ROBUSTNESS ISSUES IN PREDICTIVE CONTROL
AND SYSTEMS INCLUDING TIME DELAYS
By
KOSTAS HRISSAGIS
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
1996
ZT11 vljp TounTC(TFpao 11
ACKNOWLEDGMENTS
I express my gratitude to my advisor Professor Oscar Crisalle for his continuing encouragement, guidance, and support throughout the course of my Ph.D. program. I am grateful for the freedom he allowed me and would like to congratulate him for attracting talented graduate students to his group.
I wish to thank Professors Ioannis Bitsanis, Haniph Latchman, Spyros Svoronos, Mario Sznaier, and Rich Dickinson for serving in the supervisory committee, and for providing me with helpful suggestions, criticism, and knowledge all these years.
The friendship and support of my colleagues in the lab, Mike, Basker, Jon, and Tony, will always be appreciated. The company of my good friends and neighbors RaviKiran and Yanna has been crucial during this period. Many have contributed to my personal growth and enjoyment over this period of my life; it is unfortunate I cannot mention everyone by name. My family and friends in Athens have always been in my thoughts.
I would like to thank my teachers for imparting in me an intellectual curiosity and reinforcing my quest for knowledge. Particularly, I am grateful to Kyriacos Masavetas at NTUA and Lewis Johns at the University of Florida for their insight, challenge and freedom of thought, a source of inspiration during my academic years.
Financial support from the National Science Foundation through grant number CTS9309659 is gratefully acknowledged.
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TABLE OF CONTENTS
ACKN OW LEDGM ENTS .............................................................................................. iii
ABSTRACT .................................................................................................................... vii
I ROBUST CONTROLTHE COMMON THEME OF THIS
DISSERTATION .................................................................................................. I
1.1 M otivation......................................................................................................1
1.1.1 Plant M odels ................................................................................... 2
1.1.2 Plant Uncertainty M odels ............................................................... 2
1.1.3 Perform ance M easures .................................................................... 4
1.2 Objectives ................................................................................................. 6
1.2.1 Predictive Control .......................................................................... 6
1.2.2 Bilinear System s ............................................................................ 7
1.2.3 TimeDelay System s ...................................................................... 7
1.3 Organization of the Dissertation ............................................................... 8
2 UNCONSTRAINED ROBUST PREDICTIVE CONTROL DESIGN ........... 11
2.1 Introduction................................................................................................ 11
2.2 Nom inal Predictive Controller Design ...................................................... 14
2.3 Controller Param eterization ...................................................................... 18
2.4 Robust Predictive Design ........................................................................ 21
2.4.1 Robust Synthesis ........................................................................... 24
2.4.2 Robust Design with SteadyState Disturbance Rejection ..............28
2.5 Exam ple .................................................................................................... 30
2.6 ModelMatching Problem and Predictor Design Equations .................... 35
2.6.1 RotsteinSideris Solution to the ModelMatching Problem.......... 35
2.6.2 Design Equations for Nominal Predictive Control ....................... 38
2.7 Conclusions ............................................................................................... 39
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3 ROBUST PREDICTIVE CONTROL WITH CONSTRAINTS VIA lao/Ho.
DESIGN ........................................................................................................... 41
3.1 Introduction.............................................................................................. 41
3.2 Statem ent of the Problem ......................................................................... 43
3.2.1 Notation ......................................................................................... 43
3.2.2 Uncertainty Description and Process Constraints ......................... 44
3.2.3 Design Objectives and Proposed Approach .................................. 45
3.3 Form ulation of the Control Design Problem .......................................... 46
3.3.1 Controller Param etrization ............................................................. 46
3.3.2 Nominal Regulation and Robust Stability as an 11/H.o Problem...... 49
3.4 Synthesis Procedure .................................................................................. 51
3.5 Exam ple .................................................................................................... 54
3.5 Conclusions.............................................................................................. 60
4 ROBUST STABILIZATION OF BILINEAR SYSTEMS WITH INPUT
SATURATION ....................................................................................... .... 62
4.1 Introduction............................................................................ .... ........ 62
4.2 Preliminaries ............................................................................... ... ... 63
4.3 M ain Results ........................................................................... . .......... 64
4.4 Exam ple ...................................................................... ............. ..... 71
4.5 Conclusions....................................................................................... ... 72
5 ROBUST STABILIZATION OF DELAYED BILINEAR SYSTEMS ..........74
5.1 Introduction.................................................................................... ..........74
5.2 Prelim inaries .......................................................................................... 75
5.3 System and Assum ptions ............................................................................. 75
5.4 Constrained Control .................................................................................. 80
5.5 Exam ple .................................................................................................... 82
5.6 Conclusions............................................................................................. 83
6 DELAYDEPENDENT ROBUST STABILITY CONDITIONSCSTR
W ITH RECYCLE STREAM ........................................................................... 84
6.1 Introduction............................................................................................. 84
6.2 Problem Form ulation ............................................................................... 87
6.3 Theoretical Developments ........................................................................ 88
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6.4 Robust Stability in the Presence of Input Nonlinearities .......................... 92
6.5 E xam ple ................................................................................................. . . 96
6.6 Matrix Measure and Theorem Proof...........................................................100
7 FURTHER RESEARCHIDEAS AND FUTURE DIRECTIONS.................104
7.1 MixedOptimization Approaches to Predictive Control Design ................104
7.2 Robust Predictive Control Design for Multivariable Systems ...................106
7.3 DelayDependent Stability Conditions for Bilinear Systems .....................107
7.4 Robust Nonlinear Feedback Control of Bilinear Systems ..........................107
7.5 B ilinear Predictive C ontrol .........................................................................108
7.6 Variable Structure Control for Delayed Systems .......................................108
B IB L IO G R A P H Y ..........................................................................................................110
B IO G RA PH IC A L SK ETC H .........................................................................................117
vi
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
ROBUSTNESS ISSUES IN PREDICTIVE CONTROL AND SYSTEMS INCLUDING TIME DELAYS
By
KOSTAS HRISSAGIS
December 1996
Chairman: Dr. Oscar D. Crisalle
Major Department: Chemical Engineering
Methods for the design of predictive controllers for uncertain linear systems with guaranteed closedloop stability are developed as well as conditions for robust stability and stabilization of bilinear and timedelay systems are also derived. Specifically, a systematic technique to design robust predictive controllers for unconstrained linear systems is proposed first. The robustified controller retains the servo performance of a nominal predictive controller designed using conventional methods. In addition, a robust predictive regulator may be designed to guarantee perfect steadystate rejection of asymptotically constant disturbances. The robust predictive methodology is developed for systems affected by unstructured uncertainty, and is based on solving a discretetime modelmatching problem. It is shown that the robustified controller can be classified as a predictive controller because it minimizes the same performance functional as its nominal counterpart. The design of robust predictive controllers for a second order unstable plant illustrates the proposed method.
For systems that are subject to timedomain constraints, a new technique for the design of robust predictive regulators is proposed based on mixed l1IH. theory. The
vii
controller robustly stabilizes a system where the openloop plant model is corrupted by unstructured uncertainty, and the output is affected by bounded, but possibly persistent exogenous disturbances. Adequate nominal performance in the regulation mode is obtained by optimizing an 11 objective that minimizes the worstcase amplification of the disturbance signal. Robust stability is guaranteed by satisfying frequencydomain conditions through an H. problem. The resulting constrained predictive regulator design quantifies the achievable limit of performance with respect to the satisfaction of the timedomain constraints.
Finally, the asymptotic stability and stabilization in the face of uncertainty is examined for bilinear and timedelayed plant models. A variety of robust stability conditions is derived for those plants in the last several chapters. The obtained sufficient conditions guarantee the stability of the systems under statefeedback control in the presence of a normbounded, nonlinear, and possibly timevarying uncertainty. The effects of saturation and other input nonlinearities are examined, and finally incorporated in the continuoustime robust stability conditions. The method makes use of the matrix measure and integrodifferential inequalities to derive the robustness conditions as a function of the known uncertainty bounds. A characterization of the domain of attraction is given in the case of uncertain bilinear systems.
Suggestions for further work include the following: investigate other recently proposed mixed optimization approaches to predictive regulator design, explore further the robust stabilization of uncertain bilinear systems with time delays using linear or nonlinear feedback and slidingmode control, and finally, formulate and solve the problem of designing stabilizing predictive controllers for bilinear constrained process models.
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CHAPTER 1
ROBUST CONTROLTHE COMMON THEME OF THIS DISSERTATION
1.1 Motivation
The ultimate goal in control system design is to build a controller that will satisfactorily perform in the nonideal environment. Because the real environment may change with time or operating conditions may vary (disturbances), the control system should be able to withstand such variations. Even when the environment does not change, the quality of the model on which the design is based is of concern. The mathematical representation of physical systems often involves simplifications, if not wishful assumptions. Nonlinearities may be unmodeled or modeled but ignored later to simplify the analysis. Also, different components of the system such as actuators, sensors, etc. have dynamics, but frequently they are modeled as constant gain systems. The issue of uncertainty in control system design is therefore, of fundamental importance, as is attested by numerous reports on the modeling and control of uncertain systems.
The property that a control system must possess in order to operate properly in realistic situations (where uncertainty is prevalent) is called "robustness". This means that the compansator must perform satisfactorily not just for one "nominal" plant but for a family of plants described in the time or the frequency domain. Thus, the Robust Control problem is roughly that of analyzing and designing accurate control systems given plants which contain significant uncertainty. To define the problem more precisely, a number of problem elements must be carefully developed. In particular one should delineate the class of plant models, uncertainty models, and performance measures that are to be considered.
I
1.1.1 Plant Models
There are three common linear multivariable plant models used in the literature; they are statespace models, transferfunction matrix models, and matrixfraction models. The statespace or statevariable models are generally written in the form
x =Ax+ Bu, y = Cx+ Du (1. 1)
where x, y, and u are n, p and q dimensional state, output, and input vectors, respectively. The transferfunction matrix models are denoted G(s), where G(s) is a p xq matrix, with entries made up of rational functions in the complex Laplace transform variable s, and G(s) relates the transform of the input vector u to the transform of the output vector y. Finally, matrixfraction models are of the form, for example for a rightcoprime fraction, G(s) = N(s)D(s), where N(s) and D(s) are matrices with either polynomial entries or stable rational function entries.
1. 1.2 PlantUncertainty Models
Each plant model requires its own type of uncertainty model. In addition, uncertainty is characterized in a number of different ways. For example, uncertainty is characterized as parametric versus nonparametric, structured versus unstructured, and stochastic versus deterministic (set uncertainty). To illustrate the different types of uncertainty, consider the statespace model (1.1). The uncertainty in the term such as Ax can be modeled as follows:
(i) Ax +(A + A)x, where 3A is constrained only in norm, i.e. 1M A II a.
(ii) Ax * (A + rA )x, where l I ri 1.
(iii) Ax  Ax + Xx,Gv where vi is a white noise random process.
3
In this case, (i) and (ii) represent parametric setuncertainty models, whereas in (i), the uncertainty, represented by SA, is unstructured, and where in (ii), the uncertainty, represented by r, is structured. Essentially, structured uncertainty limits how the uncertainty enters into the matrix A. In (iii), the uncertainty is modeled stochastically, as a multiplicative white noise process vi. Examples of the use of the models in (i) and (ii) may be found in Doyle, et al. (1992), and Weinmann (1991), and the model in (iii) in Wonham (1967).
For the transferfunction matrix model, uncertainty in the transfer matrixG may be modeled as follows
(iv) G * (I + L)G, where I L(jo) I < lm(a)
(v) G * G(I + L), where 11 L(ojw) 11< l(W)
(vi) G * G+6G, where 115G 11< l(w).
In the formulas above, II L(jco) I denotes the norm of the matrix L. In (iv) and (v), L represents multiplicative unstructured nonparametric uncertainty at the plant output and input, respectively. For multivariable systems, the location of multiplicative perturbations is critical, because matrix products do not in general commute. In (vi), 3G represents additive unstructured nonparametric uncertainty. Uncertainties of these types are discussed extensively in Doyle and Stein (1981), and Doyle, et al. (1992). If only certain elements of L and 5G are variable, the uncertainty would then be structured. An important advantage of nonparametric models is that they permit the treatment of uncertainty associated with unmodeled dynamics. Thus for example, L in (iv) may be a rational matrix of degree higher than G.
Uncertainty in the plant model may have several origins:
1. There are always parameters in the linear model which are only known
approximately or are simply in error.
4
2. The parameters in the linear model may vary due to nonlinearities or changes
in the operating conditions.
3. At high frequencies, even the structure and the model order is unknown, and
the uncertainty will always exceed 100% at some frequency.
4. Measurement devices have imperfections. This may even give rise to uncertainty in the manipulated inputs, since the actual input is often measured and adjusted in a cascade manner. For example, this is often the case with valves used to measure flow. In other cases, limited valve resolution may cause input
uncertainty.
5. Even when a very detailed model is available, we may choose to work with a simpler (loworder) nominal model and represent the neglected dynamics as
"uncertainty".
6. Finally, the implemented compensator may differ from the one obtained by solving the synthesis problem. In this case, one may include uncertainty to allow
for controller order reduction and implementation inaccuracies.
Uncertainty in matrixfraction models is not considered in this dissertation. Some discussion of this type of uncertainty, where N(s) and D(s) are additively perturbed, appears in Vidyasagar (1985). There are many ways to define uncertainty from stochastic uncertainty to differential sensitivity and multimodels. Weinmann (1991) gives a good overview.
1. 1.3 Performance Measures
Control systems are designed to meet certain performance requirements. These are quantified using different indices depending on the approach followed. Of course, stability is often an implicit performance measure in control system design. In addition, however, other measures are often required, relating typically to tracking errors, disturbance
rejection, and noise suppression. The most common mathematical forms for performance measures are
(i) J = f (xTQx + uTRu)dt
(ii) J =supJIe(jo)J
(iii) J =f dlle( 1o)
The measures given by (i)  (iii) are referred to as the integral quadratic measure, H norm measure, and H2norm measure, respectively. In (ii) and (iii), the vector e represents an 'error" vector which is to be kept as small as possible. Because of the plant uncertainty, one can at best minimize an upper bound on these performance measures, or in the stochastic case, an average value. Cases where Je(jo)J is bounded at each frequency point, that is, e(jo) < l,(o) can be reduced to a normalized design condition of the form
sup e <1 (1.2)
co l, (0)
which is of the H_norm type measure. Designs which minimize an upper bound are referred to as guaranteedcost designs, while those that minimize the leastupperbound are called minimax designs. Some good introductory material for singleinput singleoutput systems design with these norms can be found in Youla, et al. (1976), and Zames and Francis (1983).
It is meaningful however, to assess quantitatively the performance of the system under control only if its stability can be assured. Otherwise, the trajectories of the system under admissible operating conditions may "blow up", thus rendering the system useless. Hence, stability is the most fundamental requirement in control system design. Motivated by this observation, design methods for controllers that guarantee closedloop stability in the face of uncertainty are sought in this dissertation. The approach is different depending
6
on whether the plant model is linear or bilinear. First, a design methodology for robust predictive controllers for unconstrained processes is developed. Then, we should extend the approach to design predictive regulators for uncertain linear systems subject to input or output constraints. After this, stability in the face of uncertainty is studied for bilinear and timedelayed plant models. A variety of robust stability conditions for those systems is derived depending on the structure of the system, the kind of uncertainty characterization, and the existence or not of constraints or nonlinearities in the input channel.
1.2 Objectives
Design techniques for predictive controllers that guarantee the stability of uncertain linear plants are developed in this dissertation. Also, the stability of uncertain bilinear or timedelay systems with saturating actuators or other input nonlinearities is examined and sufficient robust stability conditions are derived. The development of the subject is outlined in section 1.3. Before that, the basic areas upon which this dissertation builds are briefly discussed.
1.2.1 Predictive Control
The name Predictive Control describes a class of computer control schemes that utilize a process model for the following tasks:
1. To explicitly predict the future plant behavior.
2. To calculate an appropriate control action necessary to drive the predicted output
as close as possible to the desired value.
Predictive control techniques offer good performance, and are easy to understand and formulate. They have the ability to handle time delays, difficult process dynamics, and can accommodate process constraints (Ogunnaike and Ray, 1994). For these reasons, they have been popular in the chemical process industries (Seborg, 1994). Despite their
7
popularity though, there still remain challenges where more research effort is needed. Robust stability conditions for example, have been derived only for certain approaches where the openloop plant is assumed to be stable. Several of the predictive control strategies contain a substantial number of tuning parameters, and it is not always obvious how these parameters should be chosen. In contrast, this dissertation offers a design methodology that includes openloop unstable plants, guaranteeing robust stability through the calculation of a single transfer function (the Youla parameter Q) in a systematic fashion. A predictive regulator strategy is also developed for uncertain linear plants in the presence of constraints using the recently developed / 1IH, framework.
1.2.2 Bilinear Systems
As an approximation to the nonlinear plant behavior, a bilinear model can often provide a more accurate representation than a linear system. Furthermore, a bilinear model, being linear in the states and linear in the control variables but not jointly linear, presents a more tractable form than a fullscale nonlinear model. Bilinear models are also easier to obtain through an identification process than fully nonlinear models. For the above reasons, they have been studied extensively over the past two decades. However, the problem of stabilization of bilinear systems when the model is uncertain has not received much attention. We set out therefore to derive robust stability conditions for bilinear systems that are uncertain and moreover subject to actuator saturation. A state feedback approach is employed and the useful properties of matrix norms and matrix measure are utilized.
