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On measuring closed-loop nonlinearity : a topological approach using the v-gap metric Tan, Guan Tien 2003

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On Measuring Closed-loop Nonlinearity: A Topological Approach Using The z/-Gap Metric by G U A N T I E N T A N B.Eng. (Hons), Universiti Teknologi Malaysia, 1998 M.Eng., The National University of Singapore, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Chemical & Biological Engineering; Doctoral Programme) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A November 2003 © Guan Tien Tan, 2003 Abstract All chemical processes are inherently nonlinear. However, a nonlinear process does not necessarily require a nonlinear control since the feedback control itself has a certain degree of linearizing effect. This leads to an interesting and often subtle question: "When is a linear controller sufficient to control a nonlinear process?". This thesis aims to answer such a question and develop a systematic approach to quantify closed-loop nonlinearity from a controller design perspective. An immediate consequence arising from the quest of the answer for the above question leads to the two major contributions of this thesis. Firstly, a novel way to quantify closed-loop nonlinearity and a practical computational algorithm are developed. Secondly, the nonlinearity measure presented in this thesis can be used as an effective decision making tool when dealing with the question of choosing an appropriate strategy for a class of nonlinear plants that can be recast into a quasi-linear parameter varying (quasi-LPV) representation. An additional contribution of this thesis is the development of a pictorial approach that provides a better insight and understanding to the gap metric theory. In this thesis, the i^ -gap metric arising from the graph topology is used to quantify the "distance" of a nonlinear process and its linearized model in a closed-loop fashion. Since the z^ -gap metric is developed for linear time invariant (LTI) systems, the nonlinear system is recast into a quasi-LPV form. As a result, the largest ^-gap induced by the closed-loop nonlinearity together with a time variation penalty (i.e. the developed nonlinearity measure) can be obtained. The theoretical optimal stability margin is subsequently computed and compared with the largest ^-gap to check the severity of the closed-loop nonlinearity. If the developed nonlinearity measure is smaller than the theoretical optimal stability margin, then the closed-loop nonlinearity is manageable ii ABSTRACT ABSTRACT by an optimal single linear controller designed for the linear system. Otherwise, the optimal linear controller that results in a satisfactory robust stability in the linear system might show poor robust stability (or even destabilizes) the nonlinear system. Under this circumstance, a nonlinear control strategy might be needed. Since such an optimal controller might not be attainable in practice, a sub-optimal controller can be obtained via the loop shaping design procedure. Finally, the developed nonlinearity measure is applied in three different examples, a continuous stirred tank reactor control problem, an inverted cone tank control problem and a fictitious nonlinear plant control problem. Simulation results confirm the appli-cability and the reliability of the developed nonlinearity measure in tackling practical engineering control problems. iii Table of Contents Abstract i i Table of Contents iv List of Tables ix List of Figures x Acknowledgement xiv 1 Introduction 1 1.1 Background and Motivation 1 1.2 Thesis Organization and Contributions 7 2 Literature Review , 10 2.1 Nonlinearity Measures 10 2.1.1 The Statistical Approach \- 10 2.1.2 The Norm-bounded Error Approach 15 2.1.3 The Geometrical Approach 18 2.2 Overview of The Gap Metrics 19 2.3 Summary 23 iv TABLE OF CONTENTS TABLE OF CONTENTS 3 Graph Topology, The Gap Metric & Closed-loop Stability 24 3.1 Introduction 24 3.2 Finite Energy Signal Spaces 25 3.3 Operator Graphs 26 3.4 Operator Graphs and Closed-loop Stability 29 3.5 Operator Graphs and Closed-loop Robustness 30 3.6 Operator Graphs and Uncertainty Description 35 3.6.1 The Gap Metric 36 3.6.2 Homotopy and The ^-Gap Metric 40 3.6.3 The Gap Metrics and The Coprime Factor Uncertainty 46 3.7 The Gap Metrics and The Small Gain Theorem 48 3.8 Summary 51 4 A Closed-loop Nonlinearity Measure 52 4.1 Introduction 52 4.2 Formulating The Closed-loop Nonlinearity Measure 53 4.3 Theoretical Motivation 61 4.3.1 The ^-Gap Metric For Quasi-LPV Systems 61 4.3.2 The Frozen Point Nonlinearity Measure 66 4.3.3 Time Variation of Scheduling Parameter 70 4.3.4 The Choice For The Nominal Model , . . . 72 4.3.5 The Linearizing Effect of Feedback 74 4.3.6 Jffoo Loop-Shaping and Weight Selection 76 4.3.7 The Best Possible Stability Margin 81 TABLE OF CONTENTS TABLE OF CONTENTS 4.4 A Computational Algorithm 82 4.5 Summary 83 5 Design Examples 84 5.1 Example I: CSTR Control Problem 84 5.1.1 Problem Description 84 5.1.2 Design Objectives 85 5.1.3 Nonlinearity Measure 86 5.1.4 Simulation Results 91 5.1.4.1 Unity Feedback . . . 93 5.1.4.2 Setpoint Tracking Responses of CSTR Under Robust Control 94 5.1.4.3 Disturbances in Feed Concentration, CA/ 95 5.1.4.4 Disturbances in Feed Temperature, Tf 95 5.1.4.5 Concluding Remarks: CSTR Control Problem 95 5.2 Example II: Cone Tank Control Problem 98 5.2.1 Problem Description 98 5.2.2 Design Objectives 100 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank . . 100 5.3 Example III: A Fictitious Nonlinear Plant Control Problem 108 5.3.1 Problem Description 108 5.3.2 Nonlinearity Measure 108 5.3.3 Simulation Results 113 5.4 Summary 119 vi TABLE OF CONTENTS TABLE OF CONTENTS 6 Conclusions 120 6.1 Contributions 120 6.2 Recommendations 122 Nomenclature 124 Bibliography 127 A Mathematical Preliminaries 132 A . l Functional Analysis 132 A.1.1 Sets 132 A. 1.2 Mappings 133 A.l.3 Bounded Sets 134 A. 1.4 Metric Space and Completeness 134 A. 1.4.1 Metric and Metric Space 134 A.1.4.2 Completeness 135 A. 1.5 Open Set and Topology 135 A. 1.6 Normed Vector Space 137 A.2 Basic Operator Theory 139 A.3 Signals and Systems 141 A.4 Feedback Control Theory 144 A.5 Coprime Factorization 146 A.6 Quasi-Linear Parameter Varying Systems 148 A.7 Complex Analysis: Winding Number 151 A.7.1 The Argument Principle 152 vii TABLE OF CONTENTS TABLE OF CONTENTS B Proofs in Chapter 3 153 B . l Proof of Proposition 3.5.1 153 B. 2 Proof of Eq. (3.6.6) 153 C A Computational Algori thm For The f -gap Metr ic 158 C l A Computational Algorithm For The /v-Gap Metric 158 C. 2 A Computational Algorithm of Jt%o Norm 159 C.3 A Computational Algorithm of The Graph Symbol G\ 160 C.4 A Computational Algorithm of The Graph Symbol G 2 161 Index 162 viii List of Tables 1.1 Nominal operating conditions for a CSTR 4 5.1 Nominal operating conditions for an inverted cone tank 99 5.2 Scheduling space of the fictitious plant from time £ = 0 to £ = 30 s. . . . I l l ix List of Figures 1.1 A typical plot of the normalized reaction rate coefficient versus tempera-ture 2 1.2 A schematic diagram of a continuous stirred tank reactor 3 1.3 Open-loop responses of the CSTR subject to ±5K in coolant temperature Tc at three different operating points (i.e. T=300K, 320K and 350K). . . 5 1.4 Closed-loop responses of the CSTR control problem under a single linear controller. Setpoint (dashed-dotted) and reactor temperature (solid). . . 6 1.5 A reading road map 9 2.1 Nonlinearity measure proposed by Eker and Nikolaou (2002) 18 3.1 Examples of finite energy signals 25 3.2 Types of signals 26 3.3 i f 2 and 3^2 spaces 28 3.4 Standard feedback configuration 29 3.5 A pair of parallel projections, Hg^g^ and Ilgi\\gp 33 3.6 Relationship of bptc — sin# and the minimal angle (i.e. 6) between sub-spaces M and Jf 34 3.7 From plants' input-output space to the gap metric 37 3.8 The directed gap between two closed Hilbert subspaces 38 3.9 Two nonhomotopically equivalent graph spaces. . . . - 41 x LIST OF FIGURES LIST OF FIGURES 3.10 An analogy of a homotopic equivalence 41 3.11 An analogy of a nonhomotopic equivalence 42 3.12 Analogy of the homotopy condition in the z^ -gap metric and the actual closed-loop stability. 44 3.13 Stereographic projection 45 3.14 A standard M - A configuration in robust control 49 3.15 A perturbed plant with normalized coprime factor uncertainty. 50 4.1 The developed nonlinearity measure looks at closed-loop nonlinearity. . 54 4.2 A standard configuration for closed-loop nonlinearity measure 55 4.3 An ambiguity arising from applying Corollary 4.2.1 61 4.4 A graphical interpretation of Assumption 4.3.2 67 4.5 Graph spaces of a frozen-point quasi-LPV system (Gi(Qi)) and that of a linear model (Q2) 68 4.6 Homotopic and nonhomotopic analogies from a closed-loop perspective. 71 4.7 Impact of the choice of nominal model 73 4.8 Nonlinear feedback control 74 4.9 Jtffoo Loop-shaping design procedure (McFarlane and Glover, 1990). . . . 76 4.10 Specified and achieved loop-shapes (McFarlane and Glover, 1990). . . . 78 5.1 Unshaped f-gap contour. Nominal model (black dot) 87 5.2 Bode diagram for the nominal model at T=341 K 88 5.3 Specified (solid) and stabilized (dashed) loop shapes 89 5.4 Sensitivity (solid) and complementary sensitivity (dashed-dotted) functions. 90 5.5 Shaped v-g&p contour. The best shaped model (black dot) 92 xi LIST OF FIGURES LIST OF FIGURES 5.6 Top: Setpoint tracking responses of the CSTR under a unity feedback control. Reactor temperature, T (solid), setpoint (dashed-dotted). Bot-tom: Coolant temperature, TC (solid) 93 5.7 Top: Setpoint tracking responses. Reactor temperature, T (solid), set-point (dashed-dotted). Bottom: Coolant temperature, T c (solid) 94 5.8 Left: Closed-loop responses subject to ±20% in feed concentration, CA/ (Top three: +20%; Bottom three: +20%) at three operating points. Re-actor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant temperature, TC (solid) 96 5.9 Left: Closed-loop responses subject to ±5K in feed temperature, 7/ (Top three: +5K; Bottom three: +5K) at three operating points. Reactor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant tem-perature, TC (solid) 97 5.10 A schematic diagram of an inverted cone tank 98 5.11 Loop gains for the inverted cone tank. Shaped plant (solid), original plant (dashed-dotted) 102 5.12 Shaped z^ -gap contour. Nominal model (black dot) 103 5.13 Servo responses from operating point (0.2,303) to (1,333). Plant (solid) and setpoint (dashed-dotted) 105 5.14 Servo responses from operating point (1,333) to (0.2,303). Plant (solid) and setpoint (dashed-dotted) 105 5.15 Servo responses from operating point (0.2,303) to (0.2,333). Plant (solid) and setpoint (dashed-dotted) 106 5.16 Servo responses from operating point (1,333) to (1,303). Plant (solid) and setpoint (dashed-dotted) 106 5.17 Closed-loop response to +20% disturbance in inlet stream Fj at (0.2,303). Plant (solid) and setpoint (dashed-dotted) 107 xii LIST OF FIG URES LIST OF FIG URES 5.18 Closed-loop response to -20% disturbance in inlet stream Fj at (1,333). Plant (solid) and setpoint (dashed-dotted) 107 5.19 Unshaped z^ -gap contour. Nominal model (black dot) 109 5.20 Sensitivity (solid) and complementary sensitivity (dashed-dotted) functions. 110 5.21 Shaped z^ -gap contour. Nominal model (black dot) 112 5.22 The z^ -gap metric of P(-0.155, s)WlW2 and P(-0.1, s)W1CaoW2 (solid) versus the generalized stability margin with respect to [P(—0.155, s)WiC00W2, I] (dashed-dotted) over a frequency range 114 5.23 The z/-gap metric of P(-0.1, s)W1COQW2 and P(-0.05, s)WlC00W2 (solid) versus the generalized stability margin with respect to [P(—0.1, s)WiC00W2, I] (dashed-dotted) over a frequency range 115 5.24 The i>-gap metric of P(-0.05, s)W1C00W2 and P ( l x 10~5, s)W±C<x>W2 (solid) versus the generalized stability margin with respect to [P(—0.05, s) W\ COQW-2,1) (dashed-dotted) over a frequency range 116 5.25 The z^ -gap metric of P ( l x IO"5, s)WlC00W2 and P ( l x 10 - 3 , s)WlCOQW2 (solid) versus the generalized stability margin with respect to [P(l x 10 - 5 , s)WiCooW2,1] (dashed-dotted) over a frequency range 117 5.26 Time domain closed-loop simulation for the fictitious nonlinear plant. Top: Plant output, y(t) (solid), setpoint (dashed-dotted). Bottom: Con-troller output u(t) (solid). Note, the model switching sequences are given in Table 5.2 118 A . l Homotopy of / : 3£ -» <3T and g : 3C -> & . . 138 A.2 A standard feedback configuration 144 A.3 Inner- and outer-loops of a quasi-LPV feedback system 150 A.4 A standard Nyquist T contour . 152 xiii Acknowledgement First and foremost, I would like to thank my supervisors Associate Professor K. Ezra Kwok and Assistant Professor Mihai Huzmezan. Professor Kwok introduced me to the realm of nonlinear control, which later on leading me to the interesting quest of establish-ing a practical nonlinearity measure. Equally important is the persistent guidance and strong support from Professor Huzmezan, who brought me into the. fascinating world of topology. Special thanks also go to Prof. Jim Lim and Prof. Guy Dumont for their advices and constant support. Their kindness and word of wisdom have always been the sources of enlightenment to me. Their great insight and serious research attitude will always inspire me all my life. I also would like to express my gratitude to Professor Denis Sjerve, who kindly accepted the offer to be the chair of my final doctoral examination and also Professor Jimmy Feng and Professor Ruben Zamar, who not only reviewed the final draft of my thesis, but also kindly served as the university examiners during the final examination. Also, I am greatly indepted to Professor Graham Goodwin (University of Newcastle). It is my great honour to have Professor Goodwin as the external examiner of my doctoral thesis. Of course, this doctoral thesis can never reached its final state without the constructive comments and suggestions from Professor Gary Balas (University of Minnesota) and Professor Michael Cantoni (University of Melbourne). Last but not the least, I would like to express my sincere gratitude to Ching Thian Tye, Dr. Michael Chong Ping, Eman Al-atar, Lechang Cheng, Manny Sidhu, Quak Foo Lee and Dr. Nazip Suratman for their intelligence, persistence and willingness to help, which made my stay at the University of British Columbia stimulating and enjoyable. Special thanks also go to Margaret Tan, Larry Phillips and Malaysia's Consulate Office xiv ACKNOWLEDGEMENT ACKNOWLEDGEMENT for providing me a "home away from home" in Vancouver, British Columbia. As a personal note, I would like to thanks my parents for their constant love and support. Finally, financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. xv C H A P T E R 1 Introduction In this chapter the background and motivation of the thesis are presented first. Then, the contents of each chapter are briefly reviewed and the contributions of the thesis are highlighted. 1.1 Background and Motivation Almost all chemical processes are inherently nonlinear. However, a nonlinear process does not always require a nonlinear controller. This is owing to the fact that most indus-trial controllers1 are implemented in a feedback fashion in order to handle uncertainty, plant/model mismatch and noise attenuation. An interesting property of feedback is its ability to reduce open-loop nonlinearity (Desoer and Wang, 1980). It is this linearizing effect that makes a linear controller sufficient to control a nonlinear process in some cases. To see this, consider a simple first order, exothermic chemical reaction A —> B. The rate of reaction for species A (i.e. TA) is given as follows: -rA = kCA (1.1.1) where k and CA are the reaction rate coefficient and the concentration of species A , respectively. Since chemical intuition suggests that the higher the temperature, the 1Such as Proportional-Integral-Derivative controllers and model predictive controllers. 1 C H A P T E R 1. 1.1. Background and Motivation faster a given chemical reaction will proceed, k is determined by the Arrhenius equation: * = ^ e x p ( ^ ) (1-1.2) where k0, Ea, R and T denote a constant, activation energy, gas constant and the reaction temperature. Figure 1.1 shows a typical plot of the normalized reaction rate coefficient over a temperature range. Clearly, a significant degree of nonlinearity is observed as the temperature is increased from 300 K to 370 K. When this reaction rate coefficient is 300 310 320 330 340 350 360 370 T e m p e r a t u r e , T (K) Figure 1.1: A typical plot of the normalized reaction rate coefficient versus temperature. used to model the dynamics of a well-mixed continuous stirred tank reactor (CSTR), as depicted in Figure 1.2, the nonlinearity is expected to be carried forward to the CSTR dynamics, which are represented by the following ordinary differential equations (Henson 2 C H A P T E R 1. 1.1. Background and Motivation and Seborg, 1997): dCA dt dT ~dt ^{CAf - CA) - kQexp(-^)CA %)CA + ^ r ( T c - T ) RT VpCp (1.1.3) (1.1.4) where CA and Tc represent reactor effluent concentration of component A and coolant temperature, respectively. The remaining model parameters and the nominal operating conditions are given in Table 1.1. CAf, Tf CA, CB, T Figure 1.2: A schematic diagram of a continuous stirred tank reactor. Figure 1.3 shows the open-loop responses of the above mentioned CSTR subject to ±5 K step changes in coolant temperature, T c, at three different operating points (i.e. reactor temperature at T = 300 K, 320 K and 350 K, respectively). At 300 K (bottom of Figure 1.3), the open-loop responses are quite symmetric. This indicates that the system is pretty linear at this point. When the reactor temperature is increased to 320 K, see the middle of Figure 1.3, the reactor exhibits asymmetric responses. It is evident that at this point the linearity of the open-loop system deteriorates. Finally, the degree of nonlinearity becomes significant when the reactor temperature is further increased to 350 K, see the top of Figure 1.3. The oscillatory responses at this operating point is the well-known limit cycle phenomena of nonlinear systems. Unequivocally, for 3 C H A P T E R 1 . 1 . 1 . Background and Motivation Table 1.1: Nominal operating conditions for a CSTR Parameter Notation Value Feed/effluent flow rate q 100 L/min Activation energy term E/R 8750 K Feed concentration 1 mol/L Reaction rate constant k0 7.2xlO 1 0 min" 1 Feed temperature Tf 350 K Overall heat transfer term • UA 5x l0 4 J/min-K Liquid volume V 100 L Reactor temperature T 341 K Liquid density P 1000 g/L Reactant concentration cA 0.6592 mol/L Specific heat Cp 0.239 J/g-K Coolant temperature Tc 302.6 K Heat of reaction (-AH) 5x l0 4 J/mol the CSTR, the degree of nonlinearity changes as the reactor temperature is changed. Intuitively, to control the aforementioned CSTR covering a wide temperature range, one would suggest to employ a nonlinear feedback control. However, the reality is somewhat counter intuitive. Often, systems with highly nonlinear open-loop behavior look much more linear in closed-loop and sometimes the systems can be controlled by employing a single linear controller. For the reason that will become apparent in Chapter 5, the linear controller resulting from the augmentation of the weights presented in Eq.(5.1.5) and the controller given in Eq.(5.1.6) is proved to be sufficient to control the above CSTR over the temperature range of T G [303 373] K. Figure 1.4 shows that the linear controller provides a satisfactory closed-loop responses in following the successive setpoint changes over the prescribed operating range. Clearly, measuring open-loop nonlinearity neither gives sufficient information on the severity of the closed-loop nonlinearity nor tell us when a nonlinear controller is really needed. This observation leads to an interesting and often subtle question: "When is a linear controller sufficient to control a nonlinear process?\ This thesis aims to answer such a question and develop a systematic approach to quantify closed-loop nonlinearity from a controller design perspective. A topological approach is adopted in this regard. 4 C H A P T E R 1. 1.1. Background and Motivation Figure 1.3: Open-loop responses of the CSTR subject to ±5K in coolant temperature Tc at three different operating points (i.e. T=300K, 320K and 350K). 5 C H A P T E R 1. 1.1. Background and Motivation 350 10 15 20 25 Time, t (min) Figure 1.4: Closed-loop responses of the CSTR control problem under a single linear controller. Setpoint (dashed-dotted) and reactor temperature (solid). 6 CHAPTER 1. 1.2. Thesis Organization and Contributions 1.2 Thesis Organization and Contributions This thesis is organized into six chapters including this introductory chapter. In what follows, the contents of each chapters are briefly reviewed. Also to guide the readers a road map designed to link the theory with practice and contributions is presented in Figure 1.5. Chapter 2: Literature Review This chapter provides a quick glimpse on the existing techniques for nonlinearity mea-sures and a historical review of the gap metric notion. Chapter 3: Graph Topology, The Gap Metr ic &: Closed-loop Stability This chapter aims to provide a novel way of understanding the z>-gap metric. The uniqueness of this approach is that a series of diagrams instead of pure mathematical expositions are used to clearly elucidate the geometrical perspective of the z;-gap metric and its connection to control theory. This approach is intuitive and provides a better insight into the z^ -gap metric and the robust stability notions. This chapter begins with an introduction to signal space and two important graph spaces (i.e. J d 2 and M2) followed by a discussion on the connection between operator graphs and closed-loop stability. A geometrical interpretation of the relationship between the gap metric and the generalized stability margin is presented. A good understanding of the gap metrics, particularly the z>-gap metric, and the generalized stability margin has proven to be useful in understanding the materials presented in the subsequent chapters. Chapter 4: A Closed-loop Nonlinearity Measure The focal point of this chapter is to establish a closed-loop nonlinearity measure by exploiting the z/-gap metric for quasi-LPV systems and the McFarlane-Glover Jrf?^ loop shaping design procedure (McFarlane and Glover, 1990, 1992). Theoretical motivation of the developed nonlinearity measure is first presented. A state space formulation for computing the z>-gap metric for quasi-LPV systems is also given. Finally, a computa-tional algorithm, which is one of the major contributions of this thesis, is developed to evaluate the nonlinearity measure. 7 C H A P T E R 1. 1.2. Thesis Organization and Contributions Chapter 5: Design Examples In this chapter, three design examples are presented to illustrate the strength of the developed measure. The first example involving a continuous stirred tank reactor control problem shows the implementation of the developed method in a detail fashion. The second example, which concerns an inverted cone tank control problem, is used to show the practicality of the developed measure in multivariate case. The third example deals with the control problem of a fictitious'^nonlinear plant. This fictitious plant has a sign change characteristic in process gain, which is known to be challenging to control if a single linear LTI controller is used. This example not only shows that the ability of the developed measure to indicate the insufficiency of a linear controller, but also gives a good prediction on the onset of closed-loop instability. Chapter 6: Conclusions In this chapter, contributions of the thesis are highlighted. The following list provides a glimpse on the major contributions of this thesis. The detail discussion of each contri-butions is deferred to Chapter 6. • Linear or nonlinear control? A decision making tool. • A novel approach to quantify closed-loop nonlinearity. • A novel computational algorithm for nonlinearity measure. • A novel approach to explain the gap metric notion. • A Jffoc loop shaping weight selection to mitigate closed-loop nonlinearity. Lastly, future work to improve the computational aspect of the developed measure is also presented. 8 C H A P T E R 1. 1.2. Thesis Organization and Contributions A •3 © o u o C3 O A decision making tool. © A novel approach to quantify closed-loop nonlinearity. © A pictorial approach to explain the gap metric notion. 0 A novel computational algorithm for nonlinearity measure. © yCoo loop shaping weight selection to mitigate closed-loop nonlinearity. Note: • and O denote chapter and section, respectively. U5 C o O c o U Figure 1.5: A reading road map C H A P T E R 2 Literature Review Existing techniques for nonlinearity measures are briefly reviewed. These tech-niques can be categorized into three groups, namely the statistical approach, the norm-bounded error approach and the geometrical approach. In addition, a histor-ical review of the gap metric notion is also presented. 2.1 Nonlinearity Measures In designing a controller for nonlinear systems, intuition suggests that a nonlinear con-troller should be employed. Since feedback control is employed in most industrial pro-cesses, it is expected to tolerate a certain degree of nonlinearity. This implies that for mild nonlinearity a linear controller should be sufficient. Therefore, a systematic approach to quantify the degree of nonlinearity is desirable. In this section, various existing linearity tests are briefly discussed. 2.1.1 The Statistical Approach A direct approach to quantify nonlinearity is to use plant's input-output data. An important characteristic of this class of approaches is that the underlying statistical properties such as probability distribution, moment functions, conditional expectations, correlation function are exploited. In what follows, methods from this category are briefly discussed. 10 C H A P T E R 2. 2.1.1 The Statistical Approach A. The Regression Error Specification Test Proposed by Ramsey (1969), the Regression Error Specification Testis one of the popular linearity tests against an unspecified alternative. Basically, it consists of three steps: 1. A linear autoregression with exogenous input (ARX) model is first applied to plant input-output data (ut, yt)- The fitted values yt, obtained by ordinary least-squares, and the residuals it = yt — yt are then calculated. Denoted by RSSo, the residual sum of squares equals 2~2^t-2. Regress it on the set of regressors {1, y t _ x , y t - p , u t - i , u t - k , tit2, tit1} and compute the RSS. 3. The test statistic is given by, (RSS0-RSS)/(h-l) RSS/{n - p - k - h ) K ' ' ' where, n denotes the sample size. It is noted that under the hypothesis of linearity and zero expectation of u t i t - s for all s (i.e. E(utet_s) = 0 Vs), the function (h — 1)F has an asymptotic x2 distribution1. An obvious advantage of this method is that it does not depend on any assumption on the nonlinear function of the data set. B. The Bispectral Test The bispectral test of linearity was first proposed by Subba Rao and Gabr (1980) and was further improved by Hinich (1982). Generally, the bispectrum for a zero-mean process with an absolutely summable third moment function K(s,t) = ~E[ysyty0], where yo, yt and ys are the original time series y and the same time series being shifted by t and s time steps, can be written as follows: oo oo • / f l ( u ; i , W 2 ) = (2vr)-2 £ ]T K{s,t)e-**-*** (2.1.2) s=—oo £=—oo 1A theoretical distribution function for g independent squared normal distributed random variables with zero mean and unity variance (Hogg and Craig, 1995, pg. 134). Mathematically, a x2 distribution is given as: fe(x) = T(e/2)2e/^ xe^2~1e~Q^2 whenever x > 0 and zero otherwise, g £ Z + ( a set of positive integers) and T(-) denote the degree of freedom and the gamma distribution function, respectively. 