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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 GUAN TIEN 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 O F THE REQUIREMENTS FOR T H E DEGREE OF  DOCTOR OF PHILOSOPHY in T H E F A C U L T Y O F 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 O F 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 applicability and the reliability of the developed nonlinearity measure in tackling practical engineering control problems.  iii  Table of Contents  Abstract  ii  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 2.1  ,  10  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  4.3.5  The Linearizing Effect of Feedback  4.3.6  Jffoo  4.3.7  The Best Possible Stability Margin  Loop-Shaping and Weight Selection  , . . .  72 74 76 81  TABLE OF CONTENTS  TABLE OF CONTENTS  4.4  A Computational Algorithm  82  4.5  Summary  83  5 Design Examples 5.1  5.2  5.3  5.4  84  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  Example II: Cone Tank Control Problem  98  5.2.1  Problem Description  98  5.2.2  Design Objectives  5.2.3  The Nonlinearity Measure Applied To The MIMO Cone Tank . . 100  100  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  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 B.l  153  Proof of Proposition 3.5.1  153  B. 2 Proof of Eq. (3.6.6)  153  C A Computational Algorithm For The f-gap M e t r i c Cl  158  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  161  Index  2  162  viii  List of Tables 1.1  Nominal operating conditions for a CSTR  5.1  Nominal operating conditions for an inverted cone tank  5.2  Scheduling space of the fictitious plant from time £ = 0 to £ = 30 s.  ix  4 99 . . . Ill  L i s t of Figures 1.1  A typical plot of the normalized reaction rate coefficient versus temperature  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 T at three different operating points (i.e. T=300K, 320K and 350K). . . c  1.4  1.5  5  Closed-loop responses of the CSTR control problem under a single linear controller. Setpoint (dashed-dotted) and reactor temperature (solid). . .  6  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 and 3^ spaces  28  3.4  Standard feedback configuration  29  3.5  A pair of parallel projections, Hg^g^ and Ilgi\\g  3.6  Relationship of bp c — sin# and the minimal angle (i.e. 6) between sub-  2  2  p  33  t  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.  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))  .  54  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).  4.10 Specified and achieved loop-shapes (McFarlane and Glover, 1990).  ... ...  76 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)  xi  92  LIST  OF  5.6  LIST  FIGURES  OF  FIGURES  Top: Setpoint tracking responses of the CSTR under a unity feedback control. Reactor temperature, T (solid), setpoint (dashed-dotted). Bottom: Coolant temperature, T (solid)  93  C  5.7 Top: Setpoint tracking responses. Reactor temperature, T (solid), setpoint (dashed-dotted). Bottom: Coolant temperature, T (solid) c  5.8  94  Left: Closed-loop responses subject to ±20% in feed concentration, CA/ (Top three: +20%; Bottom three: +20%) at three operating points. Reactor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant temperature, T (solid)  96  C  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 temperature, T (solid)  97  C  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) xii  107  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) 5.22 The z^-gap metric of P(-0.155, s)W W l  112  and P(-0.1, s)W C W  2  1  ao  (solid)  2  versus the generalized stability margin with respect to [P(—0.155, s)WiC W , 00  I] (dashed-dotted) over a frequency range 5.23 The z/-gap metric of P(-0.1, s)W C W 1  OQ  114  and P(-0.05, s)W C W  2  l  00  versus the generalized stability margin with respect to [P(—0.1,  2  (solid) s)WiC W , 00  2  I] (dashed-dotted) over a frequency range 5.24 The i>-gap metric of P(-0.05, s)W C W 1  00  115 and P ( l x 10~ , 5  2  2  s)W±C W <x>  2  (solid) versus the generalized stability margin with respect to [P(—0.05, s) W\  COQW-2,1)  (dashed-dotted) over a frequency range  5.25 The z^-gap metric of P ( l x IO" , s)W C W 5  l  00  116  and P ( l x 10 , -3  2  s)W C W l  OQ  2  (solid) versus the generalized stability margin with respect to [P(l x 10 , s)WiCooW ,1] -5  2  (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: Controller output u(t) (solid). Note, the model switching sequences are given in Table 5.2 A.l  118  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 establishing 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  CHAPTER 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 industrial controllers are implemented in a feedback fashion in order to handle uncertainty, 1  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: -r  A  = kC  A  (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 Such as Proportional-Integral-Derivative controllers and model predictive controllers.  1  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: *= ^exp(^)  (1-1.2)  where k , E , R and T denote a constant, activation energy, gas constant and the reaction 0  a  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 T  e  340 m  p  e  r  a  350 t  u  r  e  360  370  , 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): dC  A  ^{C  dt dT  Af  - C) A  (1.1.3)  k exp(-^)C Q  A  %)C RT  ~dt  A  +  ^ ( T VpC r  c  - T )  (1.1.4)  p  where C and T represent reactor effluent concentration of component A and coolant A  c  temperature, respectively. The remaining model parameters and the nominal operating conditions are given in Table 1.1. C f, A  T  f  C, A  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 , at three different operating points (i.e. c  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  CHAPTER 1 .  1.1.  Background  and  Motivation  Table 1.1: Nominal operating conditions for a CSTR Parameter Notation Value Feed/effluent flow rate 100 L/min q E/R Activation energy term 8750 K Feed concentration 1 mol/L Reaction rate constant 7.2xlO min" k Feed temperature 350 K Tf • UA Overall heat transfer term 5 x l 0 J/min-K V Liquid volume 100 L T Reactor temperature 341 K Liquid density 1000 g/L P Reactant concentration 0.6592 mol/L c Specific heat 0.239 J/g-K C Coolant temperature 302.6 K T (-AH) Heat of reaction 5 x l 0 J/mol 10  1  0  4  A  p  c  4  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 closedloop 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  1.1. Background and Motivation  C H A P T E R 1.  Figure 1.3: Open-loop responses of the CSTR subject to ±5K in coolant temperature T at three different operating points (i.e. T=300K, 320K and 350K). c  5  1.1.  C H A P T E R 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  1.2. Thesis Organization and Contributions  CHAPTER 1.  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 measures and a historical review of the gap metric notion. Chapter 3: Graph Topology, The Gap Metric &: 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. Jd2 and M2) followed by a discussion on the connection between operator graphs and closedloop 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 computational 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 afictitious'^nonlinearplant. 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 contributions 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  1.2. Thesis Organization and Contributions  C H A P T E R 1.  A  •3  ©  o  u o  C3  O A decision making tool.  U5  © A novel approach to quantify closed-loop nonlinearity.  o  © 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.  Figure 1.5: A reading road map  C  O c o U  CHAPTER 2  Literature Review  Existing techniques for nonlinearity measures are briefly reviewed. These techniques can be categorized into three groups, namely the statistical approach, the norm-bounded error approach and the geometrical approach. In addition, a historical 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 controller should be employed. Since feedback control is employed in most industrial processes, 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  T h e Statistical A p p r o a c h  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. A.  2.1.1  The Regression  Error  Specification  Proposed by Ramsey (1969), the  The Statistical  Approach  Test  Regression  Error  Specification  one of the popular  Testis  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 (u , yt)- The fitted values y , obtained by ordinary least-squares, t  and the residuals  t  it = yt — yt  sum of squares equals  are then calculated. Denoted by  the residual  RSSo,  2~2^t-  2. Regress i on the set of regressors {1, y t  t  _  , y -  x  t  p  , u - i , u - k , t  tit ,  and  tit }  2  t  1  compute the RSS. 3. The test statistic is given by, (RSS -RSS)/(h-l) 0  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  it-  s  for all s (i.e. E(u e _ ) = 0 Vs), the function t  t  s  (h — 1)F  has an  asymptotic x distribution . An obvious advantage of this method is that it does not 2  1  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[y y y ], s  t  0  where  yo,  y  t  and y are the original time series y and the same time series being shifted by t and s s  time steps, can be written as follows: oo • /fl(u;i,W2)  = (2vr)- £ 2  oo  ]T  (2.1.2)  K{s,t)e-**-***  s=—oo £=—oo  A 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 x distribution is given as: f (x) = T(e/2)2e/^ x ^ ~ e~ ^ 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. 1  2  e  2  1  Q  2  e  +  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  (2.1.3)  j\^ ^ f {u u )dw du s  t  B  ll  2  1  2  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%  CJI  tt,  <  + U  2  < TT.  Subba Rao and Gabr (1980) and Hinich (1982) have shown that for a linear process  2  oo Vt = ^gkSt-k, fc=0  (2.1.4)  the following quantity is a constant over all frequencies. B(u^ )= . 2  In addition,  B(U>-L,LO ) 2  f  \ f f  { W (  ,  l^\  (2.1-5)  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 mentioned 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 Eq.(2.1.4) shows a moving-average time series. In which gk and e -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. 2  t  12  2.1.1 The Statistical Approach  C H A P T E R 2.  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, X . The correlation dimension is then defined by: t  ^  ..  ..  d In C(e.m)  D = limlim  ' a  e-^om^oo  2.1.6  me  where ~  m m  ' =^niE E( -  c(e m)  v  is the correlation integral.  e  e  11*  -  x  >  (2-1-7)  u  i i = i j=i+i  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 & M ), and || 1  ^|joo — max j^^ji 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 < oo, m  the BDS test is given by: = C(e,m)-[C(e,l)]  S(e,m)  Under the null hypothesis H : X is i.i.d, \/NS(e,m) 0  t  m  (2.1.8)  will have a normal distribution  with zero mean and variance that is a function of m and e. For a linearity test, one will firstfitthe 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 A R 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 y , given its past y ~k- It is noted that the best linear estimate of conditional t  t  mean, M (y - ) k  t  k  =  E[y |y _fc = y*], for a normalized series is the autocorrelation between t  t  y and yt-k, i-e. Pk- The nonlinear estimate of the conditional mean can be computed t  13  C H A P T E R 2.  2.1.1 The Statistical  Approach  by the kernel or Nadaraya-Watson estimator as follows: aw)=  {  n  -  k  ~\^  )  n where Kh(y* — yt) = h~ K(h~ (y* 1  f: : v  E =i h(y  y  ,  -  k  (2.1.9)  )  - yt)  K  t  with K being a kernel function and / i the  — y )),  1  v K  t  bandwidth. Therefore the nonlinearity index is defined as: L{M ) k  = E[{M (y _ ) k  t  -  k  (2.1.10)  y - f]  Pk  t  k  The functional L(M ) can then be estimated by computing: k  n = n " ^2{M (y ) 1  L(M ) k  k  t  (2.1.11)  - p yt} w(yt) 2  k  i=l  where u)(y ) is a weighting function. Clearly, L(M ) = 0 V7c for Gaussian process. Theret  k  fore, we may say that the linearity is rejected for large values of L(M ). k  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 E. The Higher Order Correlation  L(M ). k  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 y  t  is linear if the following higher-order correlation function  4> 2 yy  (r) equals to zero for all  time lag r.  2  (r)  =  E[{y _ - y}{y  «  N- J2{yt-r-y}{yt-y}  t  T  - y} } 2  t  N l  2  t=T  14  (2.1.12)  2.1.2 The Norm-bounded Error Approach  C H A P T E R 2.  where y denotes the sample mean of time series y . The null hypothesis for this test is t  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 ( r ) and another higher-order correlation function w  (f) 2 2 y  y  (r) at r are defined as follows:  0™(r)  Hiyt-r  - y}{yt - y}} N  N -  (j)y2 2 ( r )  =  y  «  1  E[{y _ N• t  N ~  l  (2.1.14)  Y , { v t - r - y } { y t - y }  T  - y} {y 2  Y , { y t - r - y }  t  2  -  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  2.1.2  C H A P T E R 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 Jzf norm can be used. This definition is more philosophical rather 2  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 product 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 discrepancy 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 ^ where  —  li-  (2.1.17)  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 measure . It is clear that the above equation has an interpretation of measuring the 3  Whenever 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. 