1.2.3 TimeDelay Systems
In many applications, it is often assumed that the system under consideration is governed by a principle of causality; that is, the future state of the system is independent of
8
the past states and is determined solely by the present. If it is also assumed that the system is governed by an equation involving the state and rate of change of the state, then generally, one is considering either ordinary or partial differential equations. However, under closer inspection, it becomes apparent that the principle of causality is often only an approximation to the true situation, and that a more realistic model would include some of the past states of the system. In other problems also, it is meaningless not to have some dependence on past variables. This has been known for some time, but the theory for such systems has been extensively developed only recently (Hale and Lunel, 1993). In the process industries, one often finds systems in which there is a noticeable delay between the time instant the input change is implemented and when the effect is observed, with the process output displaying an initial period of no response. Such systems are appropriately referred to as timedelay systems, and "their importance is underscored by the fact that a substantial number of processes exhibit delay characteristics" (Ogunnaike and Ray, 1994, p.245). Recognizing the importance of timelags, and recalling the common theme of this dissertation, that is, robustness, the stability of uncertain timedelay systems is treated to a considerable extent. A host of robust stabilizability and robust stability conditions are derived for different configurations of the plant model. These include a class of uncertain bilinear systems with state delay that may also be uncertain, and uncertain linear delayed models with saturation or other nonlinear effects.
1.3 Organization of the Dissertation
The dissertation is organized into seven chapters as follows. Chapter 1 contains introductory material and ideas used in subsequent chapters. In Chapter 2, we discuss the design of unconstrained robust predictive controllers. Chapter 3 clearly defines and solves the robust predictive regulator problem subject to input constraints. Chapter 4 derives sufficient conditions for the robust stabilization of inputconstrained bilinear systems. In Chapter 5, the methodology developed in Chapter 4 is extended to include time delays in
9
the state variables. Chapter 6 derives delaydependent robust stability conditions for uncertain linear systems with application to CSTRs with recycle. Finally, Chapter 7 contains proposals for future work and possible extensions of the material presented in this dissertation.
Specifically, a systematic technique to design robust predictive controllers for linear systems is proposed in Chapter 2. The approach consists of parametrizing a nominal predictive controller which is designed using conventional methods. A significant feature is the use of a Qparameterization technique that preserves the servo dynamics of the nominal controller. The method is applicable to unconstrained predictive control designs that use transferfunction models corrupted by unstructured uncertainty. It is also shown how to incorporate an integrator in the robust controller, enhancing the closed loop performance by guaranteeing steadystate rejection of asymptotically constant disturbances.
In Chapter 3, the focus is on robust regulation in the presence of input or output constraints. The predictive control technique developed in Chapter 2 is extended to encompass persistent but bounded disturbances and process constraints. Utilizing recently developed extensions of robust linear control theory, the problem of robust predictive regulation in the presence of constraints is formulated as a mixed l1/H optimization. An example using an unstable plant illustrates the results and shows the achievable limits of performance with a linear controller. A discussion on the order of the controllers obtained is also included.
Chapter 4 concentrates on stabilization of inputconstrained uncertain bilinear systems. A statefeedback controller is used to stabilize a continuoustime bilinear system with constant system matrices and subject to actuator saturation. Sufficient conditions are derived that guarantee the stability of the saturating nonlinear system in the presence of normbounded perturbations. Two different uncertainty descriptions are employed, namely, a nonlinear bounded perturbation, and linear bounded perturbations on the system
10
matrices. Examples are also given at the end of the chapter that illustrate the procedure and the simplicity of the derived conditions.
Chapter 5 extends the findings of Chapter 4 to address the robust asymptotic stabilization of a class of bilinear systems with delays in the state variables. Sufficient conditions are derived to guarantee the stability of the timedelay system in the presence of normbounded nonlinear uncertainties. Use is made of the matrix measure to yield a general robuststability condition and a characterization of the associated domain of attraction for the nonlinear system.
In Chapter 6, the stabilization of a model of an integrated reactor/separator module with recycle is studied, and robust stability conditions are derived. Linear timedelay models are employed with system nonlinearities appearing in two different terms: the first term includes perturbations which are allowed to be nonlinear and/or timevarying, and the second term represents nonlinearities in the input channel. The latter class includes input saturation as a special case. The key observation in this chapter is that nonlinearities and plant uncertainty may destabilize an otherwise stable timedelay system. A detailed model of a CSTR with recycle stream is used as an example to demonstrate the results.
Finally, Chapter 7 contains suggestions for further work.
CHAPTER 2
UNCONSTRAINED ROBUST PREDICTIVE CONTROL DESIGN
2.1 Introduction
Predictive control strategies have received much attention in the literature and have also found acceptance in industry. The popularity of these methods is due to the fact that they offer good performance, are easy to understand and formulate, and can accommodate input/output process constraints. The industrial success of the predictivecontrol techniques is apparent by the variety of commercial predictive controllers that are available to the chemical processing industry through specialized vendors. In fact, predictive control is possibly the most widely utilized modelbased control strategy in the chemical industry, and is often the only modelbased control technique supported and offered commercially by control engineering companies. Seborg (1994) reports that in oil refineries and petrochemical plants around the world, there are hundreds, perhaps thousands, of predictive controllers employed.
It is possible to design predictive controllers using different plant representations. One common approach is to use finite impulseresponse models (FIR). This is the favored approach in the earlier formulations, including Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980), Model Algorithmic Control (Mehra, et al., 1979), and the quadratic DMC formulation of Garcia and Morshedi (1986). An extensive review of FIRbased predictive control is given by Garcia, et al. (1989). The FIR methods are applicable only to stable plants, and often require large model orders, typically retaining 30 to 40 impulse response coefficients. An alternative approach is to base the controller design on transferfunction models, which are applicable to both stable and unstable plants, and lead to lowerorder
11
12
representations. Examples of transferfunction based predictive control are the wellknown Generalized Predictive Control (GPC) technique (Clarke, et al., 1987) and the MUSMAR approach (Greco, et al., 1984). A third approach is the use of statespace methods for predictive control design, a practice that has an early representative in the work of Kwon and Pearson (1977), and has recently gained popularity through multiple advocates such as the work of Muske and Rawlings (1993a). This large body of literature constitutes a rich source of knowledge to support the design and analysis of predictive controllers.
Currently, there is an increasingly visible interest in the research community to revisit the predictive control design techniques with the intention of including robustness features that guarantee stability or adequate performance when the plant model is uncertain. One interesting example is the robust quadratic DMC design including hard constraints studied by Zafiriou (1990). The author uses a contraction mapping first proposed by Economou (1985) to successfully derive timedomain conditions for robust stability with respect to uncertainty in the impulseresponse coefficients of the nominal model. Unfortunately, this approach may lead to conservative designs, and is not of practical utility because it involves a very large numerical computation effort. More recently, Genceli and Nikolaou (1993) proposed an analysis and synthesis method for predictive controllers based on FIR models, including constraints and using a linear cost functional, instead of the classical quadratic functional. These authors use a parametric model uncertainty description that bounds the maximum deviations allowed for each pulseresponse coefficient, and obtain a sufficient condition for robust closedloop stability.
The robustness of predictive controllers designed using transferfunction representations is receiving increasing attention in the literature. Kouvaritakis, et al. (1992) propose an alternative approach to GPC that employs an additional compensator that stabilizes the plant before the predictive design is carried out. These authors make use of the Qparametrization procedure popularized by Youla (Youla, et al., 1976) as the basis for a scheme that robustifies the controller with respect to unstructured perturbations. The
13
authors state rigorous necessary and sufficient conditions for robust stability; however, the approach proposed for synthesizing robust controllers is an approximate albeit practical scheme. The method consists of using polynomial or fixedorder transfer function approximations for the Youla parameter, and leastsquares methods to identify the parameters of the robust design. Robinson and Clarke (1991) analyzed the robustness of the GPC technique in an indirect fashion. They investigated two particular control designs, namely, a deadbeat and a meanlevel controller, which can be interpreted as special cases of GPC that arise for specific tuning parameters. The focus is on the effect of a polynomial prefilter T proposed by the authors to introduce stability robustness. The analysis is not strictly valid for any other choices of predictivecontrol tuning parameters, nor for the case of unstable plants. In a recent publication, Yoon and Clarke (1995) compare designs based on Tfiltering and on Qparametrization, and propose simple guidelines for robust synthesis using the T polynomials. A representative approach to robust control of constrained nonlinear systems in continuous time is reported in Michalska and Mayne (1993).
This chapter presents a systematic procedure for robustifying predictive controllers in the presence of unmodeled dynamics. A nominal predictive controller designed using the nominal plant model is parametrized to produce a robust predictive controller that ensures stability with respect to the uncertainty in the nominal model, and guarantees adequate performance. Given that so much knowledge is currently available for designing and tuning highperformance predictive controllers for the case where the plant model is assumed to be free of uncertainty, it is of fundamental importance that the robustified controller be able to preserve the performance of the nominal controller. Specifically, in the limit when the uncertainty in the plant model is negligible, the servo performance of the robust predictive controller should closely resemble the tracking behavior of the nominal predictive controller. Furthermore, in its role as a regulator, the robustified predictive controller must be able to reject the effect of asymptotically constant perturbations.
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The proposed approach consists of parametrizing a nominal predictive controller which is designed using conventional and well established methods. A significant feature is the use of a Qparametrization technique that preserves the servo dynamics of the nominal controller. The method is applicable to unconstrained predictive control designs that use transferfunction plant models corrupted by unstructured uncertainty. A solution to the robust design problem is obtained using an algorithm by Rotstein and Sideris (1992) that yields an explicit solution to an underlying Nehari extension problem. Therefore, the technique avoids the use of approximations, yet remains practical. The design also has the advantage of being able to include an integrator in the robust controller, hence enhancing the closed loop performance properties by guaranteeing steadystate rejection of asymptotically constant disturbances.
The chapter is organized as follows. Section 2 presents a concise review of the design equations for a nominal predictive controller, including the resulting control law, in transferfunction form. Section 3 derives a necessary and sufficient condition for robust stability, and Section 4 develops a comprehensive methodology for synthesizing robust predictive controllers. A design example is given in Section 5, followed by final conclusions offered in Section 6.
2.2 Nominal Predictive Controller Design
The design of nominal predictive controllers is vastly documented in the literature. In particular, a wealth of knowledge is available to resolve crucial design issues such as nominal closedloop stability, and parameter tuning (Lambert, 1987; Mohtadi, 1987). Typically, predictive controllers are employed by executing at every sampling instant an algorithm that solves a quadratic optimization problem. For analysis purposes, it is desirable to represent the algorithmic controller in terms of transfer functions, thus allowing the utilization of classical domain tools for analyzing stability and performance. This section presents a brief review of the analysis technique discussed by Crisalle, et al.
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(1989), which casts an algorithmic predictivecontrol law of the GPC type into a form involving transferfunction operators. The resulting nominal controller is used later as the basis for the design of a robust controller.
Consider the nominal process model
A(z) y(z) = B(z) u(z) (2.2.1)
where y(z) and u(z) are the process output and input, respectively, and A(z) and B(z) are the coprime polynomials
A(z) = z" + a,Z'' +...+a (2.2.2)
B(z) = b,,z"' + b,,11z"' +... + bo (2.2.3)
of order n and rn, respectively, where n>m. Predictive control involves the minimization of the quadratic cost functional
N, N,
J(t)= [r(t +i) y(t + ilt) + A [ Au(t + i)]2 (2.2.4)
t=1O
where { r(t+i) } is the sequence of future values of the set point, {y(t+ilt) } is the sequence of predicted future values of the output, { Au(t+i) } is the sequence of future control increments, A is the movesuppression parameter used to penalize excessive control energy, and parameters NY and NU are the prediction and control horizons, respectively. By definition, a predictive control law is an algorithm that at every sampling instant produces the control move u(t) that minimizes the functional (2.2.4) for the prescribed set point sequence { r(t) }. The optimal control move is naturally found by differentiating (2.2.4) with respect to the control moves, equating the result to zero, and solving for u(t). Following the development by Crisalle, et al. (1989), it is possible to write the resulting control law in terms of transferfunction operators in the form R(z) S(z)
u(z)=T(z)r(z) Y(z) (2.2.5)
which includes the polynomial operators
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R(z) = z" + _ "I+.+ r. 226
S(z) = s'z" + s'1z" +...so (2.2.7)
T(z) = tN, N + tN + + (2.2.8)
where
R(1) = 0 (2.2.9)
and
T(1) = S(1) (2.2.10)
and where the coefficients of the movingaverage polynomial S(z), the regressor polynomial R(z), and the setpoint advancement polynomial T(z) are functions of the tuning parameters N,, N,, and A, and of the model polynomials A(z) and B(z). Note that (2.2.9) implies that the predictive control law (2.2.5) includes an integrator. A blockdiagram representation of the predictive control structure is shown in Figure 2la. Specific design equations for the polynomials (2.2.6)  (2.2.8) are given in Section 2.6; further details of the derivation can be found in Crisalle, et al. (1989). A formulation equivalent to (2.2.5) is also derived in McIntosh, et al. (1991).
Note that the transfer functions operating on u(z) and y(z) in the nominal predictive controller (2.2.5) are of order n, the order of the nominal plant model. It is also significant to note that the setpoint advancement polynomial T(z) is of degree equal to the prediction horizon NV. Since N, n is a common tuning prescription (Clarke, et al., 1987), the order of T(z) may exceed the order of R(z), making the control law nonproper (noncausal) with respect to the setpoint signal. This noncausality is a natural consequence of the inclusion of future values of the set point in (2.2.4). Figure 2la shows that T(z) acts on the set point to produce the intermediate signal w(z)=T(z)r(z), which has the simple timedomain representation
17
N y N  y I
w(t) = t y r(t+N ) + t y r(t+N 1) + ... + t r(t+1) (2.2.11) It is useful to remark that the nominal model (2.2.1) and the functional (2.2.4) are simpler versions of more elaborate formulations that improve the design performance at the expense of added complexity.
d
(a)
r _* W, e U B, Y
S
(b) d
r wIV e UB
A 2 + Y NQ A
X +MQ
Figure 21. (a) Structure of a nominal predictive controller. (b) Structure of the parametrized predictive controller featuring the Youla parameter Q(z).
Typical enhancements are the inclusion of a lower predictionhorizon parameter (Clarke, et al., 1987), the inclusion of a weighted endpoint term in (2.2.4) to guarantee stability for arbitrary parameter choices (Kwon and Byun, 1989; Demircioglu and Clarke, 1993), and the use of an auxiliary (filtered) set point. Another common practice is the addition of an exogenous stochastic input to (2.2.1); however, the inclusion of residual stochastic dynamics in the model is of less significance in the context of robust control design because such dynamics may conceivably be encompassed within the uncertainty
18
model. These and other typical design choices can be accommodated within the framework proposed in this chapter through obvious modifications.
Figure 21a illustrates the closedloop system established when the nominal predictive controller (2.2.5) is connected to the process (2.2.1). In addition to the set point signal r(t), the figure also shows an additive outputdisturbance signal d(t). Note that the servo dynamics of the closed loop are fully characterized by the equations [A(z)R(z) + B(z)S(z)] y(z) = z"B(z)T(z) r(z) (2.2.12)
[A(z)R(z) + B(z)S(z)] u(z) = z"A(z)T(z) r(z) (2.2.13)
Therefore, the stability of the closed loop for a given nominal predictive controller can be easily established by calculating the roots of the characteristic polynomial A(z)R(z) + B(z)S(z). Furthermore, due to the presence of the integral action (2.2.9) in the controller and to the gain equality (2.2.10), the closedloop dynamics described by (2.2.12) are guaranteed to realize zerooffset in the servo response. The integrator also guarantees perfect steadystate disturbance rejection for all disturbance signals that reach a constant steadystate. These desirable performance characteristics of the nominal controller will be preserved in the robustified predictive controller designed in the following sections.
2.3 Controller Parameterization
In this section, the nominal predictive controller (2.2.5) is parametrized in terms of a transfer function Q(z) selected in the spirit of WienerHopf design (Youla, et al., 1976). However, a modification in the parametrization is introduced to achieve two important design requirements: (i) the parametrized controller must preserve the servo performance and the steadystate disturbance rejection properties of the nominal controller, and (ii) the parametrized controller must also be a predictive controller.