11 C H A P T E R 2. 2.1.1 The Statistical Approach where j = %/—T denotes the imaginary number. The inverse relationship is given as K(s,t)= f j\^s^tfB{ullu2)dw1du2 (2.1.3) J —TT J —TT Since K(s, t) poses several symmetry relationships, we only need to study the bispectrum over the range 0 < U)\,<JJ% < tt, CJI + U2 < TT. Subba Rao and Gabr (1980) and Hinich (1982) have shown that for a linear process2 oo Vt = ^gkSt-k, (2.1.4) fc=0 the following quantity is a constant over all frequencies. B(u^2)=f. \ f f ( { W l ^ \ , (2.1-5) In addition, B(U>-L,LO2) equals zero for a Gaussian process since it is well known that a Gaussian process can be fully characterized by its first and second moments. At first glance, Eq.(2.1.5) can be used to test the linearity and gausianity. However, research studies (Granger and Terasvirta, 1993, pg. 21) have shown that the two conditions men-tioned above may still hold even the process is nongaussian and nonlinear. In addition, the requirement of a considerable amount of observations to match the power of the best parametric tests for a given alternative limits its application (Tj0stheim, 1994; Tong, 1990). C. The Brock-Dechert-Scheinkman Test An interesting linearity test arising from chaos theory is the one proposed by Brock et al. (1987). This method is commonly known as Brock-Dechert-Scheinkman Test (BDS). For the sake of clarity, some basic concepts in chaos theory are briefly discussed. In chaos theory, the notion of correlation dimension, first introduced by Grassberger and 2Eq.(2.1.4) shows a moving-average time series. In which gk and et-k represent a weighting coefficient at time k and an independent and identically distributed (i.i.d) random noise sequence that is shifted by k amount. 12 C H A P T E R 2. 2.1.1 The Statistical Approach Procaccia (1983), quantifies the dimension for a strange attractor existing in the chaotic process. First, let Xi and Xj denote two vectors that consist of consecutive m terms from a time series, Xt. The correlation dimension is then defined by: ^ .. .. d In C(e.m) D = l iml im ' 2.1.6 e-^om^oo a me where ~ m m c ( e ' m ) = ^ n i E E e(e - 11* - x > u (2-1-7) v i i=i j=i+i is the correlation integral. 0 denotes the Heaviside step function (i.e. 0(e — \\Xi — XjWoo) = 0 if e < \\Xi — XjWao and 0(e — \\Xi — -Xj||oo) — 1 if e > \\Xi — X,||oo for an arbitrary constant 0 < e & M1), and || |^joo — max j^ j^i see Kantz and Schreiber (1997, l<i<m pg. 70). Motivated by the fact that for an i.i.d sequence, C(e,m) = [C(e, l ) ] m , m < oo, the BDS test is given by: S(e,m) = C(e,m)-[C(e, l ) ] m (2.1.8) Under the null hypothesis H0 : Xt is i.i.d, \/NS(e,m) will have a normal distribution with zero mean and variance that is a function of m and e. For a linearity test, one will first fit the data using an appropriate linear model, for instance apth-order autoregressive model AR(p). BDS test is then applied to the resulting residuals computed between the original time series and the fitted values using the pth-order AR model. If the null hypothesis is rejected, this implies that the nonlinearity is present, but its form is not known. D. The Nonparametric Test Hjellvik and Tj0stheim (1995) proposed a nonparametric linearity test based on the distance between the best linear estimate and a nonlinear estimate of the conditional mean of yt, given its past yt~k- It is noted that the best linear estimate of conditional mean, Mk(yt-k) = E[yt|yt_fc = y*], for a normalized series is the autocorrelation between yt and yt-k, i-e. Pk- The nonlinear estimate of the conditional mean can be computed 13 C H A P T E R 2. 2.1.1 The Statistical Approach by the kernel or Nadaraya-Watson estimator as follows: a w ) = { n - k ) ~ \ ^ K v f : v : y , - k ) (2.1.9) n E t=i Kh(y - yt) where Kh(y* — yt) = h~1K(h~1(y* — yt)), with K being a kernel function and / i the bandwidth. Therefore the nonlinearity index is defined as: L{Mk) = E[{Mk(yt_k) - Pkyt-kf] (2.1.10) The functional L(Mk) can then be estimated by computing: n L(Mk) = n" 1 ^2{Mk(yt) - pkyt}2w(yt) (2.1.11) i=l where u)(yt) is a weighting function. Clearly, L(Mk) = 0 V7c for Gaussian process. There-fore, we may say that the linearity is rejected for large values of L(Mk). In Hjellvik and Tj0stheim (1995), bootstrap replicas, see (Wehrens et al., 2000), of the test indices are first obtained to form the distribution. The confidence interval can then be constructed from the resulting distribution of L(Mk). E. The Higher Order Correlation Function Test While most of the methods mentioned above are found mostly in economic literature, a linearity test based on higher-order correlation functions (Billings and Voon, 1983) was designed to quantify the degree of nonlinearity for control purposes. It is assumed that under the excitation of normally distributed signals, the system generating output data yt is linear if the following higher-order correlation function 4>yy2 (r) equals to zero for all time lag r. 2 (r) = E[{y t _ T - y}{yt - y}2} N « N-lJ2{yt-r-y}{yt-y}2 (2.1.12) t=T 14 C H A P T E R 2. 2.1.2 The Norm-bounded Error Approach where y denotes the sample mean of time series y t . The null hypothesis for this test is that (f)yy2 has a Gaussian distribution for the system generating linear output sequences. The 95% confidence interval for the system generating nonlinear output sequences under the aforementioned input signals is given as follows (r) yy (o)v^v(o) > 1.96 (2.1.13) where the autocorrelation function 0 w (r) and another higher-order correlation function (f)y2y2 (r) at r are defined as follows: 0™(r) Hiyt-r - y } { y t - y } } N N - 1 Y , { v t - r - y } { y t - y } (2.1.14) (j)y2y2 (r) = E[{yt_T - y } 2 { y t - y } 2 } N • « N ~ l Y , { y t - r - y } 2 { y t - y } 2 (2.1.15) t=T A common critic of the statistical approach is that these methods involve data obtained from open-loop experiments. The conclusion of the test is difficult to carry forward to closed-loop systems. The amount and the type of process excitation is another issue. Albeit it was shown by Billings and Voon (1983) that normally distributed signals give good excitation for nonlinear systems, the implementation of the normally distributed input sequences to a real process can be difficult. 2.1.2 The Norm-bounded Error Approach The norm-bounded error approach has been widely explored in the control community. The basic idea of this approach is that the 'distance' or error between a nonlinear plant (operator) and its linearized model (linear operator) is measured based on some operator norms. An interesting feature of this class of methods is the application of functional 15 C H A P T E R 2. 2.1.2 The Norm-bounded Error Approach analysis and operator theory. In the sequel, various nonlinearity measures of this class are briefly reviewed. Desoer and Wang (1980) proposed a nonlinearity measure based on a minimization problem of the following form. c;^ inf \\N-L\\ (2.1.16) where N and L denote the nonlinear plant and its linear model. In Eq.(2.1.16), A is a set of linear models and the norm can be any suitable norm. For example, an inner-product norm or an induced Jzf2 norm can be used. This definition is more philosophical rather than practical. As pointed out by Eker and Nikolaou (2002), when the induced norm is used, the computational problem can be very complicated. To address the computational problem, Nikolaou (1993) proposed a notion of inner prod-uct and hence its associated norm for the nonlinear operator. Defining an appropriate input space and exploiting the aforementioned norm, Eq.(2.1.16) gives the distribution of c; using Monte Carlo simulations. Clearly, this distribution arises from the discrep-ancy between the real plant and its linear models. However, a major drawback of this method is that the associated norm is not an induced norm since it does not satisfy the submultiplicativity property (i.e. | |AB| | < Therefore, it is difficult to employ in the synthesis of a feedback controller. Alternatively, Allgower (1995) reformulated Eq.(2.1.16) using the following induced norm: <f = inf sup ^ — li- (2.1.17) where denotes the input space of interest. After applying the above nonlinearity measure to a number of cases, it was found that this definition is insensitive to the choice of parameterization for u and is proved to be useful in formulating the nonlinearity measure3. It is clear that the above equation has an interpretation of measuring the 3Whenever 7Y can be represented by a set of LTI models, the norm shown in Eq.(2.1.17) is equivalent to an induced norm. This fact is fully exploited in formulating the developed nonlinearity measure of this thesis. 16 C H A P T E R 2. 2.1.2 The Norm-bounded Error Approach minimum worst4 discrepancy between the nonlinear plant and its linear model. Helbig et al. (2000) extends the idea by not only considering the input space of the system, but also by including the initial conditions. The modified definition is given as: <=inf sup inf l l ^ ^ , o ) - ^ ; ^ , o ) l l ( 2 L l g ) An advantage of this approach is that <f is normalized to take values between 0 and 1. This make the results more interpretable. However, the required computational efforts for <f can be intensive. Methods discussed so far are mainly concerned with quantifying open-loop nonlinearity. To cope with closed-loop nonlinearity, Eker and Nikolaou (2002) proposed a measure based on an internal model control (IMC) structure. An incremental norm is used in this framework. The output discrepancy of two IMC closed-loops, one containing a nonlinear plant and the other consisting of the linearized version of the nonlinear plant, is obtained, see Figure 2.1. Recall that N and L denote the nonlinear plant and its linear model, the closed-loop nonlinearity measure is defined as ? = \\W(y2 - y i ) U z = \\W(NQ(I + NQ- LQ)~l - LQ)\\AZ (2.1.19) where W, Q and || • \\AZ denote a stable low-pass filter, Youla parameter and incremental norm evaluated over an input set Z, respectively. A major drawback of the methods presented in this part is that most of them consider open-loop nonlinearity only. As discussed earlier, measuring open-loop nonlinearity gives little or no information about the closed-loop one. For other cases that deal with the closed-loop nonlinearity, they only consider the discrepancy with respect to the output signals over a set of input signals. The underlying assumption of this definition is that both the nonlinear plant and the linear model are subject to the same input sequences. Obviously, this definition is quite artificial since it is known that the system's input-output signals are modified when the loop is closed. Moreover, when the nonlinear plant Owing to the terms inf and sup. 17 C H A P T E R 2. 2.1.3 The Geometrical Approach Q Q L N yi y2 Figure 2.1: Nonlinearity measure proposed by Eker and Nikolaou (2002). is different from the linear model (i.e. N L), the input-output signals of the two closed-loops can be very different. In other words, yi, y2, u\ and u2 in Figure 2.1 are different. Therefore, a more natural definition of the closed-loop nonlinearity measure is to consider the discrepancy of both input and output pairs of the two closed-loops. This new definition coincides with the graph metric notion (Vidyasagar, 1984), which is equivalent to the gap metric (Zames and El-Sakkary, 1980) and shall be discussed in 52.2. 2.1.3 The Geometrical Approach The third class consists of a differential geometry based approach proposed by Guay et al. (1995); Guay (1995), with extensions made in Guay et al. (1997a,b). Unlike the methods mentioned in the first two classes, which measure the open-loop nonlinearity, methods in this class are establishing measures for closed-loop nonlinearity, in general with a unity type controller. In Guay et al. (1997b), differential geometry interpretation of relative gain array was used to assess the degree of closed-loop nonlinearity for a given plant, however, this method is known to suffer from lack of accuracy due to process noise. 18 C H A P T E R 2. 2.2. Overview of The Gap Metrics 2.2 Overview of The Gap Metrics The aperture or gap metric of two closed Hilbert subspaces (or finite energy subspaces, denoted by J&z) is the distance quantified by the acute angle between them. This metric was first introduced by Krein and Krasnosel'skii (1947) and Sz.-Nagy (1947) into op-erator theory in order to extend the operator norm topology to unbounded operators (Feintuch, 1998). Mathematically, the gap metric for two Hilbert subspaces M\ G £ £ 2 and M2 G Jz?2 takes the following form (Krein and Krasnosel'skii, 1947): l(Jlx,J(<i) = max< sup ||n^±a:||, sup ||n^±x|| \ (2.2.1) [ x e ^ i , ||cc11=i i e ^ 2 l ]|i||=i J or alternatively (Sz.-Nagy, 1947), ( 5 ( ^ , ^ 2 ) ^ | |n^-n^ 2 1 | (2.2.2) where VLX and J ^ x denote the orthogonal projection onto subspace and orthogonal complement of Jif. The acute angle 0 between M\ and #^2 can be related to the gap metric as follows: 9 = s in - 1 l(JluJt2) (2.2.3) A few decades later, Zames and El-Sakkary (1980) and El-Sakkary (1985) exploited the gap metric notion to capture the type of perturbations of an unbounded LTI operator (i.e. an open-loop unstable system) that maintain the closed-loop stability in control theory. Alternatively, Vidyasagar (1984) defined a topological equivalent metric, the graph metric. The graph metric is found to have the same properties as the gap metric. In fact, both metrics induce the same topology, known as graph topology (Vidyasagar et al., 1982; Vidyasagar, 1984). It was shown that the graph topology is the weakest topology in which feedback stability is a robust property (Vidyasagar, 1984). This means that a variation of the elements of a stable closed-loop system within small neighborhoods with respect to their nominal values preserves closed-loop stability. Unequivocally, the use of the gap metric notion and its link to the closed-loop stabilization problem of 19 C H A P T E R 2. 2.2. Overview of The Gap Metrics LTI systems, from the graph topology perspective, have sparked a paradigm shift in the control community. Even though the gap metric provides a new paradigm to feedback stabilization, it only gained popularity after the seminal work by Georgiou (1988). In this development the gap metric is recast into a two-block optimization problem, which is computable using the commutant lifting theorem (Francis, 1987; Foias and Frazho, 1990). Subse-quently, Georgiou and Smith (1990) showed that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime fac-tor perturbations. In this work, it was shown that the radius of the uncertainty ball induced by the gap metric is equal to the radius of an uncertainty ball defined by the normalized coprime factorization. Foias, et al. (1990) and Ober and Sefton (1991) show that there is a geometrical interpretation for the gap metric notion. The relationship of closed-loop stability and the coordinatization between the graph of the plant and the inverse graph of the controller was established. In addition, they also pointed out that the generalized stability margin defined in robustness optimization for normalized coprime factor perturbation has a close connection with the parallel projections between the aforementioned two graph spaces. Doyle et al. (1993) extended the idea of a parallel projection and the feedback stability notion to nonlinear systems. El-Sakkary (1989) showed that the frequency responses5 of two single-input single-output (SISO) linear systems, g(s) E J%2 and h(s) € J$?2, can be thought as the stereographic projection onto a Riemann sphere6. The gap metric is simply the chordal distance, as defined in Eq.(2.2.4), between the images on the Riemann sphere corresponding to the two frequency responses of the two aforementioned systems in the extended complex plane. 5Ms), h(s)) = sup , J . 9 ^ : ^ , H ( , | 2 (2-2.4) An obvious shortcoming of the above definition is that the computation efforts to de-termine Sc(g(s),h(s)) can be very intensive since a search for all ( > 0 is needed. A 5Generally, a frequency response of a system L is the set of all points L(s) where s = £ + jw covers all points in the closed right half of the complex plane. 6See Figure 3.13. 20 C H A P T E R 2. 2.2. Overview of The Gap Metrics multi-input multi-output (MIMO) version of the chordal metric is defined by Qiu and Davison (1990), which is called the pointwise gap metric. In contrast to all the metrics discussed previously, Vinnicombe (1991, 1993) defined a new metric (often referred as the z^ -gap metric or the Vinnicombe metric) in the Jzf2 space. This new metric is also known to induce the same graph topology mentioned above. Since the Jz?2 space consists of both causal and anti-causal signals, care must be taken to confine oneself to causal finite energy signal space7 in order to preserve feedback stability. Topologically, this means that the two closed-loops must be homotopically equivalent. In the definition of the z^ -gap metric, the winding number8 is used to ensure that the two systems in consideration are homotopically equivalent. It can be shown that the z;-gap metric is always less than or equal to the gap metric whenever a homotopy condition is satisfied. The strength of the z>-gap metric, as compared to other metrics, which induce the same topology, lies in the fact that it gives the least conservative robust stability results whenever a homotopy condition is satisfied (Vinnicombe, 1993). In this sense if the z^ -gap between two plants is large, then a controller that gives satisfactory robust stability for one plant will show poor robust stability or even will destabilize the other plant. Likewise, if the z/-gap between two plants is small, then a controller which guarantees robust stability of one plant implies that it robustly stabilizes the other. Recently, several attempts have been made to generalize the idea of the gap metric to nonlinear systems. For instance, see (Georgiou, 1993a,b; Georgiou and Smith, 1994, 1997; Vinnicombe, 1998; Anderson and Bruyne, 1999; Vinnicombe, 1999a; James et al., 2000). Georgiou (1993a,b); Georgiou and Smith (1994) extended the idea by using the notion of a differential graph. The resulting gap is termed as differentiable gap or d-gap. Feedback stability is established using the relationship between the d-gap and the parallel projection operators. Georgiou and Smith (1997) extended the gap metric notion to nonlinear systems by using sector conditions (Zames, 1966a,b) and integral quadratic constraints (IQCs) (Megretski and Rantzer, 1997). 7For a physical realizable stable closed-loop, one needs both finite energy and causal signals in the loop. 8see §A.7. 21 C H A P T E R 2. 2.2. Overview of The Gap Metrics Similarly, the v-gap metric is used in conjunction with the IQCs to extend the existing theory to cope with nonlinear systems. In Vinnicombe (1998), the nonlinear plant P is assumed to satisfy a set of IQCs, say & . It was shown that if there exists a stabilizing controller C satisfying bPo ^ > (3 for the nonlinear plant P, where bPo ^ denotes the generalized stability margin defined by P 0 and C, and the f^ -gap metric between any nominal plant, P 0 and the IQCs (i.e. <5„(Po, &)) is smaller than /3, then one can conclude that any controller C that satisfies bp0ic > P can stabilize the nonlinear plant P. In this case, the homotopy condition is the implicit requirement of the existence of controller C. Further, in Vinnicombe (1999a) the definition of the z^ -gap metric is extended, without using IQCs, to nonlinear systems. Mathematically, the nonlinear i^ -gap metric has the following form: < W P o , Pi) = max {T%(Po, Pi), ~SJ?2(PI, Po)} (2.2.5) where fs*(PQ,Pi)± sup inf. , Po) = sup inf ^gf^ xoeSoruS-b sieging 1 1 1 0 1 1 2 xiegiOSf2 *oeS0as?2 and Qi = {[%] : y = PjU, y,uE Jz?2,ce}- Note that under this definition, all signals are defined in the extended ££i,c& signal space9. It was shown that the nonlinear i^ -gap metric is greater than the z^ -gap metric for LTI systems. In contrast to the LTI case10, the homotopy condition is only a sufficient condition for feedback stability for the nonlinear z^ -gap metric. In addition, the determination of the homotopy condition in the LTI case, which involves an evaluation of a winding number, is far more easier than the nonlinear case, where a rigorous mathematical approach of determining the homotopy condition is required. 9 A n extended space is an extension of a normed vector space. In this space, signals may not be bounded in the norm of the vector space. However, the signals are bounded under any truncation to a finite time intervals. 1 0 The homotopy condition denned in the i^ -gap metric for LTI is both sufficient and necessary con-dition for feedback stability. 22 C H A P T E R 2. 2.3. Summary 2.3 Summary This chapter discussed several existing nonlinearity quantification techniques. It is ob-vious that most of these methods are restricted to quantifying open-loop nonlinearity. Among the three classes (i.e. the statistical approach, the norm-bounded error approach and the geometrical approach), the norm-bounded error approach is particularly appeal-ing since it can be easily recast into a robust control problem. Unfortunately, most of the approaches under this category either only deal with open-loop nonlinearity or only consider output discrepancy over a set of input signals. The latter definition becomes very unnatural and quite restrictive when the systems under consideration are in closed-loop since it is assumed that both closed-loops (i.e. the one with the nonlinear plant and the other with a linear model) are subject to the same input sequences. This as-sumption is not valid since the closed-loop is known to modify the input-output signals. So, whenever the nonlinear plant is different from the linear plant, the signals in both closed-loops can be very different. As a consequence, the input sequences of interest of these two closed-loops are different as well. To cope with this, as can be seen from the discussion in §2.2, the gap metric notion provides a convenient way of measuring dis-tance between a nonlinear plant and its linear approximation. In fact, in this work, the definition of the z^ -gap metric is modified slightly to quantify closed-loop nonlinearity. However, a major stumbling block for most of the engineers who only attended a first control course is that the k'-gap metric is quite mathematically involved. In addition, background knowledge such as functional analysis, complex analysis, operator theory, topological space and modern control theory are needed to understand and appreciate the whole framework. To resolve this dilemma, the next chapter is devoted to present a novel and an easy-to-understand perspective of the philosophy behind the z/-gap metric. A pictorial approach is proposed in this regard. 23 C H A P T E R 3 Graph Topology, The Gap Metric & Closed-loop Stability This chapter presents a geometrical sense of the gap metric and the u-gap met-ric (hereafter, called the gap metrics). Several diagrams, instead of mathematical expositions, are used to give further insight into the geometrical interpretation of the gap metrics. Finally, the connection between the small gain theorem and the gap metrics is presented. 3.1 Introduction As discussed in the previous chapter, the (nonlinear) z^ -gap metric provides a convenient framework for assessing the "distance" between a nonlinear plant and a linear model, possibly the linearization of the nonlinear plant at a particular operating point. However, the concept of the z^ -gap metric can be esoteric and is quite mathematically involved. In this chapter, a new way of presenting the philosophy behind the z^ -gap metric is proposed. In contrast to the existing literature discussing about the gap metric and the z^ -gap metric (hereafter, called the gap metrics), a pictorial approach is used. The uniqueness of this approach is that a series of diagrams, instead of pure mathematical expositions, are used to clearly elucidate the geometrical perspective of the z^ -gap metric and its connection to robust control theory. The importance of this approach is that it provides an intuitive and a better insight into the z^ -gap metric and robust stability notions for engineers, particularly for process control engineers. In addition, for the 24 C H A P T E R 3. 3.2. Finite Energy Signal Spaces mathematical oriented readers and for the sake of completeness, a number of proofs, which will otherwise disrupt the flow of the exposition, are presented in Appendix B. This chapter begins with a graphical introduction to finite energy signal spaces, partic-ularly the Jz?2 and the J^f2 spaces, and the operator (i.e. system) graph spaces. Then, a discussion on the connection between the operator graphs and closed-loop stability is presented. A geometrical interpretation of the gap metric and its relationship with the generalized stability margin is discussed. 3.2 Finite Energy Signal Spaces A signal can be defined as any physical quantity that varies with time. In this thesis, we are interested in a special class of signals that have finite energy since it is often desirable for a closed-loop system to have finite energy signals in the loop in order to preserve closed-loop stability1. A typical example of such signals looks at the transient signals, which decay to zero as time progresses. Figure 3.1 shows two simple examples of such finite energy signals. Mathematically, a finite energy signal space is called the J£2 space2. oo (b) Figure 3.1: Examples of finite energy signals, (a) A signal that is defined over time interval [t0 ti], (b) A signal that has finite area between the curve and the time axis from —oo to +oo. 1In this context, the closed-loop system is said to be bounded-input bounded-output (BIBO) stable. 2See §A.3 25 C H A P T E R 3. 3.3. Operator Graphs Note that signals in the Jz?2 space can be split into two unique subclasses (i.e. an anticausal signal class and a causal signal class) with respect to the time axis. An anticausal signal is defined over the negative time axis and is zero for the positive time axis. In contrast, a signal with zero values for the negative time axis and nonzero otherwise is called a causal signal. Noncausal signals are signals that have nonzero values in both positive and negative time. Figure 3.2 depicts such anticausal, causal and noncausal signals. Mathematically, a collection of causal finite energy signals forms the J%2 (or Jz?2+) space and that of anticausal finite energy signals is termed as the J^1 (or Jzf2-) space. Clearly, the noncausal finite energy signals space is precisely the Jtf2 space. For a precise mathematical definition of these spaces, see §A.3. 