3  16  C H A P T E R 2.  2.1.2 The Norm-bounded Error Approach  minimum worst discrepancy between the nonlinear plant and its linear model. 4  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  ll^^,o)-^ ^,o)ll ;  ( 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(y  2  -  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 inputoutput signals are modified when the loop is closed. Moreover, when the nonlinear plant Owing to the terms inf and sup.  17  2.1.3 The Geometrical Approach  C H A P T E R 2.  Q  L  Q  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, y , u\ and u in Figure 2.1 are 2  2  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  CHAPTER  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 operator 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 M G Jz? takes the following form (Krein and Krasnosel'skii, 1947): 2  2  = max<  l(Jl ,J(<i) x  sup  ||n^±a:||,  [ x e ^ i , ||cc11=i  ie^  sup 2 l  ]|i||=i  ||n^±x||  \ J  (2.2.1)  or alternatively (Sz.-Nagy, 1947), (5(^,^2)  ^||n^-n^ 1| 2  where VLX and J ^ denote the orthogonal projection onto subspace x  (2.2.2) and orthogonal  complement of Jif. The acute angle 0 between M\ and ^#2 can be related to the gap metric as follows: 9 = s i n l(Jl Jt ) -1  u  2  (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). Subsequently, Georgiou and Smith (1990) showed that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor 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 responses of two single-input single-output 5  (SISO) linear systems,  g(s) E J%2  and  h(s) € J$? , 2  can be thought as the stereographic  projection onto a Riemann sphere . The gap metric is simply the chordal distance, as 6  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  (  ,  (2-2.4)  | 2  An obvious shortcoming of the above definition is that the computation efforts to determine  S (g(s),h(s)) c  can be very intensive since a search for all  (  > 0 is needed. A  Generally, 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. See Figure 3.13. 5  6  20  2.2.  C H A P T E R 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 Jzf  2  space. This new metric is also known to induce the same graph topology mentioned above. Since the Jz? space consists of both causal and anti-causal signals, care must be 2  taken to confine oneself to causal finite energy signal space in order to preserve feedback 7  stability. Topologically, this means that the two closed-loops must be homotopically equivalent. In the definition of the z^-gap metric, the winding number is used to ensure 8  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 dgap. 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). For a physical realizable stable closed-loop, one needs both finite energy and causal signals in the loop. see §A.7. 7  8  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 b ^ > (3 for the nonlinear plant P , where b ^ denotes the Po  Po  generalized stability margin defined by P and C, and the f^-gap metric between any 0  nominal plant, P and the IQCs (i.e. <5„(Po, &)) is smaller than /3, then one can conclude 0  that any controller C that satisfies bp c > P can stabilize the nonlinear plant P. In this 0i  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: <WPo,  where fs*(P ,Pi)±  Pi) = max {T%(Po, Pi), ~SJ? (PI, Po)}  sup  Q  inf.  xoeSoruS-b sieging  and Qi = {[%] : y = PjU, y,uE defined in the extended  ££i,c&  (2.2.5)  2  Jz?2,ce}-  , 1110112  Po) =  sup  ^gf^  inf  xiegiOSf *oeS as? 2  0  2  Note that under this definition, all signals are  signal space . It was shown that the nonlinear i^-gap metric 9  is greater than the z^-gap metric for LTI systems. In contrast to the LTI case , the 10  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.  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. T h e homotopy condition denned in the i^-gap metric for LTI is both sufficient and necessary condition for feedback stability. 9  10  22  C H A P T E R 2.  2.3  2.3. Summary  Summary  This chapter discussed several existing nonlinearity quantification techniques. It is obvious 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 appealing 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 closedloop 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 assumption 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 distance 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  CHAPTER 3 Graph Topology, The Gap Metric & Closed-loop Stability  This chapter presents a geometrical sense of the gap metric and the u-gap metric (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, particularly the Jz? and the J^f spaces, and the operator (i.e. system) graph spaces. Then, 2  2  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 stability . A typical example of such signals looks at the transient 1  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£ space . 2  2  (b)  oo  Figure 3.1: Examples of finite energy signals, (a) A signal that is defined over time interval [t ti], (b) A signal that has finite area between the curve and the time axis from —oo to +oo. 0  1  In this context, the closed-loop system is said to be bounded-input bounded-output (BIBO) stable. See §A.3  2  25  CHAPTER  3.  3.3. Operator Graphs  Note that signals in the Jz? space can be split into two unique subclasses (i.e. an 2  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? ) space and that of anticausal finite energy signals is termed as the J^ (or +  1  2  Jzf ) space. Clearly, the noncausal finite energy signals space is precisely the Jtf space. -  2  2  For a precise mathematical definition of these spaces, see §A.3. 1  t <  0  t >  (a)  0  t <  (b)  0  t >  0  (c)  Figure 3.2: Types of signals, (a) Anticausal signal, (b) causal signal and (c) noncausal signal.  3.3  Operator Graphs  Recall that the graph of a (possibly open-loop unstable) system is the collection of all 3  possible finite energy (or bounded) input-output pairs of signals entering and produced by the system. Mathematically, the  graph of a system P : @f C <2f G Jft -> & E 2  J^2 is defined as: CW ± & ®&  P  where  and  (3.3.1)  are the input and output spaces. These spaces belong to the Hilbert  space, J#2. Iv, @f = {u E W E Jff •• Pu E & E J^ } and 8 denote the identity 2  2  3  2  See page 133.  26  3.3.  C H A P T E R 3.  operator on  Operator  Graphs  the domain of P and the direct sum, respectively.  Likewise, the Jz? graph of a system P : 3)f  2  2  C U G Jzf —• Y € JSf can be defined as 2  2  follows: ^  (3.3.2)  A  p  where ®f = {u G U G j£f : Pu G Y G jSf } is the domain of P . 2  2  2  Similarly, the J%2 graph of a controller is given by:  (3.3.3)  c where  Q>Q  2  = {y G W : Cy  the identity operator on  G  ^ } denotes the domain of the controller and  is  . 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  y  c  —  c  0 I  (3.3.4)  I 0  Note that the inverse Jzf graph of the controller can be defined analogously. 2  The idea of J?? and ^ 2  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£f space consists of the Fourier 2  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? and the J^f 2  spaces may not generally induce the same topology.  27  2  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? space. Figure 3.3 gives a graphical representation of the above 2  discussion. The Jzf2 space consists of a Hilbert space  J ^ ,  disk, as its subset and a few "holes" punctured by the  Figure 3.3: 5£<i and ^  represented by a perfect space. {Si}, {52} and  spaces.  {^3} are three Cauchy sequences in the Jzf space. The subscripts of each sequences 4  2  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 {Si}  i—*oo  even though lim rij = n* E Jzf is bounded but the signals are noncausal. In fact, 2  l—>co  the anticausal part of {Si} induced a "hole" on the Jzf space, which makes it different 2  from the  J$?2  space. Evidently, the topologies induced by the Jzf and J%2 graphs are 2  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  3.4.  C H A P T E R 3.  3.4  Operator Graphs and Closed-loop Stability  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  U  w  (3.4.1)  2  2  J(P,C)  Note that in order for J(P, C)'  1  [ul] to have a unique and physically realizable  :[Z ] l  2  solution, Eq.(3.4.1) must be well-posed (i.e. [ ] has to have a causal inverse). The P I  feedback system [P,C] is stable if the aforementioned mapping J(P,C)~ is bounded l  (i.e. J(P,C) produces finite energy outputs from finite energy inputs). This implies that Qp@Q = W . To see this, Eq. (3.4.1) can be rewritten as: 1  c  C Ui  P  +  I  w  x  u = 2  (3.4.2)  w  2  Gp  Obviously, the decomposition of W into Qp and Q is possible if and only if ui, u , yi, l  c  2  >1 ' y , u>i and w are all unique. This means that given an arbitrary signal w = UW2 one \l can always find a unique decomposition in Qp and QQ, whenever the closed-loop shown 2  2  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  3.5. Operator Graphs and Closed-loop Robustnes  C H A P T E R 3.  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. g ng P  = {0}  c  2. Q @Q P  =  C  w,  in other words, the graph of the plant and the inverse graph of the controller induce a coordinatization  5  Proof,  ofW.  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  existence of unique decomposition.  3.4-1 establishes  a condition for the  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 Q  c  a pair of projectors that project W onto Qp and Q , c  respectively.  implies that there exist 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 See §A.1.6.  5  30  3.5. Operator Graphs and Closed-loop Robustnes  C H A P T E R 3.  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 w , this dependence can be written as 2  follows:  I P  Vi  or Ui 2/i  (I - CP)  /  (3.5.1)  -C  w  2  " (I- CP)' -V- CP)~ C P(I- • CP- )) -p(i- - CP)- C 1  L  1  Wl  L  (3.5.2)  w  2  If the closed-loop is stable, we are interested in minimizing the effects of [%] to 2  ' yi'  which means that we want to minimize the following cost function:  (i-cpy'li  7 ^  -c  (3.5.3)  which can be alternatively, defined as: -i  (I-CP)'  1  I  7  (3.5.4)  -C CO  Such a statement allows for the above minimization problem to be restated as a maximization problem (i.e. sup bp c). t  The minimization of the cost function in Eq. (3.5.3) can be seen as the minimization of closed-loop sensitivity and complementary sensitivity functions . Interestingly, this 6  norm is actually the one that needs to be minimized for the plant with normalized coprime factor type uncertainty.  Denoted by b , the alternative cost function in PiC  The entries (1,1) and (2,2) of Eq.(3.5.2) are the sensitivity and complementary sensitivity functions, respectively 6  31  3.5. Operator  C H A P T E R 3.  Graphs and Closed-loop  Robustness  Eq.(3.5.4) is also called the generalized stability margin and is used in the McFarlaneGlover  Jftfoo  loop shaping (McFarlane and Glover, 1992) procedure later employed as a  controller design technique in this thesis. Note that the b c always has it values bePi  tween 0 and 1. Small bp c means that the closed-loop stability margin is small and will t  become unstable when the b c = 0. Larger b c indicates that the system has good Pi  P>  robustness. Unequivocally, this makes the generalized stability margin easy to interpret. In addition, the bp c also turns out to have a very nice geometrical interpretation. The t  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. LT = LT). 2  In Figure 3.4, it can be easily shown that the stable closed-loop transfer function from 7  [wl] to [yi ] (i.e.  []  [i -c]) is a projection operator that maps «£f2 signals (i.e.  (I-CP)~  1  P  u>i and W2) onto the plant graph Q . Similarly, the stable closed-loop transfer function P  from  \ to [% ] (i-e. [/] 2  2  {I-PC)~  [-PI])  1  is also a projection operator that maps Jzf  2  signals onto the inverse graph of the controller. In addition, since [?] ( QQ  '--PC)  - 1  [-  p 7  ] — [fj /]>  a n  []  [i -c] +  (I-CP)~  1  P  Y signals in Jz? can be uniquely decomposed into Qp and 2  using the above two projection operators. Clearly, geometrically,  [] P  (I-CP)~  1  [i -c]  represents the parallel projection onto Qp along Q and is denoted by Hg ^. Likewise, c  denoted by Hg^g ,  [-p i) is the parallel projection onto Q along Qp.  [ j] (I-PC)~ C  p  p  1  c  Note that the expressions [j,] (I-CP)"  [I -C)  1  and [</] i -? )' 1  nonlinear systems. To resolve this, Hg ^g^ and Ug^g p  0  [~ ] are not valid for  1  p J  can be rewritten as follows, see  p  (Doyle et al., 1993)  n,QPWQ'C  I  {I-CP)'  0 /  1  [/  -c)) = [ ] (i-cpy 2  (3.5.5)  -c  P  0 0  '([^('-OP)-  i  1  -  +  /  0  0  -I  (i-cp) (i-cpy  l  1  P  i  32  J(P, C)  [i -c] =  -1  \ ]{I-CP)-  1  P  [i -c).  3.5. Operator Graphs and Closed-loop Robustness  C H A P T E R 3.  and C S{;\\Qp ~  (I - PC) - l  I  / 0 0 0  +  -P  I  0  0  I  (3.5.6)  I  J(P, C)  -1  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 Hg \\gi and Rgi \\g . p  c  c  p  he  ^  Figure 3.5: A pair of parallel projections, Ug \\g^ and TLgi^g . Under these projections, any signal in the Jz? space (i.e. h in this case) can be uniquely decomposed into m G Qp and n EQQ such that h = m + n or Jzf = Qp © QQp  p  2  2  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-g \\gi, and p  Proof. See Appendix B . l .  