19
Consider a nominal predictive controller (2.2.5) that stabilizes the closed loop system (2.2.12)  (2.2.13). Because of the stability condition, the nominal closedloop characteristic polynomial
A*(z) = A(z)R(z) + B(z)S(z)
(2.3.1)
of degree 2n is Schur. In order to parametrize the controller, consider a coprime fractional representation of the nominal plant model (2.2.1) of the form B(z) N(Z)
A(z) M(z) (2.3.2)
where N(z) and M(z) are proper and stable transfer functions that satisfy the Diophantine equation
N(z)X(z)+M(z) Y(z) =1
(2.3.3)
for some pair of stable and proper transfer functions X(z) and Y(z). (Note the use of italicized capital letters for transfer functions, while polynomials are designated with plain capital letters.) A suitable (M(z), N(z)) pair can be readily derived from the nominal characteristic polynomial (2.3.1). The procedure consists of first factoring the closedloop characteristic polynomial into the form A*(z) = Al(z)A2(z), where both A1(z) and A2(z) are of degree n. If A*(z) contains complex poles then A1(z) and A2(z) are constructed such that each complexconjugate pair is contained in either A1(z) or A2(z)to ensure that each polynomial factor has only real coefficients. Dividing both sides of (2.3.1) by the factored characteristic polynomial to obtain A(z)R(z) B(z)S(z)
+
A,(z)A2(z) A,(z)A2(z) (2.3.4)
Finally, stable and proper factorizations that satisfy (2.3.3) are easily obtained by defining A(Z) B(z)
M(z):= N(z):= B) (2.3.5)
A,(z),A,(z)
and
20
S(z) R(z)
X(z):= Y(z):= (2.3.6)
A, (Z) I A2(Z)
where X(z) and Y(z) are clearly stable and proper rational transfer functions. This result allows writing the nominal predictive control law (2.2.5) in the equivalent form Y(z) u(z) = Z(z) r(z)  X(z) y(z)
(2.3.7)
where
Z(z):= z" T(z)
A2(z) (2.3.8)
The set of all solutions of (2.3.3) can be written in terms of the transfer functions (2.3.5)  (2.3.6) and a proper and stable transferfunction Q(z) through the wellknown relations (Youla, et al., 1976)
X'(z) = X(z) + M(z)Q(z)
(2.3.9)
Y'(z)= Y(z)  N(z)Q(z)
(2.3.10)
Therefore, the set of all stabilizing controllers with the structure (2.3.7) is parametrized in the form
[Y(z)  N(z)Q(z)] u(z)= Z(z) r(z) [X(z)+ M(z)Q(z)] y(z) (2.3.11) to yield the control structure shown in Figure 2lb. Clearly, setting Q(z)=O reduces the parametrized predictive controller (2.3.11) to the nominal predictive controller (2.3.7).
Note that in contrast to the standard Youla parametrization approach, the transfer function X(z) + M(z)Q(z) appears in the feedback path of Figure 2Ib, instead of appearing in the control block immediately preceding the plant. This deliberate departure from the standard approach, in conjunction with factorizations (2.3.5) and (2.3.6) that make use of the nominal closedloop polynomial, introduces highly desirable properties in the parametrized inputoutput maps as explained in the following.
21
Proposition 3.1. The nominal control loop of Figure 2la and the parametrized control loop of Figure 2lb have identical servo transfer functions y(z)/r(z) and u(z)/r(z).
Proof The proposition is proved trivially by carrying out blockdiagram algebra on each figure to derive in both cases the servo transfer functions y(z)/r(z) and u(z)/r(z) that are immediately obtained after a rearrangement of factors in equations (2.2.12) and (2.2.13).
Corollary 3.1. Given that the nominal controller (2.2.5) is a predictive controller, then the parametrized controller (2.3.11) is also a predictive controller.
Proof. If (2.2.5) is a predictive controller, then by definition it yields a control sequence { u(t)} that minimizes the predictive performance index (2.2.4) for any prescribed setpoint trajectory { r(t)}. From Proposition 3.1 it follows that, for the given set point trajectory, the parametrized controller (2.3.11) will also produce the same control sequence due to the equality of the servo transfer function u(z)/r(z). It follows then that the parametrized controller is also a predictive controller because it yields a control sequence that minimizes (2.2.4).
Since any allowable parameter Q(z) yields the same servo transfer functions y(z)/r(z) and u(z)/r(z), the parametrized controller has the intrinsic capability of preserving the nominal servo performance. Also note that although the terms containing Q(z) effectively cancel out in the servo transfer functions, the transfer function e(z)/w1(z) = M(z)[Y(z) N(z)Q(z)] in Figure 2lb is affine in Q(z), as in the standard Youla parametrization method. This allows a shaping of the loop sensitivity while simultaneously retaining nominal performance.
2.4 Robust Predictive Design
When the nominal model (2.2.1) is not exact due to the presence of modeling errors, the plant transfer function g(z) may be written in the form
g(z)=g"(z)+A(z) (2.4.1)
B(z)
where go(z) = is the nominal plant model, and A(z) is an unstructured perturbation.
A(z)
Without loss of generality, the developments are specialized for the case of additive perturbations satisfying
A(e'")J!;W(e"')J V(D
(2.4.2)
where the uncertainty weight W(z) is a stable and proper transfer function, and the perturbation A(z) is assumed to be such that g(z) and go(z) have the same number of unstable poles. The case of multiplicative perturbations, as well as other typical unstructured uncertainty representations (Francis, 1987) can be treated in an analogous way. Of particular relevance to chemical processing systems can be the inverse multiplicative model (Maciejowski, 1989), which allows for g(z) and go(z) to have different number of unstable poles even when the perturbation A(z) is stable. The situation where the nominal model and the uncertain process have different number of unstable poles may arise in processing systems, for example, when the process evolves through a metastable steadystate in a CSTR with multiple steady states.
The objective in this chapter is to design a robust predictive controller that stabilizes the closed loop for all the members of the uncertain family of plants (2.4.1)  (2.4.2), and that in the nominal case (where A(z)=O) it recovers the performance of a nominal predictive controller (2.2.5) which is designed solely on the basis of the nominal model go(z). The stability robustness of the closed loop shown in Figure 22, which includes the parametrized controller (2.3.11) and the uncertain family of plants (2.4.1)  (2.4.2), can be analyzed using H. theory concepts as shown in Theorem 4.1 below.
Theorem 4.1. A necessary and sufficient condition for the robust stability of the closedloop system of Figure 22 is the inequality condition
W(z)C(z)4 (z)  <1 (
( 2.4.3)
23
where
C(z):= X(z)+ M(z)Q(z) (2.4.4)
Y(z)  N(z)Q(z)
and
S(z):= M(z) [Y(z)  N(z)Q(z)] (2.4.5)
Proof. Since the operators located between signals r(z) and wI(z) are stable, it suffices to determine robust stability conditions for the loop obtained by ignoring the prefilter operator z"T(z)/A2(z) and thus considering wj(z) as a bounded setpoint sequence. The proof is then completed by applying the SmallGain Theorem (Maciejowski, 1989; Doyle, et al., 1992) to the loop in question, according to the following procedure. First, write A(z) := A'(z)W(z), where A'(z) is a stable transfer function satisfying 1 A'(z) 1 1, and define the A'perturbation input variable q(z) := W(z)u(z) and output variable p(z) A'(z)q(z). Clearly, the A'perturbation variables are related through the expression q(z) =
W(z)C(z)Sj(z)p(z), where operator C(z) and the loop sensitivity function S1(Z) are as defined in (2.4.4) and (2.4.5), respectively. Finally, the necessary and sufficient robust stability condition (2.4.3) follows readily by applying the SmallGain Theorem to the transfer function W(z)C(z)S(z).
d
r w w, e I U B
 T T 1~)
A2 + YNQ A
X+MQ
Figure 22. Structure of a robust predictive controller for a plant with an additive uncertainty. An exogenous disturbance d(t) affects the plant
output.
24
2.4.1 Robust Synthesis
We propose a systematic procedure for solving the robust synthesis problem without resorting to approximations for the Youla parameter. A particular challenge to the design problem posed is the objective of including an integrator in the robustified controller in order to guarantee effective disturbance rejection. In the first subsection, we develop a design technique for the base case where the plant is unstable and has no poles on the unit circle. The next subsection treats the case where the plant is unstable but has poles on the unit circle, as in the case of an integrator. The third and final subsection treats the stable plant case.
2.4.1.1 Unstable plant with no poles on the unit circle
Consider the robust predictive controller design problem for the case where the nominal plant model go(z) is unstable but has no poles on the unit circle. The synthesis problem is attacked by rewriting the robust stability condition (2.4.3) in the equivalent modelmatching form
T<(Z)T2(Z)Q(Z)1<1 (2.4.6)
where
1(z)= W(z)X(z)M(z) (2.4.7)
T2(z) =  W(z) M(z)2 (2.4.8)
Inequality (2.4.6), which is affine in the unknown Youla parameter Q(z), is obtained by substituting equations (2.4.4) and (2.4.5) into inequality (2.4.3). The modelmatching problem is commonly approached in the context of He control theory using the giteration process (Glover, 1984), where (2.4.6) is substituted by the alternative inequality  I(Z) T2z)Q(z)11
(2.4.9)
25
where g is a positive scalar parameter selected by the designer. A robust design is obtained if a Youla parameter Q(z) is found for a specified g < 1. The key is then to be able to synthesize a Youla parameter with a reliable algorithm.
One possible venue for synthesizing parameter Q(z) is to map the operators in (2.4.6) into the Laplace domain using the bilinear transformation, and then use techniques appropriate for synthesis for continuous time systems (Glover, 1984). While this procedure is sometimes useful, it suffers from a number of disadvantages. In particular, the resulting continuoustime problem may be ill conditioned, posing serious numerical difficulties. In addition, discrete poles located at zero (such as those caused by pure delays) must be excluded in order to solve the problem in continuous time, because otherwise, the Glover procedure cannot guarantee a statespace solution (Rotstein and Sideris, 1992). In order to avoid the complications and subtleties associated with the bilinear mapping, we advocate the use of a zdomain technique proposed by Rotstein and Sideris for solving the modelmatching problem (Rotstein and Sideris, 1992; Rotstein, 1993). Their algorithm solves the problem of approximating a stable transfer function R(z) with an antistable (all poles outside the unit circle) transfer function QR(z), where the tilde superscript denotes the conjugate operation Q (z):= QR(1/ z). The problem, also known as the Nehari extension problem (Maciejowski, 1989), calls for finding an antistable function Q~(z) such that
R(z)  Q~(z) (2.4.10)
Section 2.6.1 gives relevant details of the RotsteinSideris algorithm which yields the set of all the solutions to (2.4.10). The only inputs required are the transfer function R(z) and the scalar y. A necessary condition for the existence of a solution is that y must be greater than the Hankel norm of R(z), that is, y > R(z) 1H One solution of particular significance is the central controller, denoted QR,,(z), because the set of all the solutions can then be found as an explicit function of QRZ).
26
The modelmatching problem (2.4.9) can be written in the form (2.4.10) through a series of normpreserving transformations as detailed in Section 2.6. 1. The procedure consists of first factoring T2(z) into the form T(z) = (Z) T
(2.4.11)
where T,(z) is an allpass function and T ,(z) is a stable minimumphase function, and then finding T (z)= (I/ z) and carrying out the decomposition
ap
T_ (z)7(z) = Rt,(z) + RS(z)
"p (2.4.12)
where R,(z) and R (z) are an antistable and a stable transfer function, respectively. As shown in Section 2.6.1, the transferfunction required for the Nehari problem (2.4.10) is R(z) = R (z)
il (2.4.13)
The central solution QR,,(z) is obtained from an explicit formula (see equation (2.6.4) in Section 2.6.1), and a solution to the original modelmatching problem (2.4.6) is simply recovered as
Q(z) = ,,,'(z)(R,(z)+ QR,(Z))
(2.4.14)
The final robust predictive controller design for the base case is obtained by substituting in the structure (2.3.11), the Youla parameter (2.4.14), and the factorizations (2.3.5)  (2.3.6).
2.4.1.2 Unstable plant with poles on the unit circle
When the nominal plant model go(z) has poles on the unit circle, the standard H. control theory is no longer applicable. In addition, the factorization (2.4.11) is no longer feasible because no minimumphase stable transfer function can possibly satisfy the equality. This difficulty is circumvented by introducing a change of variable that maps unitcircle poles to a circle of larger radius. Let z = c/p, where p >1 is a scalar, and define the operators
27
7T'(,) T (/p (2.4.15)
and
T(,' 7)= T2( /p) (2.4.16)
Note that the mapping z = (/p, is a special case of the well known bilinear transformation. The effect of the transformation is to map all the zplane poles located on the unit circle into 4plane poles located on a circle of radius p . Using equations (2.4.15)  (2.4.16), the basecase design problem (2.4.9) can be posed in terms of the transformed variable ; to yield
Tl' ( ) T2'(;)G QC < (2.4.17) Problem (2.4.17) and can be solved for Q'(;) using the basecase algorithm described in Section 4.1.1. The zdomain Youla parameter is simply recovered by transforming the result back to the original space, that is,
(2.4.18)
Q(z) = Q'(pz)
and the final robust predictive controller design for this case is obtained by substituting in the structure (2.3.11), the Youla parameter (2.4.18), and the factorizations (2.3.5) (2.3.6).
Using the Maximum Modulus Theorem, it follows that the transformed design problem (2.4.17) is related to the original problem (2.4.6) through the inequality ' (;) T' (;)Y ( )1 117 (z) T(z)Q(z) 1 (2.4.19)
Therefore, the transformed design represents only a sufficient condition for stability. If no Q'(;) can be found that satisfies (2.4.17), then a smaller value for p is adopted and the design is repeated.
2.4.1.3 Robust design for the case of a stable plant
When the nominal plant model go(z) is stable, the robust design is straightforward. In the normal case where the weight W(z) is minimum phase (i.e., it does not have zeros on
28
or outside the unit circle), a solution to the robust synthesis problem (2.4.6) is obtained by setting 7(z)  T2(z)Q(z) = T(z), where T3(z) is a userspecified stable biproper transfer function that satisfies the contraction condition
where Q(z) is clearly a stable transfer function because T2(z) is minimum phase. In the alternative case where the weight W(z) is nonminimum phase, the design equation used is Q(z) = T2 (z)(7(z) W(z)T(z)) (2.4.21)
which is obtained by setting Tl(z)  T2(z)Q(z) = W(z)T3(z), where 3(z) is a userspecified stable biproper transfer function satisfying W(z)T(z)JI <1. The final robust predictive controller design results by substituting in the structure (2.3.11) the Youla parameter (2.4.20) or (2.4.2 1), along with the factorizations (2.3.5)  (2.3.6).
2.4.2 Robust Design with SteadyState Disturbance Rejection
In process control applications, the controller is often required to deliver effective disturbancerejection performance. This section presents a method for introducing integral action in the robustified controllers, hence ensuring offsetfree regulation in the presence of asymptotically constant disturbances. Given that the nominal predictive controller (2.2.5) leads to the nominal regulation transfer function y(z) _ A(z)R(z)
d(z) A(z)R(z)+B(z)S(z) (2.4.22)
it follows that limy(t) = 0 for step disturbances as well as for other disturbances with a constant steady state because R(1) = 0. On the other hand, from Figure 22 it follows that the nominal regulation transfer function for the robustified predictive controller (2.3.12) is y(z) _ A(z)R(z) A(z)B(z) Q(z)
d(z) A(z)R(z)+B(z)S(z) A'(z) (2.4.23)
29
Because the synthesis procedures described in the previous section do not necessarily yield a Youla parameter satisfying Q(l) = 0, the robustified predictive controller may display unacceptable nominal regulation performance at the steady state, unless the nominal plant has an integrator (that is, A(l)=O) or is a selfregulating process (B(l) = 0). The robust predictive control design for integrating plants is carried out as indicated in Section 4.1.2, and the design for selfregulating plants is treated as in Sections 4.1.1, 4.1.2, or 4.1.3, depending on the location of the poles.
Clearly, the robust predictive controller will attain perfect steadystate disturbance rejection for all the plants belonging to the uncertain family (2.4.1) only if the Youla parameter has a zero gain, that is, Q(1) = 0. This gain constraint can be introduced in the robust predictive controller design through a simple modification of the factorizations (2.3.5)  (2.3.6). First, the integrator is extracted from the nominal predictive controller by writing R(z) = (z1)R'(z), and then (2.3.4) is rewritten in the form A(z)(z  1)R'(z) B(z)S(z) = 1 2.4.24)
A+(z)A2(z) A,(z)A2(z)
Introducing the modified coprime factorization (z  1)A(z) ) B(z) (2.4.25)
z A,(z) A(z)
and
S(z):= Y(Z):= z R'(z) (2.4.26)
A,(z)
leads to operators that satisfy the Diophantine equation k(z).(z)+1(z)Y(z)=l. The modified form of inequality (2.4.9) can then be written as
((z) t(z)O(z) < y (2.4.27)
with
T (Z) = W(z) X(z)( z) (2.4.28)
2(=  W(z)M(z)2
(2.4.29)
30
Note that the definition of M(z) in (2.4.25) is equivalent to designing a controller for a nominal plant which has been augmented by an integrator. It is now possible to proceed to solve (2.4.27) for a parameter 6(z) using the basecase algorithm, as described in Section 4.1.2. After a solution to (2.4.27) is found, the Youla parameter Q(z) used in the parametrized predictive control structure of Figure 22 is constructed by reassociating the augmentedplant integrator with the controller to obtain Q(z) = (z 1) (z) (2.4.30)
The final robust predictive controller design for this case is obtained by substituting in the structure (2.3.11), the Youla parameter (2.4.30), and the factorizations (2.3.5)  (2.3.6). The resulting controller includes an integrator since (2.4.30) satisfies the zerogain condition Q(1) = 0.
Finally, it is useful to note that when the nominal model go(z) is stable, it may be possible to include integral action in the robust predictive controller through simple specifications. In particular, this is readily accomplished by adding to the design equation (2.4.20) the gainequality specification T3(1) = TI(1) if the uncertainty weight W(z) is minimum phase, or adding the gainequality constraint T3(l) = TI(1)/W(1) to equation (2.4.21) if the weight is nonminimum phase. These specifications lead to the desired offsetfree condition Q(l) = 0. However, if TI(l) > 1, these simple designs are infeasible because T3(z) cannot simultaneously satisfy the contractionmapping and the gainequality conditions. In such a case, the robust design with integral action must be done following the general procedure (2.4.25)  (2.4.30).