1 t < 0 t > 0 t < 0 t > 0 (a) (b) (c) Figure 3.2: Types of signals, (a) Anticausal signal, (b) causal signal and (c) noncausal signal. 3.3 Operator Graphs Recall that the graph3 of a (possibly open-loop unstable) system is the collection of all possible finite energy (or bounded) input-output pairs of signals entering and produced by the system. Mathematically, the graph of a system P : @f2 C <2f G Jft -> & E J^2 is defined as: CW ± & ®& (3.3.1) where and are the input and output spaces. These spaces belong to the Hilbert space, J#2. Iv, @f2 = {u E W E Jff2 •• Pu E & E J^2} and 8 denote the identity 3See page 133. 26 P C H A P T E R 3. 3.3. Operator Graphs operator on the domain of P and the direct sum, respectively. Likewise, the Jz?2 graph of a system P : 3)f2 C U G Jzf2 —• Y € JSf2 can be defined as follows: ^ A p (3.3.2) where ®f2 = {u G U G j£f2 : Pu G Y G jSf2} is the domain of P. Similarly, the J%2 graph of a controller is given by: c (3.3.3) where Q>Q2 = {y G W : Cy G ^ } denotes the domain of the controller and is the identity operator on . Note that Eq. (3.3.1) induces a submanifold in ^ © ^ while Eq. (3.3.3) has its in <3f . Therefore, for consistency, the inverse graph of the controller C is defined as yc — 0 I c I 0 (3.3.4) Note that the inverse Jzf2 graph of the controller can be defined analogously. The idea of J??2 and ^ spaces plays a central role in the graph topology, particularly when dealing with closed-loop stability. Therefore, it is essential to distinguish between these two spaces. Recall that, in frequency domain, a j£f2 space consists of the Fourier transforms of noncausal finite energy signals, while a J%2 space contains the one-sided Laplace transform of causal finite energy signals, and is analytic in the open right half plane (RHP). Note that, when the signals are causal, its one-sided Laplace transform is equal to its two-sided (or bilateral) Laplace transform. In addition, if the ^ signal is bounded on the imaginary axis, it can be converted by means of the Fourier transform by substituting s = jco into the frequency domain. Unequivocally, the Jz?2 and the J^f2 spaces may not generally induce the same topology. 27 C H A P T E R 3. 3.3. Operator Graphs For a physically realizable stable feedback system, it is desirable to have both causal and finite energy signals in the loop. As a consequence, extra care must be taken when dealing with the Jz?2 space. Figure 3.3 gives a graphical representation of the above discussion. The Jzf2 space consists of a Hilbert space J ^ , represented by a perfect disk, as its subset and a few "holes" punctured by the space. {Si}, {52} and Figure 3.3: 5£<i and ^  spaces. {^3} are three Cauchy sequences4 in the Jzf2 space. The subscripts of each sequences denote the time index. A negative time index means that the signals are defined for the negative time and positive time index implies that the signals are defined for the positive time. Obviously, both {52} and {53} belong to since lim m-i = m* £ Ji% i—too and lim pi = p* G ^ 2 , and are causal signals. On the other hand, this is not true for i—*oo {Si} even though lim rij = n* E Jzf2 is bounded but the signals are noncausal. In fact, l—>co the anticausal part of {Si} induced a "hole" on the Jzf2 space, which makes it different from the J$?2 space. Evidently, the topologies induced by the Jzf2 and J%2 graphs are different in this case. Therefore, to be confined to the 3>i% spaces when dealing with J§?2 spaces, a certain condition needs to be imposed. Mathematically, this can be achieved by introducing the concept of a homotopy condition, which shall be discussed in §A.1.5. Since one of the primary concerns of feedback control is stability, its connection with operator graph emerges naturally and will be discussed in the next section. 4 See§A.1 .4 28 C H A P T E R 3. 3.4. Operator Graphs and Closed-loop Stability 3.4 Operator Graphs and Closed-loop Stability In what follows, the J%2 graph is assumed throughout. Consider a standard feedback configuration consisting of a system P and a controller C, as depicted in Figure 3.4. Figure 3.4: Standard feedback configuration From Figure 3.4, we have I C Ul Wi p I U2 w2 (3.4.1) J(P,C) Note that in order for J(P, C)'1 :[Zl2] [ul] to have a unique and physically realizable solution, Eq.(3.4.1) must be well-posed (i.e. [ P I ] has to have a causal inverse). The feedback system [P,C] is stable if the aforementioned mapping J(P,C)~l is bounded (i.e. J(P,C) produces finite energy outputs from finite energy inputs). This implies that Qp@Q1c = W . To see this, Eq. (3.4.1) can be rewritten as: C wx Ui + u2 = P I w2 (3.4.2) Gp Obviously, the decomposition of W into Qp and Qlc is possible if and only if ui, u2, yi, U>1 ' W2 \ l one y2, u>i and w2 are all unique. This means that given an arbitrary signal w = can always find a unique decomposition in Qp and QQ, whenever the closed-loop shown in Figure 3.4 is stable. The converse is also true. This result was shown by Doyle et al. (1993) and is presented as the next theorem: 29 C H A P T E R 3. 3.5. Operator Graphs and Closed-loop Robustness Theorem 3.4.1. Given a well-posed closed-loop [P,C], the closed-loop is stable if and only if the following two conditions hold 1. gPngc = {0} 2. QP@QC = w, in other words, the graph of the plant and the inverse graph of the controller induce a coordinatization5 ofW. Proof, see Proposition 4 of Doyle et al. (1993). • Based on the above theorem, the following remarks can be made. Remark 3.4.1. The significance of Theorem 3.4-1 is that it links two seemingly unrelated concepts (i.e. closed-loop stability and graph topology) in an elegant manner. Remark 3.4.2. The first condition in Theorem 3.4-1 establishes a condition for the existence of unique decomposition. Practically, this can be achieved by assuming that the system output y = Pu = 0 whenever u = 0. The second condition is a consequence of the stable closed-loop [P,C] and Eq.(3-4-1)-Remark 3.4.3. The unique decomposition ofW into Qp and Qc implies that there exist a pair of projectors that project W onto Qp and Qc, respectively. These two projectors turn out to have a close connection with closed-loop robustness, which will be discussed next. 3.5 Operator Graphs and Closed-loop Robustness One of the advantages of using feedback control is its ability to handle uncertainty. The extent of a closed-loop system to tolerate uncertainty before the whole system becomes unstable is called closed-loop robustness. In classical control, the gain and phase margins 5See §A.1.6. 30 C H A P T E R 3. 3.5. Operator Graphs and Closed-loop Robustness are often used to assess the robustness of a SISO closed-loop system. For multi-input multi-output (MIMO) systems, the graphic intuition in the classical gain and phase margins are lost due to the dimensionality and directionality properties. An alternative way to assess closed-loop robustness is to determine the sensitivity of the system with respect to noise and disturbances. Considering again Figure 3.4, where the plant's outputs yi and inputs U\ are subject to u>i and w2, this dependence can be written as follows: / -C w2 I Vi P (I - CP) (3.5.1) or Ui " (I- CP)'1 -V- CP)~LC 2 / i P(I- • CP-1)) -p(i- - CP)-LC Wl w2 (3.5.2) If the closed-loop is stable, we are interested in minimizing the effects of [%2] to which means that we want to minimize the following cost function: (i-cpy'li -c ' yi' 7 ^ (3.5.3) which can be alternatively, defined as: 7 (I-CP)'1 I -C - i CO (3.5.4) Such a statement allows for the above minimization problem to be restated as a maxi-mization problem (i.e. sup bptc). The minimization of the cost function in Eq. (3.5.3) can be seen as the minimization of closed-loop sensitivity and complementary sensitivity functions6. Interestingly, this norm is actually the one that needs to be minimized for the plant with normalized coprime factor type uncertainty. Denoted by bPiC, the alternative cost function in 6The entries (1,1) and (2,2) of Eq.(3.5.2) are the sensitivity and complementary sensitivity functions, respectively 31 C H A P T E R 3. 3.5. Operator Graphs and Closed-loop Robustness Eq.(3.5.4) is also called the generalized stability margin and is used in the McFarlane-Glover Jftfoo loop shaping (McFarlane and Glover, 1992) procedure later employed as a controller design technique in this thesis. Note that the bPic always has it values be-tween 0 and 1. Small bptc means that the closed-loop stability margin is small and will become unstable when the bPic = 0. Larger bP>c indicates that the system has good robustness. Unequivocally, this makes the generalized stability margin easy to interpret. In addition, the bptc also turns out to have a very nice geometrical interpretation. The rest of this section is devoted to explaining its insight by graphical means. Recall that a projection operator is defined by its idempotent property (i.e. LT2 = LT). In Figure 3.4, it can be easily shown7 that the stable closed-loop transfer function from [wl] to [yi ] (i.e. [P] (I-CP)~1 [i -c]) is a projection operator that maps «£f2 signals (i.e. u>i and W2) onto the plant graph QP. Similarly, the stable closed-loop transfer function from \ to [%22] (i-e. [/] {I-PC)~1 [-PI]) is also a projection operator that maps Jzf2 signals onto the inverse graph of the controller. In addition, since [P] (I-CP)~1 [i -c] + [ ? ] ( ' - - P C ) - 1 [ - p 7 ] — [fj /]> a n Y signals in Jz?2 can be uniquely decomposed into Qp and QQ using the above two projection operators. Clearly, geometrically, [P] (I-CP)~1 [i -c] represents the parallel projection onto Qp along Qc and is denoted by Hgp^. Likewise, denoted by Hg^gp, [Cj] (I-PC)~1 [-p i) is the parallel projection onto Qc along Qp. Note that the expressions [j,] (I-CP)"1 [I -C) and [</] i1-?0)'1 [~p J] are not valid for nonlinear systems. To resolve this, Hgp^g^ and Ug^gp can be rewritten as follows, see (Doyle et al., 1993) I n, QPWQ'C - {I-CP)'1 i -c (3.5.5) P 0 0 / 0 -1 + J(P, C) 0 / 0 -I '([^('-OP)-1 [/ -c))2 = [P] (i-cpyl (i-cp) (i-cpy1 [i -c] = \P]{I-CP)-1 [i -c). i 32 C H A P T E R 3. 3.5. Operator Graphs and Closed-loop Robustness and C - l S{;\\Qp ~ (I - PC) -P I (3.5.6) I / 0 I 0 - 1 + J(P, C) 0 0 0 I An advantage of writing the parallel projectors in the form of Eqs.(3.5.5) and (3.5.6) is that the invertibility of the transfer function J(P, C) is an explicit requirement for the existence of the aforementioned parallel projectors. Figure 3.5 illustrates the geometrical projections of Hgp\\gic and Rgic\\gp. Figure 3.5: A pair of parallel projections, Ugp\\g^ and TLgi^gp. Under these projections, any signal in the Jz?2 space (i.e. h in this case) can be uniquely decomposed into m G Qp and n EQQ such that h = m + n or Jzf2 = Qp © QQ-The existence of the aforementioned projectors has a close connection with closed-loop stability. The next theorem, which was established by Doyle et al. (1993), summarizes such a relationship. Theorem 3.5.1. The feedback interconnection [P,C] in Figure 3.4 is stable if and only if there exists a pair of stable parallel projectors N-gp\\gi, and n 0 p . he ^ Proof. See Appendix B . l . 33 • C H A P T E R 3. 3.5. Operator Graphs and Closed-loop Robustness Having defined the parallel projectors, we are now ready to see what the generalized stability margin represents geometrically. For the sake of notational simplicity, let = GP and jY ^ QQ. Define Aje^ = TL^J.^ (i.e. the restriction of orthogonal projection onto the orthogonal complement of the inverse graph of controller to the graph of plant. For the definition of this restriction, see §A.1.2). It was shown in Foias_ et al. (1990, 1993) that Q,JC,JV = A^^U.t/y± : J£>i —»• and (I — Qjz,^) : Jz?2 —> are two parallel projection operators. It is obvious that = ^gP\\gic and I — Qje^ = Tigi^gp. The graphical representation of Ajztjy and the two graph spaces M and JY are shown in Figure 3.6. Figure 3.6: Relationship of bp:c — sin.6 and the minimal angle (i.e. 0) between subspaces M and JY . From Eq.(3.5.4), Eq.(3.5.5) and the above discussion, the bptc can be expressed as fol-lows: bp,c = lin^iic/ II"1 = \\QM,A\-x = WA'J^n^w-1 (3.5.7) Since the bptc is only defined when Ugp\\gic is bounded, this implies the invertibility of Ajr.r/r- By invoking the property of the minimal modulus of an operator (i.e. if 34 C H A P T E R 3. 3.6. Operator Graphs and Uncertainty Description J?T € £ ( J z ? 2 ) is invertible, then p{J(f) = inf{||JcTx|| : = 1} — n^-iy), we can write: bp,c = fi(Ajg^) = fi(UNx\^f) (3.5.8) = inf{\\A^y^x\\ : x E ^  and = 1} (3.5.9) = inf {dist (x, : x E ^ # and \\x\\ = 1} (3.5.10) Eqs.(3.5.8) and (3.5.9) are immediate from the definition of the minimal modulus. Graphically, Eq.(3.5.9) represents the minimal distance between the origin (i.e. the intersection of JY and JV1-) and the point projected by Ajt^jv from a unit sphere in ^ onto jYx (i.e. the point Aj^^x as shown in Figure 3.6). The last equality results from Figure 3.6, where the bP>c can be interpreted as the smallest distance between a point on a unit sphere in subspace ^# to subspace JY . Clearly, the bPic can be related to the minimal angle between subspaces M and J/ as follows: sinc? = inf WA^XW = i n f \\A^x\\ = bPiC (3.5.11) xe^,\\x\\y£o \\x\\ xe^,\\x\\=i where 9 is defined by: 0 A cos"1 | ^ ' ,0^x E^,0^y EJY (3.5.12) Since 0 < bPic < 1, the above sine function is monotonically increasing in the range of [0, | ] . This means that the increasing of the bPc is proportional with the increasing of the minimum angular distance between the subspaces M and jY. A typical value for 6 is around ^ rad, which corresponds to 30% normalized coprime factor uncertainty. 3.6 Operator Graphs and Uncertainty Description As discussed in §2.2, Zames and El-Sakkary (1980) used the gap metric arising from graph topology to capture the type of perturbations of an open-loop unstable LTI system that preserves the closed-loop stability in control theory. This idea proved to be very 35 C H A P T E R 3. 3.6.1 The Gap Metric useful particularly in formulating uncertainty description for robust control problems. In this regard, this section aims to highlight the relationship between the gap metric and a special type of uncertainty, the normalized coprime factor uncertainty. The first subsection defines the gap metric in its graphical sense. Similarly, the u-gap metric and its geometrical interpretation are presented in the second subsection. The third subsection shows that the uncertainty quantified by the gap metric is equivalent to the normalized coprime factor type. 3.6.1 The Gap Metric Conceptually, the gap metric 5(P0, Pi) is quantified by the maximum of two directed gaps (i.e. S y/?2(Po, Pi) and 5 j^(Pi, Po)), which measure the maximum distance between all possible bounded input-output pairs of the two plants of interest. Mathematically, the gap metric is defined as follows: S(P0,Pi) ±max{7M(PQ}Pi),~5ytf2(Pi,P0)} (3.6.1) \\X0-Xl\\2 where 7 m ( P 0 , Pi) = sup inf "X"!,0"2,7 r A P u P 0) = sup inf .. .. and Qi = {[u] : y = PiU, t / , « e J^.}- Note that the z^ -gap metric can be defined in a similar fashion except that the signals are defined in the Jzf2 space and a homotopy condition is needed to confine oneself to causal finite energy signal space. The detail discussion of the z^ -gap metric is deferred to §3.6.2. According to Eq. (3.6.1), the computation of the gap metric involves the search of two supremums and two infimums over the graph spaces of plants Po and Pi . Obviously, a direct search of optimum solution for this problem is computational intensive. A simpler solution of this optimization problem has its root in the Hilbert space operator theory. Specifically, the orthogonal projection operator plays an essential role in formulating the solution for the optimization problem posted in Eq.(3.6.1). In contrast to the existing literature, here a novel approach is adopted to guide the readers to visualize how one can begin with two stable closed-loops and finally arrive at the optimal solution for the 36 C H A P T E R 3. 3.6.1 The Gap Metric aforementioned optimization problem. To begin, consider the following two stable closed-loop systems under unity feedback, as depicted in Figure 3.7 (a). Recall that a physically realizable stable closed-loop consists of causal finite energy signals in the loop. This implies that signals u0, ui, yo and y\ are all in 3<% space. This is consistent with the requirement of the M2 graph definition, see §3.3.1. Next by collecting all these input-output signals in the form of ordered pair8, the graph spaces represented by the dots9 in Figure 3.7 (b) for plants P 0 and P\ can be constructed. Lastly, these two graph spaces, denoted by ^ \ and ^#2, can be put together to compute the gap metric between them, see Figure 3.7 (c). Po yo _ Ml p. yi _ t' (a) (b) (c) Figure 3.7: From plants' input-output space to the gap metric, (a) Two stable closed-loops containing plants P0 and P i , respectively; (b) The collections of all possible bounded input-output pairs of the closed-loops that form two graph subspaces (denoted by the dots); (c) The two graph subspaces can be put together to form a convenient framework to quantify their similarity. Note that the directed gap defined previously can be seen as the maximum difference between all possible bounded input-output pairs of the two plants with a unity feedback. To formulate this in terms of graph topology, an appropriate definition of the distance between graphs is required. Such a definition can be obtained via the following simple geometrical observation: 8See §A.1.1 9 I n Figure 3.7 (b), for the sake of graphical clarity, only a fictitious set of bounded input-output signals is shown. In reality, the graph space should contain all the possible bounded input-output pairs. 37 C H A P T E R 3. 3.6.1 The Gap Metric The shortest distance between any point x in subspace and the subspace is the distance between that point (i.e. x € and its orthogonal projection onto J%2-In this light, consider the following diagram, which is a duplication of Figure 3.7 (c): Figure 3.8: The directed gap between two closed Hilbert subspaces To define the directed gap from M\ to JMI, let's first pick a point, say X \ , in and orthogonally project it onto The image of X\ on is given by H^2xi, where nj^ 2 denotes the orthogonal projection onto the subspace ^#2- The distance between x\ and its image on ^ 2 subspace is denoted by d\. Next, a second point, say x 2 , is picked and the similar operations are repeated again. The resulting distance between the new point x2 and its associated image U^2x2 is d 2. These operations are repeated so on and so forth until all the points in the subspace M\ are exhausted. Denoted by S (^#L,^#2), the directed gap from to ^#2 is then obtained by taking the maximum of all these distances (i.e. supj di) or in a more compact form: S (dtuJti) = sup ^ n ^ 2 ^ 2 = s u p \\x _ Yl^2x\\2 (3.6.2) i e ^ i |F||2 xe^i,\\x\\2=i where || • ||2 denotes the Euclidean 2-norm. Further, Eq.(3.6.2) can be rewritten as 5 ( ^ i , ^ r 2 ) = sup - Yl^2)x\\ = sup Hn^ixllij (3.6.3) X<=.^i,\\x\\2 = l x£^l,\\x\\2 = l where nJ{±. = I — Uy/2 is the orthogonal projection onto the complement of 38 C h a p t e r 3. 3.6.1 The Gap Metric Similarly, 8 (^#2, sup xe^2,\\x\\2 = l x - Tl^x^ s u p I  n*-^ II2 'S I I . . I I 1 1 (3.6.4) X€^2,\\x\\2 = l It is obvious that 8 ( ^ i , ^ 2 ) 7^  $ ( ^ 2 , ^ 1 ) in general. Unequivocally, this implies that the directed gap definition can not be used as a metric, see §A.1.4. To resolve this problem, a new gap function has to be defined: Remark 3.6.1. 8(^1,^2) does not generally satisfy the triangle inequality. However, 1. See Krasnosel'skii et al. (1972, pg. 206). Remark 3.6.3. Recall that ^ \ and M2 are the graphs of PQ and P 1 ; respectively. It is clear, from the comparison of Eq. (3.6.1) and Eq.(3.6.5), that the definition of the gap metric given by the above two equations are equivalent. Since we can write Eq. (3.6.5) in terms of an induced norm, the following equivalence, which was shown by Krasnosel'skii et al. (1972), arises: Theorem 3.6.1. Given the graphs of two plants (i.e. M\ and M2) and the associated orthogonal projectors (i.e. Tlje, where Jff denotes either or M^), the following relationships hold. 8(J^i,^2) — max{ 8 (J?i,JZ2), 8 (JZ2,^\)} = <5(^ 2 ,^i) (3.6.5) 8(JZi,JZ2) does satisfy the triangle inequality in a Hilbert space (Kato, 1976) which makes it useful. Remark 3.6.2. It was shown that 8 (J^i,J^2) = 8 (^#2,^1) whenever 8(M\, J£2) < 8(J{\, Jt2) (3.6.6) where \fffiM\\T^||oo and \\Il^±.TL^2\\00 are both induced norms. 39 C h a p t e r 3. 3.6.2 Homotopy and The v-Gap Metric Proof. The first equality follows immediately from Eq. (3.6.5). The second equal-ity hold since, from Eq. (3.6.4), one can establish max{ S (^±,^2), <5 (^2 ,^1 )} = maxjsup^^ j^n^ ||n^xx||2 , supxe^2tM=1\\U^±x\\2j = max { sup 8 g ^ 1 | | g | a , lin^i^lb "I lin^xn^ilb l|n^ ±n^ 2x|| 21 r suP*e^2 ||g||a r = m a x ( S U P ' s u p "iRli f = m a x \ P ^ 1 1 ^ ! Iloo, j I TT ^ j.n^2||oo|. For the last equality, an alternate proof is given in Appendix B.2. The new proof is slightly different from the existing proof and is easier to follow. • Having defined the equivalent form of the gap metric in terms of projection operators, Eq. (3.6.6) can then be recast into the following computable form proposed by Georgiou (1988). lin^xILrJoo = inf \\Gd - GiQ\U for i,j = 1,2 (3.6.7) where Gi = [M\] e is the graph symbol of plant Pi defined by its normalized comprime factorization (i.e. Pi = NiMf1). The infimum in Eq.(3.6.7) can be solved by employing a well known computational algorithm presented in Francis (1987, Chapter 8, Theorem 1). The resulting optimal value is precisely the value for the gap metric. 3.6.2 Homotopy and The zv-Gap Metr ic Likewise, the i^ -gap metric defined by Vinnicombe (1993) can be obtained following the similar logic discussed in the previous section. As mentioned earlier, the major difference between the gap metric and the z^ -gap metric is that the latter is denned in the J&2 space. Note that blindly computing the gap metric of the Jz?2 graphs is meaningless from a feedback stability perspective. Figure 3.9 clearly illustrates this idea. Recall that the requirement of a physically realizable stable feedback system in the 5^2 sense is to have causal finite energy signals in the loop. Obviously, merely evaluating the Jz?2 gap between #^1 and ^2 does not guarantee closed-loop stability since the subspace consists of noncausal signals (or has a hole caused by the anticausal signal space). Therefore, care must be taken to ensure that one only deals with causal finite energy signal in order to preserve closed-loop stability. Mathematically, a homotopy condition has to be imposed to satisfy such a requirement. 40 C h a p t e r 3. 3.6.2 Homotopy and The v-Gap Metric anticausal hole Figure 3.9: Two nonhomotopically equivalent graph spaces. Notice that there exists an anticausal hole in subspace ^#2- If no homotopy condition is imposed, the resulting i>-gap metric maybe misleading since it does not take the closed-loop stability into account. To visualize what does the homotopy condition mean, consider the following diagram presented in Figure 3.10. These two objects are said to be homotopically equivalent since there exists an action that can continuously deform the solid ball into the diamond shape without a need to punch any holes through it. Note that the hole punching action can be seen as a discontinuity in the deformation. For mathematical oriented reader, a mathematical definition of the homotopy condition is given in §A.1.5. In contrast to the homotopic equivalent, two objects are said to be nonhomotopic equiv-alent when they can not be deformed into each other without experiencing discontinuity. An analogy of the nonhomotopic relationship is depicted in Figure 3.11. It is obvious that a solid ball can not be deformed continuously into a torus (or donut) shape without creating a hole. F Figure 3.10: An analogy of a homotopic equivalence. 41 C H A P T E R 3. 3.6.2 Homotopy and The v-Gap Metric To see how the homotopy condition can be exploited to preserve closed-loop stability, consider the following diagram. Figure 3.12 shows a series of connections between the condition10 of operator graphs (i.e. the first column of Figure 3.12) and their corre-sponding feedback stability in the actual closed-loops (i.e. the second column of Figure 3.12) with respect to the increasing of magnitude in uncertainty or perturbation11. In the first column, a sharp cone is used to represent any factors that might puncture the graph space (i.e. causing instability), while the perfect surface means that the graphs only consist of causal finite energy signal. Any holes on the graphs' surface represents the space of anticausal finite energy signals. Also note that, for the sake of discussion, Pi , P 2 and P 3 are assumed to be the perturbed versions of a stable plant P 0 . In Figure 3.12 (a), since the two graph subspaces are perfect, analogous to Figure 3.10, there exists a homotopy condition between M\ and ^#2- Physically, this can be seen as continuous perturbation in P 0 to Pi = P 0 + A 0 without destroying the closed-loop stability of the second system. Figure 3.12 (b) shows that even though the magnitude of the uncertainty is further increased from A 0 to A x , closed-loop stability is preserved since the two graphs are homotopically equivalent. In contrast, when M2 hits the sharp cone representing the M'2L space and causing a puncture (i.e. anticausal hole) on it, the two graphs become nonhomotopically equivalent. This implies that further increment of the uncertainty destroys the closed-loop stability of plant P 3 = P 0 + A 2 . Unequivocally, the homotopy condition plays an important role in establishing feedback stability re-10Whether the graphs are perfect or being punctured. "That is, ||Ao|| < IIAiH < | |A 2 | | . 42 C H A P T E R 3. 3.6.2 Homotopy and The v-Gap Metric suits in the Jzf2 space. In system theory12, the winding number13 (or topological index) arising from complex analysis has been exploited as a homotopy condition since it is topologically invariant. Obviously, exploitations of a homotopy condition and the gap metric in the Jz?2 space gives the well-known Vinnicombe metric or the i^ -gap metric (Vinnicombe, 1991, 1993). Mathematically, the z^ -gap metric, which was introduced by Vinnicombe (1993), is de-fined as follows: Definition 3.6.1. Let Pt = A ^ M " 1 , i = 1,2 be two (possibly unbounded) linear opera-tors, the v-gap metric is defined as 5,(Pi,P 2) = ||G2Gi||oo if det(G;C?i)(ja;) ^ OVtu G ( - 0 0 , 0 0 ) and wno d e t ^ G i ) = 0, (3.6.8) 1 otherwise where d = [ $ ] e * , Gi = [ -Mi Ni} G Jffoo, and wno denotes the winding number of det(G2Gi)(s) as the complex variable s follows the standard Nyquist D-contour. In addition, the z^ -gap metric can be rewritten in terms of Pi and P 2 providing that the winding number condition is satisfied: ||G2C7i||oo = | | ( / + P 2 P 2 T 1 / 2 ( P 2 - Pi){I + P*Pi)-1/2\\oo (3.6.9) Note that C72C7i = [ - M 2 N2][m\] = -M2Ni + JV 2Mi = M ^ - A ^ M f 1 + M f ^ M i = (/ + P 2P 2*)- 1/ 2(P 2 - PJil + P^Pi ) - 1 / 2 . The last equality holds since N*A/\ + M * M X = I =4> (MiM{)~1 = I + P*P\ and the similar argument holds for M 2 *M 2 . For the sake of completeness, the computational algorithm of the v-gap metric is presented in Appendix C. • From the comparison of Eq.