n . 0 p  • 33  3.5. Operator Graphs and Closed-loop Robustness  C H A P T E R 3.  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^  =  (i.e. the restriction of orthogonal projection  = TL^J.^  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. y±  : J£>i —»•  t/  and (I — Qjz,^)  projection operators. It is obvious that graphical representation of Ajz jy t  are two parallel  : Jz?2 —>  = ^g \\gi and I — Qje^ P  = Tigi^g .  c  and the two graph spaces  and  M  p  JY  The  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 bp c can be expressed as folt  lows: bp,c  = lin^iic/ II" = \\QM,A\1  x  =  WA'J^n^w-  1  (3.5.7)  Since the bp c is only defined when Ug \\gi is bounded, this implies the invertibility t  of Ajr. /rr  p  c  By invoking the property of the minimal modulus of an operator (i.e. if  34  3.6. Operator Graphs and Uncertainty Description  C H A P T E R 3. J?T € £ ( J z ? 2 )  is invertible, then p{J(f) = inf{||JcTx|| :  = 1} —  n^-iy),  bp,c = fi(Ajg^) = fi(U x\^f)  (3.5.8)  N  =  inf{\\A^ ^x\\ : x E ^  =  inf {dist (x,  y  we can write:  and  = 1}  (3.5.9)  : 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 JV -) and the point projected by Ajt^jv from a unit sphere in ^ 1  onto jY (i.e. the point Aj^^x as shown in Figure 3.6). The last equality results from x  Figure 3.6, where the b c can be interpreted as the smallest distance between a point P>  on a unit sphere in subspace ^# to subspace JY . Clearly, the b c can be related to the Pi  minimal angle between subspaces M and J/ as follows: sinc? =  inf xe^,\\x\\y£o  W^W = A  \\A^x\\  X  \\x\\  i n f  = b  PiC  (3.5.11)  xe^,\\x\\=i  where 9 is defined by: 0 A cos" | ^ ' 1  ,0^x E^,0^y  EJY  (3.5.12)  Since 0 < b c < 1, the above sine function is monotonically increasing in the range of Pi  [0, | ] . This means that the increasing of the b c is proportional with the increasing of P  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  CHAPTER  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(P , Pi) is quantified by the maximum of two directed gaps 0  (i.e.  S y/? (Po, Pi) 2  and  5 j^(Pi,  which measure the maximum distance between all  Po)),  possible bounded input-output pairs of the two plants of interest. Mathematically, the gap metric is defined as follows: S(P ,Pi) 0  where 7  m  (P , 0  Pi) =  sup  (3.6.1)  ±max{7 (P Pi),~5 (Pi,P )} M  inf  Q}  ytf2  0  "X"!, " ,7 r A P 0  2  u  P) =  sup  0  inf  .. .. \\X0-Xl\\2  and Qi = {[ ] : y = PiU, t / , « e J^.}- Note that the z^-gap metric can be defined in a u  similar fashion except that the signals are defined in the Jzf space and a homotopy 2  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 P i . 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  3.6.1 The Gap Metric  C H A P T E R 3.  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 u , ui, yo and y\ are 0  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 pair , 8  the graph spaces represented by the dots in Figure 3.7 (b) for plants P and P\ can 9  0  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  Ml  t'  p.  yo _  yi _  (a)  (b)  (c)  Figure 3.7: From plants' input-output space to the gap metric, (a) Two stable closedloops containing plants P 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. 0  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: 8  See §A.1.1  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. 9  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%2In 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 orthogonally project it onto  The image of X\ on  is given by H^ xi,  and  where nj^  2  2  denotes the orthogonal projection onto the subspace ^# - The distance between x\ and 2  its image on ^  2  subspace is denoted by d\. Next, a second point, say x , is picked and 2  the similar operations are repeated again. The resulting distance between the new point x and its associated image U^ x 2  2  2  is d . These operations are repeated so on and so 2  forth until all the points in the subspace the directed gap from  are exhausted. Denoted by  M\  S (^#L,^# ), 2  to ^# is then obtained by taking the maximum of all these 2  distances (i.e. supj di) or in a more compact form: = sup ^  S (dt Jti) u  ie^i  n  ^ ^ 2  2  =  |F||2  s  u  \\  p  x  _ Yl^ x\\2 2  (3.6.2)  xe^i,\\x\\ =i 2  where || • || denotes the Euclidean 2-norm. Further, Eq.(3.6.2) can be rewritten as 2  5(^i,^r )=  sup  2  X<=.^i,\\x\\2  - Yl^ )x\\ = 2  =l  Hn^ixllij  sup x£^l,\\x\\2  =l  where n ±. = I — Uy/ is the orthogonal projection onto the complement of J{  2  38  (3.6.3)  Chapter  3.6.1 The Gap  3.  Metric  Similarly, 8  (^# ,  x  sup  2  -  Tl^x^  s  xe^2,\\x\\2 = l  It is obvious that  8  (^i,^ ) 2  u  p  X€^2,\\x\\2 'S II..  II  = l1  (3.6.4)  II n*-^ II2 1  in general. Unequivocally, this implies  7^ $ ( ^ 2 , ^ 1 )  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: — max{ 8 (J?i,JZ ),  8(J^i,^ ) 2  Remark 3.6.1. 8(^1,^2) 8(JZi,JZ )  2  2  does not generally satisfy the triangle inequality.  does satisfy the triangle  2  (3.6.5)  = <5(^ ,^i)  8 (JZ ,^\)}  2  inequality  in a Hilbert space (Kato,  However,  1976)  which  makes it useful.  Remark 3.6.2. It was shown that 1. See Krasnosel'skii  8 (J^i,J^ ) 2  =  8 ( ^ # 2 , ^ 1 ) whenever 8(M\,  J£ ) 2  <  et al. (1972, pg. 206).  Remark 3.6.3. Recall that ^ \ and M2 are the graphs of clear, from the comparison  of Eq. (3.6.1) and Eq.(3.6.5),  PQ  and  P  1 ;  respectively.  that the definition  It is  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 following  (i.e.  relationships  Tlje, where Jff denotes either  or M^),  the  hold.  8(J{\,  Jt2)  (3.6.6)  where \fffi \\T^||oo M  and \\Il^±.TL^ \\ 2  00  are both induced  39  norms.  Chapter  3.6.2 Homotopy and The v-Gap Metric  3.  Proof. The first equality follows immediately from Eq. (3.6.5). The second equality hold since, from Eq. (3.6.4), one can establish max{ S (^±,^ ),  <5 ( ^ 2 , ^ 1 ) } =  2  m a x j s u p ^ ^ j ^ n ^ ||n^xx|| , sup ^ \\U^±x\\ j lin^i^lb "I lin^xn^ilb 2  su  P*e^2  || || g  a  r  =  m  a  x  (  S U  xe  2tM=1  = max { sup ^ l|n^ n^ x|| 1 r 8g  2  P  '  s  u  ±  p  2  "iRli  2  f  =  m  a  \P ^  x  1  1  1  ||g  |  a  ,  ^ ! Iloo,  j I TT ^j.n^ ||oo|. For the last equality, an alternate proof is given in Appendix B.2. The 2  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 =  where Gi = [M\]  e  inf  \\G - GiQ\U d  for i,j = 1,2  (3.6.7)  is the graph symbol of plant Pi defined by its normalized  comprime factorization (i.e. Pi = NiMf ). 1  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  H o m o t o p y and T h e zv-Gap M e t r i c  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 ^#1 and ^2 does not guarantee closed-loop stability since the  Jz?2  gap between  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  Chapter  3.6.2 Homotopy and The v-Gap Metric  3.  anticausal hole  Figure 3.9: Two nonhomotopically equivalent graph spaces. Notice that there exists an anticausal hole in subspace ^# - If no homotopy condition is imposed, the resulting i>-gap metric maybe misleading since it does not take the closed-loop stability into account. 2  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.  F  Figure 3.10: An analogy of a homotopic equivalence. In contrast to the homotopic equivalent, two objects are said to be nonhomotopic equivalent 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.  41  3.6.2 Homotopy and The v-Gap Metric  C H A P T E R 3.  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 condition of operator graphs (i.e. the first column of Figure 3.12) and their corre10  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 perturbation . In 11  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 and P are assumed to be the perturbed versions of a stable plant P . 2  3  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 ^# - Physically, this can be seen 2  as continuous perturbation in P to Pi = P + A without destroying the closed-loop 0  0  0  stability of the second system. Figure 3.12 (b) shows that even though the magnitude of the uncertainty is further increased from A to A , closed-loop stability is preserved 0  x  since the two graphs are homotopically equivalent. In contrast, when M hits the sharp 2  cone representing the M'  L 2  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 = P + A . Unequivocally, 3  0  2  the homotopy condition plays an important role in establishing feedback stability reWhether the graphs are perfect or being punctured. "That is, ||Ao|| < IIAiH < | | A | | . 10  2  42  C H A P T E R 3.  3.6.2 Homotopy  and The v-Gap  suits in the Jzf space. In system theory , the winding number 12  13  2  Metric  (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? space 2  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 defined as follows: Definition 3.6.1. Let P = A ^ M " , i = 1,2 be two (possibly unbounded) linear opera1  t  tors, the v-gap metric is defined as  ||G Gi||oo 2  ( - 0 0 , 0 0 )  and wno d e t ^ G i ) = 0,  5,(Pi,P ) = 2  1 = [$] e * ,  where d  if det(G;C?i)(ja;) ^ OVtu G  (3.6.8)  otherwise Gi =  [ -Mi Ni}  G Jffoo, and wno denotes the winding number of  det(G Gi)(s) as the complex variable s follows the standard Nyquist  D-contour.  2  In addition, the z^-gap metric can be rewritten in terms of Pi and P providing that the 2  winding number condition is satisfied: ||G C7i||oo = | | ( / + P 2 P 2 T  1 / 2  2  Note that C7C7i 2  = [ - M N ][ \] 2  (/ + P P *)- / (P - PJil 1  2  I  =4>  2  2  (MiM{)~  2  1  2  m  ( P - Pi){I + 2  1/2  + JV Mi = M ^ - A ^ M f  = -M Ni  2  2  (3.6.9)  P*Pi)- \\oo  1  +M f ^ M i =  + P ^ P i ) - / . The last equality holds since N*A/\ + M * M X = 1  2  = 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, T h e 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. See A.7 12  13  43  3.6.2 Homotopy and The v-Gap Metric  C H A P T E R 3.  Two J#2 graphs representing two stable  Figure 3.12: Analogy of the homotopy condition in the closed-loop stability.  44  v-g&p  metric and the actual  3.6.2 Homotopy and The v-Gap Metric  C H A P T E R 3.  the v-gap metric is equal to supa(G Gi)(ju).  It has been shown that for SISO system,  2  the I'-gap metric is precisely equal to the chordal distance K between the stereographic projection of Pi(ju)  and P {ju>) onto the Riemann sphere. 2  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^f space and 2  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 Pi(%i,yi))  and P (ju>o) = x + jy 2  2  2  P\(JOJQ)  (denoted by P (x ,y )) 2  2  = X\  + jyi  (denoted by  be two frequency responses  2  of Pi(s) and P (s) at a particular frequency, say ui . A straight line is then drawn to 2  Q  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-one and 14  unique. The chordal distance of P[(£i,ipi,  £i) and P (^ , ip , C2), denoted by «(Pi, P ), is 2  2  2  2  given by K{PI, P ) = 2  ,  | P l  ~r  (3-6.10)  P  Clearly, Eq. (3.6.9) is the multivariable version of Eq.(3.6.10). The next subsection shows The corresponding one-to-one mapping are & = t+ji , 2 = 1,2 x  45  +1  d  =  a n  'i+y<i  x  +1  d ipi = j \ y f  + 1  for  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 divisor  15  (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 represented 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) = —  -^4,  S  V a 6 R > 0,  (3.6.11)  1  where g(s) and h(s) are coprime and stable transfer functions.  Note that P(s) =  g(s)h~ (s) is a right coprime factorization, while P(s) = h~ (s)g(s) is a left coprime 1  l  factorization.  Clearly, the coprime factorization is not unique.  A unique coprime  factorization can be obtained by introducing an operator M*(s) = M (—s). Then T  P(s) — g(s)h~ (s) is said to be a normalized right coprime factorization if the following 1  condition is satisfied: h*(s)h(s) + g{s)*g{s) = I.  (3.6.12)  Likewise, P(s) = h^ (s)g(s) is called a normalized left coprime factorization if: 1  h(s)h*(s) + g{s)g*(s) = I.  (3.6.13)  An interesting uncertainty description arising from the aforementioned factorization is For simplicity, consider two real numbers that can be expressed in the products of prime numbers (i.e. a = p" xP2 2 • • • x p® and b = p\ x p% • • • xp@ , 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 x 2 x 5 ). The G C D for a and 6 is then denned as Hip^ ^ '^'\ where Iii denotes the products of all i terms. Example: GCD(12,30)=1 x 2 x 3 x 5° = 6 since 12 = l x 2 x 3 x 5° and 30 = l x 2 x 3 x 5 . 15  1  n  x  2  n  1  m  1  1  1  1  ai  1  1  46  2  1  1  1  1  1  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  + Ah)'  + A )(h  : A , A e Jf*,,  1  g  g  h  A  0  <b  (3.6.14)  where A , A^ and b denote two stable unknown transfer functions, which represent the g  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 A and A ^ are stacked on top of each other (or side-by-side) to form a full g  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 theorem . 16  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  (3.6.15)  [J]-[$i]<4  So, for Pi = gih-t , 8 (P, Pi) < b, there exists a Q e J%x> (or 1  Jzfoo  satisfying a certain  homotopy condition) such that h _  hi g\_  Q  < b.  (3.6.16)  Developed by Zames (1966a,b), the small gain theorem forms the cornerstone of modern robust control theory, and shall be discussed in the next section. 16  47  3.7. The Gap Metrics and The Small Gain Theorem  C H A P T E R 3.  Next, define hi A Then, obviously  ^  9i  9  h  Q-  (3.6.17)  9  < b and P = g^T = giQQ~ h7 = (giQ^Q)1  l  l  1  x  = (g +  A )(/i + A h ) ' . The converse is also true. Note that for P = g h7 = (pi + A )(/ii + 1  l  g  1  x  ff  Ah)" , there exists a Q~ £ X o such that P = {(pi + A )Q}{(/z + A/OQ} 1  l  x  5  1  -1  is a  normalized right coprime factorization. So, it is clear that, by definition, 5 (P, Pi) can be obtained by assuming Q~ = Q £ J ^ , : l  h+  h  (5(P,Pi) = inf  9  A  ft  -(  \  A  <  H  Q)Q .  9  +  9.  A  J  oo  <b  h _9_  h+  A  H  g+A _ 9  (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  T h e o r e m 3.7.1 ( S m a l l G a i n T h e o r e m ) . Given a generalized plant M e a real positive  number 7 > 0.  well-posed and internally  Then the M - A connection  stable for all A ( s ) € ^ ^ 0 0  and  shown in Figure 3.14 is  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 )(/i + A ^ )  - 1  fl  shown in Figure 3.15, the  corresponding M and A blocks are as follows: A M  An  4 4  (3.7.1) h'^I-CP)  -1  C  (3.7.2)  -I  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] h-^I-CP)-  1  6=1/7  a n  [C -/'  ]  P  (I-CP)-  1  [-c/]||  <  < 7-  I/7,  \\h-i{i-cp)- [c x  Note that by defining  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  3.7. The Gap Metrics and The Small Gain Theorem  C H A P T E R 3.  Figure 3.15: A perturbed plant with normalized coprime factor uncertainty.  Lemma 3.7.1. Given a nominal plant P = gh~ , a unity controller C = I and a x  0  nonnegative real number b > 0, a closed-loop system containing a perturbed p (g + A )(h + A )~ with 1  g  h  a.  < b or S(P, P i ) < b if and only ifb  PC  \M\  >b,  where M is defined in Eq. (3.7.2). Proof. The result follows by invoking the small gain theorem and the equivalent property 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 c  loops were under a unity feedback. Therefore, for consistency, a unity feedbac 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 d  not affect the generality and the usefulness of the aforementioned lemma since th troller can be treated as afilterand absorbed into the nominal plant PQ.  50  3.8. Summary  C H A P T E R 3.  3.8  Summary  This chapter presents the gap metric, particularly the z^-gap metric, theory in a graphical 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 normalized 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  CHAPTER 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 generalized 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 scheduling parameter, the linearizing effects of feedback and the best choice of the nominal model, are discussed. Finally a novel, practical and easy to implement computational 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: • Ability 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. • E a s y 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 Measure  This section aims to devise a closed-loop nonlinearity measure based on the three properties previously mentioned. An overview of the resulting nonlinearity measure is presented, while the theoretical motivation finds room in the next section, to avoid disruption of the logical presentation. To formulate the aforementioned closed-loop nonlinearity measure, we begin with the following assumption: A s s u m p t i o n 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  4.2. Formulating The Closed-loop Nonlinearity Measur  C H A P T E R 4.  Assuming  that these two loops are subject to the same disturbances  the input-output  discrepancy  and noises.  Then  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 b c presented in §3.5. In the meanwhile, for the Pi  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: S (NL,  L) = inf max{ S (NL,  nl  nl  L), 5 (L,  NL)}  nl  iyfc A.  where S (NL,L) nt  and 5 (L,NL)  (4.2.1)  are the normalized input-output discrepancies be-  ni  tween the nonlinear plant, here denoted by NL, and a linear model, here denoted by L, or the directed gaps, which are defined as: U2 1 II  •ui •  S i(NL, n  L) =  inf  sup  y\]^^ [yl}^2^  U  2  2  sup  (4.2.2)  2  and 5ni(L,NL)=  V2 J 112  ||[£]||  inf  [£]efcnift[yi]esia%  •U2 1  _  ran  . V2 J L 3/1 J 2 r u2 11  (4.2.3)  [ 2/2 JI 2  2/i  NL Wi  c  n  U2  V2  L  w  2  c  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) . Eq.(4.2.1) implies that 1  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 Eq.(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. 1  55  C H A P T E R 4. 8 i(NL,L*), n  4.2. Formulating The Closed-loop Nonlinearity Measure  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{): sup  ~8 (NL,L)^ nl  inf  [NLIU,)}^^  l|[  W&2n&  y!rffi l l l a  (4.2.4)  \\[NH»I)\\\2  and sup  t (L,NL)^ nl  [LU )ZG^ 2  2  inf [ % ]eGin*2 N  Ul)  i  ^  Z  ^  ^  (4.2.5)  \\[Lu \\\ 2  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 computation 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 nonlinearity 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: A s s u m p t i o n 4.2.2. Plant's nonlinearity depends on measurable states, typically exogenous 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 transformation, the resulting closed-loop nonlinearity measure over 0 is redefined as follows: 8% (NL(a),L) LPV  ±  =  8%(NL(a),L)  mfmax{^(iVL(a),L),^(L,A/L(a))}  56  (4.2.6)  CHAPTER  4.2. Formulating The Closed-loop Nonlinearity  4.  Measure  where t°(NL( ),L)  sup  ±  a  ^ Jr!f k  ll[  inf  [^(aJuJeeiWnifbl^aJe^n^  )Ul  2]|  || [ JVL(a) J | | Ul  Vet G  fl  (4.2.7)  2  and <5S(L,iVL( ))4  S U  a  • u u2  2  L u 2  Note, the subscript  p  W2 i lll^J  inf  ]Ge2n^ [ j v i ^ u J e e i W n ^  r  u\  W ^ J l l ^  IIU« JII  2  2  ^  g  fl  (  4  ^  g  )  2  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)  Otherwise,  = Q. x  C  9\(fl)  Q. x  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^ fli) is a subset or is equal to that of the entire =1  scheduling space,  The validity of the above relationship depends heavily on the  Oi(fl).  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% (NL(a),L)±  V«G|J^  Mmax{7%(NL( ),L),7%(L,NL( ))},  LPV  a  a  (4-2.9)  i=l  where J«(NL( ),L)± a  [NL (a) U  Ul  sup ]^!(Ur =1 n )n^ t  inf 2  {Lu Y^2  57  2  ll[  7^i l; " r y-2 u2J  II [ NL(a)  Ul  J ||  2  2  (4.2.10)  CHAPTER  4.2. Formulating The Closed-loop Nonlinearity Measure  4.  and t * ( L N L ( t  a  ) ) ±  sup 2  \\[^~  inf  [% }eg njz> 2  2  [^JeffitU^W  [m^]\\  2  {  4  2  n  )  \\{LU \\\ 2  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.  ^(JVL(a),L)4  sup  sup  inf  11  [""in ^ " j ,  r  ^  «2 J  |  l  a  (4.2.12)  and -8i{L,NL{a))±  sup  sup  [w«)*]L  inf  u  Note that the above closed-loop nonlinearity measure is actually a frozen point nonlinearity 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  (NL(a),L))  is introduced, see Hyde (1991).  Therefore, the resulting total nonlinearity measure becomes nm a 6 %  S  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 provides a useful indicator for closed-loop nonlinearity measure. The above discussion can be summarized into the following definition and theorems.  58  4.2. Formulating The Closed-loop Nonlinearity Measure  C H A P T E R 4.  Definition 4.2.1. Given a quasi-LPV plant NL(a)  with a G fl ~ U i l i ^  d  an  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^ (NL(a),  L) + 6+  LPV  §  (4.2.15)  where 8+ denotes the uncertainty due to time variation and N  8f (NL{a),L)  = inf max{8 %{N L{a), L), ~8%(L, NL(a))},  PV  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^ (NL(a), n  L)(ju)  <  b c(juj), Lt  VwG[0,oo). Proof. The proof consists of two steps. For the first step, note that <5? = m  L) + 8 . f  Therefore, 8^{NL(a),L)(juj)  8f {NL{a),L){ju) pv  < b {juj), LtC  < 8% (NL(a),L){ju) LPV  S9 (NL(a), LPV  Vw G [0, oo) implies that  + 5+ < b , (ju),  \/u G [0,oo).  L c  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),  The •  a € fl and a linear model L, then:  [NL(a), C] is stable for all controllers C satisfying b c(ju)) > 8 ^ {NL{a), 1  Li  n  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  •  4.2. Formulating The Closed-loop Nonlinearity Measure  C H A P T E R 4.  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 y (NL(a) L) = ,  n  )  sup, 8^ (NL(a), L)(ju>) and b c — inf fe^cfju;). A less computation intensive robust n  L>  w  stability result is given in the next corollary.  Corollary 4.2.1. Given a linear model L and a controller C that is designed base L. Define S^ (NL(a),L) = sup, S^ (NL(a), L)(ju) and b , = i n f & ( > ) , then: m  m  L  c  w  ilC  [NL(a), C) is stable for all a E CL satisfying 5^ (NL(a), L) < b . m  L<c  Proof. From the definition of 5? (iVL(a), L) and b , it is clear that 5£ (IVZ,(a), L) < m  m  LiC  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  the stability result whenever 5^ (NL(a), L) > b c- Figure 4-3 shows the ambi n  Lt  arising when S ^ (NL(a), L) is greater than bi c- Evidently, under this circumst 1  n  t  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% (NL(a),L)(3u) m  L,C  H  Frequency, ui Figure 4.3:  An ambiguity arising from applying Corollary 4.2.1 whenever In this case, 5 ? ( N L { a ) , L ) is larger than b . 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 ' ^ ( N L ( a ) , L)(ju>) is smaller than b^cij^) over the entire frequency range. By invoking Theorem 4.2.1, the corresponding closed-loop is, in fact, stable. 5? (NL{a),L)>b . m  m  LiC  LiC  7 l  4.3  Theoretical Motivation  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 M e t r i c 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 61  CHAPTER  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 characterized by the following state space equation  ^j&  = A{a)x(t),  aefl  (4.3.2)  the system is said to be extended quadratic stable (Q stable) if there exists (3) a real e  differentiable positive-definite matrix function Q{a) = Q (a) T  ^Q(a) dt  + A(a) Q(a) T  + Q(a)A(a)  < 0,  > 0 such that  Va G Cl.  (4.3.3)  Lemma 4.3.1. Any Q stable system is exponentially stable, if 3 constants a, b > 0 e  such that a ( $ ( t , r ) ) < ae~ h{t  Q  T)  VaGO  where $ ( t , r ) denotes the transition matrix for Eq.(4-3.2) a  Proof, see (Wood, 1995, pg. 16)  •  62  CHAPTER  4.3.1 The v-Gap Metric For Quasi-LPV Systems  4.  Definition 4.3.2 (Q stabilizable). The quasi-LPV system given in Eq.(4.3.1) e  to be Q stabilizable if 3 a continuous matrix function F(a), e  is said  such that the following  system is Q stable Va G fi e  dx(t) = {A{a) + dt  B(a)F(a)}x(t).  Definition 4.3.3 (Q detectable). The quasi-LPV system given in Eq.(4.3.1) e  to be Q detectable if 3 a continuous matrix function H(a), e  is said  such that the following  system is Q stable Va G fi e  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, Q stabilizable and Q detectable state space realization e  e  P(a) :-  Note, in this thesis, P(oi) and NL(a)  A{a)  B(a)  C(a)  D(a)  are used to denote the linear quasi-LPV sys-  tems. Let F(a) and H(a) be continuous matrix functions such that B(a)F(a)}x(t)  and ^  = {A(a) + H(a)C(a)}x{t)  = {A(a) +  are Q stable Va G fi and define e  (dropping the a dependence, for notation simplicity):  N 1  v  -N  a  n  Y 1  a  A+'BF  B  -H  C + DF  D  I  F  I  0  A + HC  H  -(B + HD)  F  0  I  C  /  -D  63  (4.3.4)  (4.3.5)  C H A P T E R 4.  4.3.1 The v-Gap Metric For Quasi-LPV Systems  then Na  X  Y  M  -N  A  a  Ya  1 y  A  1  M  a  (4.3.6)  = I  X  a  A  Proof, see (Wood, 1995, pg. 149)  •  Definition 4.3.4 (Contractive right coprime factorization). Let N and M have a  a  the same number of columns. The ordered pair [N , M ] represents a contractive right a  a  coprime factorization (crcf) of P(a) over the ring SQ, if 2  1. P(a) =  NaM' ; 1  2. 3 X , Y e SQ such that X N A  A  3. [N£ M j ]  a  T  + YM  a  a  a  = I;  is a contraction in the following sense  sup  sup  (4.3.7)  | | [ £ H | < 1  aeft { e^f :||u||2<i} +  u  2  Definition 4.3.5. Define the contractive right graph symbol G  a  : Jzf  + 2  — ( > Jz? ® Jz? of +  2  +  2  an LPV system P{ct) as follows  Ga : = [ £ ] , where [N  a)  (4.3.8)  M ] is a crcf of P(a). a  Remark 4.3.1. It is obvious that G generates the set of all stable input-output pairs a  of the LPV system P(ct) by allowing G  a  to act on the whole of J z ? . 2  +  Theorem 4.3.1 (Quasi-LPV Graph). Let P(a) have a continuous, Q stabilizable e  N o t e t h a t a ring (&,+,•) m u l t i p l i c a t i o n , s u c h that: 2  1.  is a set 3£ together w i t h t w o b i n a r y o p e r a t i o n s , a n a d d i t i o n a n d a  is a n a b e l i a n g r o u p (i.e. a g r o u p for w h i c h the elements c o m m u t e . N a m e d after Niels H e n r i k A b e l , 1802-1829, a N o r w e g i a n m a t h e m a t i c i a n ) ;  2. the m u l t i p l i c a t i o n is associative (i.e. x ( y z ) —  (xy)z)\  3. the m u l t i p l i c a t i o n is d i s t r i b u t i v e (i.e. ( x + y ) z = x z + y z a n d z ( x + y) = z x + z y V x , y, z € ^*); 4. there exists a n i d e n t i t y element (i.e. e x — x = x e V x G 5 T ) .  64  4.3.1 The v-Gap Metric For Quasi-LPV Systems  C H A P T E R 4.  