2.5 Example
This section illustrates the use of the robust design technique via an example. Note that the robust synthesis technique proposed here is applicable to both stable and unstable processes. In contrast, most of the alternative robust predictive control strategies
31
documented in the literature are simply not applicable to unstable plants. In order to maximize the impact of the example, we adopt an unstable nominal plant for the illustration. Even though openloop unstable chemical processes are not predominant in the chemical industry, there is nevertheless a number of such processes that are of economic importance (Muske and Rawlings, 1993b). In particular, the unstable batch chemical reactor studied by Rotstein and Lewin (1992) is characterized by a plant model with one unstable pole. The wirecatalyzed combustion reactors investigated by Sheintuch (1989) and the network of cascaded exothermic CSTRs analyzed by Georgiou, et al. (1989) include plant models with more unstable poles. Consider the secondorder nominal plant model go(z) = 2 (2.5.1)
z~ 0.6z+1.12
which has two complexconjugate poles that lie outside of the unit circle. The nominal plant is subject to an additive uncertainty perturbation characterized by the weight
0.63z+0.6174 (2.5.2)
z +0.5
which reflects the extent of the model uncertainty measured in the frequency domain. Although the specification of appropriate uncertainty weights is a subject of current research, insight into practical procedures for its determination is given in a number of recent publications (see for example Latchman and Crisalle, 1995; Doyle, et al., 1992; Koung and MacGregor, 1993).
Three controllers are designed: (i) a nominal predictive controller (NPC), (ii) a robust predictive controller (RPC), and (iii) a robust predictive controller with integral action (RPCI). The nominal predictive controller is of form (2.2.5), and is realized using the design parameters N,, = 4, N, = 2, and A = 0, and the design equations given in Section
2.6.2, to arrive at the polynomials
R(z)=z2 0.8039z0.1961
S(z) = 0.8639Z2  1 .579z + 1.0984
32
T(z) = 0.2914Z4  0.0 156z3 + 0.366z2 +0.3243z which lead to a nominal predictive controller that stabilizes the closed loop when the uncertainty is neglected. However, the NPC controller is not robustly stabilizing because it violates the robust stability condition (2.4.3), that is 11W(z)C(z)S(z) = 2.9 > 1, where C(z) and Sl(z) are calculated using Q(z) = 0 in (2.4.4) and (2.4.5). This implies that the nominal predictive controller will fail to stabilize the closed loop for some plants belonging to the family of uncertain plants. Efforts to detune the nominal controller by extending the prediction horizon up to the value N, = 30, as well as increasing the control horizon to values greater than the number of unstable poles, also fail to yield robustly stabilizing controllers. Clearly, the lack of success of the detuned controllers in robustly stabilizing the feedback loop is due to the fact that the detuning procedure is conceived without taking into account the explicit additive uncertainty model available through the description (2.5.2).
The RPC design is of the form (2.3.11). Since the unstable nominal plant has no poles on the unit circle, the design proceeds as discussed in the base case (Section 4. 1. 1). The transfer functions Ti(z) and T2(z) are formed as prescribed in (2.4.7) and (2.4.8), using A2(z) = z2. To solve the Nehari extension problem, we use y= 0.99, which is acceptable since it exceeds the limiting Hankel norm value 11 R(z) II and is less than one as required by (2.4.6). The central controller found using the basecase algorithm leads to a Youla parameter Q(z) = NQ(z)/DQ(z) of order 8, with
NQ(z)= 0.8688z8 + 2.026z'  2.699z6 +1.515z5 +0.4916z4
2.267z3 + 2.144z2  1.297z + 0.3668
DQ(Z) = z'  0.6252z7 +1.958z6 +0.4477z5 +0.1928z4 +1.809z3 0.5414z2 +0.6926z
The RPC transfer functions Y(z)  N(z)Q(z), and X(z) + M(z)Q(z), are of order 9 in their minimal forms. The controller is robustly stabilizing because 11 W(z)C(z)S(z) = 0.35< 1.
33
Finally, the design of the RPCI is carried out as indicated in Section 4.2, using again the specification y = 0.99. Since this controller design includes an integrator, the bilinear transformation z = /p discussed in Section 4.1.2 is implemented using the value p = 1.1. The procedure leads to a Youla parameter Q(z) = (z 1)NQ(z)/(zDQ(z) ) of order 9, with
NQ(z) = 0.8648z' + 2.534z'  2.766z6 + 0.9587z + 0.6691z4
1.006z3 +0.65452  0.2434z+0.0389
DQ(z) = z  2.19z7 + 2.181z6  1.129z5 0.2604z4 +0.8927z3 0.7056z2 + 0.3307z  0.0734
The resulting RPCI transfer functions Y(z)  N(z)Q(z) and X(z) + M(z)Q(z) are of order 10 in their minimal forms, and Q(1) = 0, as desired. The RPCI controller is robustly stabilizing because 11 W(z)C(z)J(z) = 0.49 < 1.
Figure 23 shows the results of a closedloop simulation test carried out to evaluate the nominal servo and regulation performances of the three controller designs. The process is assumed to match exactly with the nominal model, that is, A(z) = 0. The servo test features step changes in the set point r(t) at instants t = 0 and t = 50, and the regulation test is carried out by introducing a unitstep disturbance d(t) (not shown in the figure) at t = 12.
Figure 23a shows that during the first 12 samples, where d(t) = 0, all three controllers display identical dynamics. This is the expected result since Proposition 3.1 guarantees that the nominal predictive controller and the robustified controllers have identical servo transfer functions, independent of the value of Q(z). Also as expected, the controlleroutput trajectories shown in Figure 23b are also identical during this interval.
The three controllers differ however in their regulation behavior. When the disturbance is introduced at t = 12, the NPC rejects the disturbance effectively, quickly returning the output to the set point, as shown in Figure 23a. In marked contrast, the RPC fails to reject the effect of the disturbance, and displays steadystate offset. The RPCI, however, succeeds in rejecting the disturbance, albeit with slower dynamics than the
3.0 2.5
2.0 1.5 1.0 0.5 0.0
0.5
0
20
40
60
80
100
Time, t
2.0
1.5 1.0 0.5 0.0
0.5
1.0
1.5
        
0
20
40
Time, t
60
80
100
Figure 23. Comparison of the performance of the nominal predictive controller (NPC), the robust predictive controller (RPC), and the robust predictive controller with integral action (RPCI) designed for a nominal plant. The set point r(t) changes at instants t = 0 and t = 50. A unitstep disturbance d(t) (not shown) is introduced at t = 12. a) Output and, b) Control action.
34
 Set point
 NPC
RPC
 RPCI
5
(a)
0
U
0
(b)
25
                
I . . . I . . ' ' I
35
nominal controller. Figure 23b shows that the NPC achieves the disturbance rejection at the expense of fairly energetic control actions that follow the onset of the disturbance.
On the other hand, the RPCI prescribes more conservative input adjustments, typical of robust controllers. In many practical situations, the smoother dynamics of the RPCI design may be highly preferable over the more aggressive behavior of the NPC.
As a final remark, one should note that all the controllers anticipate the occurrence of setpoint change at t = 50, as evidenced by the early adjustments in control action that take place starting at instant t = 46, as shown in Figure 23b. This anticipatory behavior is a characteristic of predictive controllers. Since the prediction horizon Nv has been selected equal to four samples, the controllers naturally initiate adjustments at instant t = 46 when the prediction horizon first permits detection of the upcoming setpoint change. This observation verifies that the robustified control designs can legitimately be classified as predictive controllers, as claimed in Corollary 3.1.
Figure 24 shows the results of a closedloop simulation test for a perturbed plant (A(z) # 0). As in the previous example, the setpoint changes at instants t = 0 and t = 50, and an external unitstep disturbance d(t) is introduced at t = 12. The figure shows that the NPC is unable to control the plant, causing unstable closedloop dynamics. In marked contrast, both robust predictive control designs RPC and RPCI have stable responses. As expected, the RPC controller suffers from steadystate offset due to the lack of an integrator, whereas the RPCI controller has offsetfree behavior and manages to reject the disturbance without excessive control action.
2.6 ModelMatching Problem and Predictor Design Equations
2.6.1 RotsteinSideris Solution to the ModelMatching Problem
The transformation of the modelmatching problem (2.4.9) to the form (2.4.10) can be done in a straightforward fashion as follows. First, T2(z) must be factored as shown in
36
(2.4.11). Then, using the property that 1T,,(z)G(z) = G(z) 1, it is possible to write the equalities
T l(z) T2(z)Q(z) = T,,(z)T (z)  T,,,(z)Q(z)_ =1 RZ)+R(z) T (z)Q(z) (2.6.1) Defining R~(z) := Ra(z) and QR(z) := R,(z)  Tmp(z)Q(z), and using the property G~(z) = G(z) 1, the previous equality readily reduces to the desired form
T(z)T(z)Q(z) R(z)QR(z) (2.6.2)
Note that the factorization (2.4.11) is not possible for nominal plants with poles on the unit circle. Such special cases must be treated as discussed in Section 4.1.2. A solution to the modelmatching problem (2.4.10) is proposed by Rotstein and Sideris (1992), and the main result is stated in the theorem below. Theorem 2.6.1. Let R(z) be a stable and proper transfer function with a minimal statespace realization
R(z):= C(zI  A)' B + D = (
and let the scalar y > 1R(z) II. Then the set of all antistable proper transfer functions Q~(z) satisfying IR  QR 5 y are given by the conjugate of
QR(Z) = F( QR(Z), (P(Z)) (2.6.3)
where the operator F(.,.) represents a linearfractional transformation, and cD(z) is a stable and proper transfer function satisfying IP(z)L < y. Transfer function QR,c(z) is the central solution defined as
A  B(B LA + E C)N (ALC + BE])  BD21
QR.,(z) (E C+BTL.A)N D D (2.6.4)
D TCN D TDT
where L. and L, respectively denote the controllability and observability grammians of R(z), which are the solutions to the discretetime Lyapunov equations
37
3.0
 Set point
2.5NP
 N PCRPC
2.0 RPCI
z*
1.5 3 1.0
0.5
0.0
0.5
0 20 40 60 80 100
a) Time t
2.5
1.5
 0.5
E 0.5
2.5
0 20 40 60 80 100
(b) Time, t
Figure 24. Comparison of the performance of the nominal predictive controller (NPC), the robust predictive controller (RPC), and the robust predictive controller with integral action (RPCI), designed for a perturbed plant belonging to the uncertainty description. The set point r(t) changes at instants t = 0 and t = 50, and a unitstep disturbance d(t) (not shown) is introduced at t = 12.
a) Output and, b) Control action.
38
L, = ALC AT + BBT (2.6.5)
L,= ATLOA+CTC (2.6.6)
and N = (y21  L L,)'. Rotstein (1992b) gives details on the construction of auxiliary matrices E1 I and Di, which depend on matrices A, B, C, D, L, and L,, and on the value of y.
The central controller (2.6.4) can be directly calculated from a solution to (2.6.5) (2.6.6) and the appropriate definitions for matrices E1 and DO . The use of the Hankelnorm condition y > 11 R(z)jI, stems from the fact that the minimum value of R  Q~ over the stable proper transfer functions Q(z) is exactly equal to 11R(z) *. Hence, the only legitimate values of yare those exceeding the Hankel norm.
2.6.2 Design Equations for Nominal Predictive Control
This section provides specific design equations used to synthesize a nominal predictive controller following the approach of Crisalle, et al. (1989). An equivalent formulation is given by McIntosh, et al. (1991). The final design equations for the polynomials (2.2.6) (2.2.8) that appear in the predictive control law (2.2.5) are: R(z)= zPI+ Z Nkj,(z) (2.6.7)
S(z)= k, F,(z )} (2.6.8)
T(z)= Nk z (2.6.9)
where the design operators Fi(z1) and F,(z1) and the coefficients ki, i=1, 2, ..., N are determined from the process model according the following procedure. First, rewrite the nominal plant model (2.2.1)  (2.2.3) in the equivalent form
A,(z') y(z) = z' B,(z') u(z)
(2.6.10)
39
involving inverse powers of z, where A1(z1) and B1(z1) are related to (2.2.2) and (2.2.3) in an obvious manner and are of the form
I 2
A ,(z') = I + a,,zl + a, 2Z +...+ a,,, (2.6.11)
B (z') = b +b + ... + b , z" (2.6.12)
To obtain the design operators Fi(z1), which are polynomials of degree n (the order of the plant (2.2.1)), solve the set of Diophantine equations
E,(z ')A(z ')A,(z')+ z 'F,(Z1)= 1 i = 1, 2, ..., N (2.6.13)
which also yields the intermediate polynomials Ei(z) of degree ii. The second design operators, the polynomials r';(z1) of degree n, are obtained by decomposing the product E(Z1)B(z1) in the form
Ei(z')BI(z') = G(z1) + z'F(z1) (2.6.14)
where polynomials Gi(z1) of degree iI are known as the dynamic polynomials, and are characterized by the fact that their coefficients are the sampled values of the stepresponse of the plant (2.6.10). In turn, the coefficients of the dynamic polynomials are used to define the nonzero elements of the Toeplitz matrix GN known as the truncated dynamic matrix, which contains only Nu columns. Finally, the coefficients k; , i=1, 2, ..., N are obtained as the components of the gain vector kT=[k1 k2 ... kN], calculated from the expression
T= [1 0 ... o](GGN, +A G; (2.6.15)
2.7 Conclusions
A systematic method for robustifying unconstrained predictive controllers has been proposed. The technique succeeds in preserving nominal servo performance due to the unconventional feedback configuration adopted for the parametrized controller, and also
40
due to a coprime factorization that makes use of the characteristic polynomial of the nominal closed loop. A significant feature of the proposed method is its applicability to both stable and unstable plants. Another advantage of the robust synthesis technique is that it permits the incorporation of integral action in the robustified controllers, making the resulting controllers more useful for practical applications. Although the technique is developed for singleinput singleoutput systems, we anticipate that the methodology presented in this chapter extends naturally to the multivariable case. A limitation of the proposed method is that it does not take into account the effect of input saturation. A possible venue to address this problem is to restate the H_ problem (2.4.9) as a mixed 11IH. problem that explicitly takes into consideration constraints on the manipulated variable. Research is currently in progress addressing the robustification of constrained predictive controllers.
CHAPTER 3
ROBUST PREDICTIVE CONTROL WITH CONSTRAINTS VIA l_/HDESIGN
3.1 Introduction
Predictive control has been an active area of research for almost two decades, and has received increased attention in recent years. Currently, much research emphasis is on the robustification of predictive controllers through design techniques that guarantee the stability and/or adequate performance of the closedloop system when the plant model is uncertain. Although there exists rich theory for the robust control of linear systems, few results are available concerning the robust control of systems with constraints. Such constraints are important in practical control engineering applications because safety, economics, and performance specifications often impose bounds on key process variables. The control system must therefore anticipate constraint violations and take action to correct them. Several results have appeared recently on predictive control with constraints for systems where the plant model is assumed to be known exactly. Rawlings and Muske (1993) showed that nominal stability can be guaranteed for the state feedback regulator using an infinite horizon, and Muske and Rawlings (1993) address the rejection of step disturbances and extend the nominally stabilizing regulator to use output feedback. The latter reference contains also an extensive discussion of existing linear Model Predictive Control strategies up to 1992. Different approaches to the constrained predictive control problem in the absence of disturbances are reported by Sznaier and Damborg (1993), Camacho (1993), Rossiter and Kouvaritakis (1993), Scokaert and Clarke (1994), and Gilbert, et al. (1994).
41
42
The introduction of disturbance signals complicates the design problem. In the absence of constraints, the inclusion of disturbances does not affect the loop stability, even though the overall performance may deteriorate. Disturbances may have a more detrimental effect when constraints are present, possibly affecting the stability of the system when the openloop plant is unstable. Rossiter, et al. (1994) examine the effect of disturbances on the stability of the closedloop under constrained predictive control using an exact plant model. The difficulties are further compounded when robustness with respect to modeluncertainty is also considered, and the results reported in the literature are scarce. Zafiriou (1990) investigates the robustness of a Dynamic Matrix Control scheme with hard constraints, using the contractionmapping principle. This approach leads to robust stability conditions which implies an optimization problem with mathematical tractability problems. The remaining contributions include the work of Genceli and Nikolaou (1993), and of Zheng and Morari (1993), who have derived conditions for robust stability assuming a linear performance functional (instead of the more classical quadratic functional characteristic of predictive control), and assuming an uncertainty description that places bounds on the impulseresponse coefficients.
This chapter addresses the problem of robust predictive control design for optimal regulation in the presence of input or output process constraints. In many cases, there is little information available about the nature of the disturbances, in a deterministic or stochastic sense. In this chapter, it is assumed that disturbances are persistent, but bounded, which makes good practical sense. This is an extension to the constant disturbance model commonly adopted in the Model Predictive Control literature. The proposed approach consists of parametrizing a nominal predictive controller which is designed using conventional and well established methods. The nominal plant is assumed to be affected by unstructured additive uncertainty, which may represent unmodeled dynamics. Other uncertainty models can be accommodated in an analogous fashion. Using recently developed extensions of robust linear control theory, the problem of robust
43
stability in the presence of constraints can be formulated as a mixed 11/H. optimization problem (Sznaier, 1994). In this approach, an 11 norm is used to ensure satisfaction of the constraints, while robust stability is guaranteed using H methods. The resulting design technique simultaneously addresses the problem of robust stabilization under uncertainty, and the nominal performance of the regulator under inputsaturation constraints.