(3.6.9) and Eq.(2.2.4), it is clear that the z^ -gap metric has a frequency response interpretation. Whenever the winding number condition is satisfied, 1 2 The most influential example is the generalized Nyquist stability criterion, where the count of anticlockwise encirclement of the origin can be seen as a homotopy condition. 13See A.7 43 C H A P T E R 3. 3.6.2 Homotopy and The v-Gap Metric Two J#2 graphs representing two stable Figure 3.12: Analogy of the homotopy condition in the v-g&p metric and the actual closed-loop stability. 44 C H A P T E R 3. 3.6.2 Homotopy and The v-Gap Metric the v-gap metric is equal to supa(G2Gi)(ju). It has been shown that for SISO system, the I'-gap metric is precisely equal to the chordal distance K between the stereographic projection of Pi(ju) and P2{ju>) onto the Riemann sphere. N Figure 3.13: Stereographic projection Figure 3.13 shows a unit Riemann sphere that is placed on an extended complex plane. This representation was first introduced by El-Sakkary (1989) in the J^f2 space and extended to the J??2 space by Vinnicombe (1993). .The south pole of the Riemann sphere touches the extended complex plane at the origin. Let P\(JOJQ) = X\ + jyi (denoted by Pi(%i,yi)) and P2(ju>o) = x2 + jy2 (denoted by P2(x2,y2)) be two frequency responses of Pi(s) and P2(s) at a particular frequency, say uiQ. A straight line is then drawn to connect the north pole and Pi(xi,yi). The stereographic projection of Pi(xi,yi) is the point (denoted by P{(£i, ipi, Ci)) where the straight line intersects with the Riemann sphere. Unequivocally, the projection of P i (£1,2/1) to P i ^ i , ^1, Ci) is one-to-one14 and unique. The chordal distance of P[(£i,ipi, £i) and P2(^2, ip2, C2), denoted by «(Pi, P2), is given by K{PI, P 2) = , | P l ~ Pr (3-6.10) Clearly, Eq. (3.6.9) is the multivariable version of Eq.(3.6.10). The next subsection shows 2 = 1,2 The corresponding one-to-one mapping are & = xt+ji+1, d = x'i+y<i+1 a n d ipi = j \ y f + 1 for 45 C H A P T E R 3. 3.6.3 The Gap Metrics and The Coprime Factor Uncertainty that the uncertainties characterized by the gap metrics are equivalent to those of the normalized coprime factor type. 3.6.3 The Gap Metrics and The Coprime Factor Uncertainty Two polynomials g(s) and h(s) are said to be coprime if their greatest common divisor15 (GCD) is 1. This implies that these two polynomials have no common zeros. By using the idea of coprimeness, any processes (either open-loop stable or unstable) can be rep-resented by the quotient of two stable transfer functions, yet avoiding potential unstable mode pole-zero cancellation. For instance, the unstable transfer function P(s) = can be factored into the following equivalent form: P's) = — S-^4, V a 6 R1 > 0, (3.6.11) where g(s) and h(s) are coprime and stable transfer functions. Note that P(s) = g(s)h~1(s) is a right coprime factorization, while P(s) = h~l(s)g(s) is a left coprime factorization. Clearly, the coprime factorization is not unique. A unique coprime factorization can be obtained by introducing an operator M*(s) = MT(—s). Then P(s) — g(s)h~1(s) is said to be a normalized right coprime factorization if the following condition is satisfied: h*(s)h(s) + g{s)*g{s) = I. (3.6.12) Likewise, P(s) = h^1(s)g(s) is called a normalized left coprime factorization if: h(s)h*(s) + g{s)g*(s) = I. (3.6.13) An interesting uncertainty description arising from the aforementioned factorization is 1 5For simplicity, consider two real numbers that can be expressed in the products of prime numbers (i.e. a = p"1 xP22 • • • x p®n and b = p\x x p%2 • • • xp@n, where pi, a, and /?* denote the prime numbers in ascending order and the corresponding power of a and 6, respectively. As an example, 10 = l 1 x 21 x 51). The GCD for a and 6 is then denned as Hip^m^ai'^'\ where Iii denotes the products of all i terms. Example: GCD(12,30)=11 x 21 x 31 x 5° = 6 since 12 = l 1 x 22 x 3 1 x 5° and 30 = l 1 x 21 x 31 x 51. 46 C H A P T E R 3. 3.6.3 The Gap Metrics and The Coprime Factor Uncertainty called the normalized coprime factor uncertainty. A perturbed plant, Pi(s), with a normalized coprime factor uncertainty is defined as follows: Pi(s) = {(9 + Ag)(h + Ah)'1 : Ag, Ah e Jf*,, A 0 < b (3.6.14) where Ag, A^ and b denote two stable unknown transfer functions, which represent the uncertainties in the nominal plant P(s), and a positive real number, respectively. Advantages of using the normalized coprime factor uncertainty description include: • The perturbed plant and the nominal model do not need to have the same number of unstable poles. • It allows both zeros and poles of the perturbed plant cross into the right-half plane. • When Ag and A ^ are stacked on top of each other (or side-by-side) to form a full complex perturbation block, the resulting norm-bounded stacked uncertainty can be used to establish a tight robust stability condition in terms of HMHoo, which appears in the standard M — A configuration of the small gain theorem16. To see how the gap metrics can be related to the normalized coprime factor uncertainty, recall that the directed gap can be written in the form of Eq. (3.6.7), which is restated here for convenience: 8 (P, P x) = inf Qe^ oo (or ifoc [ J ] - [$ i ]<4 (3.6.15) So, for Pi = gih-t1, 8 (P, Pi) < b, there exists a Q e J%x> (or Jzfoo satisfying a certain homotopy condition) such that h _ hi Q g\_ < b. (3.6.16) 16Developed by Zames (1966a,b), the small gain theorem forms the cornerstone of modern robust control theory, and shall be discussed in the next section. 47 C H A P T E R 3. 3.7. The Gap Metrics and The Small Gain Theorem Next, define hi Q-h A 9 9i 9 (3.6.17) Then, obviously ^ < b and Px = g^T1 = giQQ~lh7l = (giQ^Q)-1 = (g + A g )(/ i + A h ) ' 1 . The converse is also true. Note that for P1 = gxh7l = (pi + A f f)(/ii + Ah)" 1 , there exists a Q~l £ X o such that Px = {(pi + A 5)Q}{(/z 1 + A /OQ} - 1 is a normalized right coprime factorization. So, it is clear that, by definition, 5 (P, Pi) can be obtained by assuming Q~l = Q £ J ^ , : (5(P,Pi) = inf h -( h + AH Q)Q < h h + AH 9 \ . 9 + A9. J oo _9_ g + A 9 _ A f t < b (3.6.18) Evidently, the gap metrics are equivalent to the normalized coprime factor uncertainty description. In fact, the z^ -gap metric being the lower bound of the gap metric provides the smallest possible bound of such uncertainty description. The next section exploits the aforementioned property of the z^ -gap metric (or the gap metrics in general) and the small gain theorem to establish a robust stability result. 3.7 The Gap Metrics and The Small Gain Theorem Over the past few decades, substantial research efforts have been devoted to the analysis and synthesis of control systems to achieve robust stability and performance in the presence of various types of uncertainty. In this regard, the small gain theorem is a very powerful and general tool to assess the robust stability and performance of a closed-loop system. The small gain theorem is a general result which can be easily particularized for LTI systems. The small gain theorem, according to Zhou et al. (1996), is given in the following theorem: 48 C H A P T E R 3. 3.7. The Gap Metrics and The Small Gain Theorem Theorem 3.7.1 (Small G a i n Theorem). Given a generalized plant M e and a real positive number 7 > 0. Then the M - A connection shown in Figure 3.14 is well-posed and internally stable for all A(s) € ^ ^ 0 0 with 1. IIAIU < 1/7 if and only if H M ^ I U < 7; 2. IIAIU < I/7 if and only if H M ^ I U < 7 Proof. See Zhou et al. (1996, pg. 218). • Figure 3.14: A standard M - A configuration in robust control. Clearly, for a perturbed plant P — (g + A f l)(/i + A ^ ) - 1 shown in Figure 3.15, the corresponding M and A blocks are as follows: A 4 An M 4 h'^I-CP) - 1 C -I (3.7.1) (3.7.2) Therefore, according to the small gain theorem, a sufficient condition for the closed-loop system shown in Figure 3.14 to be stable is that for < I/7, \\h-i{i-cp)-x[c 'I] h-^I-CP)-1 [C -/' P] (I-CP)-1 [-c/] | | < 7- Note that by defining 6 = 1 / 7 a n d invoking the fact that the gap metrics are equivalent to the normalized coprime factor uncertainty, the following robust stability result linking the gap metrics and the generalized stability margin can be established. 49 C H A P T E R 3. 3.7. The Gap Metrics and The Small Gain Theorem Figure 3.15: A perturbed plant with normalized coprime factor uncertainty. Lemma 3.7.1. Given a nominal plant P 0 = gh~x, a unity controller C = I and a nonnegative real number b > 0, a closed-loop system containing a perturbed plant P — a . (g + Ag)(h + Ah)~1 with where M is defined in Eq. (3.7.2). < b or S(P, P i ) < b if and only ifbPC \M\ >b, follows by invoking the small gain theorem and the equivalent property Proof. The result between the gap metric and the normalized coprime factor uncertainty. Remark 3.7.1. Recall that the gap metrics were constructed by assuming that the closed-loops were under a unity feedback. Therefore, for consistency, a unity feedback controller is required to establish the result presented in Lemma 3.7.1. Remark 3.7.2. The requirement of the unity feedback controller in Lemma 3.7.1 does not affect the generality and the usefulness of the aforementioned lemma since the con-troller can be treated as a filter and absorbed into the nominal plant PQ. 50 C H A P T E R 3. 3.8. Summary 3.8 S u m m a r y This chapter presents the gap metric, particularly the z^ -gap metric, theory in a graphi-cal sense. Distinctions between the gap metric and the z^ -gap metric are highlighted. In general, both the gap metric and the z^ -gap metric are found to be equivalent to the nor-malized coprime factor uncertainty. This equivalent property leads to a powerful robust stability result by invoking the small gain theorem. In the next chapter, a systematic approach is used to formulate a reliable closed-loop nonlinearity measure by exploiting the f-gap metric and the generalized stability margin. 51 C H A P T E R 4 A Closed-loop Nonlinearity Measure The focal point of this chapter is to formulate a practical and reliable closed-loop nonlinearity measure. The developed measure extends the v-qap metric framework in the direction of formulating a nonlinearity measure and exploiting the general-ized stability margin. When these two measures are used within the nonlinearity measure, meaningful results are obtained. Various theoretical motivation such as the nature of the frozen point nonlinearity measure, time variation of the schedul-ing parameter, the linearizing effects of feedback and the best choice of the nominal model, are discussed. Finally a novel, practical and easy to implement computa-tional algorithm of the proposed nonlinearity measure is presented. 4.1 Introduction Having discussed the need of establishing a closed-loop nonlinearity measure in Chapter 1, having reviewed the literature on the existing techniques for closed-loop nonlinearity quantification in Chapter 2 and having presented, in a graphical sense, the gap metrics in Chapter 3, the next natural step is to put all these ideas together and formulate a reliable nonlinearity measure. Before proceeding further, a look on what is meant by "a reliable nonlinearity measure" is desirable. From a practical engineering point of view, a reliable nonlinearity measure must have the following properties: • Abi l i ty to capture the system nonlinearity in practical situations. The measure must reflect the actual nonlinearity while the system is in operation. For example, a process is often staying at a particular operating point for some time 52 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure before changing or switching to the next operating point related to other product grades or for start-up/shut-down. Often, the degree of nonlinearity is different from operating point to operating point. The resulting nonlinearity measure should capture such phenomena and provide a meaningful engineering measure. • Easy to compute. In order for a measure to be useful, the ease of computation and implementation is very important. This feature is particularly important in day-to-day operation, where the need of assessing plant's nonlinearity and the adequacy of an existing linear controller is required by various engineering decisions such as a tight reference control or process throughput optimization. • Easy to interpret. The results obtained using the measure must be easy to interpret. Often time, a scale between 0 and 1 is used, where 0 means "good", while 1 means "bad". In what follows, a practical formulation of closed-loop nonlinearity measure is presented. Then, in §4.3, the theoretical motivation that forms the cornerstone of the subsequent computational algorithm is discussed. In §4.4, a novel computational algorithm for the nonlinearity measure is developed. Finally, a summary is presented. 4.2 Formulating The Closed-loop Nonlinearity Mea-sure This section aims to devise a closed-loop nonlinearity measure based on the three prop-erties previously mentioned. An overview of the resulting nonlinearity measure is pre-sented, while the theoretical motivation finds room in the next section, to avoid disrup-tion of the logical presentation. To formulate the aforementioned closed-loop nonlinear-ity measure, we begin with the following assumption: Assumpt ion 4.2.1. Given two closed-loops, as shown in Figure 4-1, one consisting of a nonlinear plant and another containing a linearized version of the nonlinear plant. 53 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure Assuming that these two loops are subject to the same disturbances and noises. Then the input-output discrepancy of these two closed-loops is mainly due to closed-loop non-linearity. Nonlinear Plant Linear " Plant Nonlinearity (uncertainty) Figure 4.1: The developed nonlinearity measure looks at closed-loop nonlinearity. Based on Assumption 4.2.1, one can say that if the resulting nonlinearity is small with respect to some appropriate stability/performance measure, then a controller design for the linear plant should be sufficient, when it is implemented in the associated nonlinear plant. Likewise, if the nonlinearity is large with respect to the same measure, then either a nonlinear control is needed or the design specifications need to be redefined. In this light, the closed-loop nonlinearity measure consists of two key ingredients: • A stability or performance measure. • A metric to quantify closed-loop nonlinearity or the uncertainty. An obvious choice for the stability/performance measure, in the feedback setting, is the generalized stability margin or bPic presented in §3.5. In the meanwhile, for the appropriate metric quantifying the uncertainty, a discussion is required. It is know that the input-output signals of the two closed-loops might be different, whenever the nonlinear plant and the linear model are different. Therefore, merely measuring the output discrepancy of the two closed-loops can be unnatural and restrictive. To resolve this problem, a new approach, which considers both input and output discrepancies, needs to be developed. In this regard, a slightly modified i>-gap metric is providing an excellent framework for formulating the nonlinearity measure. Recall that the gap metric and the z^ -gap metric measure the input-output discrepancy of two LTI plants in a unity feedback fashion. Therefore, a closed-loop nonlinearity measure can be formulated in the same spirit, but with a slight modification. Here, 54 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure we are considering the system with the standard configuration as depicted in Figure 4.2. Assuming that there exists a homotopy condition between the two closed-loops, the corresponding closed-loop nonlinearity measure can then be defined as follows: Snl(NL, L) = inf max{ S nl(NL, L), 5 nl(L, NL)} iyfc A. (4.2.1) where Snt(NL,L) and 5ni(L,NL) are the normalized input-output discrepancies be-tween the nonlinear plant, here denoted by NL, and a linear model, here denoted by L, or the directed gaps, which are defined as: S ni(NL, L) = sup inf •u i • U2 1 II V2 J 112 Uy\]^^2[yl}^2^2 ||[£]||2 (4.2.2) and 5ni(L,NL)= sup inf [£]efcnift[yi]esia% •U2 1 _ r a n . V2 J L 3/1 J 2 r u2 11 [ 2/2 JI 2 (4.2.3) Wi NL 2/i c n U2 L V2 c w2 Figure 4.2: A standard configuration for closed-loop nonlinearity measure Note that the infimum in Eq.(4.2.1) is the result of a slight modification of the original definition of the gap and the z^ -gap metrics, see Eq.^.G.l) 1 . Eq.(4.2.1) implies that the minimum worst case discrepancy is obtained over all possible candidates in the membership set A. In other words, whenever the optimal solution of Eq.(4.2.1), say 1Eq.(3.6.1) is originally denned in the space. To be consistent with Eq.(4.2.6), assuming that all signals are denned in the ^2 space and a homotopy condition exists for the two operators. 55 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure 8ni(NL,L*), is obtained, the corresponding linear model L* is called the best nominal model, playing a crucial role in formulating the nonlinearity measure, as seen in §4.3.4. Observe that Eqs. (4.2.2) and (4.2.3) are equivalent to the nonlinear directed gaps and can be rewritten in the following form by assuming yi = NL(u{): ~8nl(NL,L)^ sup inf l | [ y!rffi l l l a (4.2.4) [NLIU,)}^^ W&2n& \\[NH»I)\\\2 and tnl(L,NL)^ sup inf i ^ Z ^ ^ (4.2.5) [LU2)ZG^2 [N%Ul)]eGin*2 \\[Lu2\\\2 The above two equations involve the computation of two nonlinear operator (i.e. NL(ui)) norms, which can be difficult and time consuming to evaluate. In addition, such com-putation will violate the reliability property. An appealing way to resolve this problem, without jeopardizing the model accuracy, is to transform the nonlinear system into a quasi-linear parameter varying (quasi-LPV) representation, provided that the nonlinear-ity is essentially captured by the chosen scheduling parameter(s), see §A.6 for a further discussion on the quasi-LPV transformation. By doing so, a set of LTI models can be obtained easily by merely freezing the scheduling parameter at a set of operating points of interest. Next, to enable us to exploit the quasi-LPV transformation, the following additional assumption needs to be satisfied: Assumpt ion 4.2.2. Plant's nonlinearity depends on measurable states, typically exoge-nous signals, and can be captured by selecting appropriate scheduling parameters, which are allowed to move within a prescribed scheduling space Cl. In the sequel, Assumption 4.2.2 is assumed to be satisfied. After a quasi-LPV transfor-mation, the resulting closed-loop nonlinearity measure over 0 is redefined as follows: 8%LPV(NL(a),L) ± 8%(NL(a),L) = m f m a x { ^ ( i V L ( a ) , L ) , ^ ( L , A / L ( a ) ) } (4.2.6) 56 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure where t°(NL(a),L) ± sup inf ll[^)UlJr!f2]|k Vet G fl (4.2.7) [^(aJuJee iWni fb l^aJe^n^ || [ JVL(a)U l J | | 2 and <5S(L,iVL( a))4 S U p i n f l l l ^ J W ^ J l l ^ ^ g fl ( 4 ^ g ) W2 i r u\ Luu22]Ge2n^2 [ jvi^uJeeiWn^ IIU«2JII2 • u2 Note, the subscript implies that the resulting nonlinearity measure depends on the choice of fl. If fl covers all possible input-output pairs of the original nonlinear plant, then obviously Gi(fl) = Qx. Otherwise, 9\(fl) C Qx. Next, assuming that Assumption 4.2.2 is satisfied and that the scheduling space fl can be partitioned into several subregimes fli, which are satisfying the following conditions: 2. g{NL(ai)) ~ Gifli). Satisfying these conditions mean that the graph space covered by the union of all the scheduling subspaces fli (i.e. \J^=1 fli) is a subset or is equal to that of the entire scheduling space, Oi(fl). The validity of the above relationship depends heavily on the scheduling space partitioning. The theoretical justification of these two conditions and a discussion on the scheduling space partitioning are given in §4.3 .2 . Having established the above relationship, the definition of the closed-loop nonlinearity measure can be rewritten as follows: N 5%LPV(NL(a),L)± Mmax{7%(NL(a),L),7%(L,NL(a))}, V « G | J ^ (4-2.9) i=l where r y-2 J«(NL(a),L)± sup inf l l [ 7^i l ; u 2 J " 2 (4.2.10) [NLU(a)Ul ]^!(Ur=1 nt)n^2 {Lu2Y^2 II [ NL(a)Ul J | | 2 57 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure and t * ( L t N L ( a ) ) ± sup inf \\[^~ [m^]\\2 { 4 2 n ) [%2}eg2njz>2 [ ^ J e f f i t U ^ W \\{LU2\\\2 Note, fl denotes the union of all the scheduling subspaces (i.e. ( J i l i^ i ) - Further, Eqs.(4.2.10) and (4.2.11) can be recast into the following optimization problem in terms of the v-gap metrics of the frozen point quasi-LPV transformation. [ " " i n r «2 ^ ( J V L ( a ) , L ) 4 sup sup inf 11 ^ " j , ^ J | l a (4.2.12) and - 8 i { L , N L { a ) ) ± sup sup u inf [w«)*]L Note that the above closed-loop nonlinearity measure is actually a frozen point non-linearity measure. To account for the plant time variation, homotopy conditions are assumed to be satisfied among the frozen point models and an additional 20% of the original nonlinearity measure (i.e. S 9 L P V ( N L ( a ) , L ) ) is introduced, see Hyde (1991). Therefore, the resulting total nonlinearity measure becomes Snm a 6 % L P V ( N L ( a ) , L ) + 5, (4.2.14) where the superscript nm stands for the "nonlinearity measure", while 5+ denotes the uncertainty owing to the time variation. Note that the 20% penalty of time variation depends on the characteristic of the system. This can be further finetuned (increased or decreased) by extensive simulations. Together with the generalized stability margin, the developed nonlinearity measure pro-vides a useful indicator for closed-loop nonlinearity measure. The above discussion can be summarized into the following definition and theorems. 58 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure Definition 4.2.1. Given a quasi-LPV plant NL(a) with a G fl ~ Ui l i ^ and a se^ of linear models A. The individual element of A is denoted by L, and assuming that the graphs of the scheduling subspaces are homotopically equivalent, then the closed-loop nonlinearity measure is defined as follows: §nm A s^LPV(NL(a), L) + 6+ (4.2.15) where 8+ denotes the uncertainty due to time variation and N 8fPV(NL{a),L) = inf max{8 %{N L{a), L), ~8%(L, NL(a))}, Va G [jfl,. i=i or N Note that 8%(NL(a),L) and 8 NL(a)) are given in Eqs. (4.2.12) and (4.2.13), respectively. Theorem 4.2.1. Given a linear model L and a controller C that is designed based on L, then: [NL(a),C] is stable for all a G fl satisfying 8^n(NL(a), L)(ju) < bLtc(juj), VwG[0,oo). Proof. The proof consists of two steps. For the first step, note that <5?m = S9LPV(NL(a), L) + 8f. Therefore, 8^{NL(a),L)(juj) < bLtC{juj), Vw G [0, oo) implies that 8fpv{NL{a),L){ju) < 8%LPV(NL(a),L){ju) + 5+ < bL,c(ju), \/u G [0,oo). The second step is similar to that of Theorem 4.3.2. This completes the proof. • Theorem 4.2.2. Given a quasi-LPV plant NL(a), a € fl and a linear model L, then: [NL(a), C] is stable for all controllers C satisfying bLic(ju)) > 81^n{NL{a), L)(ju), Vu £ [0,oo). Proof. The proof of this theorem follows the same logic applied in the proof of the previous theorem. The second step follows by invoking Theorem 4.3.3. See Theorem 4.3.3 for the associated proof. Hence, the proof is complete. • 59 C H A P T E R 4. 4.2. Formulating The Closed-loop Nonlinearity Measure Note that the results of the above two theorems require a search over a frequency range. This means that the computation load can be intensive. Since the computation of the Jffx, norm of the generalized stability margin and the evaluation of the i^ -gap metric have a state space formulation. The former involves a minimization problem for which the resulting Hamiltonian matrix contains no imaginary axis eigenvalues, while the latter requires solving two Riccati equations. For the corresponding computational algorithms, see Appendix C. Clearly, since the developed closed-loop nonlinearity is closely related to the u-gap metric for quasi-LPV systems, the computation of the above two quantities can be recast into two state space optimization problems by defining 5,yn(NL(a)) L) = sup, 8^n(NL(a), L)(ju>) and bL>c — infw fe^cfju;). A less computation intensive robust stability result is given in the next corollary. Corollary 4.2.1. Given a linear model L and a controller C that is designed based on L. Define S^m(NL(a),L) = sup, S^m(NL(a), L)(ju) and bL,c = in f w & i l C (>) , then: [NL(a), C) is stable for all a E CL satisfying 5^m(NL(a), L) < bL<c. Proof. From the definition of 5?m(iVL(a), L) and bLiC, it is clear that 5£m(IVZ,(a), L) < bL,c is valid over the entire frequency range. Then, by invoking Theorem 4.2.1, the robust stability result follows immediately. • Remark 4.2.1. Corollary Jj.,2.1 is only a sufficient condition. It says nothing about the stability result whenever 5^n(NL(a), L) > bLtc- Figure 4-3 shows the ambiguity arising when S1^n(NL(a), L) is greater than bitc- Evidently, under this circumstance, a frequency-by-frequency test is needed. In short, this new formulation not only captures the essential nonlinearity of the system, but also provides a computational viable and easy to interpret characteristic. In the next section, several technical issues arising from the developed measure are discussed further. 60 C H A P T E R 4. 4.3. Theoretical Motivation 6%m(NL(a),L)(3u) HL,C Frequency, ui Figure 4.3: An ambiguity arising from applying Corollary 4.2.1 whenever 5?m(NL{a),L)>bLiC. In this case, 5 ? m ( N L { a ) , L ) is larger than bLiC. Blindly apply Corollary 4.2.1 results in making an incorrect conclusion that the resulting closed-loop is unstable. A frequency-by-frequency test reveals that S ' ^ 7 l ( N L ( a ) , L)(ju>) is smaller than b^cij^) over the entire frequency range. By invoking Theorem 4.2.1, the corre-sponding closed-loop is, in fact, stable. From the Definition 4.2.1, the developed closed-loop nonlinearity measure shows a close relationship to the i^ -gap metric for LTI systems. In this section, the i^ -gap metric for quasi-LPV systems is first presented. This is followed by a series of discussions on various technical and practical issues arising from the nonlinearity measure formulation. Such issues include the frozen point nonlinearity measure, the time variation of scheduling parameter, the linearizing effect of feedback, 3%^ loop-shaping and weight selection for controller synthesis, the choice of the best nominal model and the best possible stability margin. 4.3.1 T h e i/-Gap Metric For Q u a s i - L P V Systems Similar to its counterpart for LTI systems, the i^ -gap metric for quasi-LPV systems begin with an appropriate construction of the normalized coprime factorizations. Note that, for 4.3 Theoretical Motivation 61 C H A P T E R 4. 4.3.1 The v-Gap Metric For Quasi-LPV Systems completeness, most of the results in this subsection are cited from Wood (1995), except Theorems 4.3.2 and 4.3.3, which are analogous to the LTI z^ -gap metric of Vinnicombe (1993) whenever the scheduling parameter is frozen. In the sequel, we will consider a quasi-LPV system which has the following state-space realization = A{a)x(t) + B(a)u(t) (4.3.1) y{t) = C{a)x{t) + D{a)u{t) where a C x(t) is the scheduling parameter residing in the scheduling space Q. Definition 4.3.1 (Extended Quadratic Stability). For a dynamic system charac-terized by the following state space equation ^j& = A{a)x(t), aefl (4.3.2) the system is said to be extended quadratic stable (Qe stable) if there exists (3) a real differentiable positive-definite matrix function Q{a) = QT(a) > 0 such that ^Q(a) + A(a)TQ(a) + Q(a)A(a) < 0, Va G Cl. (4.3.3) dt Lemma 4.3.1. Any Qe stable system is exponentially stable, if 3 constants a, b > 0 such that a($ Q ( t , r )) < ae~h{t-T) V a G O where $ a (t ,r) denotes the transition matrix for Eq.