and Q detectable realization, then a contractive right graph symbol of P(a) is e  BS- *  A + BF G :—  1  C + DF DS' *  s- *  F where F = -S~ (B X 1  + D C),  T  1  S = I + D D,  T  l  (4.3.9)  1  a  T  R = I + DD  T  and X  is a solution of  x  the generalized control Riccati inequality (GCRI) dXi  + (A-  BS- D C) X 1  T  +X,(A-  T  l  -X BS~ B X l  l  + C R~ C < 0  T  1  BS~ D C)  T  1  (4.3.10)  T  \/aeQ  1  Proof, see (Wood, 1995, pg. 150)  •  R e m a r k 4.3.2. The results, as stated here, are for the right coprime factorizatio The dual results can be easily obtained for the left coprime factorizations.  R e m a r k 4.3.3. Analogous to Vinnicombe (1993), the quasi-LPV graph in Eq.(4-3. used in the sequel to define the corresponding quasi-LPV v-gap metric. The quasi-LPV z^-gap metric can be defined as follows: Definition 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 IIG^GJIoo 6Q  L p  v p {  { a i ) ) P { a j ) )  if det(G* .G )(jcu) ^ 0 V w 6 (-oo, oo)and a  ai  wno det(G* .G )(jcj) = 0, Va*, ctj e fl  : =  a  1 where G  ai  and G  aj  ai  otherwise  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  = 5 whenever oti,ctj are frozen. Together with the bp c, the following U  t  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 scheduling parameter P(ctj),  G  fl  and a controller C, then: [P(a^),C] is stable for all plants  Va, G fl if and only if 6^ (P( ), LPV  ai  P( ))(ju) aj  < b  {juj),  Va; G [0,oo).  P{a%)tC  Proof. Since 5®  = 8 whenever a^a^ are frozen, the proof follows from that of  Vinnicombe  Theorem  LFV  (1993),  V  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^ (juj) tC  (ju) Vctj efl,Vu>£  > 5® (P(ai), LPV  [0, oo).  Proof. Also see Vinnicombe ( 1 9 9 3 ) , Theorem 4 . 5 .  4.3.2  P(a.j))  •  T h e 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. I n this regard, a quasi-LPV model is obtained by freezing the scheduling parameter at a particular operating point, says ctj. 3  66  CHAPTER  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(a )) 0  Figure 4.4: A graphical interpretation of Assumption 4.3.2  is that the i^-gap between NL(ai)  implies that S (NL(cti), U  N'L(a.i ±  4.3.2.  and NL(oci ± e) must be sufficiently small. This  e)) < r  —> 0 . +  Therefore, the following relationship,  depicted in Figure 4.5, can be established:  CC7(ft)  In addition, if N —> oo, then G ^Ui=i  ~ G{fl)-  (4.3.12)  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 r. Assuming  G fl and a positive  real number  ( J i = i ^ — ^> that fli intersects with all its adjacent scheduling 67  subspaces  4.3.2 The Frozen Point Nonlinearity Measure  C H A P T E R 4.  Figure 4.5: Graph spaces of a frozen-point quasi-LPV system linear model (<? ).  (Gi{tii))  and that of a  2  flj, forijt  j = 1,...,N,  implies that G(ai) Proof.  —•  and S (NL(a.i),  £7(fij).  u  N'L{pn ± e)) < r, Va* ± e G  /n addition, if-N — > oo, t/ien  (7 (Ui=i^)  The proof follows the above discussion.  Then r  0  ~*• ^(^)-  •  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 partitioned. 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.  8 {NL{cti), u  adjacent frozen points of af).  6 8  NL(aj)),  where j denotes all the  CHAPTER 4.  4.3.2 The Frozen Point Nonlinearity Measure  5. Check if6 (NL(oi), v  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 jthfrozen points. Then, check 5„(iVX(ai), NL(ay)) < r and 5 (NL(ctj>),NL(ctj)) < v  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% (NL(a),L)  Va e  = mf max{7%(NL(a),L)Ji (L,NL(a))}  LPV  A  l  (Jfi*  (4.3.13)  i=i  where -Z*(NL(a) L)±  sup  t  inf  — ^  NL(a)ui  r  »  i  J e S i O J . ^ n , ) ^ Uu2]s&as«b  ^ \[ II T  [n  (4.3.14)  ff"  )u  Ul  111  II [NL(a)  \\\  Ul  2  and 1«{L,NL{«))±  sup  inf IIL^J Ly^Jlb [jvLWuxjeeiCUiliniJn^, lllz* JII  [^92^2  2  (  4  3  1  5  )  2  Finally, from Figure 4.5 and Assumption 4.3.2, it can be shown that the following equation holds. .N QLPV  S  "  ( (^) NL  a  L  =  i  n L  f  s  u  p  5  u  (NL(oi), L ) , Va G I I  eAi=i,...,Ar '  a  (4.3.16)  ~ 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 nonlinearity measure when the union of the scheduling space is considered. 69  CHAPTER  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 continuous 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 requiring 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 a to cti — a + e. If a homotopy condition exists for the graphs of the two closed-loops, then one can infer that the second closedloop 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. 0  71  0  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 measure. 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 characteristics is maintained. In conclusion, to cope with time variations of the scheduling parameter, the following additional conditions need to be satisfied: • Assuming that  (Ji=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  8 (N L(a.i), L) U  to account  for time variations. Remark  4.3.4. The amount of 5+ really depends on system's dynamics.  purpose, based on the author's experience, a value corresponding to 20%  For practical  o/supj 8 (N U  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 nonlinearity 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 operating 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 P i (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 (similarly, the quasi-LPV 0  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 and PQ). Thus, care must be taken to reduce the conservatism of the 0  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: D e f i n i t i o n 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. ^  e.fl,  S^  P V  (P(  A I  = sup^  ) , P( )) AJ  Vo; € [0, oo)). Repeat for all on 6 fl.  L W  ( P ( a ) , P(ajWu)> i  V  a  i^  The best nominal model is then defined as  follows:  (4.3.17)  R e m a r k 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.  73  4.3.5 The Linearizing Effect of Feedback  C H A P T E R 4.  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 ) + ^-(e 0  - e) + ^ ^ ( e 0  - e ) + • • • + HOT 2  0  (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  (4.3.19)  2  By defining x (0) = ^^y, open-loop nonlinearity distortion, we have: OZ/  A(e) ^ -  x (0) + ^ - f i e OL  e  2  (4.3.20)  2  It is obvious that if A(e) is linear (or nearly linear), x (0) i Oi  sz e r o  (  or  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) /  =  dr  dr  dr  dr  dT(r)\ dr J  c  \  A'(e) 1 +  where A'(e') =  (4.3.22)  CA'(e)  ^Jf^.  d  Similarly, the second derivative of T(r) is r  ,  _ d T(r)  d A(e)  2  d A(e)  2  =  dr  dr  2  dr  2  =  dT(r)  2  c  2  dA(e)  c  dr  dr  d T(r) 2  dr  2  "< > (l + CA'(e)) A  (4 3 23) • '• •  e  l4  ;;  i  J,  The closed-loop nonlinearity distortion x ( 0 ) can then be obtained as follows: CL  v  X  C L  r  y  m  =  )  nO)  A"(0)  T'(0)  X (0) Since r = e in open-loop «,  + CA'(O) A'(0)  (1 + CA'(0)) A"(0) 1 A'(0) l-f-CA'(O) X  CL  1 2  O L  (0)  5(0)  = X° (0)-5(0)  (4.3.24)  L  =  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, (i.e. L(0) = CA'(0))  one would try to keep the loop-gam  as large as possible.  Proof. Consider Figure 4.8. Since x ( 0 ) = x° (0)5(0) CL  L  follows immediately.  and 5(0) =  J ^ J ^ ,  the result •  75  CHAPTER  4.  Loop-Shaping and Weight Selection  4.3.6  Loop-Shaping and Weight Selection  4.3.6  Jf?oo  The  loop shaping procedure of McFarlane and Glover (1990, 1992) has proved to  Jffoo  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  w  2  Stage I: Open-loop Shaping  p  w  2  Ps Coo Stage II: Jif^ Robust Stabilization  1  0  ^  a  P  #  b C  W  x  W  Coo  2  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 M ^ controller (i.e. Coo)- The advantages 3  76  4.3.6  C H A P T E R 4.  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 that : 4  o ((/ + PC)" ) , o ({I + PC)~ P) , o ((/ + CP)' ) , a (C(I + PC)' ) 1  l  1  (4.3.25)  1  be made small in some low frequency range, typically UJ  G  (0,0;;).  Lemma 4.3.4, it is also desirable to keep a ((I + PC)~ ) and & ((I + l  Also, according to 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)' ) , a (CP(I + CP))  (4.3.26)  1  be made small in some high frequency range (i.e. UJ G (o; ,co)). The requirements u  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,  cr(CP)  <  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 W PWi. Therefore, the actual loop shape is expected to deviate from the specified one 2  4  See §A.4 and Zhou et al. (1996, § 5.5).  77  4.3.6 M'OQ Loop-Shaping and Weight Selection  C H A P T E R 4.  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  III,  point b), which is given by  PWiCooWjj.  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{PW C W ) 1  00  2  = aCW^WiPW^Wz)  >  g(W2P^l)g(Coo)  c{W )  (4.3.29)  2  and a(CP)  = QLiWiC^w.p)  = a(w c w w wr ) 1  1  00  a  1  where c(-) = ^ | denotes the condition number. depend on  cr(Coo).  >^ P ^ i k ( O o ) c{Wi)  (  4  3  3  0  )  Clearly, Eqs.(4.3.29) and (4.3.30)  The following result, which was derived by McFarlane and Glover  (1990), shows that a ^ ) is bounded by 7 = b  PCoo  78  and a{P ) = s  qiW^PWi).  4.3.6 Jffoo Loop-Shaping  C H A P T E R 4.  Theorem 4.3.4. Any controller, C^,  and Weight  Selection  satisfying  C  n  P.Coo)" ^-  (I -  1  <  1  (4.3.31)  7  where [N , M ] is a normalized left coprime factor of P , and P is assumed square, also s  s  s  s  satisfies o(P (ju))  - V T ^ T  s  for all co such that a(P (ju>)) s  (4.3.32)  > \Ay — 1. 2  Proof. See McFarlane and Glover (1990, Theorem 6.2, pg. 111-114)  It is obvious that when a(P (ju)) s  > ^/  2 7  - 1, ^(C^ju))  asymptotically greater than or equal to as a(P (ju)) ^  > -7==, where > denotes ~ Vr-i ~  -» oo . This implies that g_{PC) >  s  »  •  * 1 (respectively, g_{CP) > - ^ 2 = » ^  1). Note that  c(VKi) and c(W ) are selected by the designer, and are normally close to one. Analogously for a (PC) and a(CP), we have 2  a(PC)  = otPV^CooWa) < a ( W P W ) a ( C ) c ( i y )  (4.3.33)  o(CP)  = aiWiCMP)  (4.3.34)  2  1  2  0O  and  Like g_(PC) and a (CP),  <  o(PC) and a (CP)  o(W PW )a(C )c(W ) 2  1  0O  1  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  cr(Coo).  Theorem 4.3.5. Any controller, C o o , satisfying Eq.(4-3.31),  a(CooO))  <  ^f^l 1-  +  a(P (ju)) s  ^^lo(P (ju)) s  79  also satifies  (4.3.35)  4.3.6 M'oo Loop-Shaping  C H A P T E R 4.  for all u such that a(P (ju>)) < s  and Weight Selection  ^—. V7 -i 2  Proof. See McFarlane and Glover (1990, Theorem 6.4, pg. 114-116) Likewise, if o-(P ) <§;  J-—, then a(Coo(juj)) ;$ Jy  — 1, where % denotes asymptoti-  1  s  • v / 7 —1  ~  2  •  v  ~  cally less than or equal to as a(P ) — > 0. Obviously, a (PC) <^ 1 and a (CP) <C 1. s  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 , W , W , and P. x  Theorem  2  4.3.6. Let P be the nominal plant and let C — WiCooW be the associated 2  controller obtained from the J^,  loop-shaping procedure. Then if  I  (I +  <  PsC^M-  1  Cor  (4.3.36)  7  we have  (4.3.37)  a(C(I + PCY )  <  o((I + PCy )  <  min { a(M )c(M/ ), 1 + a(7V )c(W )  (4.3.38)  a(C(I + PC)-^)  <  m i n [ a ( i V ) c ( W i ) , l + 70 (M )c(V7i)  (4.3.39)  l  l  o(M )o(W )o(W )  1  s  1  7  2  s  2  7  s  2  :  7  a  a  (4.3.40) odl + CPy ) 1  < min ^o(M )c(W ),  1+  <  I+  s  o(P(I + CPy C) l  1  min {jo(N )c(W ), s  2  7 ^ ) 0 ( ^ 1 )  (4.3.41)  jo(M )c(W )  (4.3.42)  s  2  where  o(N ) s  =  o(N ) =  °\Ps) l +  o(M ) s  =  (4.3.43)  s  a(M ) s  ° (P ) 2  s  1  1 + <L (P ) 2  S  80  (4.3.44)  CHAPTER  and  [N ,M ] S  4.3.7  4.  (respectively,  S  [N ,M ]) S  S  of P  =  Stability  factorization  Margin (respec-  W2PW1.  right coprime  Proof.  See Zhou et al. (1996, Theorem 18.11, pg. 493)  s  Best Possible  left coprime  tively,  4.3.7  factorization)  is a normalized  The  •  The Best Possible Stability Margin  Recall that in the developed nonlinearity measure formulation, the closed-loop nonlinearity, 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 linear 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 P . s  bpc,ma.x(P ) s  Defined by  the optimal value can be obtained via the following formula:  — supbp c, Si  c  (4.3.45) where || •• \\ and [g h ] denote Hankel norm and a pair of normalized left coprime H  s  factorization of plant  s  P  s  (i.e.  R e m a r k 4.3.6. Eq.(4-3.45) priate  weightings,  of the application ibility  or conflict  P  s  =  hj g ). 1  s  only depends  on the plant  information.  which  incorporate  of Eq.  (4-3-45). Often time, small bpc,max(P ) indicates  of  the process specifications,  However,  are crucial s  specifications.  81  appro-  to the  success  poor  compat-  CHAPTER  4.4  4.  4.4. A Computational  Algorithm  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 scheduling 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:. QLPV  5  =  {  S  Q L P V  {  p  {  a  i  )  j  p  {  a  3  )  )  j  a  V  3  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 specifications and forms the shaped plant (i.e. L = W L*Wi). s  2  6. Repeat step 3, but applying W\ and W to all P(ai) and P{a ) at this time. 2  3  Obtain the new L* according to step 4 and subsequently the new L . s  7. Compute b ,  PC 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^f  pv  = {5^ {P(ai)C, LPV  V a Gfi}in-  P{OLJ)C),  3  stead, where C = W^iCooM^10. Compute b c,i, where / denotes a unity feedback. P  11. By employing Corollary 4.2.1, the closed-loop nonlinearity is manageable by the designed linear controller if b ,i > ^{P{a)W C W , PC  l  82  00  2  L*W C W ). 