The chapter is organized as follows. Section 2 states the elements of the design problem, including relevant notation, and the model and uncertainty descriptions adopted. Section 3 formulates the constrained regulation design as a mixed 11/H. optimization problem. Details of the synthesis procedure proposed are presented in Section 4. Section
5 gives a design example that illustrates the theory, followed by conclusions in Section 6.
3.2 Statement of the Problem
3.2.1 Notation
Let q = { qk} represent a real sequence. Then l. denotes set of bounded real sequences with the norm IqI :=sup ik 1, and 11 denotes the set of real sequences with
k
finite norm 11q : Jqk . Let G(z) represent a scalar complexvalued transfer function with real coefficients. Then L. denotes the Lebesgue space of complexvalued transfer functions that are essentially bounded on the unit circle and have the associated norm JG(z)II. := supg11 G(z)l. Furthermore, H. denotes the set of stable (that is, analytic in lz 1) transfer functions G(z) e L., and H: denotes the set of antistable (that is, analytic in LzI 1) transfer functions G(z) e L.. Let Q(z) denote the ztransform of a sequence q e l, and recall the fact that q E l if and only if Q(z) E H.. Finally, a tilde superscript is used to indicate the conjugate operation G(z) = G(1/z).
44
3.2.2 Uncertainty Description and Process Constraints
Consider the openloop process
y(z) = g(z)u(z) + d(z) (3.2.1)
where y(z) , u(z), and d(z) are the process output, input, and disturbance signals, respectively, and g(z) is a rational transfer function that represents the plant. The exogenous signal d(k) represents a persistent but bounded disturbance such that dLJJ 1, where, without loss of generality, the unity bound represents the result of a signal normalization. The input u(k) is subject to an operational constraint of the type I1uL /P. Although this paper explicitly deals with input constraints, output constraints may be treated using an analogous procedure.
The plant in (3.2.1) is uncertain in the sense that a nominal model go(z) is known such that
g(z) = go(z) + 'A(z) (3.2.2a)
or
g(z) = go(z)[l + 'A(z)] (3.2.2b)
where go(z) = is characterized by the coprime polynomials
A(z)
A(z) = z" + a,_z + ...+ ao (3.2.3)
B(z) = b,z"' + b,,_,z"' +... + bo (3.2.4)
and where A(z) represents the uncertainty in the nominal model. The model uncertainty is quantified through the frequencyresponse bound A(e") :! IW(e0)1 Vo) (3.2.4b)
where the known uncertainty weight W(z) is a stable and proper transfer function. Following the standard approach in Francis (1987) and Doyle, et al. (1992), the unstructured perturbation A(z) is assumed to be stable, so that g(z) and go(z) have the same
45
number of unstable poles. For simplicity of exposition in this paper, it is assumed that the uncertainty is additive. Multiplicative unstructured perturbations are also treated in the sequel. Other uncertainty types, such as inverse multiplicative or more complicated ones, can be treated by the synthesis procedure developed in this chapter in a similar fashion. For more details on different uncertainty descriptions in the frequency domain, the reader is referred to Maciejowski (1989) and Doyle, et al. (1992).
3.2.3 Design Objectives and Proposed Approach
The objective in this chapter is to design a predictive regulator that satisfies two specifications. The first specification is the nominal regulation performance of the controller, requiring the effective rejection of the effect of exogenous disturbances while satisfying the input constraints in the case when the plant model is exact (that is, A(z) = 0). The second specification is the robust stability of the closed loop, requiring the stabilization of the system for all the plants belonging to the uncertain family (3.2.2)  (3.2.4b). Hence, the predictive controller design focuses on robust regulation under timedomain constraints.
A mixed l1/H. optimization approach is adopted to meet the problem specifications. The nominal performance specification is addressed through an 11 optimization problem that seeks to minimize the effect of the disturbances on the magnitude of the input. In addition, the robust stabilization specification is addressed in terms of an H. problem that seeks to appropriately shape the loop frequencyresponse in order to guarantee stability with respect to the modeling uncertainty.
The control design procedure consists of first producing a nominal predictive regulator, designed ignoring the presence of modeling uncertainty and exogenous disturbances. Then, the nominal controller is rewritten in terms of a free Youla parameter, and finally, the realization of a Youla parameter that meets both design objectives is
46
obtained by solving an l1/H. optimization problem. The details of the problem formulation and its solution are given in Sections 3.3 and 3.4, respectively.
3.3 Formulation of the Control Design Problem
3.3. 1 Controller Parametrization
The cornerstone of predictive control design is the minimization of a quadratic performance functional
N, N,
J(t)= r(t+ i)  y(t + ilt)]2+A [Au(t+i)]2 (3.3.1)
i=O
where { r(t+i) } is the sequence of future setpoint values (assumed to be known), {y(t+ilt)} is the sequence of predicted future values of the output, {Au(t+i)} is the sequence of future control increments, A is the movesuppression parameter, and parameters Ny and Nu are the socalled prediction and control horizons, respectively. The terms in the first summation penalize future predicted errors, and the terms in the second summation penalize excessive control energy. Of the sequence of predicted control increments, only the first is applied, and the optimization problem is solved again in the next time instant.
When modeling uncertainties; process constraints, and the presence of exogenous disturbances are ignored, and the following nominal predictive control law in feedback form results as a consequence of the minimization of the quadratic functional (3.3.1) (Crisalle, et al., 1989)
R(z) u(z) = T(z) r(z)  S y(z) (3.3.1b)
where R(z), S(z), and T(z) are polynomial operators. The nominal closedloop system with the RST configuration (3.3.1b) is shown in Figure 3.1, where the plant is assumed to be g(z) = gj(z). For the purpose of this chapter, it is useful to remark that polynomials
47
R(z) and S(z) are of degree n, that is, of degree equal to that of the nominal plant denominator polynomial A(z). Explicit formulas for the three predictive control polynomials can be found in Crisalle, et al. (1989) and in Section 2.6.1.
d
T
S
Figure 31. Structure of a nominal predictive controller.
Following the procedure discussed in the previous chapter, it is possible to parametrize a stabilizing nominal RST predictive controller of the form (3.3. lb) according to WienerHopf theory (Youla, et al., 1976) to yield a family of nominally stabilizing predictive controllers. The resulting parametrized structure is shown in Figure 32, where the Youla parameter Q(z) is an arbitrary proper and stable transfer function. Furthermore, the stable and proper transfer functions N(z), M(z), X(z), and Y(z) shown in Figure 3.2 satisfy the Diophantine equation N(z) X(z) + M(z) Y(z) =1, and are functions of the nominal predictive controller and the nominal plant through the fractional representations A(z) N(z):= B(z)
A,(Z) A,(Z) (3.3.2)
and
R(z) S(z)
A2(z) ' A2(z) (3.3.3)
where polynomials A,(z) and A2(z) have real coefficients, are both of degree n, and are obtained by factoring the characteristic closedloop polynomial in the form A(z)R(z)+B(z)S(z)= A,(z)A1(z). If the characteristic polynomial contains complex poles, then Ai(z) and A2(z) are constructed such that each complexconjugate pair is contained in either AI(z) or A2(z) to ensure that each polynomial has only real coefficients.
48
Finally, the operator Z(z) = znA2(z)/T(z) shown in Figure 32 is the setpoint advancement transfer function. In Chapter 2, the following theorem characterizing the robust stability of the parametrized control loop was derived.
X +MQ
6 d
r re e B +
S Y YNQ A
X+MQ
Figure 32. Structure of a parameterized predictive controller for a plant
with a) additive uncertainty, and b) multiplicative uncertainty.
Theorem 3.1. A necessary and sufficient condition for the robust stability of the closedloop system of Figure 32 is the inequality condition W(z)C(Z)_ (Z) 11 < 1 (3.3.4)
where
X(z) + M(z)Q(z)
Y(z) N(z)Q(z) (3.3.5)
and
S (z):= M(z) [Y(z)  N(z)Q(z)] (3.3.6)
Proof. See Chapter 2, and for the standard H_ approach, consult Francis (1987) and Doyle, et al. (1992). Also, in the case of multiplicative uncertainty, the condition (3.3.4)
49
should read 11 W(z)T7(z) 1 < 1, where T,(z) = 1  4(z) is the complementary sensitivity of the loop.
The dependence of the robust stability condition on the parameter Q(z) is made more evident by writing (3.3.5) in the modelmatching form T (W  T(W)Q(Z) 11 < (3.3.7) where
T (z)= W(z)X(z)M(z) (3.3.8a)
T2(z) = W(z)M(z)2
for the additive uncertainty case and
T(z)= W(z)N(z)X(z)
(3.3.8b)
T2(z) = W(z)N(z)M(z)
where a multiplicative uncertainty description is used.
Hence, a robustly stabilizing controller may be obtained by solving the modelmatching problem (3.3.7) for Q(z) e H., and then employing the controller as shown in Figure 32. This is the approach taken in Chapter 2 to produce robust predictive control designs. Although the resulting parametrized controller is shown to preserve the nominal servo (setpoint tracking) performance of the original RST predictive controller, it offers no guarantees on the nominal regulation (disturbance rejection) performance of the controller in the presence of input constraints. This shortcoming is addressed in the following subsection by combining the H. problem (3.3.7) with an '1 problem which takes into account the effect of the disturbance on the constrained variable.
3.3.2 Nominal Regulation and Robust Stability as an l1/H_ Problem
In this section, the problem of robust stabilization of a predictive control loop under persistent disturbances and manipulatedvariable constraints is formulated as an 11/Hproblem. Although the formulation is made for the case of a constrained plant input, this
50
choice fully illustrates the design procedure, which can be used as a paradigm for the case of a similar constraint on the output variable. To emphasize the focus on the regulator design, set r(k) = 0 in the structure shown in Figure 32. Let T,4(z) represent the nominal closedloop transfer function between the manipulated variable and the disturbance, that is, u(z) = Ted(z)d(z) when A(z) = 0. After carrying out elementary blockdiagram algebra on the structure shown in Figure 32, it follows that Td(Z) = Td,I(Z) + ,2(Z)Q(Z) (3.3.9)
where
T..(z) = M(z)X(z) (3.3.10)
and
'.2 (z) = M(z)2 (3.3. 1Ob)
Notice that both u(k) and d(k) are /.signals because they are bounded. Consider the operator 1norm = Jtjl, where { ti } is the impulseresponse sequence of Tud(z). Obviously TI = Jt, hence the operator 1norm is equal to the 11 norm of the impulse response sequence. Using the fact that I 1l depends on the infinity norms of the signals u(k) and d(k) through the relationship (cf Dahleh and DiazBobillo, 1995) Na = sup IIuIL (3.3.11)
it follows that infinitynorm constraints on the input can be addressed by the regulator design strategy by seeking to minimize (3.3.11). Hence, the problem of robust stability with constraints on the input can be stated as the following l1/H. design problem: o inf Tud(Z)+Tud,2()Q( (3.3.12)
subject to
T< (Z)  T1()Q(z) < (3.3.13)
Clearly, the l problem (3.3.12) addresses the nominal regulation performance of the controller, where tO is the maximal possible 1. norm of the input., that is, luII pto for
51
all bounded disturbances satisfying ydiL_ 1. However, from Theorem 3.1, it follows that the H problem (3.3.13) addresses the robust stability of the closedloop with respect to the modeling uncertainty.
The 11/H. problem (3.3.12)  (3.3.13) must be solved for an optimizing Youla parameter Q(z) e H. that minimizes the 1norm of Td and simultaneously satisfies the infinitynorm constraint (3.3.13). If the optimal solution satisfies 0 < /, then the robust predictive regulator satisfies the constraint specifications on the input. Conversely, if tO >P, it can be conclusively stated that there is no linear robustlystabilizing predictive regulator that can satisfy the input constraints. Hence, the 11/H. design paradigm establishes the limit of performance of the robust linear predictive control structure.
3.4 Synthesis Procedure
A solution to the l1/H_ problem of the type (3.3.12)  (3.3.13) has been proposed by Sznaier (1994), who shows that a solution can be obtained by truncating the 1i norm after a finite horizon N to obtain IIN 'tj. This simplification leads to great
computational simplifications because the truncated l problem and the H. problem can be decoupled and solved sequentially. A subtlety of the approach lies in ensuring that the truncated 11 performance measure also leads to adequate performance at times beyond the truncation horizon. This is accomplished by introducing a bilinear transformation that places the poles of the Youla parameter inside a circle of radius 3< 1, hence guaranteeing the rapid decay of the response modes. Thus, the original problem (3.3.12)  (3.3.13) must be transformed using the mapping z = 3, to yield operators of the form O) = G(&). The resulting transformed problem is denoted as l1/H_.8.
The l1/H_,8 problem considered in Sznaier (1994) is of the form
6= inf 7; W+7CV)QC)) (3.3.14)
52
subject to
R(z)  Q (z) < y1
(3.3.15)
The accuracy of the solution po depends on the truncation horizon N. Sznaier (1994) shows that as 3  1, the modified solution satisfies o < go < + E provided that
N No = [log E(1J)  log K] where K:= 7j(z) + T2(z) +(ly) R(z) . Let
yAR ybR yARbR ... y R'bR
CRARN1x dR cRb R ... CRA RN2 R 0 L(q) :=0 dR + 0
CRA~x CRbR 0
cRx 0 0 ... dR 0
/ log 8
0 q0
0
0
(3.3.16)
0 q0
0
qN1 qN2
q0
(3.3.17)
where h(2) := cR(ZJ AR)bR +dR, x:= L 2, y :=L 2, and L controllability and observability grammians of R( ), respectively.
and L, are the discrete
Theorem 3.2. A suboptimal solution to the /1/H, problem (3.3.14)  (3.3.15) with cost po p< p< + E is given by Q(~)= QO~)~NU(~)(3.3.17b) where 0"(2) = 2 qt' with q0 = (q,,..qN )T solves the finitedimensional convex
optimization problem:
q" = argmin 11t, + 10,
qe9tN
IL(q)J2 < y
and where Q;"() solves the unconstrained Nehari extension problem
") argmin R( )_ ()N()1 QReH.
(3.3.18) (3.3.19)
with t, and T given by
53
'tio' t20 0 ... 0
t( , t2 t20 0
t. 0 (3.3.20)
tINI t2N ' 21 t 20
where R( )e H, and th, t2i, and q;, are the first N impulse response coefficients of TId.1( T and QR(j), respectively, and N is selected according to (3.3.16).
Proof. See Theorem 2 in Sznaier (1994).
Hence, from Theorem 3.2, it is possible to construct a solution to the l1/H_,5 problem by solving the finitedimensional constrained optimization problem (3.3.18) using a numerical method for nondifferentiable optimization. Then the unconstrained Nehari extension (3.3.19) is solved using, for example, the technique proposed by Rotstein and Sideris (1993). The separate solutions are combined as indicated in (3.3.17b).
The l1/H. design problem (3.3.12)  (3.3.13) for robust predictive regulation must be put in the standard l1/H_,8 form (3.3.14)  (3.3.15) in order to exploit Theorem 3.2 for arriving at a solution. First, one must rewrite (3.3.12)  (3.3.13) in terms of the operators T,(i), T2(), Td.i(2), and Td,2 U) obtained through the bilinear transformation z = 3Z. Second, transform the modelmatching problem (3.3.13) into the Nehari extension problem (3.3.15) by first writing ()= (()(() where t,(() is an is an allpass function and TP() is a stable minimumphase function. Then find T_ (z) = 7p,(I/ ) and carry out the ap
stable/antistable decomposition
(,U) () =( ) + s (U) (3.3.21)
where R(z) e H and Rs(z) e H,. Using the property that t()0(u) = O(6) , it is possible to write the equalities
tC)7 0(2)) _ = 1~p(Z)7 (Z) p (Z)$(Z) = 1(;)+ ( ((z) nip 1(3.3.22)
Defining
(3.3.23)
O' ( ) . k Z( )  I ,(2)^(U)
54
equality (3.3.15) reduces to the desired form
()  T(j)Q(u) u(2)  ' (Z) (3.3.24)
Third, using the definition (3.3.17), rewrite the operators of the l problem (3.3.12) to establish the equality
ud. +u+d,2(Z)Qf ) = ufl(Z)+( (2) (3.3.25)
where
4d() I (z)+ d2(z)T (+) t(d) (3.3.26)
and
(U) Td()T (u) (3.3.27)
Hence, the desired l1/H_, problem in the standard form (3.3.14)  (3.3.15) follows from (3.3.25) and (3.3.24) using the maps (3.3.23), (3.3.36) and (3.3.27).
The synthesis technique proceeds as follows. Theorem 3.2 is used to yield a solution S(?) = ~ (z)+ tN,$(Z), which combined with (3.3.23) yields ]) . ( )" + N (3.3.28)
and recover the original Youla parameter associated with problem (3.3.12)  (3.3.13) using the inverse bilinear transformation Q(z) = QR(Z/3) (3.3.29)
Finally, the robust li/H_ predictive regulator is implemented as shown in Figure 32, making use of the Youla parameter (3.3.29) and the control operators N(z), M(z), X(z), and Y(z) defined in (3.3.2)  (3.3.3).
3.5 Example
To illustrate the applicability of the design technique developed in the previous sections, consider a secondorder nominal plant model that has two poles outside of the unit circle
55
z+0.2
z2 0.6z+1.12 (3.5.1)
To show the flexibility of the method, a multiplicative uncertainty description is used here to characterize the family of plants to be stabilized. The uncertainty weight is given below as
z 0.1111
W(z)= +0.117 (3.5.2)
Two regulators are designed: (i) a nominal predictive controller (NPC) in the standard RST form, and (ii) a constrained robust predictive controller (CRPC) with integral action, synthesized using the 1i/H technique discussed in the previous section.