(4-3.2) Proof, see (Wood, 1995, pg. 16) • 62 C H A P T E R 4. 4.3.1 The v-Gap Metric For Quasi-LPV Systems Definition 4.3.2 (Qe stabilizable). The quasi-LPV system given in Eq.(4.3.1) is said to be Qe stabilizable if 3 a continuous matrix function F(a), such that the following system is Qe stable Va G fi dx(t) dt = {A{a) + B(a)F(a)}x(t). Definition 4.3.3 (Qe detectable). The quasi-LPV system given in Eq.(4.3.1) is said to be Qe detectable if 3 a continuous matrix function H(a), such that the following system is Qe stable Va G fi dx{t) dt {A{a) + H{a)C(a)}x(t). Lemma 4.3.2 (Quasi-LPV Coprime Factorizations). Let P(a) have a continuous, Qe stabilizable and Qe detectable state space realization P(a) :-Note, in this thesis, P(oi) and NL(a) are used to denote the linear quasi-LPV sys-tems. Let F(a) and H(a) be continuous matrix functions such that = {A(a) + B(a)F(a)}x(t) and ^ = {A(a) + H(a)C(a)}x{t) are Qe stable Va G fi and define (dropping the a dependence, for notation simplicity): A{a) B(a) C(a) D(a) N Y 1 v a 1 a A+'BF B -H C + DF D I F I 0 ( 4 . 3 . 4 ) -Nn A + HC H -(B + HD) F 0 I C / -D ( 4 . 3 . 5 ) 6 3 C H A P T E R 4. 4.3.1 The v-Gap Metric For Quasi-LPV Systems then XA YA Ma -Na Proof, see (Wood, 1995, pg. 149) • Definition 4.3.4 (Contractive right coprime factorization). Let Na and Ma have the same number of columns. The ordered pair [Na, Ma] represents a contractive right coprime factorization (crcf) of P(a) over the ring2 SQ, if 1. P(a) = NaM'1; 2. 3 XA, YA e SQ such that XaNa + YaMa = I; 3. [N£ M j ] T is a contraction in the following sense sup sup | | [ £ H | < 1 (4.3.7) aeft {ue^f2+:||u||2<i} Definition 4.3.5. Define the contractive right graph symbol Ga : Jzf2+ (—> Jz?2+ ® Jz?2+ of an LPV system P{ct) as follows Ga : = [ £ ] , (4.3.8) where [Na) Ma] is a crcf of P(a). Remark 4.3.1. It is obvious that Ga generates the set of all stable input-output pairs of the LPV system P(ct) by allowing Ga to act on the whole of J z ? 2 + . Theorem 4.3.1 (Quasi-LPV Graph). Let P(a) have a continuous, Qe stabilizable 2 N o t e that a ring (&,+,•) is a set 3£ together w i t h two b ina ry operat ions, an addi t ion and a mul t ip l i ca t ion , such that: 1. is an abel ian group (i.e. a group for which the elements commute . N a m e d after Niels Henr ik A b e l , 1802-1829, a Norwegian mathemat ic ian) ; 2. the mul t ip l i ca t ion is associative (i.e. x ( y z ) — (xy)z)\ 3. the mul t ip l i ca t ion is d is t r ibut ive (i.e. ( x + y ) z = x z + y z and z ( x + y) = z x + z y V x , y, z € ^*); 4. there exists an ident i ty element (i.e. e x — x = x e V x G 5 T ) . N Y 1 y a 1 a Ma XA = I (4.3.6) 64 C H A P T E R 4. 4.3.1 The v-Gap Metric For Quasi-LPV Systems and Qe detectable realization, then a contractive right graph symbol of P(a) is given by A + BF BS-1* Ga :— C + DF DS'1* F s-1* (4.3.9) where F = -S~1(BTXl + DTC), S = I + DTD, R = I + DDT and Xx is a solution of the generalized control Riccati inequality (GCRI) dXi + (A- BS-1DTC)TXl +X,(A- BS~lDTC) -X1BS~lBTX1 + CTR~1C < 0 \/aeQ (4.3.10) Proof, see (Wood, 1995, pg. 150) • Remark 4.3.2. The results, as stated here, are for the right coprime factorizations. The dual results can be easily obtained for the left coprime factorizations. Remark 4.3.3. Analogous to Vinnicombe (1993), the quasi-LPV graph in Eq.(4-3.9) is used in the sequel to define the corresponding quasi-LPV v-gap metric. The quasi-LPV z^ -gap metric can be defined as follows: Defini t ion 4.3.6 (The q u a s i - L P V z^-gap M e t r i c ) . The quasi-LPV v-gap 6$LPV is given by 6 Q L p v { p { a i ) ) P { a j ) ) : = IIG^GJIoo if det(G*a.Gai)(jcu) ^ 0 V w 6 (-oo, oo)and wno det(G*a.Gai)(jcj) = 0, Va*, ctj e fl 1 otherwise where Gai and Gaj denote the contractive right graph symbol of P(cti) and the contrac-tive left graph symbol of P(atj), respectively as defined in Theorem 4.3.1. It is obvious that the 8®LPV = 5U whenever oti,ctj are frozen. Together with the bptc, the following theorem is one of the main results arising from the z^ -gap metric notion. 65 C H A P T E R 4. _ 4.3.2 The Frozen Point Nonlinearity Measure T h e o r e m 4.3.2. Given a nominal plant P(cti) G P{OL) obtained by freezing the schedul-ing parameter G fl and a controller C, then: [P(a^),C] is stable for all plants P(ctj), Va, G fl if and only if 6^LPV(P(ai), P(aj))(ju) < bP{a%)tC{juj), Va; G [0,oo). Proof. Since 5®LFV = 8V whenever a^a^ are frozen, the proof follows from that of Vinnicombe (1993) , Theorem 4 .5. • T h e o r e m 4.3.3. Given a nominal plant P(«i) G P(a) and perturbed plants P(aj) G P(a) V«j G fl obtained by freezing the scheduling parameter at ai,ctj G fl respectively, [P(ctj), C] is stable for all controllers, C if and only if bp^tC(juj) > 5®LPV(P(ai), P(a.j)) (ju) Vctj efl,Vu>£ [0, oo). Proof. Also see Vinnicombe (1993), Theorem 4 .5. • 4.3.2 The Frozen Point Nonlinearity Measure From the discussion in the previous subsection, the i^ -gap metric for quasi-LPV systems at various frozen points gives a powerful tool to assess closed-loop robustness. Therefore, a natural first step of formulating the closed-loop nonlinearity measure is to consider that based on a frozen point approach. Assuming that Assumption 4.2.2 is satisfied and that the scheduling space fl can be partitioned into several subregimes fli, which satisfies the following condition: N {Jfli~fl (4 .3.11) i=i The next two assumptions are essential to establish the subsequent arguments: A s s u m p t i o n 4.3.1. The i-th scheduling subspace (i.e. fli) intersects its adjacent ones. For each scheduling subregimes fli, an appropriate frozen quasi-LPV mode? NL{af) representing local dynamics around ai G fli is selected. 3 I n this regard, a quasi-LPV model is obtained by freezing the scheduling parameter at a particular operating point, says ctj. 66 C H A P T E R 4. 4.3.2 The Frozen Point Nonlinearity Measure Assumption 4.3.2. The local model NL(cti) is assumed to have the graph space that is approximately equal to that of the scheduling subregime fli = [a* — e, a* -f- e], or mathematically, g(NL{oi)) « G{fli) = G(NL(ai - e)) U Q(NL(ai)) U G(NL(ai + e)). Figure 4.4 shows the graphical interpretation of the assumption made for a system with a one-dimensional scheduling space. Clearly, an immediate consequence of Assumption g(NL(a0)) Figure 4.4: A graphical interpretation of Assumption 4.3.2. 4.3.2 is that the i^ -gap between NL(ai) and NL(oci ± e) must be sufficiently small. This implies that SU(NL(cti), N'L(a.i ± e)) < r —> 0 + . Therefore, the following relationship, depicted in Figure 4.5, can be established: C C 7 ( f t ) (4.3.12) In addition, if N —> oo, then G ^ Ui=i ~ G{fl)- An implicit requirement of the above relationship is that there exist a homotopy condition among all G{fli)- Note that the overlapping requirement of the scheduling subspaces is to guarantee smooth transition when the scheduling parameter is moving from one subspace to another subspace. The following lemma summarizes the results presented in the above discussion: Lemma 4.3.3. Given a quasi-LPV plant NL(a) with a G fl and a positive real number r. Assuming ( J i = i ^ — ^> that fli intersects with all its adjacent scheduling subspaces 67 C H A P T E R 4. 4.3.2 The Frozen Point Nonlinearity Measure Figure 4.5: Graph spaces of a frozen-point quasi-LPV system (Gi{tii)) and that of a linear model (<?2). flj, forijt j = 1,...,N, and Su (NL(a.i), N'L{pn ± e)) < r, Va* ± e G Then r 0 implies that G(ai) —• £7(fij). /n addition, if-N —> oo, t/ien (7 (Ui=i^) ~*• ^(^)-Lemma 4.3.3 establishes the relationship between the operator graph of the quasi-LPV plant and that resulting from the union of all scheduling subspaces. The validity of the developed nonlinearity measure clearly depends on how the scheduling space is parti-tioned. To achieve this, the following algorithm can be employed: 1. Set an initial radius of the frozen point neighborhood, r. 2. Set an initial number of frozen points, says N points. 3. Grid the scheduling space accordingly. 4. For each ith-frozen point, compute the z^ -gap metric between the ith-frozen point to all its adjacent frozen points (i.e. 8u{NL{cti), NL(aj)), where j denotes all the adjacent frozen points of af). Proof. The proof follows the above discussion. • 6 8 CHAPTER 4. 4.3.2 The Frozen Point Nonlinearity Measure 5. Check if6v(NL(oi), NL(aj)) < r. If "YES", proceed to step 6. If "NO", add inter-mediate points, which are called j' points, between the ith-frozen point and all jth-frozen points. Then, check 5„(iVX(ai), NL(ay)) < r and 5v(NL(ctj>),NL(ctj)) < r. If the new i/-gaps still larger than r, add more intermediate points or consider to have a finer grid. 6. Go to the next frozen point (i.e. aij+i) and repeat step 4. 7. Repeat for all z = 1,..., N. Having defined the graph space for the frozen-point quasi-LPV system, the resulting closed-loop nonlinearity measure can be written as follows provided that a homotopy condition is satisfied: N 8%LPV(NL(a),L) = mfAmax{7%(NL(a),L)Jil(L,NL(a))} Va e (Jfi* (4.3.13) i = i where -Z*(NL(a)tL)± sup inf [n^)u\[ ff" NL(a)ui — ^ r » i II T Ul 111 J e S i O J . ^ n , ) ^ Uu2]s&as«b I  [NL(a)Ul \ \ \ 2 and (4.3.14) 1«{L,NL{«))± sup inf IIL^J L y ^ J l b ( 4 3 1 5 ) [^92^2 [jvLWuxjeeiCUiliniJn^, lllz*2JII2 Finally, from Figure 4.5 and Assumption 4.3.2, it can be shown that the following equation holds. . N SQLPV (NL(a^L) = i n f s u p 5 u (NL(oi), L ) , Va G I I a (4.3.16) " L e A i = i , . . . , A r ~ ' ' i = i In other words, as can be seen from Figure 4.5, the maximum over all r^ -gaps between individual subregime NL(cxi) and the linear model L is equal to the frozen point non-linearity measure when the union of the scheduling space is considered. 69 C H A P T E R 4. 4.3.3 Time Variation of Scheduling Parameter A major drawback of the frozen point nonlinearity measure is that it does not take into account the time variation of the scheduling parameter. The controller synthesized using this method is likely to suffer from degradation in its stability margin and performance when the actual plant is time-varying. The next section attempts to incorporate this observation into the nonlinearity measure. 4.3.3 Time Variation of Scheduling Parameter A unique feature of the quasi-LPV transformation is that the plant's nonlinearity is captured by the scheduling parameter, which is normally time varying and is allowed to evolve within a prescribed scheduling space. For slowly time varying process, the frozen point approach provides a convenient way to assess closed-loop robustness of the nonlinear plant, when the scheduling parameter is moving from one point to the other. However, when the time variation effect is significant, no guarantee can be made to ensure that the controller will perform adequately. This subsection aims to establish additional conditions for frozen point approach to allow for the incorporation the time varying effects. Recall that two closed-loops satisfying a homotopy condition can be seen as a contin-uous perturbation of one closed-loop to another one, while preserving the closed-loop stability, as depicted in Figure 4.6. Note that from Assumption 4.2.1, it is assumed that both closed-loops are subject to the same noise and disturbances. Hence, the sole perturbation that goes into the two closed-loops is essentially owed to the changes of the scheduling parameter. The existence of a homotopy condition, as shown in Figure 4.6, merely implies the preservation of closed-loop stability under continuous perturbation and gives no information on how the rate of change of the scheduling parameter affects the controller performance. In Hyde (1991), when designing a scheduling controller based on the frozen parameter quasi-LPV model approach, the time variations of the scheduling parameter are treated as an additional perturbation covered by the stability margin. This can be done by re-quiring a certain level of performance, in terms of the generalized stability margin, over 70 C H A P T E R 4. 4.3.3 Time Variation of Scheduling Parameter Figure 4.6: Homotopic and nonhomotopic analogies from a closed-loop perspective, (a) An originally stable quasi-LPV closed-loop is continuously perturbed by gradually scheduling the scheduling parameter from a0 to cti — a0 + e. If a homotopy condition exists for the graphs of the two closed-loops, then one can infer that the second closed-loop is stable, (b) Assuming that the initial condition is similar to (a), however, this time the scheduling parameter is scheduled by e'. Since there is no homotopy condition for the two graphs, the second closed-loop is unstable. 71 C H A P T E R 4. • 4.3.4 The Choice For The Nominal Model all frozen points. A similar idea can be employed for the developed nonlinearity mea-sure. By adding a certain amount of perturbation to the resulting i^-gap metric for the quasi-LPV system, accountability of the plant scheduling parameter time varying char-acteristics is maintained. In conclusion, to cope with time variations of the scheduling parameter, the following additional conditions need to be satisfied: • Assuming that ( J i = i ^ — >^ there exists a homotopy condition between fij and flj, for allz ^ j, i,j = 1,2,..., A/. • An additional amount of uncertainty, says 5+, is added to 8U(N L(a.i), L) to account for time variations. Remark 4.3.4. The amount of 5+ really depends on system's dynamics. For practical purpose, based on the author's experience, a value corresponding to 20% o/supj 8U(N L(af), L) is introduced. 4.3.4 The Choice For The Nominal Model The choice of the nominal plant model plays a crucial role in the proposed nonlinear-ity measure since the quasi-LPV gap quantifies the radius of the uncertainty, which is induced by the closed-loop nonlinearity, centered around the nominal model. Figure 4.7 shows a nonlinear trajectory of a plant over a range of operating points. The oper-ating regime is decomposed into a set of operating sub-regimes. For each sub-regime, the scheduling parameter is frozen to obtain a local linear model representing the local dynamics. To construct the uncertainty ball induced by the nonlinearity, a nominal model is first chosen among the local models. Assuming that Assumptions 4.3.1 and 4.3.2 are satisfied, the radius of the uncertainty ball is given by the distance between the nominal model and the linear model that gives the maximum quasi-LPV gap. As can be seen in Figure 4.7, the radius of the aforementioned uncertainty ball is clearly affected by the choice of the nominal model. For example, the radius of the uncertainty ball centered around Pi (i.e. the distance between Pi and P[ measured by the f-gap for quasi-LPV systems) in Figure 4.7 is larger than that of P 0 (similarly, the quasi-LPV 72 C H A P T E R 4. 4.3.4 The Choice For The Nominal Model Figure 4.7: Impact of the choice of nominal model gap between P 0 and PQ). Thus, care must be taken to reduce the conservatism of the proposed nonlinearity measure by an adequate choice of the nonlinear plant nominal model. To resolve this, the following definition for the best linear model is adopted, to assist in selecting the nominal model: Def in i t ion 4.3.7. For a quasi-LPV system P(a), a set of local models is obtained by freezing the scheduling parameter a. Select a nominal model, say P ( a i ) , from the above membership set. The v-gaps of the chosen nominal model and all other members in the set are computed (i.e. S ^ P V ( P ( A I ) , P(AJ)) = s u p ^ L W ( P ( a i ) , P(ajWu)> V a i ^ ^ e.fl, Vo; € [0, oo)). Repeat for all on 6 fl. The best nominal model is then defined as follows: Remark 4.3.5. Note, often time the nominal model is pre- and post-compensated to give the required loop shape. Experience shows that when the same compensators are used for all the members in the model set, the choice of the best nominal usually remains the same. However, this statement is not generally true and is greatly dependent on the systems. (4.3.17) 73 C H A P T E R 4. 4.3.5 The Linearizing Effect of Feedback 4.3.5 The Linearizing Effect of Feedback It is well recognized that feedback has a linearizing effect when the loop gain is sufficiently large. To see this, consider a SISO standard feedback system consisting of a continuous nonlinear function A(e) and a linear compensator C, see Figure 4.8. - e y A(e) U C Figure 4.8: Nonlinear feedback control Recall that the Taylor series expansion of A(e) is: A(e) = A(e 0) + ^-(e - e0) + ^ ^ ( e - e 0) 2 + • • • + HOT (4.3.18) where HOT denotes the higher order terms. By assuming A(0) = 0 and neglecting the higher order terms, Eq.(4.3.18) is reduced to: A(e) « A'(0 ) e - f - ^ ^ e 2 (4.3.19) By defining xOZ/(0) = ^^y, open-loop nonlinearity distortion, we have: A(e) xOL(0) 2 ^ - e + ^ - f i e 2 (4.3.20) It is obvious that if A(e) is linear (or nearly linear), x O i(0) i s z e r o ( o r negligible). From Figure 4.8 it follows that: y = T(r) = A(e), e = r — cy = r — cT(r) 74 (4.3.21) C H A P T E R 4. 4.3.5 The Linearizing Effect of Feedback where T(r) denotes the closed-loop nonlinear function of r. The first derivative of T(r) is given as T = dA(e) de ^ dA(e) / cdT(r)\ dr dr dr dr \ dr J A'(e) 1 + CA'(e) (4.3.22) where A'(e') = d^Jf^. Similarly, the second derivative of T(r) is r , _ d2T(r) = d2A(e) cd2A(e) dT(r) cdA(e) d2T(r) dr2 dr2 dr2 dr dr dr2 = A"<e> (4 3 23) (l + CA'(e)) l 4 • ; ; ' • i • J , The closed-loop nonlinearity distortion x C L (0) can then be obtained as follows: v C L r m = nO) A"(0) 1 + CA'(O) X y ) T'(0) (1 + CA'(0)) 2 A'(0) A"(0) 1 A'(0) l-f-CA'(O) X O L ( 0 ) 5(0) X C L (0) = X ° L (0 ) -5 (0 ) (4.3.24) Since r = e in open-loop «, = Unequivocally, the closed-loop nonlinearity distortion is reduced by the sensitivity function 5(0) of the system linearized around 0. The next lemma summarizes the above observation. Lemma 4.3.4. To reduce closed-loop nonlinearity, one would try to keep the loop-gam (i.e. L(0) = CA'(0)) as large as possible. Proof. Consider Figure 4.8. Since x C L (0) = x°L(0)5(0) and 5(0) = J ^ J ^ , the result follows immediately. • 75 C H A P T E R 4. 4.3.6 Loop-Shaping and Weight Selection 4.3.6 Jf?oo Loop-Shaping and Weight Selection The Jffoo loop shaping procedure of McFarlane and Glover (1990, 1992) has proved to be an effective and intuitive method for MIMO control systems design. This procedure involves three major stages as depicted in Figure 4.9. In the first stage, the open-loop p w2 Stage I: Open-loop Shaping p w2 Coo Ps Stage II: Jif^ Robust Stabilization 1 0 ^ P a # b C Wx Coo W2 Stage III: Final Controller Figure 4.9: Loop-shaping design procedure (McFarlane and Glover, 1990). plant is shaped by applying a pre- and a post-compensators to give a desirable shape to the singular values of the open-loop frequency response. The resulting shaped plant is then robustly stabilized with respect to normalized coprime factor uncertainty using Jt^oo optimization in the second stage. In the last stage, the final controller is formed by combining the compensators and the resulting M3^ controller (i.e. C o o ) - The advantages 76 C H A P T E R 4. 4.3.6 Jif^ Loop-Shaping and Weight Selection of this procedure are that there is no problem specific uncertainty modeling required, and that the J ^ o robustness optimization problem is non-iterative. This means that an optimum robustness of the controller can be obtained explicitly. As one may notice a crucial step in the first stage is to appropriately shape the singular values of the open-loop plant. Recall that, good performance controller design requires that4: o ((/ + PC)"1) , o ({I + PC)~lP) , o ((/ + CP)'1) , a (C(I + PC)'1) (4.3.25) be made small in some low frequency range, typically UJ G (0,0;;). Also, according to Lemma 4.3.4, it is also desirable to keep a ((I + PC)~l) and & ((I + C P ) - 1 ) small in low frequency range in order to minimize closed-loop nonlinearity. Good closed-loop robustness requires that: a {PC(I + PC)'1) , a (CP(I + CP)) (4.3.26) be made small in some high frequency range (i.e. UJ G (o;u,co)). The requirements in Eqs. (4.3.25) and (4.3.26) can be rewritten in terms of loop-gains in the appropriate frequency ranges. Therefore, for low frequency range, a(PC) > 1, a{CP) » 1, a{C) > 1 (4.3.27) and for high frequency range, a(PC) < 1, c r ( C P ) < 1, a(C) < M (4.3.28) where M > 0 is not too large. Note that the actual achieved loop shape at plant input (Figure 4.9, stage III, point a) is given by W i C o o W ^ - P , which is quite different from the specified desired loop shape W2PWi. Therefore, the actual loop shape is expected to deviate from the specified one 4See §A.4 and Zhou et al. (1996, § 5.5). 77 C H A P T E R 4. 4.3.6 M'OQ Loop-Shaping and Weight Selection HPWiCooWi) Figure 4.10: Specified and achieved loop-shapes (McFarlane and Glover, 1990). because of the presence of C ^ . A similar conclusion can be drawn for the achieved loop shape at plant output (Figure 4.9, stage I II , point b), which is given by P W i C o o W j j . Figure 4.10 shows the possible discrepancies that may occur between the specified and the achieved loop shapes. Fortunately, as showed by McFarlane and Glover (1990, 1992), the deterioration of the loop shape, at either input or output, is limited at those frequencies where the specified loop shape is sufficiently large or sufficiently small. To see this, note that a(PC) = a{PW1C00W2) = aCW^WiPW^Wz) > g ( W 2 P ^ l ) g ( C o o ) c{W2) (4.3.29) and a(CP) = QLiWiC^w.p) = a(w1c00waw1wr1) > ^ P ^ i k ( O o ) ( 4 3 3 0 ) c{Wi) where c(-) = ^ | denotes the condition number. Clearly, Eqs.(4.3.29) and (4.3.30) depend on c r ( C o o ) . The following result, which was derived by McFarlane and Glover (1990), shows that a ^ ) is bounded by 7 = bPCoo and a{Ps) = qiW^PWi). 78 C H A P T E R 4. 4.3.6 Jffoo Loop-Shaping and Weight Selection Theorem 4.3.4. Any controller, C^, satisfying Cn (I - P . C o o ) " 1 ^ - 1 < 7 (4.3.31) where [Ns, Ms] is a normalized left coprime factor of Ps, and Ps is assumed square, also satisfies o(Ps(ju)) - V T ^ T for all co such that a(Ps(ju>)) > \Ay2 — 1. (4.3.32) Proof. See McFarlane and Glover (1990, Theorem 6.2, pg. 111-114) • It is obvious that when a(Ps(ju)) > ^ / 7 2 - 1, ^(C^ju)) > -7==, where > denotes ~ V r - i ~ asymptotically greater than or equal to as a(Ps(ju)) -» oo . This implies that g_{PC) > » ^ * 1 (respectively, g_{CP) > - ^ 2 = » ^ 1). Note that c(VKi) and c(W2) are selected by the designer, and are normally close to one. Analogously for a (PC) and a(CP), we have a(PC) = otPV^CooWa) < a ( W 2 P W 1 ) a ( C 0 O ) c ( i y 2 ) (4.3.33) and o(CP) = aiWiCMP) < o(W2PW1)a(C0O)c(W1) (4.3.34) Like g_(PC) and a (CP), o(PC) and a (CP) are bounded from above by the maximum singular values of the specified loop shape, the controller and the condition numbers of the compensators. Has been proven by McFarlane and Glover (1990), the following result gives the upper bound for c r ( C o o ) . Theorem 4.3.5. Any controller, C o o , satisfying Eq.(4-3.31), also satifies a ( C o o O ) ) < ^ f ^ l + a(Ps(ju)) 1 - ^^lo(Ps(ju)) (4.3.35) 79 C H A P T E R 4. 4.3.6 M'oo Loop-Shaping and Weight Selection for all u such that a(Ps(ju>)) < ^—. V 7 2 - i Proof. See McFarlane and Glover (1990, Theorem 6.4, pg. 114-116) • Likewise, if o-(Ps) <§; J-—, then a(Coo(juj)) ;$ Jy1 — 1, where % denotes asymptoti-• v / 7 2 —1 ~ v ~ cally less than or equal to as a(Ps) —> 0. Obviously, a (PC) <^ 1 and a (CP) <C 1. As shown in McFarlane and Glover (1990) and Zhou et al. (1996), the next theorem shows that all closed-loop objectives are guaranteed to be bounded and the bounds depend only on 7 , Wx, W2, and P. T h e o r e m 4.3.6. Let P be the nominal plant and let C — WiCooW2 be the associated controller obtained from the J^, loop-shaping procedure. Then if I Cor (I + PsC^M-1 < 7 (4.3.36) we have a(C(I + PCYl) < 1o(Ms)o(W1)o(W2) o((I + PCyl) < min { 7a(M s)c(M/ 2), 1 + 7a(7V s)c(W 2) a(C(I + PC)-^) < min[ 7 a(iV a )c(Wi) , l + 70 :(M a)c(V7i) odl + CPy1) < min ^ o(Ms)c(W1), 1 + 7 ^ ) 0 ( ^ 1 ) o(P(I + CPylC) < min {jo(Ns)c(W2), I+ jo(Ms)c(W2) (4.3.37) (4.3.38) (4.3.39) (4.3.40) (4.3.41) (4.3.42) where o(Ns) = o(Ns) = o(Ms) = a(Ms) °\Ps) l + °2(Ps) 1 1 + <L2(PS) (4.3.43) (4.3.44) 80 C H A P T E R 4. 4.3.7 The Best Possible Stability Margin and [NS,MS] (respectively, [NS,MS]) is a normalized left coprime factorization (respec-tively, right coprime factorization) of Ps = W2PW1. Proof. See Zhou et al. (1996, Theorem 18.11, pg. 493) • 4.3.7 The Best Possible Stability Margin Recall that in the developed nonlinearity measure formulation, the closed-loop nonlinear-ity, which is measured by the summation of the largest i^ -gap metric over all a, G fl and a time variation penalty, is compared with the generalized stability margin associated with the linear model and a linear controller. Since the generalized stability margin is a function of both the plant and the controller, there may exist another controller that gives satisfactory stability margin. Clearly, if one uses a controller that provides an optimal stability margin, then one can conclude that a linear controller is insufficient to control the nonlinear plant whenever the closed-loop nonlinearity is larger than the stability margin. Otherwise, no conclusion can be made about the abilities of the lin-ear controller since there is a chance that one might be able to design another linear controller which provides a sufficient stability margin. In this regard, Glover and McFarlane (1989) derived such an optimal value of the generalized stability margin for a given (shaped) linear plant, says Ps. Defined by bpc,ma.x(Ps) — supbpSic, the optimal value can be obtained via the following formula: where || •• \\H and [gs hs] denote Hankel norm and a pair of normalized left coprime factorization of plant Ps (i.e. Ps = hj1gs). Remark 4.3.6. Eq.(4-3.45) only depends on the plant information. However, appro-priate weightings, which incorporate the process specifications, are crucial to the success of the application of Eq. (4-3-45). Often time, small bpc,max(Ps) indicates poor compat-ibility or conflict of specifications. c (4.3.45) 81 C H A P T E R 4. 4.4. A Computational Algorithm 4 . 4 A Computational Algorithm Having discussed the formulation and several theoretical motivations of the developed closed-loop nonlinearity measure. In what follows, a practical computational algorithm which allows quick evaluation of the nonlinearity measure is presented. 1. Recast the nonlinear system into a quasi-LPV form and adequately grid the schedul-ing parameter space by using the partitioning algorithm presented in §4.3.2. 2. A set of linear models is obtained by freezing the scheduling parameter at each grid point. 3. For each model, the quasi-LPV gaps to all other models are obtained as:. 5QLPV = { S Q L P V { p { a i ) j p { a 3 ) ) j V a3 e fi}. 