1  QO  2  CHAPTER 12.  4.5.  4.  If b ,i  employ Theorem  < ^(P(a)W C W ,L*W C W ),  PC  l  00  2  l  00  2  Summary  Then, the  4.2.1.  closed-loop nonlinearity is said to be manageable by the designed linear controller if bpcAju) 13.  > 8^(P(a)W C W ,L*W C W ){ u )^u l  00  2  1  Likewise, by using Theorem  4.2.2,  00  2  3  e [O.oo).  J  the closed-loop nonlinearity is larger than what  the designed linear controller can cope with if b c,i(juj) P  <  S'^ (P(a)WiC W , n  00  2  L W C W ) ( j a ; ) , V a ; € [0,oo). 1  14.  0 0  2  In addition, if  6p  C  l  max!»  <  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  CHAPTER 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 nonlinear 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  5.1.2 Design Objectives  C H A P T E R 5.  process dynamics are restated here. dC dt dT ~dt A  q & y{C f - C ) - k exp(- — )C  (5.1.1)  +kC +  (5.1.2)  A  k T  f  - T )  A  0  2  A  k (T -T) 1  A  c  with k\ = vUA 'k exp(-^). Where C , T, and T represent reactor c and k = effluent concentration of component A, reactor temperature, and coolant temperature, H  2  P  c  0  A  c  p  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 nonlinearity 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 T . In c  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 C f and ± 5K in A  feed temperature Tf). 4. Closed-loop bandwidth is limited by the actuator bandwidth (i.e. 20 rad/min). 85  5.1.3 Nonlinearity  C H A P T E R 5.  5.1.3  Measure  Nonlinearity Measure  To check the adequacy of a linear control strategy, it is desirable to measure the closedloop 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 T  T  d_ di  CA  T  ~  CA,eq  —T  ; i + /c exp(-^)) 0  dTc,eq  —dr  h. 2  fc  dCA, dT  dT dT , c  dT  CA  ~  CA,eq  T c —T - c,eq ±  1  (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 function , appropriate 1  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.  ^ e e §4.3.5.  87  CHAPTER  5.1.3 Nonlinearity  5.  Frequency  Figure 5.2: Bode diagram for the nominal model at T=341 K  88  Measure  C H A P T E R 5.  5.1.3 Nonlinearity  Next, to shape the open-loop plant (i.e. P = W PQWI) S  SO  2  Measure  that it gives the desirable  closed-loop properties, the following two compensators are applied to the nominal plant: 7830s+ 78300 l y  ' = ^  +  8  0.  +  900-  T I W  ,  > =  , 1  '  ( 5  1 5 )  Note that there are no hidden RHP pole and zero cancellations in the above compensators, 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  10"  1  .  I  1  ,  ,  10°  10  1  ,  i  J  10  2  Frequency  Figure 5.3: Specified (solid) and stabilized (dashed) loop shapes.  89  10  3  CHAPTER  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. M ) is 1.6521, which is slightly larger than the typical value 1.25. Therefore a slightly T  larger total variability (TV) in the output response is expected . However, as long as 2  the overshoot is less than 20%, the output response is acceptable. 10  1  1  1  1  1  •  1 1  '—  10  ..,  1 111  /  /  " \  ^  /  \  /  \  /  \  /  io"  2  \  /  e  •  \  /  10  1  X  /  \  \  : \  : \  10  \  \  : \ \  10  \  1 \ \  V  <  10 10  ; \  10"  10  10  1  • • 1  10  10'  10"  Frequency  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 linear using the compensators 3  shown in Eq.(5.1.5). Alternatively, this means that the radius of the coprime factor Since M pg. 34) 2  T  3  <TV  < (2n + 1)M , T  where n is the order of T , see Skogestad and Postlethwaite (1996,  Recall that the degree of closed-loop nonlinearity is quantified by the f-gap values.  90  CHAPTER  5.1.4 Simulation Results  5.  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 a unity feedback (i.e.  &PC,I)  Pn^iCooW^  with respect to  is 0.31. This value is then compared to the nonlinearity  measure (i.e. S^ (P(a), P )). Since b ,i = inf o,oo) b ,i(M m  0  PC  we[  PC  and 5^ (P(a),P ) 4. m  0  Pu>e[o,oo) <^ (-P( )) Po)(jv), the designed controller, according to the Corollary 4.2.1,  su  m  a  is sufficient to cope with the closed-loop nonlinearity when the plant is pre- and postcompensated, 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 designed robust control is shown. Finally, simulation results of the same system subject to various disturbances (i.e. ± 20% in feed concentration C f and ± 5K in feed temperA  ature Tf, respectively) are presented. The setpoint tracking and disturbance rejection  91  CHAPTER  5.1.4 Simulation  5.  I 300  10  1  310  L—i L LJ I 1 I i 320 330 340 350 Selected nominal at i-th operating point, T.(K)  /  i 360  i_  370  Figure 5.5: Shaped i^-gap contour. The best shaped model (black dot).  92  Results  CHAPTER  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, T (solid). c  93  CHAPTER  5.1.4.2  5.  5.1.4 Simulation  Results  Setpoint T r a c k i n g Responses of C S T R U n d e r R o b u s t C o n t r o l  The setpoint tracking performance of the closed-loop is shown in Figure 5.7. The scheduling 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, T (solid). c  94  5.1.4 Simulation Results  C H A P T E R 5.  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 closedloop 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  5.1.4 Simulation Results  C H A P T E R 5. g  372.0  — I 0)  §  CL  371.0  E  hQ)  o  1 CC  g  370.0 0  2  4  6  8  10  2  4  6  8  10  2  4 6 Time, t (min)  8  10  341.3  H  3  cd  341.2  (5  CL  |  341.1  o  S  341.00  _ 300.008 | 300.004 CL  E  c|c300.0000  4 6 Time, t (min)  4 6 Time, t (min)  Figure 5.8: Left: Closed-loop responses subject to ±20% in feed concentration, C (Top three: +20%; Bottom three: +20%) at three operating points. Reactor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant temperature, T (solid). 96 Af  c  5.1.4 Simulation Results  C H A P T E R 5.  302  S  8. £ 300 H  c ro  10  3 298 _275 5 273 E  o 271  4 6 Time, t (min)  4 6 Time, t (min)  10  4 6 Time, t (min)  10  306  o 302 „278 tf°277 ii  1276 £  ~ 275 re  <r 299.6  3 274  4 6 Time, t (min)  0  Figure 5.9: Left: Closed-loop responses subject to ±5K in feed temperature, T (Top three: +5K; Bottom three: +5K) at three operating points. Reactor temperature, T (solid), setpoint (dashed-dotted). Right: Coolant temperature, T (solid). f  c  97  5.2. Example II: Cone Tank Control Problem  C H A P T E R 5.  5.2  Example II: Cone Tank Control Problem  This section illustrates the usefulness of the developed nonlinearity measure for a multiinput 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 nonlinearity measure works equally well for stable/unstable SISO or M I M 0 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 q has the inlet temperature T c  ci  and outlet temperature T at 289K and 313K, respectively. co  F  Pi i Ii  T-  H  F, T 0  Figure 5.10: A schematic diagram of an inverted cone tank.  98  5.2.1 Problem Description  C H A P T E R 5.  Assuming that the outlet flow characteristic is given by: F  0  (5.2.1)  KV7L  =  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:  * - -w -? • Fi  dh  f  + FA 1  K  ,  -  -  (5 22)  T -  „  ^<?<- )-$IT„-TJ  3J  (5.2.3)  T  I  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 Cone radius Cone height Outlet valve coefficient Inlet stream Coolant inlet temperature Coolant outlet temperature  99  Notation R H K  Value 0.798m lm 50 m t m i n 10m min 289K 313K 3  T T x  CO  _1  -1  5.2.2 Design Objectives  C H A P T E R 5.  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 q . The c  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  T h e Nonlinearity Measure A p p l i e d T o T h e M I M O C o n e 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—, -imax  Fi = „  F i  "i.max  , Fj = J  ^  ',  "^max  q  — —. The scaled version of the inverted cone tank 2s  =  r  °  9c,max  can then be rewritten as follows: dh  _  (F + l  F )F j  dT  =  3(Fi  K  h  -2 + F ^ F ^  (5.2.5)  -=-1.5 ^ - _  _ 3g g c  C|m  ax ^  _ ^  (  5  2  6  )  Next, by selecting the liquid height and the liquid temperature as scheduling parameters,  100  5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank  C H A P T E R 5.  Eq.(5.2.7) shows the resulting quasi-LPV model: h T  d_ It  A(h,T)  F - FQc  h  0 0  T  0 0  +  Fi - F-  V2  0 1  9c, eg  Qc,eq  (5.2.7)  1 0  where Fi,  0 0 A(h,T) =  3Ji,max(^i-^)  0 0 0 0 0 0  0 3q , max c  <Ph ^max i, max V^max F-  Ti  T)  dF  •f)  9q ,  dh  iieq  3g , max {Tpo  dT  d  <Ph ''max  c  _  Tj  (5.2.8)  c  ^ h j ^ ~ c eq  )  c  /]4ax  i,eq  Qc,eg <fh ^max dh  (Tco  dT  .max  dT  By using 20 grid points on the scheduling parameter pair (h, T) and employing the 4  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  (5.2.9)  = h  In this regard, the associated optimal generalized stability margin, bpc, ., for the aforemax  mentioned compensated plant (i.e. P = W PoWi) is 0.5433. Figures 5.11 and 5.12 show s  2  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? (P(a), Po)) is 0.4255. m  4  W h i c h guarantees less than 7% discrepancy from one scheduling subspace to its adjacent ones.  101  CHAPTER  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  CHAPTER  I—I  0.2  5.2.3 The Nonlinearity  5.  1  1—I  0.3  L_J  0.4  »  Measure Applied  i l_ I I i 0.5 0.6 0.7 Liquid Level, h(m)  To The MIMO  I  i  0.8  I  J  0.9  Figure 5.12: Shaped ^-gap contour. Nominal model (black dot).  103  Cone Tank  I 1  CHAPTER  5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank  5.  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'^ (P(a) P ) is less than the b cj (i.e. n  1  0  P  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. Simulation 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  CHAPTER  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  CHAPTER  5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank  5.  Figure 5.15: Servo responses from operating point (0.2,303) to (0.2,333). Plant (solid) and setpoint (dashed-dotted)  0.985 f 0  0  . 1  1  . ,  2  2  . 3  .  1  I  4  0  1  3 4 Time, t (min)  0  1  2  3  4  2 3 4 Time, t (min)  Figure 5.16: Servo responses from operating point (1,333) to (1,303). Plant (solid) and setpoint (dashed-dotted) 106  CHAPTER  5.2.3 The Nonlinearity Measure Applied To The MIMO Cone Tank  5.  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  0  1  2  3  0  1  2 3 Time, t (min)  0  1  2  2 3 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  CHAPTER  5.3  5.  5.3.  Example III: A Fictitious Nonlinear Plant Control Problem  Example III: A Fictitious Nonlinear Plant Control Problem  In this section, a fictitious nonlinear plant is used to illustrate the strength of the closedloop nonlinearity measure in predicting the insufficiency of a linear controller, particularly in anticipating the onset of closed-loop instability.  5.3.1  P r o b l e m 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 scheduling parameter is evolved in the scheduling space. Note that, the sign change characteristic 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 adjacent 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  CHAPTER  ^  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 preand post-compensators are augmented to the nominal plant: W = 4682 x  S  +l  s(s + 30)  and W = I. 2  (5.3.3)  The resulting stabilized sensitivity and complimentary sensitivity functions are shown in Figure 5.20. The associated co-norm of the complementary sensitivity function M  T  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  5.3.2 Nonlinearity Measure  C H A P T E R 5.  110  CHAPTER  5.3.2 Nonlinearity Measure  5.  The resulting  controller obtained via the McFarlane-Glover Moo loop shaping pro-  cedure is as follows: -49.4930 -64.4058 Coo —  32.0000  -7.8613  7.0914  -93.4669 -11.6834  16.4839  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-compensators 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) Grid point, k  0 < t < 10 -0.155  10 < t < 15 -0.1  Figure 5.22 shows the z/-gap metric of  15 < t < 20 -0.05  P(-0.l55,s)W C W 1  oo  2  versus the associated generalized stability margin with respect  20 < t < 25 1 x 10~ 5  t > 25 1 x 10~  3  P(-0.l,s)W C W to [P(—0.155, s)WiC W , and  1  oo  2  0O  2  I}. Since the generalized stability margin is greater than the z^-gap over the entire frequency range, the resulting closed-loop remains stable when the scheduling parameter 111  CHAPTER  5.3.2 Nonlinearity Measure  5.  -5  -4  -  3 2 1 0 Selected nominal at i-th operating point, y.(t)  1  Figure 5.21: Shaped z^-gap contour. Nominal model (black dot).  112  2  UHAPTER  b.  5.3.3 Simulation Results  is evolved from k = —0.155 to —0.1. Similarly, Figure 5.23 indicates that the corresponding 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~ ), the z^-gap values are greater than the generalized stability 5  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~ to 5  1 x 10 ). Note, by assuming that the time variation effect of the nonlinear plant over the -3  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 closedloop stability of the fictitious nonlinear plant when the scheduling parameter is evolved in the sign change region (i.e. [-0.155 1 x 10 ] from t = 0 to t = 30 s), see Table 5.2. 3  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~ , s). Even though the process output seems to be stable 5  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  CHAPTER  5.3.3 Simulation Results  5.  