of the feedback loop of Figure 7.3 is achieved if and only if the weighted output remains bounded by unity, that is, ||W y (s)y(s )||2 < 1 , for all disturbances in the set {d € l 2 , MU < 1 }, for all other external inputs to the system equal to zero, and for all uncertain plants of the family P(s,e). Accordingly, we have the following robust performance condition from Equations (7.2) and (7.5): |||W,(»)S(»)Wi(»)| + Mw,e)H(»)HL < 1. (7.6) We obtain two weighting functions and an augmented design criterion. In order to use the Hoo design knowledge, some future work is needed in this direction.

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102 Figure 7.3: Augmented feedback loop with performance weights.

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REFERENCES [1] S. A. Al-Shamali, H. A. Latchman, 0. D. Crisalle, B. Ji, The Nyquist Robust Sensitivity Margin of Uncertain Closed-Loop Systems, Accepted for publication in International Journal of Robust and Nonlinear Control , December 2003. [2] S. A. Al-Shamali, H. A. Latchman, O. D. Crisalle, B. Ji, A Margin for Robust Stability and Robust Performance, Proceedings of American Control Conference, Denver, Colorado, USA, June 2003. [3] C. T. Baab, Robust Stability Analysis Methods for Systems with Structured and Parametric Uncertainties , Ph.D. Dissertation, University of Florida, 2002. [4] C. T. Baab, H. A. Latchman, J. C. Cockburn, and O. D. Crisalle, “Generalization of the Nyquist Robust Stability Margin and its Application to Systems with Real Affine Parametric Uncertainties”, International Journal of Robust and Nonlinear Control , December 2001. [5] C. T. Baab, H. A. Latchman, J. C. Cockburn, and O. D. Crisalle, “Robust Controller Synthesis for Systems with Nonconvex Value Sets using an Extension of the Nyquist Robust Stability Margin”, Conference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, July 2001. [6] G. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, p-Analysis and Synthesis Toolbox, The MathWorks Inc., 2002. [7] B. R. Barmish, New Tools for Robustness of Linear Systems, Macmillian, New York, NY, 1993. [8] B. R. Barmish, and P. P. Khargonekar, “Robust Stability of Feedback Control Systems with Uncertain Parameters and Unmodelled Dynamics,” Proceedings of American Control Conference, Georgia, USA, 1988. [9] B. R. Barmish, and Z. Shi, “Robust Stability of a Class of Polynomials with Coefficient Depending Multilinearly on Perturbations,” IEEE Transactions on Automatic Control, vol. 35, 1990. [10] A. C. Bartlett, C. V. Hollot, and H. Lin, “Root Location for a Polytope of Polynomials: it Suffices to Check the Edges” , Mathematics of Control, Signals and Systems, vol 1, pp. 61-71, 1988. 103

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BIOGRAPHICAL SKETCH Baowei Ji was born in China in 1971. He obtained his BachelorÂ’s degree at Henan Normal University in 1993, and his MasterÂ’s degree at Beijing University of Aeronautics and Astronautics in 1996, both from departments of electrical engineering. From 1996 to 1999, he worked as a system engineer at the China Unmanned Aircraft Vehicle (UAV) Research and Development Center. He enrolled in the PhD program at the University of Florida in 1999, and has been doing research on robust control under the direction of Dr. Haniph A. Latchman and Dr. Oscar D. Crisalle. Meanwhile, he also participated in many other advanced projects conducted in the Laboratory for Information Systems and Telecommunications (LIST: www.list.ufl.edu ). He graduates in May 2004. 109

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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 Haniph A. Latchman, Chair 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. Oscar D. Crisalle, Cochair 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. Tan FrAVong Assistant Professor of Electrical and Computer Engineering dissert ation for the degre eof Doctor of Philosophy. 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^egre©-o| Doctor of Philosophy. Norman GTFitz-Coy Associate Professor of Mechanical and Aerospace Engineering 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 2004 1 /NuCWwt? Pramod P. Khargonekar Dean, College of Engineering Kenneth J. Gerhardt Interim Dean, Graduate School