4. Denote by L*, the best nominal model for closed-loop control which is the one that has the smallest oo-norm in <5j, V i. 5. Choose appropriate pre- and post-compensators for L* to reflect the design speci-fications and forms the shaped plant (i.e. Ls = W2L*Wi). 6. Repeat step 3, but applying W\ and W2 to all P(ai) and P{a3) at this time. Obtain the new L* according to step 4 and subsequently the new Ls. 7. Compute bPC,max = \-8. Exploiting the information of 6pc,max, a suboptimal robust controller is de-signed using X o loop shaping design procedure. 9. Repeat step 4, but compute 5^fpv = {5^LPV{P(ai)C, P{OLJ)C), V a3 G fi} in-stead, where C = W i^CooM -^10. Compute bPc,i, where / denotes a unity feedback. 11. By employing Corollary 4.2.1, the closed-loop nonlinearity is manageable by the designed linear controller if bPC,i > ^ {P{a)WlC00W2, L*W1CQOW2). 82 C H A P T E R 4. 4.5. Summary 12. If bPC,i < ^(P(a)WlC00W2,L*WlC00W2), employ Theorem 4 . 2 . 1 . Then, the closed-loop nonlinearity is said to be manageable by the designed linear controller if bpcAju) > 8^(P(a)WlC00W2,L*W1C00W2){3uJ)^u e [O.oo). 13. Likewise, by using Theorem 4 .2 .2 , the closed-loop nonlinearity is larger than what the designed linear controller can cope with if bPc,i(juj) < S'^n(P(a)WiC00W2, L W 1 C 0 0 W 2 ) ( j a ; ) , V a ; € [0,oo). 14. In addition, if 6 p C l m a x ! » < S^Pi^W^M, L ^ O ^ ) ^ ) , Vo, 6 [0,oo), then a linear controller corresponding to the design specifications is insufficient to control the nonlinear plant. In this regard, either the design specifications need to be redefined or a nonlinear control strategy is needed. 4.5 Summary A closed-loop nonlinearity measure exploiting the generalized stability margin and the z/-gap metric has been developed. Theoretical motivations of the proposed closed-loop nonlinearity measure have been presented. The resulting nonlinearity measure asserts that if the closed-loop nonlinearity quantified by the z^ -gap metric and a time-variation penalty is smaller than the optimal generalized stability margin, then there exists a linear controller that can stabilize the nonlinear plant. Otherwise, either the plant's performance specifications need to be redefined or a nonlinear control strategy is needed. In the latest part of this chapter, a practical engineering algorithm was developed to compute the nonlinearity measure. In the next chapter, this algorithm will be employed to three design examples to illustrate the strength of the developed measure. 83 C H A P T E R 5 Design Examples This chapter presents three design examples, namely a continuous stirred tank reactor (CSTR) temperature control problem, an inverted cone tank control prob-lem and a fictitious control problem, by using the developed closed-loop nonlinearity measure. All examples exhibit open-loop nonlinear behavior. The first example is SISO, while the second one is MIMO. The third example involving a fictitious non-linear plant is used to illustrate the ability of the developed nonlinearity measure in predicting the inadequacy of a linear controller. Analysis and simulation results show that the developed nonlinearity measure is able to assess the adequacy of a linear controller. 5.1 Example I: C S T R Control Problem The purpose of this SISO example is to demonstrate the strength of the developed nonlinearity measure. The linearizing effect of feedback is also clearly illustrated. 5.1.1 Problem Description A schematic diagram of a CSTR is depicted in Figure 1.2. Consider again the continuous stirred tank reactor temperature control problem presented in §1.1. For the sake of completeness, the nonlinear ordinary differential equations, which describe the CSTR 84 C H A P T E R 5. 5.1.2 Design Objectives process dynamics are restated here. dCA q & y{CAf - CA) - k0exp(- — )CA k T f - T ) + k2CA + k1(Tc-T) (5.1.1) dt dT ~dt (5.1.2) with k\ = UA and k2 = cH'k0exp(-^). Where CA, T, and Tc represent reactor vPcp effluent concentration of component A, reactor temperature, and coolant temperature, respectively. The remaining model parameters and the nominal operating conditions are similar to those given in Table 1.1. As discussed in §1.1, the open-loop nonlinearity of a CSTR is closely related to the operating points. The higher the reactor temperature, the higher the open-loop non-linearity is. Therefore, intuition suggests that a nonlinear control strategy is needed. However, as mentioned in §1.1, the outcome of putting the CSTR in closed-loop is quite counter-intuitive. The degree of nonlinearity of the CSTR turns out to be much more manageable in closed-loop, as opposed to its open-loop one. In what follows, a detail closed-loop nonlinearity assessment of the CSTR is presented. 5.1.2 Design Objectives The primary objective of this control problem is to maintain closed-loop stability as the reactor temperature T is changing in the range of [300, 373] K by manipulating Tc. In addition, the control system is required to satisfy the following specifications over the above mentioned operating range: 1. Zero steady state offset. 2. Less than 20% overshoot. 3. Good disturbances rejection (i.e. ± 20% in feed concentration CAf and ± 5K in feed temperature Tf). 4. Closed-loop bandwidth is limited by the actuator bandwidth (i.e. 20 rad/min). 85 C H A P T E R 5. 5.1.3 Nonlinearity Measure 5.1.3 Nonlinearity Measure To check the adequacy of a linear control strategy, it is desirable to measure the closed-loop nonlinearity. To achieve this, the algorithm proposed in §4.4 will be employed. The first step is to recast Eqs.(5.1.1) and (5.1.2) into a quasi-LPV representation via a state transformation. In this example, the reactor temperature is chosen as the scheduling parameter. Eq.(5.1.3) gives the resulting quasi-LPV model d_ di T CA ~ CA,eq T —T ; i + /c 0 exp(-^)) dTc,eq h. —dr fc2 dCA, dT dT dTc, dT T CA ~ CA,eq T —T ± c -1 c,eq (5.1.3) By employing the scheduling space partition algorithm presented in §4.3.2, 66 grid points are required to guarantee that the discrepancy from one scheduling subspace to the adjacent ones is less than 5%. The best model Po(T), according to Definition 4.3.7, is the one corresponding to T = 341 K after employing the aforementioned computational algorithm. The z^ -gap contour plot of all possible combinations in the membership set is shown in Figure 5.1. Note that the Tj-axis and Tj-axis represent the selected nominal model at z-th operating point and the other models in the membership set, respectively. The contour is determined by the i^ -gap between Ti and Tj. Evidently the chosen nominal model is the most similar to other members in the set from a closed-loop perspective. In addition, as can be seen from Figure 5.1, no matter which nominal model is chosen, the maximum z^ -gap between the nominal model and the others exceeds 0.9. For instance, consider the nominal model at Ti = 341 K (black dot). Draw a vertical line, which is parallel to the ?/-axis, through the black dot. The intersections of this line with the contour lines give the corresponding i^ -gap metrics between the nominal model and the model at Tj. Obviously, the largest z^ -gap is greater than 0.9. This means that a unity feedback fails to maintain closed-loop stability at some points in the scheduling 86 C H A P T E R 5. 5.1.3 Nonlinearity Measure space Cl. However, since the nonlinearity is scaled by sensitivity function1, appropriate modifications of the loop shape are expected to bring the z^ -gap values down . 300 300 310 320 330 340 350 360 Selected nominal model at i—th operating point, T. (K) 370 Figure 5.1: Unshaped z^ -gap contour. Nominal model (black dot). The transfer function of the nominal model is given as follows: 2.092(s + 1.517) ft W = . ( . - 1 . 2 b ) ( , +0.526) ( 5 X 4 ) Figure 5.2 shows the bode diagram of the nominal plant. Clearly, the selected nominal model's bandwidth is 1.74 rad/min, and the model is open-loop unstable. Therefore, care must be taken when choosing the gain crossover bandwidth. If the bandwidth is set too high, the closed-loop might lose controllability due to input saturation. In this case, the gain crossover bandwidth is set equal to the actuator bandwidth. ^ee §4.3.5. 87 C H A P T E R 5. 5.1.3 Nonlinearity Measure Frequency Figure 5.2: Bode diagram for the nominal model at T=341 K 88 C H A P T E R 5. 5.1.3 Nonlinearity Measure Next, to shape the open-loop plant (i.e. PS = W2PQWI) SO that it gives the desirable closed-loop properties, the following two compensators are applied to the nominal plant: 7830s+ 78300 T I , , l y ' = ^ + 8 0 . + 900- W > = 1 ( 5 ' 1 5 ) Note that there are no hidden RHP pole and zero cancellations in the above compen-sators, and that the slope of the loop-shape at the crossover frequency (i.e. 20 rad/min) is -20dB/decade as required by the loop shaping procedure. For this shaped plant, the optimal generalized stability margin 6pc,max is found to be 0.34. Figure 5.3 shows the specified and stabilized loop shapes. Obviously, the stabilized loop shape only slightly deteriorates from the specified one. Bode Diagram -90 -1801= 1 . I 1 , , , i J 10"1 10° 101 102 103 Frequency Figure 5.3: Specified (solid) and stabilized (dashed) loop shapes. 89 C H A P T E R 5. 5.1.3 Nonlinearity Measure The resulting stabilized sensitivity and complementary sensitivity functions are also shown in Figure 5.4. The oo-norm of the sensitivity (Ms) is 1.8463, which translates to GM > 2.2 and PM > 31.5°. The oo-norm of the complementary sensitivity (i.e. MT) is 1.6521, which is slightly larger than the typical value 1.25. Therefore a slightly larger total variability (TV) in the output response is expected2. However, as long as the overshoot is less than 20%, the output response is acceptable. 10 10 10 e i o " 2 10 10 10" 10 10 1 1 1 1 1 1 1 • 1 111 . . , 1 • '— / " \ / \ / ^ / \ / \ -/ \ / \ / X / \ -\ : \ : \ \ -\ : \ \ \ 1 \ \ ; < \ V 1 • • 1 10 10 10 Frequency 10' 10" Figure 5.4: Sensitivity (solid) and complementary sensitivity (dashed-dotted) functions. Note that the closed-loop bandwidth is around 10 rad/min, which is much greater than the crossover frequency of the nominal loop. It is interesting to see how this affects the nonlinearity (i.e. the i^-gap values). Figure 5.5 shows the corresponding contour plot. Clearly, there is a significant reduction in the i^ -gap values (i.e. from 0.97 to 0.25). This implies that the closed-loop system is much more linear3 using the compensators shown in Eq.(5.1.5). Alternatively, this means that the radius of the coprime factor 2Since MT <TV < (2n + 1)MT, where n is the order of T , see Skogestad and Postlethwaite (1996, pg. 34) 3Recall that the degree of closed-loop nonlinearity is quantified by the f-gap values. 90 C H A P T E R 5. 5.1.4 Simulation Results type uncertainty ball induced by the nonlinearity is reduced to a level that is more manageable. Finally, by adding an additional 20% of the computed v-gap to account for time variation, the resulting closed-loop nonlinearity is equal to 0.30. A by-product obtained using the proposed computational algorithm is a robust linear controller C — W&ooWt, where -82.7441 -19.7057 -4.4243 -1.6377 -0.2390 -0.0171 128.0000 0 -13.5719 -9.7690 -1.6082 -13.5749 0 64.0000 -75.9028 -54.6345 -8.9941 -75.9196 0 0 -38.8646 -39.4913 -6.5012 -54.8768 0 0 -0.2813 7.7975 -0.0333 -0.2814 11.7498 6.5169 2.2476 0.8150 0.1185 0 Finally, the associated stability margin corresponding to Pn^iCooW^ with respect to a unity feedback (i.e. &PC,I) is 0.31. This value is then compared to the nonlinearity measure (i.e. S^m(P(a), P0)). Since bPC,i = infwe[o,oo) bPC,i(M and 5^m(P(a),P0) 4. suPu>e[o,oo) <^m(-P(a)) Po)(jv), the designed controller, according to the Corollary 4.2.1, is sufficient to cope with the closed-loop nonlinearity when the plant is pre- and post-compensated, and the closed-loop stability is guaranteed VT e Cl. Simulation results are presented in the next section. 5.1.4 Simulation Results In this subsection, the system's servo responses under a unity feedback control are first presented. This is to illustrate that without appropriately shaping the loop gain, the result can be disastrous. Next, setpoint tracking responses of the CSTR under the de-signed robust control is shown. Finally, simulation results of the same system subject to various disturbances (i.e. ± 20% in feed concentration CAf and ± 5K in feed temper-ature Tf, respectively) are presented. The setpoint tracking and disturbance rejection 91 C H A P T E R 5. 5.1.4 Simulation Results 10 I 1 L—i L LJ I 1 I i / i i_ 300 310 320 330 340 350 360 370 Selected nominal at i-th operating point, T.(K) Figure 5.5: Shaped i^-gap contour. The best shaped model (black dot). 92 C H A P T E R 5. 5.1.4 Simulation Results responses of the system are subsequently used to evaluate the time-domain reponses of the designed system. 5.1.4.1 U n i t y Feedback Figure 5.6 shows the closed-loop responses of the CSTR under a unity feedback control. The responses are unacceptable. In fact, the system is unstable and the simulation is beyond the range of validity (i.e. the coolant temperature is below 0 K, which violates the physical rule). Note that this observation is consistent with the results obtained from the z^ -gap metric calculation. Time, t (min) Figure 5.6: Top: Setpoint tracking responses of the CSTR under a unity feedback control. Reactor temperature, T (solid), setpoint (dashed-dotted). Bottom: Coolant temperature, Tc (solid). 93 C H A P T E R 5. 5.1.4 Simulation Results 5.1.4.2 Setpoint Tracking Responses of C S T R U n d e r Robus t Con t ro l The setpoint tracking performance of the closed-loop is shown in Figure 5.7. The schedul-ing parameter T is moving over the entire operating range (i.e. T 6 fi = [300, 373] K). As can be seen from this figure that the closed-loop remains stable throughout fi. Even though the overshoot of the system becomes larger and larger as the reactor temperature approaching 373K, simple calculation reveals that the overshoot remains less than 20% even at its worst case at T = 373K. The increasing of the overshoot is expected since the process nonlinearity becomes dominant as the reactor temperature exceeds 350K, see §1.1. Moreover, zero steady state offset is observed for all successive step changes in setpoint. Clearly, specifications 1 and 2 in §5.1.2 are satisfied. 350 20 25 Time, t (min) Figure 5.7: Top: Setpoint tracking responses. Reactor temperature, T (solid), setpoint (dashed-dotted). Bottom: Coolant temperature, Tc (solid). 94 C H A P T E R 5. 5.1.4 Simulation Results 5.1.4.3 Disturbances in Feed Concentration, C U / According to the design specification, the final closed-loop must be able to tolerate ± 20% changes in CA/, see §5.1.2. Figure 5.8 shows the disturbance rejection responses of the closed-loop subject to ± 20% changes in CAJ- Evidently, the closed-loop poses good disturbance rejections at all three operating points (i.e. 300K, 341K and 370K) representing the low, medium and high regimes of the operating space. 5.1.4.4 Disturbances in Feed Temperature, T/ As can be seen in Figure 5.9 that the closed-loop has a satisfactory disturbance rejection performance when subject to ±5K changes in Tf. 5.1.4.5 Concluding Remarks: C S T R Control Problem Clearly, the developed measure not only provides a reliable quantification of the closed-loop nonlinearity, but also produces as a by-product, a robust linear controller, that satisfies all the design specifications. The next subsection presents a brief design on a different control problem, an inverted cone tank control problem. 95 C H A P T E R 5. 5.1.4 Simulation Results g 372.0 I— 0) § 371.0 CL E Q) h-o 1 370.0 CC g 341.3 H 3 341.2 cd (5 CL | 341.1 o S 341.0 _ 300.008 | 300.004 CL E | 300.000 c  0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Time, t (min) 4 6 Time, t (min) 4 6 Time, t (min) Figure 5.8: Left: Closed-loop responses subject to ±20% in feed concentration, CAf (Top three: +20%; Bottom three: +20%) at three operating points. Reactor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant temperature, Tc (solid). 96 C H A P T E R 5. 5.1.4 Simulation Results <r 299.6 4 6 Time, t (min) 4 6 Time, t (min) S 302 8. £ 300 H c ro 3 298 _275 5 273 E o 271 306 o 302 „ 2 7 8 tf°277 i i 1276 £ ~ 275 re 3 274 0 4 6 Time, t (min) 4 6 Time, t (min) 10 10 10 Figure 5.9: Left: Closed-loop responses subject to ±5K in feed temperature, Tf (Top three: +5K; Bottom three: +5K) at three operating points. Reactor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant temperature, Tc (solid). 97 C H A P T E R 5. 5.2. Example II: Cone Tank Control Problem 5.2 Example II: Cone Tank Control Problem This section illustrates the usefulness of the developed nonlinearity measure for a multi-input multi-output (MIM0) system via an inverted cone tank example. This example together with others tested and not presented here allows the statement that the non-linearity measure works equally well for stable/unstable SISO or MIM0 plants. 5.2.1 Problem Description Considered as a relevant MIM0 example an inverted cone tank (Tan and Chiu, 2001), as depicted in Figure 5.10, where two feed streams, Fi and Fj, enter the tank with the inlet temperature Tj at 350K and the cooling stream qc has the inlet temperature Tci and outlet temperature Tco at 289K and 313K, respectively. Pi i Ii F T-H F0, T Figure 5.10: A schematic diagram of an inverted cone tank. 98 C H A P T E R 5. 5.2.1 Problem Description Assuming that the outlet flow characteristic is given by: F0 = KV7L (5.2.1) where K and h denote the outlet valve coefficient and the liquid level, respectively. The process dynamics of the inverted cone tank can be described by the following ordinary differential equations: dh Fi + FA K , T - „ * - -w1-? • (5-22) f - 3J^<?<-T)-$IT„-TJI (5.2.3) where R and H are the cone radius and cone height, respectively. The nominal values of the process parameters are given in Table 5.1. Table 5.1: Nominal operating conditions for an inverted cone tank Parameter Notation Value Cone radius R 0.798m Cone height H lm Outlet valve coefficient K 50 mtmin - 1 Inlet stream 10m 3min _ 1 Coolant inlet temperature T 289K Coolant outlet temperature T x CO 313K 99 C H A P T E R 5. 5.2.2 Design Objectives 5.2.2 Design Objectives The control objective of this example is to have the liquid level h and the effluent temperature T operating within the prescribed operating range, i.e. h € [0.2, 1] m and T 6 [303, 333] K. This objective can be attained by manipulating Fi and qc. The actuator bandwidth is given as 20 rad/min. The typical control requirement, in the context of this problem, include additionally: 1.) Zero steady state offset; 2.) Less than 20% overshoot; and 3.) Good disturbance rejection. 5.2.3 The Nonlinearity Measure Appl ied To T h e M I M O Cone Tank As previously illustrated through the example presented in §5.1, the algorithm proposed in §4.4 has been employed to test the adequacy of a linear control strategy. As pointed out in Skogestad and Postlethwaite (1996, pg. 5-8), when dealing with a MIMO problem, an appropriate scaling of the system is important for sensitivity analysis and singular value computation. Therefore, prior to implementing the algorithm in §4.4, the state variables and the other associated variables are first scaled by defining h = -r^—, T = 7?r—, Fi = „ F i , Fj = ^ ', qr = —2s—. The scaled version of the inverted cone tank - imax " i . m a x J " ^ m a x ° 9c,max can then be rewritten as follows: dh _ (Fl + Fj)Fh K -2 -=-1.5 (5.2.5) dT = 3(Fi + F ^ F ^ ^ - _ _ 3g c g C | m ax ^ _ ^ ( 5 2 6 ) Next, by selecting the liquid height and the liquid temperature as scheduling parameters, 100 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank Eq.(5.2.7) shows the resulting quasi-LPV model: d_ It h T F - F-Qc Qc,eq A(h,T) h 0 0 T 0 0 + Fi - F- 1 0 9c, eg 0 1 V2 (5.2.7) where A(h,T) = 0 0 0 0 0 0 0 0 Fi, 3 J i , m a x ( ^ i - ^ ) <Ph ^max i, max i,eq T) dFiieq V ^ m a x dh dT F- dQc,eg •f) 9qc,eq <fh ^max dh <Ph ''max dT 0 3qc , max (Tco Tci ) / ] 4 a x _ 3g c , max {Tpo Tcj ^ h j ^ ~ dT .max (5.2.8) By using 20 grid points4 on the scheduling parameter pair (h, T) and employing the proposed algorithm, the best nominal model is found to be the one at h = 0.4526 and T = 0.8657 which corresponds to liquid level h = 0.4526 and liquid temperature T = 303. To cope with the design specifications, the corresponding pre- and post-compensators are chosen as: 506 s+30 0 0 506 s+30 and W2 = h (5.2.9) In this regard, the associated optimal generalized stability margin, bpc,max., for the afore-mentioned compensated plant (i.e. Ps = W2PoWi) is 0.5433. Figures 5.11 and 5.12 show the resulting open-loop gain and the contour plot of the shaped //-gaps, respectively. From Figure 5.12, the most dissimilar model for this example is the one at h=l and T = 0.9514 which corresponds to liquid level h = 1 m and liquid temperature T = 333 K with the f-gap value equals to 0.3546. After adding the 20% penalty of time variation, the resulting closed-loop nonlinearity measure (i.e. 5?m(P(a), Po)) is 0.4255. 4 Which guarantees less than 7% discrepancy from one scheduling subspace to its adjacent ones. 101 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank Singular Values Frequency (rad/sec) Figure 5.11: Loop gains for the inverted cone tank. Shaped plant (solid), original plant (dashed-dotted). 102 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank I—I 1 1—I L _ J » i l_ I I i I i I J I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Liquid Level, h(m) Figure 5.12: Shaped ^-gap contour. Nominal model (black dot). 103 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank Based on the compensators given in Eq.(5.2.9), the following Jtf^ linear controller is obtained via the loop-shaping technique. -137.7 -19.2 -0.9 -0.1 -0.002 0.0004 0.008 0.0006 256.0 0 -4.4 0 0 -0.1 -2.4 -0.4 0 256.0 -85.7 0 0 -1.7 -45.5 -6.8 -47.3 -24.2 -20.2 -3.5 -0.02 -1.3 -9.3 -5.2 0 0 -220.7 512.0 0 -84.1 -117.1 -81.3 0 0 -96.2 0 256.0 -582.9 -51.1 -35.9 34.0 17.2 1.9 0.2 0.004 -0.0004 0 0 5.7 2.7 0.2 -1.9 -0.006 0.0007 0 0 By using this controller, the generalized stability margin with respect to a unity feedback (i.e. bpcj) is found to be 0.4798. Note that the resulting bpc,i is slightly larger than a typical value, which is normally in the range of [0.3, 0.4]. This is mainly owing to the limit imposed by the current actuator bandwidth (i.e. 20 rad/min). If the actuator bandwidth constraint is removed, the value of the bpcj can be further reduced by pushing the MIMO closed-loop bandwidth to a higher frequency range. Of course, by doing so, the control action will become more aggressive and the closed-loop become less robust to uncertainty. Clearly, since the closed-loop nonlinearity 8'^n(P(a)1 P0) is less than the bPcj (i.e. 0.4255<0.4798), Corollary 4.2.1 ascertains that the designed controller is sufficient to handle the closed-loop nonlinearity when the plant is pre- and post-compensated. Sim-ulation results shown in Figures 5.13 to 5.18 confirm the resulting controller is able to provide satisfactory performance in both servo and load responses over the entire prescribed operating space. 104 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank Figure 5.13: Servo responses from operating point (0.2,303) to (1,333). Plant (solid) and setpoint (dashed-dotted) E \ u . -Figure 5.14: Servo responses from operating point (1,333) to (0.2,303). Plant (solid) and setpoint (dashed-dotted) 105 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank Figure 5.15: Servo responses from operating point (0.2,303) to (0.2,333). Plant (solid) and setpoint (dashed-dotted) 0.985 f . . . . 1 I 0 1 2 , 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Time, t (min) Time, t (min) Figure 5.16: Servo responses from operating point (1,333) to (1,303). Plant (solid) and setpoint (dashed-dotted) 106 C H A P T E R 5. 5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank Figure 5.17: Closed-loop response to +20% disturbance in inlet stream Fj at (0.2,303). Plant (solid) and setpoint (dashed-dotted) 43 0 1 2 3 0 1 2 0 1 2 3 0 1 2 3 Time, t (min) Time, t (min) Figure 5.18: Closed-loop response to -20% disturbance in inlet stream Fj at (1,333). Plant (solid) and setpoint (dashed-dotted) 107 C H A P T E R 5. 5.3. Example III: A Fictitious Nonlinear Plant Control Problem 5.3 Example III: A Fictitious Nonlinear Plant Con-trol Problem In this section, a fictitious nonlinear plant is used to illustrate the strength of the closed-loop nonlinearity measure in predicting the insufficiency of a linear controller, particu-larly in anticipating the onset of closed-loop instability. 5.3.1 Problem Description In what follows, a slowly time-varying fictitious plant whose local transfer functions are assumed to be. dependent on a scheduling parameter, k, is considered and has the following structure. k P(k, s) = , -5 < k < 2 (5.3.1) Obviously, the aforementioned plant experiencing a sign change in gain when the schedul-ing parameter is evolved in the scheduling space. Note that, the sign change characteris-tic in process gain is not uncommon in process industry. A typical example of processes that involve the aforementioned sign change characteristic is the pressurized refiner in mechanical pulp production system. The control objective of this problem is to achieve offset free setpoint tracking with less than 20% overshoot while maintaining closed-loop stability. The actuator bandwidth is given as 20 rad/sec. 5.3.2 Nonlinearity Measure By employing the algorithm proposed in §4.4, 60 grid points are needed in order to achieve a maximum 5% discrepancy level between one scheduling subspace to all its ad-jacent scheduling subspaces. Figure 5.19 shows the plot of the unshaped ^-gap contour. Obviously, the best nominal model is the one corresponding to k = —0.155, which has 108 C H A P T E R 5. ^ ^ 5.3,2 Nonlinearity Measure the following transfer function: 0.155 ~s + 1 (5.3.2) Figure 5.19: Unshaped z^ -gap contour. Nominal model (black dot). To satisfy the design specification presented in the previous section, the following pre-and post-compensators are augmented to the nominal plant: Wx = 4682 S + l and W2 = I. (5.3.3) s(s + 30) The resulting stabilized sensitivity and complimentary sensitivity functions are shown in Figure 5.20. The associated co-norm of the complementary sensitivity function MT is 1, while that of the sensitivity function Ms is equal to 1.75. In addition, the optimal value of the generalized stability margin, with respect to the shaped plant, is found to be 0.5803. 109 C H A P T E R 5. 5.3.2 Nonlinearity Measure 110 C H A P T E R 5. 