The nominal predictive controller is realized adopting the design parameters Ny = 7, Nu = 3, and A = 0 to yield the RST polynomials R(z)= z2  0.8008z  0.1992 (3.5.3)
S(z)=1.4480z2 1.6909z+1.1155 (3.5.4)
T(z)= 0.1148z7 +0.3196z6 +0.1965z'  0.0523Z4  0.0757z3 +0.1339z2 +0.236z
(3.5.5)
It can be readily verified that the NPC design stabilizes the closed loop when the uncertainty A(z) is neglected. However, the NPC controller is not robustly stabilizing, because it violates the robust stability condition (3.3.13), that is, 1(z) T(z)Q(z)lj = 5.3 > 1 calculated using Q(z) = 0. This implies that the nominal predictive controller will fail to stabilize the closed loop for some plants belonging to the family of uncertain plants. The nominal performance of the NPC regulator is characterized with a peaktopeak norm d1 = 12.14, which indicates that bounded disturbances exist with norm 11dI 1, which cause input sequences {uk} displaying peaks with amplitude values as large 12.14. This shows that when the worstcase disturbance is affecting the plant, the NPC design can only cope
56
with input constraints of the form ju # only for # > 12.14. Tighter constraints will lead to controlaction saturation.
The CRPC design is carried out as indicated in Section 4. The control operators M(z), N(z), Y(x) and X(z) are produced using (3.3.2) and (3.3.3). A slight modification is introduced to augment the plant with a fictitious integrator which is later associated with the final Youla parameter, in order to produce a controller with integral action. Details have already been given in the previous chapter. The design choice 6 = 0.9852 is used for the bilinear transformation and the truncation horizon is selected as N = 150. Solving the l1/H.S, problem yields a high order Youla parameter Q(z), as expected from the large value of truncation horizon that was selected to achieve better accuracy. The CRPC design obtained by combining the Youla parameter with the operators defined in (3.3.2)  (3.3.3) is robustly stabilizing because 1T (z) T(z)Q(z)l 0.94 < 1. Furthermore, the nominal regulation performance is greatly enhanced with respect to the NPC case because feTJ, = 1.72, allowing the CRPC design to deliver saturationfree dynamics for much tighter constraints (P > 1.72). The tolerable constraint limit is improved by a factor of 7 over the NPC design.
The Youla parameter of the CRPC design was simplified using simple modelorder reduction techniques (Safonov and Chiang, 1988) to yield an implementable transfer function Q(z) of order 6. The reducedorder CRPC controller displays no appreciable performance degradation with respect to the fullorder controller in terms of the values of the 11 and H norms. Furthermore, dynamic simulations revealed that the two controllers display virtually indistinguishable closedloop responses.
In Figure 33, the performance of the nominal (NPC) and robust (CRPC) designs is compared by carrying out a regulation test. In these simulations, the nominal model is assumed to match exactly the actual plant, that is, A(z) = 0. A persistent disturbance d(k) shown in Figure 33c is introduced at timeinstant k = 0, and then has a constant nonzero value at instant k = 50. Figure 33b shows the superior performance on the CRPC design
57
(a)
0 . . ...
0
2 0
2
4
0 50 100 150 200
12
(b)
8 
4
0
4
8
12
0 50 100 150 200
2
(C)
1
0
0 50 100 150 200
Time, k
Figure 33. Comparison of the performance of the NPC regulator and the robust (CRPC) regulator designed for an exactlyknown plant. a) output, b) input and, c) disturbance.
58
which produces a much more attenuated sequence of control actions { u(k) I as a consequence of the smaller value of the norm JTdJ.d ' Figure 34b shows that the NPC develops peaks of magnitude as high as 12 whereas the CRPC design leads to much smaller magnitudes which are no larger than 1.5. This effect is clearly a consequence of the 11 part of the CRPC design which deliberately attenuates the magnitude of the maximal input peak. Hence, the CRPC design is inherently capable of satisfying tighter saturation constraints. Finally, Figure 34a shows the output responses where the CRPC controller also displays farimproved performance. The superior output performance is a side benefit of the CRPC controller since the design was obtained without explicitly aiming to minimize the 11 norm of the output. Note that the CRPC regulator rejects the effect of the constant disturbance after instant k = 50 due to the builtin integral action.
Further insight into the simulation results reported in Figure 33 can be gained with another regulation test where the manipulated variable u(k) must satisfy the saturationtypeconstraints 1.8 u(k) 1.8. The same disturbance d(k) shown in Figure 33c is introduced in the output of the plant. The superiority of the CRPC design is demonstrated in Figure 34a where it is shown that the introduction of the disturbance produces an unfavorable output response for the NPC design with peaks of magnitude between 2 and 7 over a prolonged period of time. However, the CRPC regulator successfully rejects the disturbance quickly and efficiently, demonstrating peaks with magnitude less than 2. The unacceptable performance of the NPC controller is a consequence of the input saturation (Figure 33b), and the fact that the plant is openloop unstable. Furthermore, Figure 33b shows that the CRPC regulator satisfies the input constraint, avoiding saturation. The results can be even more dramatic if the saturation limits are lowered to values less than 1.78. In that case, the NPC is not able to stabilize the system in the presence of the saturation constraints and the disturbance shown in Figure 33c. The CRPC design still performs well with saturation limits between 1.2 and 1.8 for the same disturbance. Finally on the last regulation test, a different unstable plant belonging to the uncertain family
59
8
6 (a)
4 2 0
2
4 
6
0 50 100 150 200
2
1.5 (b)
1 .
0.5
0
S0.5
11
0 50 100 150 200
Time, k
Figure 34. Comparison of the performance of the NPC, and the CRPC regulator, when the input u(k) satisfies the saturation constraints 1.8!! u(k)
:! 1. 8.
characterized by the weight in (3.5.2) is used. The same disturbance as in the previous two regulation tests is used to compare the two designs. The results are shown in Figure 35 where it is clear that the NPC design is destabilizing while the CRPC regulator successfully rejects the disturbance. The instability of the NPC design in this case is due to the fact that a plant different than the nominal one was used and shows the inability of the NPC
60
regulator to be robustly stabilizing. The CRPC regulator efficiently stabilizes the plant with mild control action as shown in Figure 35b.
For completeness, it is reported here that a purely 11 design may yield a lower value for the one norm (JjTd,= 1.27), but the resulting controller is not internally stabilizing because the Youla parameter is such that Q(z) + . However, an unconstrained H. design may yield a better value for the c norm (17,(z) T2(z)Q(z)I_= 0.84), but the II norm is too large (ITjd 1 = 3.82).
3.5 Conclusions
A systematic 11/H method for robustifying predictive regulators is proposed for control system design in the presence of constraints on the manipulated variable. Experience with these methods shows that while the unconstrained design technique leads to controllers of reasonable order, the constrained design is likely to result in very highorder controllers. Although model reduction techniques may be used to successfully approximate the 11/H regulator, the model reduction process may not be required given that large controller orders are deemed acceptable in predictive control schemes where highorder convolution models are used to represent the nominal plant.
. . . . . . . . .. . .
50
. .
50
Time, k
Figure 35. Comparison of the performance of the NPC and the CRPC regulator, designed for a perturbed plant belonging to the uncertainty description.
(6 I
10 7.5
5
2.5
0
2.5
5
7.5
10
0
10
150
200
(b)
5
0
5
0
0
U
100
100
10
0
150
200

CHAPTER 4
ROBUST STABILIZATION OF BILINEAR SYSTEMS WITH INPUT SATURATION
4.1 Introduction
Bilinear systems are a special class of nonlinear systems that has attracted much attention in the last two decades (Mohler, 1991). This interest is due to the fact that many engineering, biological, ecological, socioeconomic, and nuclear processes have been shown to be more accurately described in terms of a bilinear model (Mohler and Kolodziej, 1980). Over the years, a number of publications have been devoted to the analysis of the structural properties and the problems of identification, estimation, and stabilization of bilinear systems (Benallou, et al., 1988; Mohler, 1991; and their references). Some of the different stabilizing control approaches include Genesio and Tesi (1988), where output feedback is used to stabilize a SISO bilinear system and Benallou, et al. (1988), where quadratic state feedback is used to stabilize a MIMO bilinear system and optimize a special quadratic cost. In Longchamp (1980), sliding modes are used to derive a discontinuous control law for SISO bilinear systems. The local stabilization of the origin is guaranteed if certain conditions are satisfied. Finally, in Yang, et al. (1989), sufficient stabilizability conditions are given for timevarying bilinear systems via output feedback.
Many practical control problems involve constraints, such as actuator saturation. The stability of linear systems with saturating actuators assuming exact knowledge of the plant has been studied extensively (Bernstein and Michel, 1995; Bitsoris and Vassilaki, 1988; Chen and Wang, 1988; Krikelis and Barkas, 1984; Lin and A. Saberi,1993; Sussmann, et al., 1994), however, this is not the case for bilinear systems. In addition, the mathematical
62
63
modeling of physical systems always involves some kind of uncertainty associated with the nominal model. For uncertain linear systems, modified Riccati equations (Kim and Bien, 1994) and Lyapynov functions with a subsequent quadratic optimization (Tarbouriech, et al., 1994) have been used to treat the saturation problem. Apparently, there is a lack of results in the literature on the stabilization of uncertain bilinear systems with input saturation (when no uncertainty is present, see Niculescu, et al. (1995)). We are thus motivated to study the robust stabilization of inputconstrained bilinear systems in the presence of uncertainty.
In this chapter, a statefeedback control law is used to locally stabilize a continuoustime multipleinput multipleoutput (MIMO) bilinear system subject to actuator saturation. Then, robust stability tests are developed in the form of sufficient conditions that guarantee the stability of the saturating nonlinear system in the presence of normbounded perturbations. The derived conditions, given in terms of scalar matrixnorm inequalities, are characterized by their simplicity and ease of calculation. Two uncertainty descriptions are included, namely, a nonlinear bounded perturbation, and linear bounded perturbations on the system matrices. The detailed stability analysis is carried out for the general bilinear regulator problem, avoiding any simplifications on the structure of the system (homogeneous or dyadic models, for example).
4.2 Preliminaries
The matrix measure is a function p: C"'" + R p(A)= lim
E+0* E
where 1 is an induced matrixnorm on C"". For the usual 1, 2 and infinity induced norms, the matrix measure is given by the following simple formulas, respectively:
,,(A) = max Re (ajj + Y , I j
64
p2(A)= AmA*jA
p(A)= max(Re(a,i)+I a,1 I
where A* is the hermitian of matrix A, and A,,. represents the maximum eigenvalue. Some matrix measure properties that will be of use are:
(i) p() is a convex function,
(ii) u((A) = 3p(A), and
(iii) Re A(A) p(A),
where A is any eigenvalue of the matrix A. A class of matrix pairs (A, B), A e Rnxn ,B e R"x", is introduced for the stabilization problem, defined as
'9:= {(A, B)13 F e R"', w e [0, 1): p(A + L BF) < 0
The matrix classU corresponds to the class of systems for which there exists a statefeedback law u(t) = Fx(t) such that p(A + (+w) BF) < 0.
4.3 Main Results
Consider an inputconstrained uncertain bilinear MIMO system described by the equations
x(t)= Ax(t)+ Bsatu(t)+ Nix(t)satui(t)+ g(x(t),t) (4.1)
y(t)=Cx(t) (4.2)
where x(t) = [x/ x2 . X]T e R" is the state vector with initial state x(0) = xO, u(t) = [U 2 ...u,,]T E R' is the input vector, and y(t) = [y, y2 ... y, E RP is the output vector, and A, B, Ni, and C are constant real matrices of appropriate dimensions. The
65
vector function g(x(t),t) e R" represents a nonlinear modeling uncertainty with the property
g(x(0),t):! yAx(t) (4.3) where y is a positive real constant and the operator 11 represents a vector norm. The saturation function sat u(t) is defined as follows (see Figure 41): sat u(t )= [sat u, (t) sat u,(t) ... sat u,, (t)]' (4.4)
where
sat ui(t)= u , u,
and ui and , are real scalars representing lower and upper saturation limits, respectively.
The objective is to find the state feedback control law
u(t) = Fx(t) (4.6)
where F is a constant matrix, that renders the closedloop (4.1)  (4.2) and (4.6) asymptotically stable for all modeling uncertainties that satisfy the norm bound (4.3).
Based on the definition of the vectornorm function, it is easy to verify that
sat u(t)  u(t) Iu(t) (4.7)
2 2
It should be noted that inequality (4.7) is used here because of its simplicity to facilitate the exposition of the ideas in this chapter (see also Chen and Wang, 1988; Niculescu, et al., 1996; Krikelis and Barkas, 1984 where (4.7) is used). Other possibly less conservative conditions may be used in conjunction with the developments below. The interested reader is referred to Kim and Bien (1994); Niculescu, et al. (1995); and Tarbouriech, et al. (1994) where alternative conditions to (4.7) are used. Theorem 4.1 gives conditions that guarantee the asymptotic stability of the closed loop:
66
sat u,
slope =r
/ssopp= =I
Figure 41. Actuator saturation with bounds u and i, and sectors [0,1]
(slope = 1), and [r,1] (slope = r).
Theorem 4.1. Suppose that the nonlinear perturbation in the constrained bilinear system (4.1)  (4.2) satisfies (4.3). Let the robust statefeedback matrix F be chosen to satisfy the inequality
pu(A  1BF )+jBjF+ y < 0 2+ (4.8)
Then, the uncertain constrained bilinear system is asymptotically stable if the initial state belongs to the region of attraction defined by the inequality p( A BF )+ JBF+ y
x1 011 <  2(4.9) Proof From (4.1) and (4.6), the closedloop system is written as
i(t)= Ax(t) + Bu(t) + B sat u(t)  !u(t) + XNAx(t)sat uj(t) + g(x(t),t) (4.10)
= Xx(t) + B sat u(t)  ju(t) + XNix(t) sat u,(t) + g(x(t),t) (4.11)
67
where the term iBu(t) has been added and subtracted in (4.10), and where X = A +BF. The solution to (4.11) for t 0, can be expressed as the integral equation x(t) = e'xot+ ' eA(){B[sat u(s)ku(s)]+ ' Nix(s)sat(Fix(s))+g(x(s),s) ds(4.12) where ui(t) = Fix(t) where F is the ith row of matrix F. Taking the norm of both sides in (4.12), using the inequality (Vidyasagar, 1993) e^ e (^I, t 0 (4.13)
and invoking (4.7) yields
x(t) eA'Ixo+ f eP(A( {'" B=u(s)+ 'N x(s)sat(IFx(s)) + g(x(s), s)Ids
eu*'\tx + f e(A'")( JBIIFI+ y) x(s)jds + Je '(A)() N JIFX(S)2ds(4.14) where we have used the inequalities N x(s)sat(fx(s)) = 'XINSIII (S )II (4.15)
n'lNilllFx(s)l lx(s)ll t INj  F\\Ix(s)2 Now consider the scalar differential equation
(tO [M(X) + LJBIFII + y]z(t) +  NIIFz(t)2, t 0
2 (4.16)
z(O) =x1
The solution of (4.16) is unique and is given by the expression (Bellman, 1970) z(t) = ep)'z(0) + Ie ( '4( B(+ IBF11+ y)z(s)ds + I eP ( t 11NJI1F1z(s)2ds (4.17) From inequality (4.14), it follows that Ix(t)j z(t), for t 0 (4.18)
therefore, asymptotic stability for (4.16), that is, z(t) > 0 when t * co, implies asymptotic stability for the original system (4.11). From the Poincar6Lyapunov theorem (cf Bellman, 1970), it follows that (4.16) is asymptotically stable if P(J) + L1 B111F 11+ y < 0 and if z(0) is sufficiently small. The asymptotic behavior of z(t) as a function of z(O) is examined in
68
detail in the next paragraph (see case (i)), where it is shown that for a nonlinear differential equation of the form of (4.16) z(t) > 0 as t a  when the initial state satisfies
z(0)< u(ALBF)+
The asymptotic behavior of a nonlinear equation of the type of (4.16) is examined in some detail in this paragraph. Consider the nonlinear differential equation i(t) = az(t) + bz(t)2 (4.16A)
where b > 0, a # 0, and z(O) = zo. The solution z(t) of (A) can be obtained using separation of variables and integrating to get
z(t) azoe (4.16B)
a + bzo  bzoe"
The equilibrium points of (4.16A) are found by setting the derivative of z(t) equal to zero to obtain z, = 0 and z, = a/b. The asymptotic behavior of the solution (as t  o) depends on the sign of a, and on the initial condition zo as follows:
(i) If a < 0, then z(t) + 0 for zo < b/a, and z(t) +  in finite time for all zo> b/a.
(ii) If a > 0, then z(t) > b/a for zo < 0, and z(t)   in finite time for all zo> 0.
The following remarks are in order:
1. The bounds obtained using the sufficient conditions (4.8) and (4.9) vary with the
chosen norm and the corresponding matrix measure (Vidyasagar, 1993). It is possible therefore, that with a given norm and matrix measure, one can conclude stability, while with others the stability condition may not hold. In particular, adopting the 1 or infinity vector norms simplifies the computations, avoiding possibly more expensive singularvalue calculations. The problem of choosing a suitable norm and matrix measure to improve the stability condition resembles that of finding
69
an appropriate Lyapunov function in the well known and widely used Lyapunov
techniques for determining stability.