Frequency  Figure 5.22: The u-gap metric of P(-0.155, s)W C W and P(-0.1, s)W C W (solid) versus the generalized stability margin with respect to [P(—0.155, s)WiC W , I) (dashed-dotted) over a frequency range. 1  OQ  2  1  00  2  00  114  2  CHAPTER  5.3.3 Simulation Results  5.  Frequency  Figure 5.23: The v-gap metric of P(-0A,s)W C W and P(-0.05,s)W C W (solid) versus the generalized stability margin with respect to [P(—0.1, s)WiC W , I] (dasheddotted) over a frequency range. 1  oo  2  1  00  115  oo  2  2  5.3.3 Simulation Results  C H A P T E R 5.  Frequency  Figure 5.24: The u-gap metric of P(-0.05, s)W C W and P ( l x 1CT , (solid) versus the generalized stability margin with respect to [P(—0.05, s) W\ (dashed-dotted) over a frequency range. 5  1  116  0O  2  s)W C W L  OQ  COQWI, I]  2  CHAPTER  5.3.3 Simulation Results  5.  Frequency  Figure 5.25: The v-gap metric of P ( l x 1CT , s)W C W and P ( l x 10~ , s)WiC W (solid) versus the generalized stability margin with respect to [P(l x 1CT , s)WiC W ,1] (dashed-dotted) over a frequency range. 5  3  1  OQ  2  OQ  5  OQ  117  2  2  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 I O to 1 x IO ), the unstable response of the closed-loop system becomes prominent. -5  -3  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  i  i  1  4 £  3  -  /  V CO  fx  onset of instability  0 -1 -2  500 0 3 Q.  I  L  10  15  I  1  1  v  |  1  .  1  20  25  30  1  • -  -500 -  O  "o -1000 c  o O -1500 -2000 -  1  1  10  —  1  15 Time (s)  t  20  1  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  5.4. Summary  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  CHAPTER 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 nonlinear control is mandatory. In fact, feedback control is known to modify open-loop 120  CHAPTER  6.1. Con tri b u tions  6.  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 nonlinear 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 presence 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 easyto-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  CHAPTER  6.2. Recommendations  6.  explaining the concept of homotopic relationship of the operator graphs and the causality of the J^f and the J$? spaces. 2  •  2  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 sensitivity 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 determining 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 homotopy 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 discontinuity (or nonlipschitz) functions: The developed technique assumes that the nonlinear equations describing the process 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 integral quadratic constraints (IQCs) can be embedded to the developed nonlinearity measure. A possible way to achieve this is to embed the aforementioned nonlipschitz nonlinearity using a set of IQCs and carry out the rest of the procedures as discussed in this thesis.  123  Nomenclature 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^  inner product spaces or J'I spaces  2  £  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  NOMENCLATURE V  NOMENCLATURE  for all elements in a set if and only if  Ai  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  fl  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  A R X autoregressive with exogenous input model 125  NOMENCLATURE  RHP  right half plane  RSSQ,  RSS  NOMENCLATURE  residual sum of squares  126  Bibliography 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 theory, 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£ are two sets, and if every element of 5£ is an element of SC\, then SCi is a 2  subset  2  of SC\ or 3t C SC\. S£\ and 3Ci are said to be equal, denoted 3C\ = 3C , if and 2  2  132  A. 1.2  APPENDIX A .  only if  « £ i C <2T  2  and SC  union of JTi and ^  2  Denoted by  C  « # i U «2T =  :  2  x  or x €  G #i  Mappings ^2},  the  is the set of elements which lie in SC\ and 3C - The intersection of 2  and 5£ , or 3£iC\ 3£ = {x x E 5£\ and a; G ^ 2 } , is the set of elements which lie in 2  2  both f%\ and SC . The complement of a set 2  «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£ , or 3C\ x i2T  2  2  =  {[«]'• ^ 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  A . 1.2  ^ 2 7^  X  Mappings  Denoted by / : JTi —»  a mapping (or function) from set  of each element of  to a single element of  or @(f), and y  is called the image of x  G i£~2  to <5T is an association 2  Where SC\ is called the domain of / G J2TI .  The range of the function / (i.e.  7£(/) = {f(x) '• x G SC\}) is the set of elements in <^T that are images of elements in 2  8£\.  The graph of / , denoted by Q , is defined as f  Of = {{y=f(x)}-x e  ^1}  A function / : S£\ —> ^ 2 is surjective or onio if and only if every y G S£ is the image 2  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\ —> SC and let 2  C S^i, the restriction of / to S£* is defined as / | ^ r « : iST/ —> ^ 2 . The inverse  133  APPENDIX A.  mapping of / , denoted  A.1.3 Bounded Sets  maps elements of f(SC\) onto SE\ and if / is bijective, then  f~ is a well-defined mapping on the whole of S£ l  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 subsequently the concept of completeness of a metric space is presented.  A.1.4.1  Metric and Metric 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.1.5 Open Set and Topology  APPENDIX A .  A.1.4.2  Completeness  To define completeness, we first need the following definition. A sequence  {x }i° n  is a Cauchy sequence if lim d(x , x ) = 0 (i.e. for any e > 0, 3 an m  n  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  O p e n Set a n d 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  & (x , e) = {x e SE : d(x, x ) < e}. £  0  0  (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 SS (xQ,e) C srf'. e  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)  an open set if every element x G srf is an interior point.  135  is said to be  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 3T = {0, X}, 3T = {</>, {1}, XT}, ST, = {0, {2}, JT} and ^ = {</>, {1}, {2}, SC). X  2  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 <^ (l, §) and 3 § ( 2 , §). These neighborhoods generate the e  £  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 J ) are performed on the robe in order 7  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  APPENDIX A .  A. 1.6 Normed  Figure A . l : Homotopy of / : SE -»• <& and g : SE 2. For every pair  (ei,t>)  with  ei € K  Vector  Space  ^  and v £ Y, there is a corresponding vector  t\v, in such a way that lv = v, ei(e v) = €it v (where e £ K), and such that 2  2  2  C\(v + u) — c\V + c.\U and (e + e )v — e v + e v hold. x  A  2  x  2  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.  l  ||et;|| = |e|||u||  if v £ Y and e £ K is a scalar,  3. \\v + u < v\ + \\u\\ Vu.ti £ 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  special class of Banach spaces are the Hilbert spaces, denoted by  Operator  Jz? , 2  Theory  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,  2.  (u,v)  3.  (eu, v) = e(u,  4.  (v,v)  = (v,u),  w),  where (-, •) denotes the complex conjugate, 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 complement  of ^ C J5f is given by J£ 2  2  Q%  — {he 3£  elements u in the closed Hilbert subspaces  2  n^x)  2  of this decomposition are called  0 Vu £  orthogonal  For any  there exists a unique decomposition, such  that h = ui + u where U i G ^ and u £ 2  '• (u, h) =  the  orthogonal  The associated projections (HV and projections,  and are defined as follows:  In addition, ^ and *% are said to induce a coordinatization if ^ fl <&'- = {0}. L  A.2  L  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  APPENDIX  A.  A.2. Basic Operator Theory  Definition A.2.1 (Linear operator). Given two vector spaces "V\ and %, a functio  P which maps V\ into % is called a linear operator if for all x, y £ ~fi and a £ C, th 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^ —> Y is 2  called bounded if nr. 11  „„ „  llPzll  ||P|| = S U p ||Px|| = S U p ||Fx|| = S U p ——— < OO. ||z||<l  ||z||=l  x#0  where \\ • \\ denotes the norm of P. The set of bounded linear operators which map "V\ to "V is denoted by H{fV\,'V ). In 2  2  the above, it is assumed that the domain of the linear operator is the whole Jzf space. 2  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. Jzf space) that has its images in bounded 2  output space (e.g. Jzf or a Banach space). Mathematically, this means: 2  *2>P  = {u £ Y\ : Pu £  f} 2  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 = |  where  \®pcy ®y  ,i  l  P  denotes the identity operator on 140  2  APPENDIX  A.3. Signals and Systems  A.  Some special types of operators include • self-adjoint operator: P* = P, where * denotes complex conjugate transpose. • normal operator: P*P = PP*. • projection operator: P = P (idempotent) and P* = P (self-adjoint). 2  • 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. Jzf~ and Jzf ) with +  2  2  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.3. Signals and Systems  A P P E N D I X A.  and hence the norm  is defined as ||/|| = y  ||/||2  2  (/,/),  where * and the hat denote  complex conjugate transpose and the Fourier transform of the signals, respectively. 1  Since the Fourier transform is a Hilbert space isomorphism from 2  -S?2(— joo, joo), Jtfp, p  =  it maps  1,2,...,00,  Jzf  + 2  onto  and Jz? ~ onto  M2  2  M?A-  J z ?  2  onto  ( — 0 0 , 0 0 )  (Francis, 1987). Note that  represent the Hardy spaces , which consist of all complex-valued 3  functions f(s) of a complex variable s = C + ju> that are analytic and bounded in the 4  open right half plane (RHP). Clearly,  is a (closed) subspace of Jz? (—joo, 2  joo).  The  corresponding norm is defined as:  ll/lll  = s u p | ^ J°°  (A.3.1)  f (C + ju>)f(C + Ju) ^ } m  where tilde denotes the (bilateral) Laplace transform of the signals f(i). By invoking 5  the  Maximum  it is easy to see that the norm for  Modulus Theorem , 6  can be rewritten  as: 1 1 / 1 , 2  Remark A.3.1.  Throughout  explicitly  distinguish  frequency  domain.  P  1 W  f°° ~ rtiu)fUu)du-  (A.3.2)  this thesis, whenever there is no confusion,  between the notation For instance,  that are defined on positive  A system  =  of functions  we use <Jt\ and «5f  +  2  axis and zeros  in the time domain interchangeably  we do not and in the  to denote signals  otherwise.  is an operator mapping between signal spaces. In this thesis,  the Banach space of matrix-valued functions defined on jui  : u> G E f l 0 0 .  Jzfoo  denote  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. A n isomorphism is a bijective homomorphism. A homomorphism of a group & into a group S' is a mapping <p : & -> 9 <p(ab) = <p(a)ip(b) V a,b£&, see (Grillet, 1999, pg. 13). N a m e d after a famous English mathematician Godfrey Harold Hardy, 1877-1947. /(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. Pierre-Simon Laplace, 1749-1827, a French mathematician. See (Zhou et al., 1996, pg. 97, Theorem 4.3). 2  C  3  4  5  6  142  A.3. Signals and Systems  A P P E N D I X A.  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  .{P^h  IPI  -CDC  \\  o^ue.m IPII2  where o denotes the maximum singular value. The singular values of a matrix A  mxn  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:  T h e o r e m A . 3 . 1 . Let A e K  M  X  N  .  There exist unitary  U =  Ui,U ,  V  v ,v ,  =  1  r  e  ...,v  2  7  e  ...,u ,  2  matrices  n  K  mxm  K  nxn  such that  A = UEV\ E =  Si 0 0 0  where  0 ••• 0  o~ \  0  a  0  0  2  •••  0  and o\ > a > • • • > a > 0, p = min m, n. 2  v  Proof. See (Zhou et al., 1996, pg. 32, Theorem 2.11).  •  Note that cr(A) = o~\ and a (A) = o represent the maximum singular value and the v  minimum singular values, respectively. The condition number of a matrix A is defined 7  See unitary operator in §A.2, pg. 141.  143  APPENDIX  A.4. Feedback Control Theory  A.  as:  ^> S  <^>  4  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  d  u  P  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 L are defined as 0  follows: Li = CP, L = PC, 0  144  A.4. Feedback Control Theory  APPENDIX A .  respectively. The associated input and output sensitivity matrices are  Si  =  (I  +  L i ) '  :  1  dt -»• u ,  S  p  =  0  (I  +  L o ) -  : d ^ y .  1  Since S + T = I, the corresponding input and output complementary sensitivity are given by Ti  =  Li{I  +  L i ) -  1  ,  T  0  =  L (I  +  0  L o Y  1  ,  respectively. Figure A.2 together with the above definitions of senstivity and complementary sensitivity matrices, the following equations can be easily obtained.  y  r - y  u u  p  - n) +  =  T (r  =  S  =  CS (r-n)-CS d-Tidi  =  CS (r  0  S  ( r - d ) + T  0  0  0  P d i  +  n -  S  Sd  (AA.l)  P d i  (A.4.2)  0  0  (A.4.3)  0  0  0  - n) -  CS d  +  0  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 small. Likewise, from Eq.(A.4.4), 0  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 and T small. However, as 0  0  discussed earlier, this contradicts with the closed-loop limitation given by S + T = I. 0  0  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. 5. Coprime Factorization  APPENDIX A .  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, frequency where a(S^) (or respectively a(Sn)) first crosses -j=  UB  is the  = 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 ui , which is defined c  as the frequency where o(L^(ju )) first crosses 1 from above. c  A.5  Coprime Factorization  Any stable or unstable plant P € S&M'oo operators  (i.e. P = NM'  1  oo  c  a  n  be expressed as a quotient of two coprime  = M~ N). l  The coprimeness implies that the two  quotient operators do not contain common RHP zeros, hence no unstable mode polezero 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$?oo such that  X  Y  M  =XM+YN=I  N  146  (A.5.1)  APPENDIX  A.  A.5.  Coprime  Factorization  Note that it is equivalent to saying that [$] 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  (A.5.2)  = MX + NY = 1  Y  Similarly, ones can say that [M fr] is right-invertible in  Recall that P 6 @3foo, and P = NM'  and P = M' N  1  l  3?^^.  denote a right-coprime and  a left-coprime factorizations of P, respectively. The following lemma gives the doublycoprime factorization of P. Lemma A.5.1. For each proper real-rational matrix P there exist eight MJi? satisfying  ^-matrices  the equations  P = NM'  = M^N  X  -Y  M  Y  -N  M  N  X  1  (A.5.3)  Definition A.5.3 (Normalized coprime factorization). A coprime factorization called normalized  coprime factorization  M~M  is  iff  + N~N = I  or MM~ + NN~ = /  147  (A.5.5)  APPENDIX  A.6  A.  A.6. Quasi-Linear Parameter Varying Systems  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: dx A(a)x + B(a)u ~dt y = C(a)x + D(a)u  (A.6.1) (A.6.2)  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 L P V 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 timevarying 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 148  APPENDIX  A.6. Quasi-Linear Parameter Varying Systems  A.  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 presentation 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)  =  Assume that there exist differentiable functions z  eq  0 0  =  4>i ( « )  (f> (a)  +  A (a)  A (a)  A {a)  A {a)  n  and u , such that for every a, eq  a  12  + B(a)u (a) eq  21  2  (A.6.3)  + B(a)u  (A.6.4)  22  In the above, z (a) and u (a) denote a family of equilibrium points obtained by setting eq  eq  the derivatives in Eq.(A.6.3) to zero. Next, by subtracting Eq.(A.6.4) from Eq.(A.6.3), 0 A (a)  d_ a dt z  a  12  0 A {a) 22  z -  + B(a){u -  u (a)) eq  (A.6.5)  z {a) eq  Since dz (a)  da  da  dt'  eq  dt ^ Z  =  (A.6.6)  substituting ^ from Eq.(A.6.5) into Eq.(A.6.6), yields dz {a) eq  dt  dz {a) eq  da  {A (a)(z 12  - z (a)) eq  149  + B (a){u x  -  u (a))} eq  (A.6.7)  A. 6. . Quasi-Linear Parameter Varying System  APPENDIX A .  Next, substracts Eq.(A.6.7) from § in Eq.(A.6.5), dz dz (a) eq  ~E  A (a)-^^A (a)\  dt  22  (z - z (a))  12  eq  B {a) - ^LB (a)] 2  d_ di  +  z - z (a)  -12(a)  :  eq  (A.6.8)  a  12  0 A  z - z (a)  eq  A {a)  0  a  (u - u (a))  1  eq  (u — u, eq(a)) l  5 (a)-^Pi(a)_  (A.6.9)  2  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 u (a) eq  (i.e. u = u + u (a), where u denotes the controller output) as depicted in Figure A.3, c  eq  c  where P, C and T are the nonlinear plant, a controller and a transfer function that r+  C  u + u c  p  a T  Figure A.3: Inner- and outer-loops of a quasi-LPV feedback system. produces the trim condition u (a). Any incorrect estimation of u (a) may jeopardize eq  eq  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 u (a). Thus the resulting quasi-LPV representation can be rewritten as follows: eq  150  APPENDIX  A.l.  A.  a z-  dt  u-  z (a)  0  A (a)  0 A (a)  -  B (a)  l2  22  eq  Complex Analysis: Winding Number  dueq(a)  u (a)  a  x  -^A (*)  d  12  B (a) 2  ^ B ^ a )  ,  eq  +  z-  z (a)  u-  u (a)  eq  eq  (A.6.10)  v  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 realization for all a G 0 throughout this thesis. 8  P(a) =  A(a)  B{a)  C(a)  D(a)  C(a) (si - A)'  1  (A.6.11)  B(a) + D(a  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 g(s).  axis poles of  Then the winding number of g(s) with respect to this contour, denoted by  is the number of counterclockwise  encirclements  wno(g),  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~ ) - n(g); l  8  That is, the pair (A(a), B(a)) is stabilizable and the pair (A(a),C(a)) is detectable.  151  APPENDIX  A.  A. 7.1 The Argument Principle  Figure A.4: A standard Nyquist T contour c. vmo(g~) = -wno(g)  d. wno(l + g)  = 0 if  - rj (g- )  + 770(0);  1  0  g  e  < 1;  and^g^  e. wno det(7 + F) = 0 if F e ^ i f o o and ||F||oo < 1.  where n{G) and i] (G) denote the number of open right-half plane and imaginary axis 0  poles of G(s).  A.7.1  T h e A r g u m e n t Principle  Definition A.7.2 (The Argument Principle). Let T be a closed contour in the complex plane. Let f(s) on T. Assume  f(s)  be a function  analytic along the contour, i.e. 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 of the  origin.  152  has no poles  encirclements  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. P r o p o s i t i o n 3.5.1  The feedback interconnection [P, C] in Figure 3.4 is stable if and only if there ex pair of stable parallel projection operators Tig^gi and ITgi ^g . p  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( ) and QQ, respectively, see also §A.1.6. a  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 E q . (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  APPENDIX  B.  B.2. Proof of Eg. (3.6.6)  Show that 5(^,^2)  where l i n ^ x l l ^  ||oo  and  ||IT^xll^lloo  = ||ru-n^Hoo  (B.2.1)  are both induced norms.  Proof. First we proof an equality which will be useful for the proof in the sequel.  (LU - n.^ )x 2  / • (ru -  (ru + / - n ^ ) ( n ^ - r u j s (B.2.2)  ru)z =  = (n^ + n^±)(n^ - r i ^ j x = n.^(n^ - n^ )x + n^j.(n^ - u^ )x 2  2  = n^n^i-n^n^x  (B.2.3)  Also note that:  iW(n^  - ) x  =  ( n ^ - r u  =  n^(/-n^ )x  x  ) x  2  =  n^IUL*  (B.2.4)  and:  To show that ||n^ -  II^Joo  =  (I - U^Il^x  =  (U  =  -U^U^ x  y/l  -  - U \ ) x -  YL fflj x M  h  U^±l\jc x 2  (B.2.5)  2  = max { l i n ^ n ^ x ^ , l i n ^ x I I ^ J o o } , we need to show  both sides of the equality hence that \\U^-TL^  2  ||oo  < max j || 11,^11 ^x ||oo, \\Ii^±U^  and iin^ - n^iioo > max jiin^n^xiioo, iin^xn^iiooj. 154  2  ||oo|  A P P E N D I X B.  B.2. Proof of Eg. (3.6.6)  Part I: To show that  lln^  -  I i ^ J Loo < S  rnax {lin^n^iioo, | | n ^ x n y | | o o } /2  a  =  f3  ((a-P)x (a-P)x) )  =  x*(a*a  + (3*[3 ~2a*p)x  <  x*(a*a  +  P*P)x  0  a  0  a  X  .  0 P_ X )  13  a  0  0  P_  0 \ ol  2  X 2  n^n^x  o  n^n^  0  (B.2.6)  From Eq. (B.2.6),  — n^,  n  oo  ^ i  <  An  A  Ai  A22  Ai  A2  2  m  and let each A  {i  be an appropriately  n . x n  matrix  •  12  o  (B.2.7)  o  L e m m a B . 2 . 1 . Let A be a block partitioned  A =  ^ x  n  with  A, A  2q  A 13  A, mq  m  dimensioned  155  matrix.  Then for any induced  matrix  B.2. Proof of Eq. (3.6.6)  APPENDIX  B.  the p-norm  is defined as:  \\MP<  Pllllp  U121  ll^2i||  ||^22||  P  II A n l | | p  ll^llp ll^llp  P  || A n 2 | | p  ' '  -  (B.2.8)  mq\\p  Further, the inequality becomes an equality if the Frobenius-norm  Proof.  is used.  See Zhou et al., 1996, pg. 30.  •  From Lemma B.2.1 it is clear that Eq. (B.2.7) can be rewritten as  liru - IL  l|n^j.n^ ||  0 =  Part II: To show that  lliun^.  <  2  0  (B.2.9)  max {|| n ^ n ^ a ||oo, lin^xii^ ||oo  - U^W^ > max { i j n ^ n ^ j . I U , | | n ^ x l l ^ J 2  = ||n^ — n^n^Jioo = P ^ - n^n^Hoo 2  =  1 1 1 1 ^ ( 1 1 ^ — n^ )  <  11  2  || oo  1100 • l|n^i - n^r2|  = •lln.^',  156  —11.^  linn-  (B.2.10)  A P P E N D I X B.  B.2. Proof of Eg. (3.6.6)  Similarly,  lin^n^jioo  =  ||(/ - n ^ j n ^ J i o o  =  l|n^  =  ll ^2  2  — n^n^iioo  n  = ll(rL#  2  <  n^n^Hoo  -  —  n^-jn^Hoo  lin^ — n^Hoo • iin^Hoo  = l|n.^-n^iu. Hence,  - ILrJoo > max |  Obviously, l{JC ,Jfa) x  = max  (B.2.11)  ||n^n^x H^, l i n ^ n ^ H o o j .  j u n ^ n ^ ||oo,  completes the proof.  11Il^rJ-II^r11oo} = P ^ i 2  - ILrJoo. This •  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).  Cl  A Computational Algorithm For The z^-Gap Metric  Consider a plant with the following realization: A.  A  B  C D where P e @  >vxq  D€R . pxq  ^P(s)  = C(sl  - A)' B  + D.  l  with McMillan degree deg(P) < n, A e R  n x n  , B G R , nxq  Ce R  p x n  ,  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  C.2. A Computational Algorithm of J^o Norm  A P P E N D I X C.  A  Bi  x  (a) First obtain the minimal realization of Pi =  Ci  -(A G  1  2  B W Bf 2  where W := (I + P ^ i ) "  C\YC  2  A -  T  °  D  2  2  - BiWDlCif  x  =  1  2  2  .  1  A{ * G )-  B  2  2  (b) Next construct the "A" matrix of (G* G )~ 1  A  and P =  (C.1.1)  B DfYC  2  2  2  and Y := (I + D D'[)~ .  1  1  2  (c) wno det(G^G2) = 0  A ( * ) - i has precisely deg(Pi) eigenvalues with a G  Ga  positive real part. 3. Define eig(A( » )-i) =Number of eigenvalues with positive real part of G  +  G2  (a) If e i g ( A j ( G  G a )  -i)  +  A (  *  G  G  2  ) - i .  ^ deg(P ), 8 {P P ) = 1. 1  V  U  2  (b) If e i g ( A ( * ) - i ) = deg(Pi), 5„(Pi, P ) can be computed as follows G  G 2  2  +  i. If 8„(Pi, P ) is a priori known to be less than UNITY , then 2  8 (Px, P ) can be caculated via the following equation: v  2  Pi  yjl-5 (P^P f v  2  I  -l -l  (I + PZPi)  P  2  I  (C.1.2)  ii. If <5„(Pi, P ) not know, then it can be calculated using the following equa2  tion: ^(Pi.Pa^llGaGJoo.  C.2  (C.1.3)  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 A + BR- D C  r  1  - C 5r C T  7  ~{BR- B  T  1  l  -(A + 159  T  BR~ D C) 1  T  T  (C.2.1)  APPENDIX  where  C.  C.3. A Computational Algorithm of The Graph Symbol G  y  := ( / - D D) 2  T  7  and  S  7  ^  11-P1100 <  Thus, calculating the  :=  7  (  # 7  2 7  /  -  Then for 7  DD ). T  > CT(D),  has no jui axis eigenvalues.  norm of a transfer matrix involves searching for the smallest  Jtffoo  7 for which H has no imaginary axis eigenvalues. 1  C.3  A Computational Algorithm of The Graph Symbol Gi  1. Solve the following Generalized Control Algebraic Riccati Equat (A, - B^DldYX  where R  x  + X{A  X  := I + D D? X  - B^Djd)  - XB S{ B X l  1  T  ion: + CfR^Ci  =0  (C.3.1) Note that the minimality of  and Si := I + DjD . x  [Ai, Bi, Ci, D{) is sufficient to ensure that there exist unique solutions X = X >0. r  2. Define a generalized control gain, F as  F:=-S{\D Ci+B X) T  T  (C.3.2)  3. G i can then be obtained from the following equation:  Gi =  Ai + BiF  Ni  X  Ci + DiF DiS~  Mi  F  for some unitary matrix U €  B S~  R . qxq  160  5-1 u  (C.3.3)  APPENDIX  C.4  CA. A Computational Algorithm of The Graph Symbol G  C.  2  A Computational Algorithm of The Graph Symbol G  2  1. Solving the following Generalized Filtering Algebraic Riccati Equation: {A - B S7 DlC ) Z l  2  2  + Z(A - B S- D^C )  T  - ZB S BlZ  1  2  2  2  + C^Rr C = 0  x  2  2  1  2  2  (C.4.1) where R  2  := I + D D\ 2  and S  := I + D\D .  2  Note that the minimality of  2  (A , B , C , D ) is sufficient to ensure that there exist unique solutions Z = 2  Z  2  2  2  > 0.  T  2. Define a generalized filter gain, H as H := ~(B D2  + ZC )R ~ T  2  2  (C.4.2)  l  2  3. G can then be generated from the following equation: 2  G = -M 2  N.  A + HC 2  2  UR^C  2  for some unitary matrix U G R  p x p  -H  B + HD  -UR-*  UR^D  .  Note, for the sake of simplicity, U = I can be assumed.  161  2  2  2  (C.4.3)  Index Symbols T distribution function X distribution function 2  F  11 11  feedback function  A anticausal signal Arrhenius equation ARX  G generalized stability margin 32 graph 26, 27, 133 inverse 30 of a controller 27 greatest common divisor 46 greatest lower bound 134  see signal 2 11  B Banach space bandwidth Bezout identity bijective bounded from above bounded from below bounded operator bounded set  see space 146 147 see mapping 134 134 see operator 134  H  Hilbert space see space, see space homomorphism 142 homotopy 137 homotopy condition 137 I  C  identity operator image index infimum see greatest injective inner product inner product space input space interior point intersection inverse causal inverse graph isomorphism  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 distance function domain  see operator see mapping 137 lower bound see mapping 139 see space see space 135 see set see mapping 29 see graph 142  K  11 134 27, see mapping  kernel  see operator L  least upper bound supremum limit cycle  E  eigenvalue  29 see mapping  143  162  134 134 3  INDEX  INDEX  linear operator loop shaping lower bound  see operator 32 134  orthogonal complement orthogonal projection output space  M  R range see mapping RESET 11 residual 11 residual sum of squares 11 resilience index . . . . see Morari resilience index restriction see mapping  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  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  N neighborhood noncausal signal norm norm topology normed space null set Nyquist contour O one-to-one one-to-one and onto onto open set operator bounded linear identity kernel of linear ordered pair  139 139 see space  135 see signal 138 see topology 138 see set 151 see mapping see mapping see mapping see set 140 ,27 140 140 133  T  topological space topology 163  136 136  INDEX  INDEX  norm weakest  138 136 U  union upper bound  see set 134 V  vector zero vector space subspace vector subspace Vinnicombe metric  137 137 137 138 see vector space see metric  W  weakest topology well-posed winding number  see topology 29 137, 151 Z  zero vector  see vector  164  

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