5.3.2 Nonlinearity Measure The resulting controller obtained via the McFarlane-Glover Moo loop shaping pro-cedure is as follows: -49.4930 -64.4058 -7.8613 7.0914 32.0000 -93.4669 -11.6834 16.4839 Coo — 0 -27.7768 -4.4721 6.3096 4.6233 5.8147 0.7088 0 Unlike the previous two design examples, the augmentation of the pre- and post-compen-sators and the controller to the local models does not bring down the z^ -gap values significantly. Moreover, it can be seen from Figure 5.21 that the i^ -gap values exceed 0.6 whenever k is moved beyond the range of [—1,-0.5]. However, it is known that merely applying the Corollary 4.2.1 is insufficient since it only guarantees closed-loop stability whenever the generalized stability margin is greater than the i^ -gap metric. But it says nothing when the v-g&p value is greater than the generalized stability margin. Under this circumstance, as proposed in the algorithm presented in §4.4, Theorem 4.2.1 or Theorem 4.2.2 needs to be employed to exploit the frequency-by-frequency property of the z^ -gap metric. In this case, if the z^ -gap at a certain frequency is greater than the generalized stability margin at that particular frequency, then the instability condition at that frequency can be pinpointed. For a slowly time-varying system, the low frequency region is often of major interest. In what follows, assuming that the nonlinear plant is scheduled according to Table 5.2: Table 5.2: Scheduling space of the fictitious plant from time t = 0 to t = 30 s. Time (s) 0 < t < 10 10 < t < 15 15 < t < 20 20 < t < 25 t > 25 Grid point, k -0.155 -0.1 -0.05 1 x 10~5 1 x 10~3 Figure 5.22 shows the z/-gap metric of P(-0.l55,s)W1CooW2 and P(-0.l,s)W1CooW2 versus the associated generalized stability margin with respect to [P(—0.155, s)WiC0OW2, I}. Since the generalized stability margin is greater than the z^ -gap over the entire fre-quency range, the resulting closed-loop remains stable when the scheduling parameter 111 C H A P T E R 5. 5.3.2 Nonlinearity Measure - 5 -4 - 3 - 2 - 1 0 1 2 Selected nominal at i-th operating point, y.(t) Figure 5.21: Shaped z^ -gap contour. Nominal model (black dot). 112 U H A P T E R b. 5.3.3 Simulation Results is evolved from k = —0.155 to —0.1. Similarly, Figure 5.23 indicates that the corre-sponding closed-loop is stable when the scheduling parameter k is scheduled from —0.1 to —0.05. However, when the nonlinear plant is experiencing a sign change (i.e. from k — —0.05 to 1 x 10~5), the z^ -gap values are greater than the generalized stability margin around the low frequency region, see Figure 5.24. This indicates the onset of instability. Finally, as can be seen from Figure 5.25, the closed-loop remains unstable when the magnitude of the sign change is further increased (i.e. from k — 1 x 10~5 to 1 x 10 - 3). Note, by assuming that the time variation effect of the nonlinear plant over the aforementioned scheduling subspaces is negligible, the z^ -gap metric is equivalent to the developed closed-loop nonlinearity measure. Therefore, one can say that the designed linear controller fails to cope with the sign change dynamics of this fictitious plant. To resolve this, either the linear controller needs to be redesigned or a nonlinear control strategy is needed. Time domain simulation results of the above analysis is presented in the next section. 5.3.3 Simulation Results In the previous section, a number of theoretical analyses were done to assess the closed-loop stability of the fictitious nonlinear plant when the scheduling parameter is evolved in the sign change region (i.e. [-0.155 1 x 103] from t = 0 to t = 30 s), see Table 5.2. In this section, the corresponding time domain simulation is presented to illustrate the ability of the developed closed-loop nonlinearity measure in anticipating the onset of closed-loop instability and also the insufficiency of a linear controller. The setpoint is initially zero and is changed to 1 at t = 5 s. Evidently, as can be seen from Figure 5.26, the closed-loop remains stable and tracks the setpoint nicely up to t — 20 s, regardless of the fact that there are a couple of model switching at t — 10 s and £ = 15 s. At £ = 20 s, there is a model switching from P(—0.05, s) to P ( l x 10~5, s). Even though the process output seems to be stable from £ = 20 s to t = 25 s, but a close examination on the controller output reveals that the closed-loop is at the brink of instability (i.e. the controller output is unbounded). 113 C H A P T E R 5. 5.3.3 Simulation Results Frequency Figure 5.22: The u-gap metric of P(-0.155, s)W1COQW2 and P(-0.1, s)W1C00W2 (solid) versus the generalized stability margin with respect to [P(—0.155, s)WiC00W2, I) (dashed-dotted) over a frequency range. 114 C H A P T E R 5. 5.3.3 Simulation Results Frequency Figure 5.23: The v-gap metric of P(-0A,s)W1CooW2 and P(-0.05,s)W1CooW2 (solid) versus the generalized stability margin with respect to [P(—0.1, s)WiC00W2, I] (dashed-dotted) over a frequency range. 115 C H A P T E R 5. 5.3.3 Simulation Results Frequency Figure 5.24: The u-gap metric of P(-0.05, s)W1C0OW2 and P ( l x 1CT5, s)WLCOQW2 (solid) versus the generalized stability margin with respect to [P(—0.05, s) W\ COQWI, I] (dashed-dotted) over a frequency range. 116 C H A P T E R 5. 5.3.3 Simulation Results Frequency Figure 5.25: The v-gap metric of P ( l x 1CT5, s)W1COQW2 and P ( l x 10~3, s)WiCOQW2 (solid) versus the generalized stability margin with respect to [P(l x 1CT5, s)WiCOQW2,1] (dashed-dotted) over a frequency range. 117 C H A P T E R 5. 5.3.3 Simulation Results When the model is switched again at t = 25 s towarding a more positive gain k (i.e. 1 x IO - 5 to 1 x IO - 3), the unstable response of the closed-loop system becomes prominent. Unequivocally, the developed closed-loop nonlinearity measure when applied to slowly time varying systems, whose time variation effect is negligible, gives an interesting result. It not only indicates the insufficiency of a linear controller when implementing to a nonlinear process, but also provides an accurate anticipation of the onset of closed-loop instability. 5 4 £ 3 CO fx 0 -1 -2 500 -0 -3 Q. -500 -O "o -1000 -c o O -1500 --2000 -i i 1 -onset of instability V / 1 . I L 10 15 20 25 I 1 1 1 v | 1 • -1 1 — 1 t 1 30 10 15 Time (s) 20 25 30 Figure 5.26: Time domain closed-loop simulation for the fictitious nonlinear plant. Top: Plant output, y(t) (solid), setpoint (dashed-dotted). Bottom: Controller output u(t) (solid). Note, the model switching sequences are given in Table 5.2. 118 C H A P T E R 5. 5.4. Summary 5.4 Summary The developed nonlinearity measure is obviously a reliable tool for the assessment of closed-loop nonlinearity. Simulation results confirm the observation claimed in Chapter 4 including the linearizing effect of feedback and the reduction of the z/-gap values after appropriately pre- and post-compensated. The proposed computational algorithm also proved to be easy to use and provided better insight into the nature of the system's closed-loop nonlinearity. 119 C H A P T E R 6 Conclusions This chapter highlights the contributions of this thesis and suggests some future directions to improve on the proposed nonlinearity measure. A closed-loop nonlinearity measure exploiting the i^ -gap metric, the special structure of a quasi-LPV transformation and the J ^ , loop shaping design procedure is developed. In this approach, a nonlinear plant is first recast into a quasi-LPV form via a state transformation. Then a set of linear models is obtained by freezing the scheduling parameter(s). The z^ -gap metric is then used to select an appropriate nominal model. Subsequently, the McFarlane and Glover loop shaping design procedure is employed to design a linear controller. Resulting from the developed measure are a generalized stability margin and the largest radius induced by closed-loop nonlinearity and the time variation effect in the sense of the z^ -gap metric. If the radius is less than the stability margin, then the designed linear controller is sufficient. Otherwise, a nonlinear control strategy might need to be considered. The major contributions of this thesis and some future directions for future research are outlined below. 6.1 Contributions • Linear or nonlinear control? A decision making tool: All chemical processes are inherently nonlinear. This does not mean that a nonlin-ear control is mandatory. In fact, feedback control is known to modify open-loop 120 C H A P T E R 6. 6.1. Con tri b u tions nonlinearity. Therefore, a reliable tool is required to achieve this. In such a regard, the developed closed-loop nonlinearity measure acts as an effective decision making tool for the control engineers when they are faced with the problem of deciding whether to stick to a linear control strategy or use a nonlinear control approach in solving their daily control problems. • A new approach to quantify closed-loop nonlinearity: Existing nonlinearity measures based on norm-bounded distance between a non-linear plant and its linear counterpart are mainly focused on the deviation of plant output only. The underlying assumption of these approaches is that the nonlinear plant and the linear model are subject to the same input sequences. However, practically, this assumption is very restrictive especially when the loops are closed. Even though both closed-loops are implementing the same linear controller, the resulting nonlinear plant's (respectively, model's) input-output signals can be very difference from those of the model (respectively, nonlinear plant) due to the pres-ence of nonlinearity. A more natural and less restrictive way is to quantify the different of both input and output signals between the nonlinear plant and the linear model. This novel way of quantifying closed-loop nonlinearity is parallel with the v-g&p metric notion. • A pictorial approach to explain the gap metric notion: In general, the gap metric notion seems to be a framework developed by the experts for the experts. This misconception often arises due to the fact that the level of mathematics involved beyond that offered by the typical first course in control and can be esoteric to some engineers. In contrast, this thesis presents a novel and easy-to-understand perspective of the gap metrics. To achieve this, a pictorial approach is proposed. This new approach of presenting the gap metric and the f^ -gap metric notions is important since it not only gives better insight and understanding to the gap metric framework, but also provides a graphical perspective to various robust stability results arising from the gap metric notion. In addition, the distinction between the gap metric and the i^ -gap metric is clearly illustrated by carefully 121 C H A P T E R 6. 6.2. Recommendations explaining the concept of homotopic relationship of the operator graphs and the causality of the J^f2 and the J$?2 spaces. • loop shaping weight selection to mitigate closed-loop nonlinearity: It was shown in Chapter 4 that the mitigation of closed-loop nonlinearity is closely related to the sensitivity function of the linearized plant. The smaller the sensitiv-ity function the more linear the closed-loop. Of course, in practice, it is impossible to keep the sensitivity function small over the entire frequency range. Since the low frequency range is of main interest for most chemical processes, an obvious choice is to push the bandwidth up to the limit defined by actuators or other process performance limitations such as the nonminimum phase zeros. • A novel computational algorithm for the proposed nonlinearity measure: A novel computational algorithm is developed to quantify closed-loop nonlinearity. This algorithm allows one to find the best nominal model that guarantees the smallest radius of the uncertainty ball induced by closed-loop nonlinearity. Using the developed algorithm, the assessment of the closed-loop nonlinearity is simple and easy to implement as treated in a number of relevant design examples. 6.2 Recommendations • A n extension to the winding number condition: In the developed nonlinearity measure, the winding number condition needs to be checked for each combination in the set of linear models obtained by freezing the scheduling parameter(s). It is known that in the realm of operator theory the winding number condition has a close relation with the Fredholm operator index. Recently, Jonckheere (1997) showed that a single index can be established for a family of Fredholm operators by exploiting K-theory (Karoubi, 1978; Wegge-Olsen, 1993). In this light, it is believed that a single index can be formulated to test the homotopy condition needed to evaluate the i^ -gap metric for quasi-LPV systems. This will greatly reduce the computational load of the developed measure. 122 C H A P T E R 6. 6.2. Recommendations • A fast computational algorithm for the z^-gap metric The existing computational algorithm of the z^ -gap metric (Date, 2000) involves de-termining the normalized coprime factorizations (NCF). This includes solving two Algebraic Ricatti Equations (i.e. one to design a stabilizing controller and another to determine the observer), of the system and check for the satisfaction of the ho-motopy condition. Unequivocally, the computation load of the existing algorithm is intensive. Since the z/-gap metric has a nice frequency domain interpretation, it is anticipated that an accurate estimation of the z^ -gap can be obtained directly from the process input-output data. • A n extension to the nonlinearity measure to cope with local disconti-nuity (or nonlipschitz) functions: The developed technique assumes that the nonlinear equations describing the pro-cess dynamics are differentiable with respect to the scheduling parameter in order to transform the nonlinear system into a quasi-LPV representation. For systems exhibit local discontinuity (such as hysteresis and saturation), a set of the inte-gral quadratic constraints (IQCs) can be embedded to the developed nonlinearity measure. A possible way to achieve this is to embed the aforementioned nonlips-chitz nonlinearity using a set of IQCs and carry out the rest of the procedures as discussed in this thesis. 123 d distance function 38 Banach spaces C set of complex numbers @ domain T homotopy j£?2 Hilbert spaces DC vector field, either C or E . J^2 inner product spaces or J'I spaces £ bounded linear operator n sample size IR set of real numbers 1Z range "V vector space Z + set of positive integers Mathematical symbols 3 there exist E[-] expectation 124 N o m e n c l a t u r e NOMENCLATURE NOMENCLATURE V for all elements in a set if and only if A i i-th largest eigenvalue of a matrix. —> maps to x cartesian product of two sets 9 such that G is an element of $ is not belong to f l intersection of two set C is a subset of U union of two sets inf infimum of a set sup supremum of a set Greek letters it residual at time t T a standard Nyquist contour T(-) The Gamma distribution function <f> null set Abbreviation AR autoregressive model ARX autoregressive with exogenous input model 125 NOMENCLATURE NOMENCLATURE R H P right half plane RSSQ, RSS residual sum of squares 126 Bib l iography Allgower, F. (1995). Definition and computation of a nonlinearity measure. 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Prentice-Hall, New Jersey. 131 APPENDIX A Mathematical Preliminaries In this chapter terms and basic results from functional analysis, operator the-ory, signal and systems, feedback control theory and complex analysis are presented. These results are collected to facilitate understanding of the materials presented i n this thesis. Notations used in this thesis are also defined. Prerequisites are knowledge of linear algebra and a first course in control theory. A . l Functional Analysis This section presents some basic elements that form the cornerstone of modern control theory (i.e. signal & systems, feedback control theory, and etc.). Notations used in this thesis are also defined in this section. These materials are collected without proof. For further details, the reader is referred to (Oden, 1979; Ramkrishna and Amundson, 1985; DeVito, 1990; Rudin, 1991; Lang, 1997). A.1.1 Sets A set SC is a collection of objects that share a common property. The notation x 6 S£ means that x is an element of SC. Whereas, y £ 3£ means "y does not belong to SC\ Denoted by f>, a set that contains no elements at all is called an empty or null set. If 3C\ and S£2 are two sets, and if every element of 5£2 is an element of SC\, then SCi is a subset of SC\ or 3t2 C SC\. S£\ and 3Ci are said to be equal, denoted 3C\ = 3C2, if and 132 A P P E N D I X A . A. 1.2 Mappings only if «£ i C <2T2 and SC2 C Denoted by «# i U «2T2 = : x G # i or x € ^ 2 } , the union of JTi and ^ is the set of elements which lie in SC\ and 3C2- The intersection of and 5£2, or 3£iC\ 3£2 = {x x E 5£\ and a; G ^ 2 } , is the set of elements which lie in both f%\ and SC2. The complement of a set «52Ti (with respect to some universal set U) is the set of elements, denoted S£{ = [x : x G U; x £ •%].}, which do not belong to 3£. The Cartesian product of SC\ and 3£2, or 3C\ x i2T2 = { [ « ] ' • ^ G ^ i ; y G 3£2}, is the set made up of all ordered pairs (x,y). Note that two ordered pairs are equal if their respective components are equal, i.e. (x, y) = (a, b) 4 = ^ x — a and y = 6 Also, in general, X ^ 2 7^ X A . 1.2 Mappings Denoted by / : JTi —» a mapping (or function) from set to <5T2 is an association of each element of to a single element of Where SC\ is called the domain of / or @(f), and y G i£~2 is called the image of x G J2TI . The range of the function / (i.e. 7£(/) = {f(x) '• x G SC\}) is the set of elements in <^T2 that are images of elements in 8£\. The graph of / , denoted by Qf, is defined as Of = {{y=f(x)}-x e ^1} A function / : S£\ —> ^ 2 is surjective or onio if and only if every y G S£2 is the image of some element of SC\ and it is said to be infective or one-to-one if for every y G rZ(f), there is exactly one x G such that y = f(x). The above mapping is called bijective or one-to-one and onto if it is both injective and surjective. For / : SC\ —> SC2 and let C S^i, the restriction of / to S£* is defined as / | ^ r « : iST/ —> ^ 2 . The inverse 133 A P P E N D I X A . A.1.3 Bounded Sets mapping of / , denoted maps elements of f(SC\) onto SE\ and if / is bijective, then f~l is a well-defined mapping on the whole of S£2-A.1.3 Bounded Sets For a given set SE and a mapping / : SE —> srf C R (real number),'srf is said to be bounded from above if there exists (3) a real number a such that (9) a < a for all (V) a € srf. In this case, a is said to be the upper bound of srf. Analogously, s& is bounded from below if 3/? € IR s.t. (3 <, a Va £ /? is said to be the /ower bound of If ^  is bounded from above and below, then srf is said to be bounded . Let srf be a nonempty set that is bounded from above, and let U denote the set of all upper bounds of srf. The minimum of U is the least upper bound or supremum of srf and is denoted by sup $0. Likewise, if srf is nonempty and bounded from below, and let L denote the set of all lower bounds of srf. The maximum of L is the greatest lower bound or infimum of srf and is denoted by inf srf. A . 1.4 Metric Space and Completeness In this subsection, the notion of metric and metric space are first defined and subse-quently the concept of completeness of a metric space is presented. A.1.4.1 Metr ic and Metr ic Space A metric space is a set SE which is equipped with a metric or distance function d : SE x M —> R satisfying the following conditions: 1. d(x,y) > 0 Vx,y G SE. (Nonnegativity) 2. d(x, y) = 0 if and only if x = y. (Nondegeneracy) 3. d(x,y) = d(y,x) Vx,y e SE. (Symmetry) 4. d(x, y) < d(x, z) + d(z, y) \/x, y,z £ SE. (Triangle inequality) 134 A P P E N D I X A . A.1.5 Open Set and Topology A.1.4.2 Completeness To define completeness, we first need the following definition. A sequence { x n } i ° is a Cauchy sequence if lim d(xm, xn) = 0 (i.e. for any e > 0, 3 an m,n—KX> N 3 d(xm,xn) < e V m,n > N). Hence completeness means: Definition A . 1.1 (Completeness). A metric space (SE ,d) is complete if every Cauchy sequence in SE converges. A.1.5 Open Set and Topology In this section, the notion of neighborhood is introduced followed by the definition of an open set, an important element in topology. The topology is then constructed using the open set. Definition A.1.2 (e-neighborhood). Let (SE,d) be a metric space and a real scalar e > 0. An e-neighborhood about XQ € SE is defined as the set &£(x0, e) = {x e SE : d(x, x0) < e}. (A.l . l) The above definition implies that a neighborhood can be seen as an open ball with an arbitrary radius e centered around any element in SE. Definition A.1.3 (Interior point). Let (SE,d) be a metric space and srf C SE. XQ E srf is said to be an interior point of srf if exists e > 0 3 SSe(xQ,e) C srf'. This definition means that an interior point of a set is one which has a sufficiently small neighborhood that is entirely contained in the set. Having introduced the interior point, the open set is defined as follows: Definition A.1.4 (Open set). A set srf C SE of a metric space (SE,d) is said to be an open set if every element x G srf is an interior point. 135 A P P E N D I X A. A.1.5 Open Set and Topology Definition A . 1.5 (Topological space). A topological space is a pair ( $ T , £f) consist-ing of a set S£ and a set 5? of subsets of S£ (called open set), such that the following axioms hold: 1. Any union of open sets is open. 2. The intersection of any two open sets is open. 3. <p and are open. 5? is called the topology of the topological space (SP, 3?). Obviously, the set 3£ = {1,2,3,4} together with ST = {0, {1}, {1, 2}, {1, 2, 4}, «£"}, a collection of subsets of ^T, is a topological space since any unions or intersections of these subsets belongs to 3T. However, the collection of subsets <§ — {<f>, {1}, {1, 2}, {2, 3, 4}} is not a topological space since {1,2} fl {2,3,4} = {2} ^ S. It is obvios at this point that a given set may give rise to several distinct topological space. For example, given a set 3£ = {1, 2}, there exists four distinct topologies on it, namely 3TX = {0, X}, 3T2 = {</>, {1}, XT}, ST, = {0, {2}, JT} and ^  = {</>, {1}, {2}, SC). This implies that there are four associated topological spaces (3^, Sty, [3£, Sty and («3T, Sty. In this example, any metric d defined on the set 3£ must satisfy the four conditions in §A.1.4.1. Assuming that d(l,2) = d(2,1) = s > 0, the neighborhoods around 1 and 2 are given by <^e(l, §) and 3 § £ ( 2 , §). These neighborhoods generate the two subsets consisting of the single elements 1 and 2 respectively. Together with <f> and 2£, an associated topology is given by the set SK^ = {</>, {1}, {2}, S£). In this case, the corresponding topological space (3£, Sty is said to be metrizable, while the other three, are not. Therefore, a topological space is said to be metrizable when it can be associated with a metric space. A topology is said to be the weakest topology if it contains the fewer open sets such that the continuity property is preserved. 136 APPENDIX A . A.1.6 Normed Vector Space The idea of homotopy plays an important role in topology. Given two topological spaces 5€ and <3f, two continuous functions / and g that map SE to & . Homotopy T means / (respectively, g) can be continuously deformed into g (respectively, / ) . Mathematically, this means: F : [0,1] x SE -+ ^ with ^(0, p) = f(p) G p) = g(p) pe SE (A.1.2) To visualize this, consider the left hand side of Figure A . l where an elastic robe is losely encircling a pole twice (i.e. topological space SE). Assume that the pole has infinite length. Next, a series of actions (represented by J7) are performed on the robe in order to change its shape (topological space '&"). The actions at t = 0 and t — 1 are denoted by / and g, respectively. It is obvious that albeit the shapes of the robe are different from t = 1, t = tk up to t = 1, the number of encirclement of the robe around the pole remains the same. This means that no extra effort has been made to cut and glue the robe in order to change the number of encirclements. The actions of cut and glue can be seen as a discontinuity. Therefore, the actions from t = 0 to t = 1 can be continuously changing from one into another. In this case, T is called a homotopy of / and g. Topological objects that have the homotopy property are called homotopic equivalent. Another interesting observation is that the number of encirclement is clearly invariant and can be used as a homotopy condition. For two homotopically equivalent topological spaces, a winding number, see §A.7, or index, can be exploited as a homotopy condition. A . 1.6 Normed Vector Space A vector space over IK (let K stand for either C or K ) is a set V, whose elements are called vectors, and in which the following two algebraic properties (i.e. vector addition and scalar multiplication) are defined: 1. For every u,v,w G Y there corresponds a vector v + u = u + v and u + (v + w) = (u + v) + w; Y contains a zero vector 0 such that v + 0 = v for every v € Y\ and to each v G Y corresponds a unique vector — v such that v + (—v) = 0. 137 A P P E N D I X A . A. 1.6 Normed Vector Space Figure A . l : Homotopy of / : SE -»• <& and g : SE ^ 2. For every pair (ei,t>) with ei € K and v £ Y, there is a corresponding vector t\v, in such a way that lv = v, ei(e2v) = €it2v (where e2 £ K), and such that C\(v + u) — c\V + c.\U and (ex + e2)v — exv + e2v hold. A set W £ Y is called a subspace of y if it remains in Y under the vector addition and scalar multiplication operations defined on Y. A vector space Y is said to be a normed space if for every v £ Y there is an associated nonnegative real number \\v\\, called the norm of v, such that the following conditions hold: 1. ||v|| > 0 if v ^ 0 and ||v|| = 0 v = 0, 2. ||et;|| = |e|||u|| if v £ Y and e £ Kl is a scalar, 3. \\v + u < v\ + \\u\\ Vu.t i £ Y. Obviously, a normed vector space is also a metric space, and induces a norm topology, which is metrizable. Denoted by B, a Banach space is a complete normed space. A 138 APPENDIX A. A.2. Basic Operator Theory special class of Banach spaces are the Hilbert spaces, denoted by Jz? 2 , whose norms are induced by an inner product. An inner product on a vector space Y over K is a mapping defined by {-,•): Y xY —> K such that for all u,v,w £ V and each e £ IK the following axioms hold. 1. (u + v, w) = (u, w) + (v, w), 2. (u,v) = (v,u), where (-, •) denotes the complex conjugate, 3. (eu, v) = e(u, v), 4. (v,v) > 0 if v ^ 0. Clearly, an inner product induces the following norm on Y: \\v\\ = yj(v,v). Vector spaces associated with an inner product are called inner product spaces, denoted by J^2. Thus, a complete inner product space is a Hilbert space. Denoted by the orthogonal complement of ^ C J5f2 is given by J£2 Q% — {he 3£2 '• (u, h) = 0 Vu £ For any elements u in the closed Hilbert subspaces there exists a unique decomposition, such that h = ui + u2 where U i G ^ and u2 £ The associated projections (HV and n^x) of this decomposition are called orthogonal projections, and are defined as follows: In addition, ^ and *%L are said to induce a coordinatization if ^ fl <&'-L = {0}. A .2 Basic Operator Theory Some important classes of linear operators are introduced in this section. Throughout this section, all vector spaces are assumed to be Hilbert spaces unless stated explicitly. Let's begin with what is meant by a linear operator: 139 A P P E N D I X A. A.2. Basic Operator Theory Definition A.2.1 (Linear operator). Given two vector spaces "V\ and %, a function P which maps V\ into % is called a linear operator if for all x, y £ ~fi and a £ C, the following two properties are satisfied: 1. P{x + y) = P{x) + P{y), 2. P(ax) = aP(x) Definition A.2.2 (Bounded linear operator). The linear operator P : 7^ —> Y2 is called bounded if nr . 11 „ „ „ llPzll | |P|| = S U p ||Px|| = S U p ||Fx|| = S U p ——— < O O . ||z||<l ||z||=l x#0 where \\ • \\ denotes the norm of P. The set of bounded linear operators which map "V\ to "V2 is denoted by H{fV\,'V2). In the above, it is assumed that the domain of the linear operator is the whole Jzf2 space. The domain @p for a linear (possibly ubounded) operator P : ffip C Yi —> % is defined as the subset of the total input space (e.g. Jzf2 space) that has its images in bounded output space (e.g. Jzf2 or a Banach space). Mathematically, this means: *2>P = {u £ Y\ : Pu £ f2} The range of P is Tip = {Pu : u £ 3>P}. The kernel of P, denoted by ker(P), is defined as follows: ker(P) = {Pu — 0 : u £ &>p} The graph of an operator P is defined as: QP = | ,i\®pcyl®y2 where denotes the identity operator on P 140 A P P E N D I X A. A.3. Signals and Systems Some special types of operators include • self-adjoint operator: P* = P, where * denotes complex conjugate transpose. • normal operator: P*P = PP*. • projection operator: P2 = P (idempotent) and P* = P (self-adjoint). • unitary operator: P*P = PP* = I. • isometry operator: P*P = I. A . 3 Signals and Systems Normed vector space provides a convenient framework for system analysis. Under this framework, signals can be seen as a vector space corresponding to a mapping of time interval (either discrete or continuous) to a normed vector space. If the signals are finite energy, the resulting normed vector space is obviously a Hilbert space. The inner product on continuous time signal spaces 5£<i{—oo, co) is defined as Note that ^{—oo, oo) can be decomposed into two unique parts (i.e. Jzf2~ and Jzf2+) with respect to time axis (—oo,0] and [0, oo). Signals with zero values for all negative time are called causal signals, while those with zero values for all positive time are anticausal. Noncausal signals are signals that have nonzero values in both positive and negative time. The associated inner products are Alternatively, in frequency domain, the inner product of ££i(— joo, joo) is given by 141 A P P E N D I X A. A.3. Signals and Systems and hence the norm | | / | | 2 is defined as | | / | | 2 = y ( / , / ) , where * and the hat denote complex conjugate transpose and the Fourier1 transform of the signals, respectively. Since the Fourier transform is a Hilbert space isomorphism2 from J z ? 2 ( — 0 0 , 0 0 ) onto -S?2(— joo, joo), it maps J z f 2 + onto M2 and Jz? 2~ onto M?A- (Francis, 1987). Note that Jtfp, p = 1 , 2 , . . . , 0 0 , represent the Hardy spaces3, which consist of all complex-valued functions f(s) of a complex variable s = C + ju> that are analytic4 and bounded in the open right half plane (RHP). Clearly, is a (closed) subspace of Jz? 2(—joo, joo). The corresponding norm is defined as: l l / l l l = s u p | ^ J°° fm(C + ju>)f(C + Ju) ^ } (A.3.1) where tilde denotes the (bilateral) Laplace5 transform of the signals f(i). By invoking the Maximum Modulus Theorem6, it is easy to see that the norm for can be rewritten as: 1 f°° ~ 1 1 / 1 , 2 = W rtiu)fUu)du- (A.3.2) Remark A . 3 . 1 . Throughout this thesis, whenever there is no confusion, we do not explicitly distinguish between the notation of functions in the time domain and in the frequency domain. For instance, we use <Jt\ and «5f2+ interchangeably to denote signals that are defined on positive axis and zeros otherwise. A system P is an operator mapping between signal spaces. In this thesis, Jzfoo denote the Banach space of matrix-valued functions defined on jui : u> G E f l 0 0 . The Jzfoo norm is given by | |P|| = ess sup o(P(jcu)). weMnoo P is said to be stable if it maps finite energy inputs onto finite energy outputs. The *Jean Baptiste Joseph Fourier, 1768-1830, a French mathematician. 2 A n isomorphism is a bijective homomorphism. A homomorphism of a group & into a group CS' is a mapping <p : & -> 9 <p(ab) = <p(a)ip(b) V a,b£&, see (Grillet, 1999, pg. 13). 3 Named after a famous English mathematician Godfrey Harold Hardy, 1877-1947. 4/(s) is said to be analytic at a point ZQ e S C C if it is differentiable at ZQ and also at each point in some neighborhood of ZQ. 5Pierre-Simon Laplace, 1749-1827, a French mathematician. 6See (Zhou et al., 1996, pg. 97, Theorem 4.3). 142 A P P E N D I X A. A.3. Signals and Systems Hardy space C Jzfoo is the space of transfer functions of stable linear, time-invariant, continuous time, systems. Since we are dealing with rational transfer functions, denoted by ^Jftfoo C J$?oo, the norm (or the system gain) is IPI .{P^h -CDC \ \ o^ue.m IPII2 where o denotes the maximum singular value. The singular values of a matrix Amxn are the positive square roots of the p = min(m, n) largest eigenvalues Aj of both A*A and AA* (i.e. ^ = y/\{A*A) = 'y/\(AA*)), and can be obtained by employing singular value decomposition (SVD) as shown in the following theorem: Theorem A .3 . 1 . Let A e K M X N . There exist unitary matrices7 U = V = Ui,U2, ...,ur, v1,v2, ...,vn e Kmxm e Knxn such that where A = UEV\ E = 0 0 and o\ > a2 > • • • > av > 0, p = min m, n. Si 0 0 0 o~ \ 0 ••• 0 0 a2 ••• 0 Proof. See (Zhou et al., 1996, pg. 32, Theorem 2.11). • Note that cr(A) = o~\ and a (A) = ov represent the maximum singular value and the minimum singular values, respectively. The condition number of a matrix A is defined 7See unitary operator in §A.2, pg. 141. 143 A P P E N D I X A. A.4. Feedback Control Theory as: ^>4S <a>^ The condition number has a close relationship with matrix invertibility. A matrix with a large condition number is said to be ill-conditioned. The resulting matrix inversion of an ill-conditioned matrix can be misleading. Also, from a control perspective, a large condition number may signifies control problems. As pointed out by Skogestad and Postlethwaite (1996, pg. 87), a large condition number for a given plant G maybe be due to small a(G), which may indicate poor controllability. Typically, if a(G) < 1, poor control performances are expected. a(G) is also called the Morari Resilience Index, see (Morari, 1983). A.4 Feedback Control Theory In practice, feedback control is primarily employed to handle uncertainty. The focal point of this section is to discuss the properties of feedback control and some definitions arising from the feedback control theory. Figure A.2 shows a standard feedback configuration. \di r •+• e C u P d y + + + A' n Figure A.2: A standard feedback configuration For a multivariable system, PC and CP do not commute with each other. Therefore, the input loop transfer matrix Li and the output loop transfer matrix L0 are defined as follows: Li = CP, L0 = PC, 144 A P P E N D I X A . A.4. Feedback Control Theory respectively. The associated input and output sensitivity matrices are Si = (I + L i ) ' 1 : dt -»• up, S 0 = (I + L o ) - 1 : d ^ y . Since S + T = I, the corresponding input and output complementary sensitivity are given by Ti = L i { I + L i ) - 1 , T0 = L 0 ( I + L o Y 1 , respectively. Figure A.2 together with the above definitions of senstivity and complementary sensi-tivity matrices, the following equations can be easily obtained. y = T0(r - n) + S 0 P d i + S0d (AA.l) r - y = S 0 ( r - d ) + T 0 n - S 0 P d i (A.4.2) u = CS0(r-n)-CS0d-Tidi (A.4.3) up = CS0(r - n) - CS0d + Sid{ (AAA) These equations show some interesting properties of a feedback system that can be exploited to achieve good disturbance rejection and good closed-loop robustness. For instance, to alleviate the effects of disturbance d, Eq.(A.4.1) suggests that this can be achieved by making the output sensitivity function S 0 small. Likewise, from Eq.(A.4.4), one would try to keep Si small in order to mitigate the effects of disturbance di on the plant input. Typically, for servo control, one would like to track the setpoint as close as possible. One way to achieve this, as suggested by Eq.(A.4.2), is to keep r — y small. This implies that one would try to make both S 0 and T0 small. However, as discussed earlier, this contradicts with the closed-loop limitation given by S 0 + T0 = I. Fortunately, n is the measurement noise that is normally dominant in high frequency range, while the disturbance d normally affects system's behavior in the low frequency range. The idea of carefully shaping the loop transfer matrix in a different frequency range is the central idea of the classical frequency domain controller design. The loop 145 A P P E N D I X A . A. 5. Coprime Factorization shaping notion (Doyle et al., 1992) and the mordern J^o loop shaping design procedure (McFarlane and Glover, 1990, 1992) are natural extensions. The notion of smallness presented above can be achieved by using singular values. To gain a full benefit from the loop shaping method, the concept of bandwidth plays an important role. In general, a large bandwidth corresponds to a faster rise time since high frequency signals are more easily passed on to the outputs. In contrast, if the bandwidth is small, the time response will generally be slow, and the system is usually more robust. Bandwidth in feedback control can also be interpreted as the frequency range over which control is effective (Skogestad and Postlethwaite, 1996). Definition A.4.1 (Closed-loop bandwidth). The closed-loop bandwidth, UB is the frequency where a(S^) (or respectively a(Sn)) first crosses -j= = 0.7071 ~ —3dB from below, where H denotes either i (input) or o (output). Note for the purpose of loop shaping, the gain crossover frequency uic, which is defined as the frequency where o(L^(juc)) first crosses 1 from above. A.5 Coprime Factorization Any stable or unstable plant P € oo c a n be expressed as a quotient of two coprime S&M'oo operators (i.e. P = NM'1 = M~lN). The coprimeness implies that the two quotient operators do not contain common RHP zeros, hence no unstable mode pole-zero cancellation. In what follows, the concept of right and left coprime factorization are defined. Definition A.5.1 (Right coprimeness). Two matrices M and N in&M'00 are right-coprime if they have equal number of columns and there exist matrices X, Y 6 &J$?oo such that X Y M N =XM+YN=I (A.5.1) 146 A P P E N D I X A. A.5. Coprime Factorization Note that it is equivalent to saying that [$] is left-invertible in SftJiV^. Eq.(A.5.1) is also called Bezout identity. Definition A.5.2 (Left coprimeness). Two matrices M and N in ^Jf^ are left-co-prime if they have equal number of rows and there exist matrices X, Y € ^Jif^ such that M N X Y = MX + NY = 1 (A.5.2) Similarly, ones can say that [M fr] is right-invertible in 3?^^. Recall that P 6 @3foo, and P = NM'1 and P = M'lN denote a right-coprime and a left-coprime factorizations of P, respectively. The following lemma gives the doubly-coprime factorization of P. Lemma A.5.1. For each proper real-rational matrix P there exist eight MJi? ^-matrices satisfying the equations P = NM'1 = M^N (A.5.3) X -Y M Y -N M N X Definition A.5.3 (Normalized coprime factorization). A coprime factorization is called normalized coprime factorization iff M~M + N~N = I or MM~ + NN~ = / (A.5.5) 147 A P P E N D I X A. A.6. Quasi-Linear Parameter Varying Systems A.6 Quasi-Linear Parameter Varying Systems The application of LTI systems is prevalent in system analysis and controller design. Unfortunately, owing to the time invariant assumption, systems with rapid time variation might not be well represented by switching between a set of LTI models or by gain scheduling. If the time variations are known a priori, bounds on its magnitude and rate of change lead to linear time varying (LTV) systems. If the coefficients of a linear system are known to depend on an exogenous time-varying parameter, a linear parameter varying (LPV) can be formulated. In general, a state space realization of an LPV system takes the following form: where a is the time-varying parameter. In the above equations, A(-), B(-), C(-) and £>(•) are state space matrices. Often a is unknown a priori but can be estimated or measured during the system operation. Exploiting the special structure of an LPV system, a gain scheduling controller can be devised by freezing the time-varying parameter. As reported in Shamma and Athans (1990), this approach has guaranteed robustness and performance properties provided the parameter time variations are sufficiently slow. To accommodate fast time variations, a quasi-linear parameter varying or quasi-LPV system was introduced by Shamma and Cloutier (1993). Historically, the quasi-LPV transformation was first introduced to obtain a set of linear models that can describe missile dynamics, particularly that of missile endgame (i.e. the moment before the missile hits the target). This often involves large and rapid time-varying acceleration and also fast changing angle of attack. Linear models obtained from linearization about a single operating point cause deterioration of the autopilot performance, see (Shamma and Cloutier, 1993). An elegant approach to address the above issues is to convert the nonlinear system into a quasi-LPV system via a state transformation. The sole assumption of this conversion is that the system's nonlinearity dx ~dt A(a)x + B(a)u (A.6.1) y = C(a)x + D(a)u (A.6.2) 148 A P P E N D I X A. A.6. Quasi-Linear Parameter Varying Systems is captured by measurable state variables. Obviously, by doing so, a more accurate model that not only captures system's nonlinearity, but also easily reduces to a set of LTI models by merely freezing the scheduling parameter can be obtained. Since this thesis depends heavily on the quasi-LPV representation, the following presen-tation follows a step-by-step transformation of a given nonlinear plant, whose nonlinear dynamics are captured by state variables, to a quasi-LPV representation. Let a and z denote the scheduling state and unscheduling state, respectively. Assuming that a is measurable, the following equation represents a state dependent nonlinear system. d dt = 4>(a) + A(a) + B(a)u (A.6.3) Assume that there exist differentiable functions zeq and ueq, such that for every a, + B(a)ueq(a) (A.6.4) 0 4>i ( « ) = + 0 (f>2 (a) An(a) A12(a) A21{a) A22{a) a In the above, zeq(a) and ueq(a) denote a family of equilibrium points obtained by setting the derivatives in Eq.(A.6.3) to zero. Next, by subtracting Eq.(A.6.4) from Eq.(A.6.3), d_ dt a 0 A12(a) a z 0 A22{a) z - zeq{a) + B(a){u - ueq(a)) (A.6.5) Since dtZ^ = dzeq(a) da da dt' (A.6.6) substituting ^ from Eq.(A.6.5) into Eq.(A.6.6), yields dzeq{a) dzeq{a) dt da {A12(a)(z - zeq(a)) + Bx(a){u - ueq(a))} (A.6.7) 149 A P P E N D I X A . A. 6. . Quasi-Linear Parameter Varying Systems Next, substracts Eq.(A.6.7) from § in Eq.(A.6.5), dz dzeq(a) ~E dt A22(a)-^^A12(a)\ (z - zeq(a)) B2{a) - ^LB1(a)] (u - ueq(a)) (A.6.8) d_ di a z - zeq(a) 0 0 A: A12{a) -12(a) a z - zeq(a) + 5 2 ( a ) - ^ P i ( a ) _ (u — u, leq(a)) (A.6.9) A major drawback of using the quasi-LPV plant described in Eq.(A.6.9) in a feedback control synthesis is that the resulting control action u involves a trim condition ueq(a) (i.e. u = uc + ueq(a), where uc denotes the controller output) as depicted in Figure A.3, where P, C and T are the nonlinear plant, a controller and a transfer function that r + C uc + u p T a Figure A.3: Inner- and outer-loops of a quasi-LPV feedback system. produces the trim condition ueq(a). Any incorrect estimation of ueq(a) may jeopardize the closed-loop robustness even though the outer-loop may have guaranteed robustness properties. To avoid this pitfall, the plant input can be augmented with an integrator (i.e. u = f v dt). Clearly, this allows a controller design without involving the trim condition ueq(a). Thus the resulting quasi-LPV representation can be rewritten as follows: 150 A P P E N D I X A. A.l. Complex Analysis: Winding Number dt a z - zeq(a) u - ueq(a) + 0 Al2(a) Bx(a) 0 A22(a) - d-^A12(*) B2(a) - ^ B ^ a ) v dueq(a) , a z - zeq(a) u - ueq(a) (A.6.10) Finally, by defining an appropriate output matrix C(a) and a feed through matrix D(a), the quasi-LPV system in Eq.(A.6.10) is assumed to have the following minimal state space realization8 for all a G 0 throughout this thesis. P(a) = A(a) B{a) C(a) D(a) C(a) (si - A)'1 B(a) + D(a (A.6.11) where Q denotes the scheduling space. A.7 Complex Analysis: Winding Number Definition A.7.1 (Winding Number). Let g(s) be a scalar transfer function and let T denote a Nyquist contour indented around the right of any imaginary axis poles of g(s). Then the winding number of g(s) with respect to this contour, denoted by wno(g), is the number of counterclockwise encirclements around the origin by g(s) evaluated on the Nyquist contour T, see Figure A.4-Lemma A.7.1 (Properties of Winding Number). Let g and h be biproper rational scalar transfer functions and let F be a square transfer matrix. Then a. wno(<?/i) = wno(g) + wno(/i); b. wno(g) = n(g~l) - n(g); 8That is, the pair (A(a), B(a)) is stabilizable and the pair (A(a),C(a)) is detectable. 151 A P P E N D I X A. A. 7.1 The Argument Principle Figure A.4: A standard Nyquist T contour c. vmo(g~) = -wno(g) - rj0(g-1) + 770(0); d. wno(l + g) = 0 if g e and^g^ < 1; e. wno det(7 + F) = 0 if F e ^ i f o o and ||F||oo < 1. where n{G) and i]0(G) denote the number of open right-half plane and imaginary axis poles of G(s). A.7.1 Th e Argument Principle Definition A.7.2 (The Argument Principle). Let T be a closed contour in the complex plane. Let f(s) be a function analytic along the contour, i.e. f(s) has no poles on T. Assume f(s) has Z zeros and P poles inside V. Then f(s) evaluated along the contour T once in an anticlockwise direction will make Z — P anticlockwise encirclements of the origin. 152 APPENDIX B Proofs in Chapter 3 B . l Proof of Proposition 3.5.1 This section provides a short proof for the Proposition 3.5.1. Propos i t ion 3.5.1 The feedback interconnection [P, C] in Figure 3.4 is stable if and only if there exists a pair of stable parallel projection operators Tig^gi and ITgi ^gp. Proof. For the "if" part, it is clear that from Proposition 3.4.1 that a stable system always induces a coordinatization between the graph of the plant and the inverse graph of the controller. This implies that there exists a pair of projectors that map the total space W = % © onto Qp(a) and QQ, respectively, see also §A.1.6. For the "only if" part, as can be seen from Eqs.(3.5.5) and (3.5.6), the existence of the projectors implies that J(P(a),C) is invertible, and therefore the system is stable. Hence this completes the proof. • B.2 Proof of Eq. (3.6.6) The relationship between gap metric in the graph topology and the norm topology is presented in this section. This provides an alternative proof, which is easier to follow, in the existing literature. 153 A P P E N D I X B. B.2. Proof of Eg. (3.6.6) Show that 5 ( ^ , ^ 2 ) = | | r u - n ^ H o o (B.2.1) where l i n^x l l^ ||oo and | | IT^xll^lloo are both induced norms. Proof. First we proof an equality which will be useful for the proof in the sequel. (LU - n.^2)x - / • ( ru - ru)z = (ru + / - n^)(n^ - rujs (B.2.2) = (n^ + n^±)(n^ - r i^ jx = n.^(n^ - n^ 2)x + n^j.(n^ - u^2)x = n ^ n ^ i - n ^ n ^ x (B.2.3) Also note that: i W ( n ^ - ) x = ( n ^ - r u x ) x = n ^ ( / - n ^ 2 ) x = n ^ I U L * (B.2.4) and: = (I - U^Il^x - YLMffljhx = (Uy/l - U \ ) x - U^±l\jc2x = -U^U^2x (B.2.5) To show that | |n^ - I I ^ J o o = max {l in^n^x^, l in^xII^Joo} , we need to show both sides of the equality hence that \\U^-TL^2 ||oo < max j || 11,^11 ^ x ||oo, \\Ii^±U^2 ||oo| and iin^ - n^iioo > max jiin^n^xiioo, i in^xn^iiooj. 154 A P P E N D I X B. B.2. Proof of Eg. (3.6.6) Part I: To show that l l n ^ - I i ^ J L < oo S rnax {lin^n^iioo, | |n^xny / 2 | |oo} a f3 = ((a-P)x)(a-P)x) = x*(a*a + (3*[3 ~2a*p)x < x*(a*a + P*P)x ol \ X ) a 0 X a 0 0 13. 2 0 P_ a 0 X 0 P_ 2 n^n^x o 0 n ^ n ^ (B.2.6) From Eq. (B.2.6), — n^, oo < n ^ i n ^ x o o n . x n L e m m a B . 2 . 1 . Let A be a block partitioned matrix with • A, A = An A12 A2i A22 Ami Am2 A2q A, mq A 13 (B.2.7) and let each A{i be an appropriately dimensioned matrix. Then for any induced matrix 155 A P P E N D I X B. B.2. Proof of Eq. (3.6.6) the p-norm is defined as: \\MP< Plll lp U121 ll^2i||P | | ^ 2 2 | | P ll^llp ll^llp II A n l | | p || A n 2 | | p ' ' - mq\\p Further, the inequality becomes an equality if the Frobenius-norm is used. Proof. See Zhou et al., 1996, pg. 30. (B.2.8) • From Lemma B.2.1 it is clear that Eq. (B.2.7) can be rewritten as liru - IL < lliun .^ 0 l|n^j.n^2||0 = max {|| n^n^a ||oo, l in^xii^ ||oo Part II: To show that - U^W^ > max { i j n ^ n ^ j . I U , | | n^xl l^ 2 J (B.2.9) = | |n^ — n^n^Jioo = P 2 ^ - n^n^Hoo = 1111^(11^ — n^2) || oo < 11 1100 • l|n^i - n^r2| = •lln.^', —11.^ l i n n - (B.2.10) 156 A P P E N D I X B. B.2. Proof of Eg. (3.6.6) Similarly, l in^n^jioo = ||(/ - n^jn^Jioo = l|n^2 — n^n^i ioo = l l n ^ 2 - n ^ n ^ H o o = ll(rL#2 — n^-jn^Hoo < lin^ — n^Hoo • iin^Hoo = l|n.^-n^iu. (B.2.11) Hence, - ILrJoo > max | ||n^n^x H ,^ l i n ^ n ^ H o o j . Obviously, l{JCx,Jfa) = max j un^n^ ||oo, 11Il^ rJ-II^ r211oo} = P ^ i - ILrJoo. This completes the proof. • 157 APPENDIX C A Computational Algorithm For The z^ -gap Metric A computational algorithm for the z^ -gap metric is presented in this appendix. Several subalgorithms that are needed to compute the z^ -gap metric are also presented. This algorithm is adopted from (Vinnicombe, 1999b). C l A Computational Algorithm For The z^-Gap Met-ric Consider a plant with the following realization: ^P(s) = C(sl - A)'lB + D. A. A B C D where P e @>vxq with McMillan degree deg(P) < n, A e R n x n , B G Rnxq, C e R p x n , D € Rpxq. The quadruple (A, B, C, D) is called minimal if n =deg(P). Given two plants, Pi(s) and P2(s), the z^ -gap metric can be obtained using the following algorithm: 1. First, determine the McMillan degree of Pi(s), deg(Pi). 2. Next, obtain the winding number of det(GiG 2). 158 A P P E N D I X C. C.2. A Computational Algorithm of J^o Norm (a) First obtain the minimal realization of Pi = (b) Next construct the "A" matrix of (G*1G2)~1 Ax B i Ci and P 2 = A2 B2 . °2 D2 A{G*1G2)-1 = -(Ax - BiWDlCif C\YC2 B2WTBf A2 - B2DfYC2 (C.1.1) where W := (I + P ^ i ) " 1 and Y := (I + D2D'[)~1. (c) wno det(G^G2) = 0 A ( G * G a ) - i has precisely deg(Pi) eigenvalues with a positive real part. 3. Define eig(A( G» G 2)-i) +=Number of eigenvalues with positive real part of A ( G * G 2 ) - i . (a) If e i g ( A ( G j G a ) - i ) + ^ deg(P1), 8V{PU P2) = 1. (b) If e i g ( A ( G * G 2 ) - i ) + = deg(Pi), 5„(Pi, P2) can be computed as follows i. If 8„(Pi, P2) is a priori known to be less than UNITY , then 8v(Px, P2) can be caculated via the following equation: yjl-5v(P^P2f Pi I (I + PZPi) - l P2 I -l (C.1.2) ii. If <5„(Pi, P2) not know, then it can be calculated using the following equa-tion: ^ ( P i . P a ^ l l G a G J o o . (C.1.3) C .2 A Computational Algorithm of Jt%o Norm For (A, B, C, D), a minimal realization of a stable P, a Hamiltonian matrix is defined as rA + BR-1DTC ~{BR-lBT - 7 C T 5 r 1 C -(A + BR~1DTC)T (C.2.1) 159 A P P E N D I X C. C.3. A Computational Algorithm of The Graph Symbol Gy where := ( 7 2 / - DTD) and S 7 : = ( 7 2 / - DDT). Then for 7 > CT(D), 11-P1100 < 7 ^ # 7 has no jui axis eigenvalues. Thus, calculating the Jtffoo norm of a transfer matrix involves searching for the smallest 7 for which H1 has no imaginary axis eigenvalues. C .3 A Computational Algorithm of The Graph Sym-bol Gi 1. Solve the following Generalized Control Algebraic Riccati Equat ion: (A, - B^DldYX + X{AX - B^Djd) - XB1S{lBTX + CfR^Ci = 0 (C.3.1) where Rx := I + DXD? and Si := I + DjDx. Note that the minimality of [Ai, Bi, Ci, D{) is sufficient to ensure that there exist unique solutions X = X r > 0 . 2. Define a generalized control gain, F as F:=-S{\DTCi+BTX) 3. G i can then be obtained from the following equation: (C.3.2) Gi = Ni Mi Ai + BiF BXS~ Ci + DiF DiS~ F 5-1 u (C.3.3) for some unitary matrix U € Rqxq. 160 A P P E N D I X C. CA. A Computational Algorithm of The Graph Symbol G2 C .4 A Computational Algorithm of The Graph Sym-bol G2 1. Solving the following Generalized Filtering Algebraic Riccati Equation: {A2 - B2S7lDlC2)TZ + Z(A2 - B2S-1D^C2) - ZB2S2xBlZ + C^Rr1C2 = 0 (C.4.1) where R2 := I + D2D\ and S2 := I + D\D2. Note that the minimality of (A 2, B2, C2, D2) is sufficient to ensure that there exist unique solutions Z = ZT > 0. 2. Define a generalized filter gain, H as H := ~(B2D2 + ZC2T)R2~l (C.4.2) 3. G 2 can then be generated from the following equation: G2 = -M N. A2 + HC2 -H B2 + HD2 UR^C2 -UR-* UR^D2 (C.4.3) for some unitary matrix U G R p x p . Note, for the sake of simplicity, U = I can be assumed. 161 Index Symbols T distribution function 11 X2 distribution function 11 A anticausal signal see signal Arrhenius equation 2 ARX 11 B Banach space see space bandwidth 146 Bezout identity 147 bijective see mapping bounded from above 134 bounded from below 134 bounded operator see operator bounded set 134 C cartesian product 133 Cauchy sequence 135 causal inverse see inverse causal signal see signal closed-loop bandwidth... see bandwidth closed-loop stability see stability complementary sensitivity 31 completeness 135 complex conjugate 139 condition number 143 coordinatization 30 coprime 46 D degree of freedom 11 distance function 134 domain 27, see mapping E eigenvalue 143 F feedback 29 function see mapping G generalized stability margin 32 graph 26, 27, 133 inverse 30 of a controller 27 greatest common divisor 46 greatest lower bound 134 H Hilbert space see space, see space homomorphism 142 homotopy 137 homotopy condition 137 I identity operator see operator image see mapping index 137 infimum see greatest lower bound injective see mapping inner product 139 inner product space see space input space see space interior point 135 intersection see set inverse see mapping causal 29 inverse graph see graph isomorphism 142 K kernel see operator L least upper bound 134 supremum 134 limit cycle 3 162 INDEX INDEX linear operator see operator loop shaping 32 lower bound 134 M mapping 133, 139 bijective 133 domain 133, 140 image 133 injective 133 inverse 133 one-to-one 133 one-to-one and onto 133 onto 133 range 133, 140 restriction 133 surjective 133 maximum singular value . . . see singular value metric 134 Vinnicombe 21 metric space 134-136 metrizable 136, 138 minimal modulus 34 Morari resilience index 144 N neighborhood 135 noncausal signal see signal norm 138 norm topology see topology normed space 138 null set see set Nyquist contour 151 O one-to-one see mapping one-to-one and onto see mapping onto see mapping open set see set operator bounded linear 140 identity ,27 kernel of 140 linear 140 ordered pair 133 orthogonal complement 139 orthogonal projection 139 output space see space R range see mapping RESET 11 residual 11 residual sum of squares 11 resilience index . . . . see Morari resilience index restriction see mapping S sensitivity 31 set 132 complement 133 intersection 133 null 132 open 135 subset 132 union 133 set complement see set signal anticausal 141 causal 26, 27, 141 noncausal 27, 141 singular value maximum 143 minimum 143 space Banach 138, 142 Hilbert 26, 139 inner product 139 input '. 26 output 26 stability of a quasi-LPV system 29 stable 142 subset see set supremum see least upper bound surjective see mapping T topological space 136 topology 136 163 INDEX INDEX norm 138 weakest 136 U union see set upper bound 134 V vector 137 zero 137 vector space 137 subspace 138 vector subspace see vector space Vinnicombe metric see metric W weakest topology see topology well-posed 29 winding number 137, 151 Z zero vector see vector 164 

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