2. An alternative to the original problem formulation (4.1)  (4.2) would be to consider
X1Nxsatu +g(x,t) as a normbounded perturbation, which eliminates the bilinearity of the system and thus apply results developed for linear uncertain systems with saturation (Kim and Bien, 1994; or Niculescu, et al., 1996). Of course, this would yield rather conservative results, since the model would allow for
perturbations that may never occur in the actual system.
3. In order to reduce conservatism in the developed sufficient conditions, one could use
a different condition in place of (4.7), as in the approaches of Kim and Bien (1994) and Tarbouriech et al., (1994). It should be noted, however, that the conditions developed here are in a sense simpler to use than solving modified Riccati equations (Kim and Bien, 1994) or following a Lyapunov function approach that includes
solving a quadratic optimization problem (Tarbouriech et al., 1994).
4. Theorem 4.1 concerns a nonlinear saturation that lies in the sector [0, 1] as defined in
(4.4)  (4.5). The results of Theorem 4.1 can be improved when the saturation nonlinearity lies inside a finite sector [r, 1], 0 5 r 1 as shown in Figure 41
(Vidyasagar, 1993). In this case, (4.7) becomes
1 1
satu(t) (1+ r)u(t) 5 (1  r)jju(t) (4.19)
2 2
and conditions (4.8)  (4.9) of Theorem 4.1 are written as
p[A {(l + r)BF]+ (1  r)IBF+ y <0 (4.20)
with the domain of attraction
p[A  '(1 + r)BF] + 4(1  r)BF+ (4.21)
Besides the wellknown 1, 2, and infinity norms, other induced norms and matrix measures involving weighting parameters may be utilized to reduce conservativeness in the
70
stability conditions. As an example, consider the following matrix norm and corresponding measure (Mori, et al., 1983)
AL = xAmax ax a, + aj, (4.14)
Optimization with respect to the arbitrary weighting parameters w, is likely to yield less conservative robust stability conditions. This topic merits further investigation and is not pursued here. Also, similarity matrix transformations can be used to reduce conservativeness in the robustness conditions. Consider a transformation matrix T such that T' exists and the pair (T'AT, TB) belongs to the class U. Then, conditions (4.8) and (4.9) of Theorem 4.1 can readily be written as p(T'AT  L T'BF)+LT'BlFI+ y'<0 (4.22)
and
pu(T'AT  'T'BF)+ B +F+ ' Ix01 < 2 I Bll l+(4.23)
I," IT'NTF
where y' is the bound in T'g(x,t) < y'Ix(t)jj. Thus, matrix T can be selected in order to satisfy (4.22), and simultaneously increase the region of attraction if T'NT
Corollary 4.1. Consider the bilinear system with perturbed matrices A, B and N, given by
x(t) = (A + AA)x(t) + (B + AB)satu(t) + (N, + AN, )x(t)satu, (t) (4.24) where the matrix perturbations satisfy the norm bounds AA l sa, ABJ /, jjAN
and
71
1'[A '(l+ r)BF]+a+ (l r)(JIBj+#)IIF(4 IX0 < ~(h 2 (4.27)
Proof It is analogous to the proof of Theorem 4.1 and follows readily after invoking (4.19) in place of (4.7), and using the bounds (4.23) instead of (4.3).
It is useful to accompany the sufficient stability conditions derived thus far for an uncertain bilinear system with a Design Procedure for the selection of an appropriate feedback matrix F.
From the previous analysis, an iterative procedure can be proposed as a guideline for selecting the matrix F to satisfy the robust stability conditions. (see Chen and Wang, 1988; Kim and Bien, 1994 for the linear case)
Step 1: Check the norm bound of the plant uncertainty and select distinct negative eigenvalues A, i = 1, 2, ... n.
Step 2: Find the control law matrix F using a standard eigenvalueassignment technique. Then, check whether condition (4.8) is satisfied. If so stop; a robust matrix F has been obtained, otherwise, continue with Step 3.
Step 3: Shift the system eigenvalues to the left using A, = A,  AA, i =1, 2, ... n. where AA >0; then go back to step 1.
4.4 Example
Consider an uncertain bilinear system defined as in (4.1) with dynamics described by A 0.1 0.2 ], b = ~N =0 0.2]
0.8 2.1_ 0.2 _0.1
and the nonlinear uncertainty bound y= 0.3. The operational range of the saturating actuator lies in the sector [1/3, 1], and therefore, inequalities (4.20) and (4.21) will be used as prescribed by Remark 4. Notice that the openloop system is unstable since matrix A has one positive eigenvalue.
72
By a standard poleplacement technique, we take the eigenvalues of X=Aj(l+)BF to be 2.09 0.4i and then find the feedback matrix
F=[2.95 0.25]. Using condition (4.20) now, the usual 1, 2, and infinity norms are obtained.
For the 1norm, p[A2BF]+jJB IjFj+y = 0.03 > 0, therefore nothing can be concluded concerning the stability of the system.
For the 2norm, p[A  BF] + kIIBIIFI +)y = 0.658 with a corresponding region of attraction given by 11x0 1 1.11 while for the onorm, j[A  BF]+11BF+y =
0.298 with 11x011 0.506. Therefore, it follows from Theorem 4.1 and remark 4 that the bilinear system is asymptotically stable with a region of attraction which is at least 11x011 <
1.11.
When there is no uncertainty present, the results are even better; using the 2norm which for this example gives the largest bound, the following is obtained: p[A BF]+1jBjjFI+ y = 0.9581 with an adequate region of attraction given by 11x011 < 1.618. It should be noted that the approach of Lin and Saberi (1993) and Sussmann, et al. (1994) developed for openloop stable linear systems cannot be applied in this example since A is not hurwitz. Also, in Longchamp (1980), an example is given with a much smaller region of attraction (11x011 0.255), which is claimed to be large enough for many practical applications.
4.5 Conclusions
In this chapter, results derived for linear systems are extended to bilinear systems with a subsequent characterization of the region of attraction. Theorem 4.1 and Corollary 4.1 yield sufficient conditions for the robust stabilization of general bilinear systems via state feedback. The easily calculable conditions are expressed in terms of matrix norms and measures, and do not require the solution of Riccati or Lyapunov equations.
73
A central requirement is that the matrix measure of the closed loop matrix can be made negative enough by appropriate linear feedback to overcome the effect of perturbations. Current and future research will focus on extending the results to the case in which some of the states are not available, and therefore must be estimated via an appropriate extended Kalman filter. Finally, a trialanderror design procedure has been proposed to find the feedback matrix, and an example is given to illustrate the results.
CHAPTER 5
ROBUST STABILIZATION OF DELAYED BILINEAR SYSTEMS
5.1 Introduction
Bilinear systems as a simple class of nonlinear systems have received continuous attention in the literature for approximately twenty five years (Mohler and Kolodziej, 1980). Such interest may be due to the fact that various processes in engineering and science can be modeled as bilinear systems. The earlier publications in the field are mostly concerned with the analysis and structural properties of bilinear systems (Mohler, 1991). Later, optimal control theory and quadratic performance indices were used for the design of controllers for bilinear systems (cf. Benallou, et al., 1988; and included references). The stability of exact bilinear models has also been studied using various methods (Mohler, 1991; Genesio and Tesi, 1988; Longchamp, 1980; and Yang, et al., 1989).
It is well known that various engineering systems involve time delays in the state and/or the control variables. These time delays, which are often ignored to make the theoretical analysis simpler, can be a source of instability. For this reason, much work has concentrated on the analysis of linear systems with delays (cf. Choi, 1994; Chen and Latchman, 1995; and their references). In contrast, the corresponding problem of analyzing the stability of bilinear timedelay models has not been given comparable attention. In a recent publication, Lu and Wey (1993) examine the stability of a bilinear system with delay in the state, and derive a sufficient condition using the Lyapunov direct method. The LuWey approach assumes that the system model is exactly known, however, the mathematical modeling of physical systems always involves uncertainties
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associated with the nominal model. The treatment of uncertain bilinear systems with time delays and their stability properties appears to be completely absent from the literature.
In this chapter, a statefeedback controller approach is employed to stabilize a continuoustime multipleinput multipleoutput bilinear system with delay in the state variables. The nominal bilinear model used by Lu and Wey (1993) is extended to include new elements representing nonlinear, possibly timevarying modeling uncertainty.
5.2 Preliminaries The matrix measure is a function p : C""  R p(A) = lim II+E1
E+0*, E
where 1 is an induced matrixnorm on C"". For the usual 1, 2, and infinity induced norms, the matrix measure is given by the following simple formulas:
pu(A) = max Re(a) + ,I aj I A ( AjJ
JU2 (A) A* + A
2
/p_(A) =max Re(aii) + YIaj I where A* is the Hermitian of matrix A, and Ama represents the maximum eigenvalue.
5.3 System and Assumptions Consider the uncertain MIMO bilinear system with timedelay in the state represented by the equations
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i(t)= Ax(t)+ Ax(t , )+ Bu(t)+ Nix(t)u,(t)+ gj(x(t),t) + g2(x(t  r),t) =1 (5.1)
x(0)=4(6) , e[r,0]
y(t) = Cx(t) (5.2)
where x(t) e R" is the state vector with initial state x(O) = xO; u(t) e R' is the input vector; y(t) E RP is the output vector, and A,, B, Ni, and C are constant matrices of appropriate dimensions. g(t) is a continuous vectorvalued initial function; and r> 0 is the time delay. The vector functions gl(x(t),t) E R" and g2(x(t'r)t) e R" represent nonlinear modeling perturbations that depend on the current state x(t) and the delayed state x(tr) of the system, respectively. It is assumed that the modeling uncertainties satisfy the bounds
1g, (x(0)1~ 0 ik Y' 1x(0) (5.3)
and
g2(x z),t) yj21x(t  ')11 (5.4)
where y, and y2 are known positive real constants, and the operator 1 may be any vector norm. We will assume that the following inequality is satisfied for all 0 e [r,0], and all real q > 0 (Mahmoud, 1996):
x(t + ) qx(t) (5.5)
Similar assumptions have been used in the context of Razumikhintype theorems where Lyapunov functionals are employed for stability analysis. For an extensive discussion, see Hale and Lunel (1993). Note that condition (5.5) is not very restrictive since we allow q to adopt any value greater or smaller than one.
Using a state feedback control law
u(t) = Fx(t)
(5.6)
where F is a constant matrix, the objective is to find sufficient conditions that F must satisfy in order to asymptotically stabilize the closedloop (5.1)  (5.2) and (5.6) for any
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modeling uncertainties that satisfy the norm bounds (5.3)  (5.4). Any matrix F that stabilizes the uncertain bilinear system is said to be robustly stabilizing.
Theorem 5.1: Suppose that the bilinear system (5.1)  (5.2) satisfies the uncertainty bounds (5.3)  (5.4) and inequality (5.5). Then, (5.6) is a robustly stabilizing state feedback control if
( A, + BF)+ qA2+ y, + qY2 < 0 and the initial state lies in the domain of attraction defined by the inequality x p(A, +BF)+q)A2 l+y, +q)2 I n 1  11F1 (5.8)
Proof: Let ui(t) = Fix(t), where Fj is the ith row of matrix F, denote a component of (5.6). From (5.1), the closedloop system is written as
(t)= Xx(t)+ Ax(t  z )+ XNA x(t)x(t) + gl(x(t),t) + g2(x(t  T),t) (5.9)
1.=1
where X = A + BF. The solution to (5.9) for t > 0, is readily expressed as the integral equation
x(t) = eAtx0 + J ex(t){A2x(t  r) + Ni F1x(t)x(t)+g,(x(t),t)+g2(x(t  r),t) ds (5.10)
Taking the norm of both sides in (5.10), using the inequality (Vidyasagar, 1993) e A ep(A)t t 0 (5.11)
and invoking the bounding inequalities (5.3)  (5.5) yields
Jx(t) eP'(A)txOjj+ feP()(t')jA2 \1 x(s  T)1 + 11g (x(s), s) + 1g2 (x(s  r), s)jI}ds
+ ep()(1s'~ NiFx(t)x(t) ds
e")'jx0 +eP()(S)(qA 1+y, +qY2x(s)ds (5.12)
+ e u(A)(t,) I INiFIx(s)2ds Now consider the scalar differential equation
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W(t = [y() + qA2 + + qy2 ]z(t) + I'l NjFz(t)2, t 0
z(0) =x0 1 (5.13)
The solution to (5.13) is unique and is given by the integral expression (Bellman and Cooke, 1963)
z(t = e( 'z(0) + f e " (qA2+ y,+ qy2 )z(s)ds +fJ e (')( INiFIz(s)2 ds (5.14) From inequality (5.12), it follows that
IIx(t) z(t), for t 0 (5.15)
therefore, asymptotic stability for (5.13), that is, z(t)  0 as t > oo, implies asymptotic stability for x(t). From the PoincardLyapunov theorem (cf. Bellman and Cooke, 1963, pg. 335), it follows that (5.13) is asymptotically stable if p(X)+ qjjA211+,y, + qY2 < 0 and if z(0) = zo is sufficiently small. A characterization of the "smallness" of zo can be obtained by examining the asymptotic behavior of z(t) as a function of zo. First, the equilibrium points of (5.13) are found by setting the derivative of z(t) equal to zero to obtain z, = 0 and p(X)+qA2+y, +qy2
I n, 111F  11(5.16) The asymptotic behavior of the solution depends on the sign of pJ(X)+qllA2jf+y1+qY2 and on the initial condition zo. In particular, when p() + qjA2 1+ y, + q72 < 0, then z(t)  0 for all zO < z2, and z(t)  oo in finite time for all zO> z2. Using the fact that zO = xk 1 and recognizing (5.15), it immediately follows that when the inequality condition (5.7) is satisfied, then x(t)  0 as t * provided that the initial state in turn satisfies the condition
141 < (A) + q A2 11+ YI + qy2
N JFJ (5.17)
As can easily be verified, the bounds obtained using the sufficient conditions (5.7) and (5.8) vary with the chosen norm and the corresponding matrix measure (Vidyasagar,
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1993). It is then possible that for a given norm and matrix measure, one can conclude stability while with other matrix norms, the stability condition may not hold. Note also that the adoption of the 1 or infinity norms may simplify the calculations, avoiding possibly more expensive singular value computations. In this respect, the problem of choosing a suitable norm and matrix measure to improve the stability condition is similar to the problem of finding an appropriate Lyapunov function candidate in the wellknown and widely used Lyapunov techniques for determining the stability of control systems.
Theorem 5.1 can be further specialized to the case where the uncertainty can be described as linear perturbations of the system matrices. The following result can be obtained.
Corollary 5.1: Consider the uncertain bilinear system given by
.(t) = (A, + 1)x(t)+ (A2+ AA2)x(t T)+ (B+ AB)u(t)+ X(N, + AN,)x(t)uj(t) i=1 (5.18)
with nominal matrices A, B, and N, affected by respective perturbations A4,,A4,,AB,AN,, i = 1,2,...,n, where the perturbations satisfy the norm bounds
JAA,11ja1, IA4211!a2, AB 3, IANi< Vi , i =1,2,...,m
(5.19)
Then, the closedloop system is asymptotically stable if p(A, + BF) +a, +q(A2+a2) <0 (5.20)
and
p(A, + BF)+a, +q(IIA21+ a2)
o (11 11+ v,)lFj (5.21)
Proof: The proof is analogous to the one given for Theorem 5.1 and follows readily after invoking the bounds given in (5.19) instead of (5.3).
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5.4 Constrained Control
The case of control saturation is considered in this section. An uncertain bilinear timedelayed system with input saturation can be described by the following equations x(t)= Ax(t) + A2x(t  T) + Bsatu(t) + YN,x(t)satu+(t)+ 1 t),t) + g2(x(t  r),t) x()= 0(0) ,  r 0 (5.22)
y(t) = Cx(t) (5.23)
where all the symbols are as defined in Section 5.3, and the saturation function sat u(t) is given by (see Figure 4.1)
sat u(t) = [sat u (t) satu2(t) ... satu (t)]T (5.24)
where
sat u(t)=I u,, u.
_I (5.25)
and where u* and Wi are real scalars representing lower and upper saturation limits, respectively. Equation (5.25) defines a saturation function that lies in the sector [0,1]. Also of interest is the case where the saturation nonlinearity lies in a sector [r,1], 0 < r < 1, as shown in Figure 1. The definition of the saturation nonlinearity for this case is obvious from the figure.
Using the properties of the vectornorm function it is easy to verify that
satu(t)  i(l + r)u(t)jjI <(1  r)Iu(t)II (5.26)
Theorem 5.2. Suppose that the nonlinear perturbation of the constrained bilinear system (5.22)  (5.23) satisfies (5.3)  (5.4). Let the robust statefeedback matrix F be chosen to satisfy the inequality
(A)+ qIA211+}(l  r)IBIF+ y, + qY2 <0 (5.27)
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Then, the uncertain constrained bilinear system is asymptotically stable if the initial state belongs to the region of attraction defined by the inequality p() + qA2+1 (1  r)BF+ ,+ qY2
1N F 1 (5.28)
Proof From equations (5.22) and (5.6), the closedloop system is written as
i(t)= A x(t)+(I+r)Bu(t)+ A2x(t  + B sat u(t)(1+r)u(t)
2 .2.
+ Nx(t)sat ui(t) + g, (x(t),t) + g,(x(t  T), t) (5.29)
or equivalently
i(t)= Xx(t) + Ax(t 'r) + B sat u(t)  I(1 + r)u(t) + Nx(t) satui(t) + g1 (x(t),t) + g2(x(t where the term I(1 + r)Bu(t) has been added and subtracted in (5.29), and matrix X is defined as A = A+  (1 + r)BF. Now proceeding on the same vein as in the proof of Theorem 5.1, and taking the norm of the integral solution of (5.29) using (5.26) and (5.11) and applying the Comparison Theorem (Lemma 1), the conditions (5.27) and (5.28) of the theorem readily follow.
Corollary 5.2. Consider the bilinear statedelayed system with unstructured perturbations in matrices A1, B and N, given by i(t) = (A, + AA )x(t ) + A2 + A2 )x(t  z) + (B + AB)satu(t) + (Ni + AN )x(t)satui (t) t1
x(0)= 0(0) ,  0 z 0 (5.30)
with matrix perturbation bounds given by AA a,, AB$P, AN,1
Then the closedloop system is asymptotically stable if
u[A, +I(1 + r)BF]+ a, + q(A2 + a11)+ (1  r)(11B11+ )F 1 < 0
(5.32)
and the initial condition satisfies
pu[Al +(I + r)BF]+ a, + q(A2 + a21)+ I(1  r)(11B11+ P)1 F 11
NI + vi 11) + 1 F (5.33)
Proof. It is elementary to verify conditions (5.32) and (5.33) by simply following identical steps as in the proof of Theorem 5.1.
5.5 Example
Consider an uncertain bilinear system defined as in (5.1) with dynamics described by [0. 1 0.2 ].90
A=[ 21 A2=~O~
0.8 2.1 = 01 0. 1]
1.1] N 0.2]
0.2 0.1
and the nonlinear uncertainty bounds y, = 0.3 and 2 = 0. The operational range of the saturating actuator lies in the sector [1/3, 1], and therefore, inequalities (5.27) and (5.28) will be used as prescribed by Theorem 5.2. Notice that the openloop system is unstable since matrix A has one positive eigenvalue.
By a standard poleplacement technique, we take the eigenvalues of X=A!(1++)BF to be 2.09 0.4i, and then find the feedback matrix
F = [2.95 0.25]. Using condition (5.27) now, we obtain for the usual 1, 2 and infinity norms:
For the 1norm. p(X)+qjjA2 j+1jlBlllFj+y, = 0.15 which is greater than zero; therefore nothing can be concluded for the stability of the system.
For the 2norm, p(X) + q1A2 1+ 11B1F1 +,y, = 0.54 with a corresponding region of attraction given by 11x011 0.91 while for the onorm, p(X)+q11A2+jj+BjjFII+y, =
0.18 with Ixo1 0.30. Therefore, it follows from Theorem 5.2 that the delayed bilinear
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system is asymptotically stable when the initial state belongs to the region of attraction which is at least 1xoII 0.91.
When there is no uncertainty present, the results are even better, as is expected. Using the 2norm, which for this example gives the largest bound, the following is obtained: u(X)+qllA2I+ JlBIIFII+y, = 0.84 with a region of attraction given by lx, < 1.41, which is adequate for practical applications. It should be noted that the approach of Lin and Saberi (1993) and Sussmann, et al. (1994) developed for exactly known stable linear systems would not be applicable to this example since A is not Hurwitz. Also, in Longchamp (1980), an example is given with a much smaller region of attraction (1IxoII <
0.255), and the author claims that it is large enough for practical applications.
5.6 Conclusions
This chapter establishes sufficient conditions for the robust stabilization of uncertain bilinear systems. The results presented are applicable to continuous time models that include delayed states as well as nonlinear uncertainty descriptions that are normbounded. An example has been given that illustrates the simplicity as well as the effectiveness of the derived robust stability conditions.
CHAPTER 6
DELAYDEPENDENT ROBUST STABILITY CONDITIONSCSTR WITH RECYCLE STREAM
6.1 Introduction
Many processes have dynamic behavior that is significantly affected by time delays due to transportation lags and measurement delays. Systems with time delays are known to occur in diverse areas. Delays, for example, are inherent in a variety of biochemical, optical transmission, electric or hydraulic networks and photochemical systems (Scell and Ross, 1986). Two types of timedelay processes can be distinguished. Natural time delays such as those occurring in biochemical processes and medicine in which transport of reactants across a membrane, and transmission of signals by the circulation of hormones or slow transcription of ribonucleic acid, are examples of events that can induce a delayed outcome on regulation of reaction paths. Imposed time delays caused by illuminated thermochemical reactions in the presence of delayed feedback and reactors which recycle unreacted feed material is another type of timedelay process..
Chemical process control systems are conventionally designed using the unit operations approach. That is, controllers are designed for each piece of equipment or unit in a plant, and then any conflicts between control loops are reconciled (Stephanopoulos, 1983). As Price and Georgakis (1993) demonstrate in their plantwide control design framework, two of the candidate structures for use in this modular control procedure are materials recycle and the chemical reactor/separator module, both of which may introduce time lags in the model. Both operations are used extensively in the chemical processing industry. The dynamics and stability of continuous stirred tank reactors (CSTR) has been
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the topic of numerous publications and several books in the last forty years (Aris and Amundson, 1958). Several of the published works include the effects of a recycle stream on the dynamic behavior of the reactor. Because recycling reduces waste of reagents, and hence the cost of reaction, its use is widespread in industry. A difficulty though with much of the literature on recycling is that the model almost always assumes no time delay in the recycle stream (Uppal, et al., 1974). This means that the separation process and the return time to the reactor is assumed to be instantaneous. While this assumption makes the analysis simpler, it is highly unrealistic. The process of recycling requires a finite amount of time which introduces a delay into the system since both the concentration of the reactants and the temperature in the reactor depend on some past time.
The description of timedelay systems leads to differentialdifference equations, the solutions of which require knowledge of past values of the system variables. The response of a system with a time delay can be quite complex. For example, studies of isothermal reactions indicate that delayed feedback may stabilize unstable stationary states or may destabilize an already stable steady state (Inamdar, et al., 1991). Other phenomena like multistationarity, periodic oscillations and chaos are possible at longer time delays. From this discussion, it is evident that the existence of time delays may cause major difficulties in the design and implementation of control and can deteriorate the system performance. A variety of deadtime compensation techniques have been proposed. Much attention has been given to the Smith predictor which effectively removes the time delay from the characteristic equation if the process model is perfect. However, it is wellknown that this technique can give unacceptable closedloop responses in the presence of plant/model mismatch (Wong and Seborg, 1986).
Because the introduction of time delays makes the analysis much more complicated, convenient methods to determine stability have long been sought. Lyapunov theory has played a central role in the stability analysis of ordinary dynamic systems. The second method of Lyapunov has also been extended to deal with stability analysis of time delay
86
systems (Hale and Lunel, 1993). While this method has physical meaning in certain cases and other advantages; until recently, there has not been general systematic procedures to construct the appropriate Lyapunov function. Also, a complicated Lyapynov matrix equation may not be a trivial task to solve. Of the existing stabilizing approaches, perhaps the most appealing one is to use differential inequality techniques. These techniques have features useful for design, and have been used to analyze ordinary as well as timedelayed systems (Mori, et al., 1981; and Mori, 1985). Two types of criteria have been developed:
(i) conditions that are independent of the size of time delay, and (ii) delaydependent stability criteria (Chen, et al., 1988). For an extensive list of references and discussion, see Hale and Lunel (1993) and Chen and Latchman (1995).
Although linear control theory has a wide range of applicability, there are always some nonlinearities that must be considered in practice. Actuators, for example, have physical limitations and saturation may result in their operation. If such nonlinearities are not taken into account during control system design, integral wind up or limit cycles may occur (Krikelis and Barkas, 1984). The stability of linear systems with saturating actuators has been studied extensively (Bitsoris and Vassilaki, 1995; Bernstein and Michel, 1995); however, most of the stabilization methods proposed so far are not directly applicable to timedelay systems with series nonlinearities. There are few reports available on the robust stabilization of time delay systems with input saturation, let alone chemical reactors with modeling uncertainties. Hence, the problem of designing a robust controller to stabilize uncertain CSTR models with time delay and series nonlinearities in the presence of uncertainty is well motivated.
In this chapter, the stabilization of a linear model of an integrated reactor/separator module with recycle is studied. Nikolaou and Hanagandi (1994) have shown that there are nonlinear systems that are virtually linear for a nonvanishingly small range of inputs and others that can be approximated by linear models. In this work, linear time delay nominal models are used with system nonlinearities appearing in two different terms: the first term
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includes perturbations which are allowed to be nonlinear and/or timevarying, and the second term represents nonlinearities in the input channel. The latter includes input saturation as a special case. The chapter is organized as follows: first, the mathematical model of the chemical reactor with recycle is developed as an illustrative example of systems that include delays and uncertainty. For any given plant that contains delays in the state variables and plant uncertainty, robust stabilization conditions are derived in Section 3. The case of series nonlinearities is considered next in Section 4. The key observation is that nonlinearities and plant uncertainty may destabilize the time delay system. An algorithm to find a robustly stabilizing feedback matrix is given in Section 5 followed by an example that illustrates the findings and necessary conclusions.
6.2 Problem Formulation
Consider a continuous stirred tank reactor (CSTR) in which a first order reaction A * B occurs. The dimensionless equations describing the conservation of mass and energy in the CSTR with recycle stream are given by
V2(M (6.1)
k1(t)f= J(x):= x1(t)+ (11 )x,(t  h)+ D,,e +"'/(1  x,(t))
x,(t )
k2(t) = f2(x):= X2(t)+(1  A)x2(t  h)+ B De '+"' (1  x1(t))  (x2(t)  x2,(t)) (6.2) where the dimensionless variables x, and x2 refer to the extent of conversion and the temperature in the reactor, respectively. The remaining dimensionless groups are defined in the Notation section. It is useful to remark that equations (6.1) and (6.2) require knowledge of past values of variables x, and x2. In the absence of time delay, but with the recycle stream still operating, equations (6.1)  (6.2) reduce to
j (tj)= x, (tj)+ Del**" (f,/ (  x, (t")) (6.3)
22 0"(
.i2 (t,) X2 (t )+ BDe" (1(  X1 (tn) On 3(X2(t X2 x2(t,) (6.4)
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where t, = Xt, D = D,, /k, and / = /k. Equations (6.3)  (6.4) are the wellknown equations for a CSTR given by Uppal, et al. (1974).
Suppose that the control objective is that of regulating the extent of conversion of the reactant (x1), by adjusting the temperature of the cooling stream (u = x2). Expanding (6.3)
 (6.4) in a Taylor series and linearizing around a steadystate operating point a linear delaydifferential equation in matrix form i(t) = Ax(t) + Adx(t  h) + bu(t) results, where x = [x, x21 ]f = f1, f2l, A = [df dx],,, A. = [df ldx(t  h)],, and b = [df /du],.
When designing a control system, one should take into account modeling uncertainties related to the linearization process above, or originating from various other sources such as identification error, model reduction for design purposes, variations of the plant parameters during operation, and other inaccuracies. With the issue of robustness being of particular importance currently, the model considered in the present work is given by the state equation
i(t) = Ax(t) + Adx(t  It) + Bu(t) + g(x(t),t) + gd(x(t  h),t) (6.5) and belongs to the class of uncertain timedelay systems where g(x(t),t) and g,(x(th),t) represent nonlinear, possibly timevarying, modeling perturbations.
6.3 Theoretical Developments
In the previous section, we discussed, through an example, how a statespace time delayed model can be derived from first principles, for a CSTR with recycle stream. Here, a robust stability analysis of uncertain systems with delay in the state variables is developed. It is assumed that there are no constraints on the manipulated variable. The robust stability analysis in the presence of input nonlinearities is deferred for the next section.
In the analysis that follows, the concept of the matrix measure is used extensively. For this reason, a formal definition and a few important properties are given next. For a
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more extensive discussion, the reader is referred to the excellent book by Vidyasagar (1993).
Definition 6.1. The matrix measure is a function p R""" + R
pu(A) = lim II+Ml
EO* e
where is an induced matrixnorm on R"'. The matrix measure is also known in the
literature as logarithmic derivative. The following properties are of relevance here:
(i) p(.) is a convex function.
(ii) p(S A) = 6p(A), and
(iii) Re A(A) p(A), where A is any eigenvalue of matrix A.
Consider the uncertain linear system with timedelay in the state represented by the equations
i(t)= Ax(t) + Adx(t  h) + Bu(t)+ g(x(t),t) + gd(x(t  h),t) (6.6)
x(0)= p(O) , e [h, 01
y(t) =Cx(t) (6.7)
where x(t) e R" is the state vector with initial state x(O) = xO; u(t) e R' is the input vector; y(t) e RP is the output vector; A,, B, and C are constant matrices of appropriate dimensions; p(t) is a continuous vectorvalued initial function; and h > 0 is the time delay. The vector functions g(x(t),t) e R" and gd(x(th),t) E R represent nonlinear modeling perturbations that depend on the current state x(t) and the delayed state x(th) of the system, respectively. No statistical information is required about the uncertainty vectors g and gd; it is only assumed that the modeling uncertainties satisfy the following normbounds:
ng(x(t),t)d kx(t) (6.8)
and
jgd(x(t  h),t)j kdI x(t  h)II
(6.9)
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where k and kdare a priori known positive real constants, and the operator may be any appropriate vector norm.
In this chapter, a state feedback control law of the form
u(t)= Fx(t) (6.10)
where F is a constant matrix, is used to derive the robust stability conditions. The objective can be stated as follows: find conditions that F must satisfy in order to asymptotically stabilize the closedloop (6.6)  (6.7) and (6.10) for all modeling perturbations that conform with the norm bounds (6.8)  (6.9). Any matrix F that stabilizes the uncertain delayed system is said to be robustly stabilizing. It should be pointed out that dynamic output feedback compensation is also possible in the context of this approach, but is not attempted here.
Theorem 6.1: Delaydependent robust stability conditions
Suppose that the plant uncertainties satisfy conditions (6.8) and (6.9) and the following inequality holds:
Iu(A + A)kkdhM>0 (6.11)
where A= A+BF and M= AA + AA,+IJAdI(k+kd). Then, the uncertain time
delayed system (6.6)  (6.7) is asymptotically stable under state feedback control given by (6.10).
The proof of Theorem 6.1 is contained in Section 6.5. When the time delay h is uncertain, Theorem 6.1 can alternatively be stated in the following way to find an upper bound on h. Let the feedback (6.10) be implemented where F is a known matrix, and then the closed loop system (6.6)  (6.7) is asymptotically stable if the delay Ih is bounded by 0
M
which is simply a rearrangement of inequality (6.11).
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Some remarks concerning the derived conditions follow:
* The tightness of the bound in (6.12) or (6.13) varies with the chosen norm and the corresponding matrix measure (Vidyasagar, 1993). In other words, it is possible to determine stability with a given norm and matrix measure, while with other choices the stability condition may not hold. The largest bound computed for the 1, 2, or infinity norms should be selected.
* When checking the asymptotic stability of a given uncertain delay system, one should try the 1 or infinity vector norms first, dispensing the troublesome eigenvalue computations associated with the 2norm. The freedom in choosing a suitable norm and matrix measure to improve the stability condition resembles that of constructing an appropriate Lyapunov function candidate in the wellknown and widely used Lyapunov approach for determining stability.
* For the nominal case when the uncertainty is negligible (that is, g and go are identically zero) and also h = 0, inequality (6.11) of the theorem reduces to p(A + Ad) < 0 which implies that X + Ad is asymptotically stable, since Re A(X + Ad) p( + Ad)<0. When h # 0, condition (6.11) simply means that X + Ad should be stable enough to overcome the difficulty posed by the time delay in the system. It is thus evident that time delay can destabilize an otherwise stable closed loop.
* It has been known (Mori, 1985) that delay independent criteria are conservative due to lack of information on the delay, especially when delays are small. It is then reasonable when checking the stability of uncertain delay systems to start with delay independent criteria and if they fail to turn to delaydependent ones.
* Besides the wellknown 1, 2, and infinity norms, other induced norms and matrix measures involving weighting parameters may be utilized in the stability conditions. As an example consider the following matrix norm and corresponding measure (Strbm, 1975 and Mori, 1981):
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A=max a , A= maxai+ a (6.13)
I 1 WiiL W
Other weighted norms can be defined similarly. Now, a simple optimization problem with respect to the arbitrary weighting factors will yield less conservative robust stability conditions. This is a topic that merits further investigation and is not pursued further in this work.
Corollary 6.1. Consider the uncertain timedelay system with uncertain matrices A , Ad, and B, represented by
= (t)=(A + AA)x(t) + (Ad + AA )x(t  h) + (B + AB)u(t)
x(0)= q(O) , O e [h, 0] (6.14)
where the matrix perturbations satisfy the norm bounds A al, lAAg! ad, AB1 (6.15)
Then, the closedloop system is asymptotically stable if 0< h<1 = +p( Ad)a ad  F (6.16)
M
where the denominator M in this case is given by M = AdA +AdAdII+jjAdI(a + ad).
Proof It is analogous to the proof of Theorem 6.1 and follows readily after invoking the bounds (6.15) instead of (6.8)  (6.9). (see Section 6.5).
6.4 Robust Stability in the Presence of Input Nonlinearities
In this section, a stability analysis is presented for time delay systems affected by nonlinearities in the input channel. Prior to the discussion of robust stability, some useful concepts are depicted.
Definition 6.2. For a continuous nonlinear mapping N: R" 4 R", and for two real numbers p and q such that , N is said to lie inside a sector [p, q] (see Figure 2) if N satisfies the following two properties:
