Control Theory for Engineers

Brigitte d’Andréa-NovelMichel De Lara

Control Theory for Engineers

A Primer

123

Brigitte d’Andréa-NovelCentre de RobotiqueMINES ParisTechParis Cedex 06France

Michel De LaraCERMICSEcole des Ponts ParisTechMarne la Vallée Cedex 2France

ISBN 978-3-642-34323-0 ISBN 978-3-642-34324-7 (eBook)DOI 10.1007/978-3-642-34324-7Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013931751

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Foreword

We are surrounded by electronic devices: mobile phones, computers, traffic lightregulators, etc. Many of them work automatically with inputs provided by sensorsscrutinizing the environment. Plane trips contain less and less manual drive.Electrical systems are managed with automatic circuits ensuring stability. Exam-ples abound, and our technological society will use more and more such automaticdevices under the pressure of various factors. Technological innovation and costsreduction increase the penetration of so-called ‘‘smart’’ devices. The worldincrease in demand for communications technologies and for energy requires moreand more coordination in a global economy. Environmental protection fosters theneed for soberness in the use of resources, hence of an optimized management.Control Theory is at the heart of Information and Communication Technologies ofcomplex systems; it can contribute to answer such challenges.

We aim at providing students and engineers with basic tools and methods totackle dynamical systems control problems. The book is organized as an engineerclassically proceeds to solve a control problem, that is elaboration of a mathe-matical model capturing the process behavior, analysis of this model, and design ofa controller to accomplish desired objectives.

Central to Control Theory are the notions of feedback and of closed-loop. In hisautobiography, Eye of the Hurricane (see [29]), the famous applied mathematicianRichard Bellman recalls how he was led from open-loop solutions of controlproblems (‘‘functions of time’’) to closed-loop solutions (‘‘policies’’): Again theintriguing thought: A solution is not merely a set of functions of time, or a set ofnumbers, but a rule telling the decision maker what to do; a policy.1 The notion offeedback, as a rule mapping the state space (and the time) onto the control space, isdeveloped throughout this text.

As with heat regulation by a thermostat, many systems are regulated by feed-back and the way to elaborate such feedback controllers needs some appropriatemathematical tools which are highlighted here. This is why we are particularly

1 To make these notions more concrete, consider traffic regulation. Fixed cycle gears for trafficlights are examples of open-loop controls, whereas closed-loop traffic lights controls adapt theirtiming and phasing according to current traffic conditions identified by means of sensors.

v

interested in systems which can be represented by a mathematical model whichembodies the reactions to external inputs.

The book is divided into three Parts.In Part I, we start by exploring a graded approach of modeling in Chap. 1. We

shine the spotlight on some general principles such as mass and energy conserva-tion laws, illustrated by many examples coming from various fields (electricalengineering, mechanics, chemistry, biological processes, etc.). In the process ofelaborating a mathematical model for control purposes, we highlight particulardynamical structures and related classes of variables, like state, control, and outputvariables. The notion of state of a system is emphasized, namely a finite number ofquantities which, being known at a given time instant, allow us to determine thefuture evolution of the system. A state is a summary of past history, sufficient forprediction. From a mathematical point of view, we focus our attention ondynamical systems whose main properties are highlighted in Chap. 2. Then, weshift our gaze from internal to external mathematical descriptions. Chapter 3 isdevoted to the frequency-domain approach and the associated input–output rep-resentation, in which the system is embraced as a sort-of ‘‘black box’’ reacting to aset of input signals. Graphical representations are used in the case of a scalar system(one input, one output), highlighting natural definitions of robustness degrees suchas gain or phase margins. Let us emphasize that the input–output approach is welladapted to the case of systems for which it is difficult, or sometimes impossible, toobtain mathematical dynamical models from physical laws.

Part II is devoted to system analysis to ensure stability in the neighborhood of aset point, a classical problem in control science (take off of a rocket, unstablechemical reactor. . .). An equilibrium point of a dynamical system represents asteady state of the system’s model, and Chap. 4 is dedicated to their stabilityproperties. From a practical point of view, the local stability property is expressedfrom a simpler model than the original one, namely the linearized or tangent modelaround the equilibrium point, which constitutes the first-order approximation. Wewill show how control laws can be elaborated from this linear model.

The difference between a knowledge model—devoted to better understanding,at the price of a detailed description of the ‘‘piece of reality’’ under scrutiny, andgenerally nonlinear—and a control model—simpler, often linear and used toelaborate a control law—is emphasized.2

2 As an illustration of the differences between knowledge and control models, let us consider theclimate change issue. Climatologists develop large models to improve their knowledge of theclimate mechanisms under increases in greenhouse gas emissions. Such models, embodied incomputer codes, integrate a huge number of physical and chemical relations. They are oftenobtained from discretizing partial differential equations on a planetary grid: the finer the grid, thesharper the model. The mythical 1/1 map may be seen as the ultimate form of such knowledgemodels. Now, climate change economists also develop models. However, the most basic of thesemodels have a few lines and are written on a spreadsheet. Such a difference stems from differentperspectives. Economists do not care to understand and embrace the physics of climate change,but they do care in how to decide about the issue: should we start reducing now (at a sure costnow, for possible, but uncertain, benefits tomorrow), or should we wait for more information and

vi Foreword

Both knowledge models and control models have their part in Control Theory.Whereas control laws or policies are designed on the basis of a control model, theyare tried and simulated on the more precise knowledge model.

To solve a local stabilization problem of a system described by a model havinginput variables (controls or disturbances) and output variables (measurements), auseful approach consists in computing first its tangent linearization at the desiredequilibrium set point. Doing so, we reveal the model skeleton tailored for stabil-ization purpose. In Chap. 5, the structural controllability and observability prop-erties of this tangent linear system are highlighted. The controllability propertymeans that the system can be driven from an arbitrary state to another one bymeans of the control inputs. The observability property expresses that it is possibleto reconstruct the entire state of the system only from the past history of partialknowledge of the state vector, that is, the output variables or measurements. Weprove that a controllable and observable system can be locally asymptoticallystabilized by means of a linear feedback control law. This study is done in Chap. 5for continuous-time dynamical systems and in Chap. 6 for discrete-time systems.The latter, represented by difference equations, are of great importance from apractical point of view, since the digital character of computers implies that thecontrol is fed into the system only at discrete instants. We show how to discretizecontinuous-time linear systems at a given sampling period. In Chaps. 5 and 6, weilluminate the links between state space and input–output representations,respectively, in continuous and discrete time.

The closed-loop system is characterized by its stability and precision propertiesand some sort of compromise must be achieved. The quadratic synthesis casted inChap. 7 proposes a benchmark to display the tradeoffs, under the form of a qua-dratic intertemporal criterion. Moreover, we introduce a probabilistic frameworkto take into account perturbations and measurement errors as random variables, sothat the estimation problem can be solved as a linear filtering problem whosesolution is the well-known Kalman-Bucy filter.

At the end of Parts I and II, the local stabilization problem is solved, at leasttheoretically. However, some degrees of freedom are left in the control law whichcould be used to deal with transient behaviors. Indeed, besides stability, the controlshould also guarantee some robustness. In other words, the control should preservethe stability property in spite of modeling errors or unknown disturbances, such asbias on sensors or actuators, oldness of components, etc. Some answers have beengiven in Chap. 3 in the scalar case and will be developed in Part III.

More precisely, to take into account the effect of disturbances, the state spaceapproach consists of adding some state variables to estimate the perturbations;

(Footnote 2 continued)postpone costly decisions? Such a formulation is a caricature of the climate change debate,discarding details, and emphasizing other traits: economists say their models are fables. Many‘‘fables models’’ have been developed to try and grasp the key economic features of the climatechange issue, thus shedding light onto decision making: time preferences, discount rate, riskaversion, uncertainty, learning, etc.

Foreword vii

however, this increases the complexity of the control law. In Chap. 8, we introducethe polynomial approach enabling us to obtain suitable polynomial representationswhich are well adapted to express the links between the outputs and the distur-bances. In the multivariable case, after stabilization, degrees of freedom are left inthe control law, whereas they could be used for disturbance rejection withoutincreasing the complexity of the feedback control law.

Throughout the text, different examples are developed, both in the chapters andin the exercises. The inverted pendulum on a cart deserves special mention. Thistoy problem runs throughout the book because it is a prototype of a naturallyunstable system that can be stabilized with the different techniques that we exposein the book.

We would like to acknowledge our institutions, Mines ParisTech and École desPonts ParisTech, respectively, for the confidence and the liberty that they grant us.This book has benefited from valuable comments from close colleagues, with aparticular mention to Jean-Philippe Chancelier. Thanks to him, we have been ableto elaborate a series of computer practical works http://cermics.enpc.fr/scicoslabwith the free scientific software Scicoslab that he has largely contributed todevelop [16]. We feel honored that Professor Jean-Michel Coron has accepted tointroduce our book, and we warmly thank him.

References

1. S. Dreyfus, Richard Bellman on the birth of dynamic programming. Oper. Res. 50(1), 48–512002

2. S. Campbell, J.-P. Chancelier, R. Nikoukhah, Modeling and Simulation in Scilab/Scicos withScicosLab 4.4, 2 edn. (Springer-Verlag, New York, 2010)

viii Foreword

Preface

Control Theory is one of the most important branches of engineering science. Thereare already good books on this theory. But this book is an outstanding one. Theauthors, within less than 280 pages, successfully cover the key concepts and resultsof the theory. This includes, in particular, the input–output representation, Routh’scriteria, the PID compensator, gain and phase margins, stability and asymptoticstability, Lyapunov functions, controllability, linear state feedback, observer, out-put regulator, quadratic optimization, Kalman—Bucy filter, polynomial represen-tation, and disturbance rejection. The book deals with linear and nonlinear controlsystems and the control systems may be continuous or discrete in time.

I appreciate very much the excellent balance between the state space and thefrequency-domain approaches. These two approaches are very important, but, inall the books I know in Control Theory, one of the two approaches is alwaysessentially ignored. Here, one clearly sees the usefulness of each approach andhow they can be used together. A special emphasis is put on modeling, which isindeed a crucial step for the application of Control Theory to real life. The proofsof the theorems and propositions are very clearly detailed. The notions and resultsare driven from applications. Their importance is illustrated with numerous andtutorial examples borrowed from various domains of engineering and science, as,for example, fluid mechanics, thermodynamics, classical mechanics, continuummechanics, chemistry, electricity, and biology. The inverted pendulum on a cart,which is present throughout the whole book, allows the reader to grasp quickly thebig picture on many control issues. At the end of each chapter, there are illumi-nating exercises. They are also issued from various domains of engineering andscience. They show the power of the concepts, results, and tools described in thisbook.

This book is primarily intended for engineers. However, it is also very inter-esting for many other scientists, including, mathematicians, physicists, chemists,and biologists.

In conclusion, I cannot praise this book too much. This is a must have text.It will become soon a classical textbook in Control Theory.

Professor Jean-Michel CoronSenior member of the Institut universitaire de France

ix

Contents

Part I Modelling, Dynamical Systems and Input-OutputRepresentation

1 Basics in Dynamical System Modelling . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Balance Equations and Phenomenological Laws . . . . . . . . . . . 3

1.2.1 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Phenomenological Laws . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Basic Laws and Principles of Physics . . . . . . . . . . . . . . . . . . . 61.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Principles of Thermodynamics. . . . . . . . . . . . . . . . . . 71.3.3 Point Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.4 Electromagnetism Equations . . . . . . . . . . . . . . . . . . . 8

1.4 Applications in Solid Mechanics, Fluid Mechanicsand Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.3 Elementary Models of Electrical Circuits . . . . . . . . . . 16

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Finite Dimensional State-Space Models . . . . . . . . . . . . . . . . . . . . . 172.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Definitions of State-Space Models . . . . . . . . . . . . . . . . . . . . . 172.3 Examples of Modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 The Inverted Pendulum. . . . . . . . . . . . . . . . . . . . . . . 212.3.2 A Model of Wheel on a Plane . . . . . . . . . . . . . . . . . . 232.3.3 An Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Vibrations of a Beam . . . . . . . . . . . . . . . . . . . . . . . . 282.3.5 An RLC Electrical Circuit. . . . . . . . . . . . . . . . . . . . . 292.3.6 An Electrical Motor . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.7 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.8 Growth of an Age-Structured Population . . . . . . . . . . 332.3.9 A Bioreactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xi

2.4 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Linear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Input-Output Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Input-Output Representation . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . 463.2.2 Characteristic Responses and Transfer Matrices . . . . . . 47

3.3 Single-Input Single-Output l.c.s. Systems . . . . . . . . . . . . . . . . 503.4 Stability and Poles: Routh’s Criteria. . . . . . . . . . . . . . . . . . . . 523.5 Zeros of a Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 Controller Synthesis: The PID Compensator . . . . . . . . . . . . . . 55

3.6.1 First-Order Open-Loop System . . . . . . . . . . . . . . . . . 573.6.2 Open-Loop Second-Order System . . . . . . . . . . . . . . . 57

3.7 Graphical Methods: Gain and PhaseMargins—Stability-Precision Dilemma . . . . . . . . . . . . . . . . . . 57

3.8 Lead and Lag Phase Compensators . . . . . . . . . . . . . . . . . . . . 633.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Part II Stabilization by State-Space Approach

4 Stability of an Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . . . 714.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Stability and Asymptotic Stability of an Equilibrium Point . . . . 714.3 The Case of Linear Dynamical Systems . . . . . . . . . . . . . . . . . 734.4 Stability Classification of the Zero Equilibrium

for Linear Systems in the Plane . . . . . . . . . . . . . . . . . . . . . . . 754.5 Tangent Linear System and Stability . . . . . . . . . . . . . . . . . . . 804.6 Lyapunov Functions and Stability . . . . . . . . . . . . . . . . . . . . . 844.7 Sketch of Stabilization by Linear State Feedback. . . . . . . . . . . 894.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Continuous-Time Linear Dynamical Systems . . . . . . . . . . . . . . . . 975.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3 Stability of Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Controllability. Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.2 Systems Equivalence. Controllable Canonical Form . . . 1045.4.3 Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

xii Contents

5.5 Observability. Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.6 Observer-Regulator Synthesis. The Separation Principle . . . . . . 1145.7 Links with the Input-Output Representation . . . . . . . . . . . . . . 116

5.7.1 Impulse Response and Transfer Matrix . . . . . . . . . . . . 1165.7.2 From Input-Output Representation

to State-Space Representation . . . . . . . . . . . . . . . . . . 1185.7.3 Stability and Poles . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.8 Local Stabilization of a Nonlinear Dynamical Systemby Linear Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.9 Tracking Reference Trajectories. . . . . . . . . . . . . . . . . . . . . . . 1225.9.1 Stabilization of an Equilibrium Point

of a Linear Dynamical System . . . . . . . . . . . . . . . . . 1225.9.2 Stabilization of a Slowly Varying Trajectory. . . . . . . . 1235.9.3 Stabilization of Any State Trajectory . . . . . . . . . . . . . 125

5.10 Practical Set Up. Stability-Precision Dilemma . . . . . . . . . . . . . 1255.10.1 Steps for the Elaboration of a Control Law. . . . . . . . . 1255.10.2 Sensitivity to Model Parameter Uncertainty:

Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.10.3 Sensitivity to Input Delay: Stability . . . . . . . . . . . . . . 129

5.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Discrete-Time Linear Dynamical Systems . . . . . . . . . . . . . . . . . . . 1336.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 Exact Discretization of a Continuous-Time Linear

Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3 Stability of Discrete-Time Classical Dynamical Systems. . . . . . 137

6.3.1 Stability of an Equilibrium Point . . . . . . . . . . . . . . . . 1376.3.2 Case of Discrete-Time Linear Dynamical Systems . . . . 139

6.4 Stability of Controlled Discrete-Time LinearDynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5 Controllability. Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.6 Observability. Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.7 Observer-Regulator Synthesis. Separation Principle . . . . . . . . . 1486.8 Choice of the Sampling Period . . . . . . . . . . . . . . . . . . . . . . . 1506.9 Links with the Input-Output Representation . . . . . . . . . . . . . . 151

6.9.1 Impulse Response, Transfer Matrix and Realization . . . 1516.9.2 Stability and Poles. Jury Criterion . . . . . . . . . . . . . . . 1536.9.3 Zeros of a Discrete-Time Scalar l.c.s. System . . . . . . . 1546.9.4 Relation Between an l.c.s. System in Continuous-Time

and the Exact Discretized . . . . . . . . . . . . . . . . . . . . . 1556.10 Local Stabilization of a Nonlinear Dynamical System

by a Control Law in Discrete-Time. . . . . . . . . . . . . . . . . . . . 159

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6.11 Practical Set Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7 Quadratic Optimization and Linear Filtering . . . . . . . . . . . . . . . . 1657.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.2 Quadratic Optimization and Controller Modes Placement . . . . . 166

7.2.1 Optimization in Finite Horizon . . . . . . . . . . . . . . . . . 1667.2.2 Optimization in Infinite Horizon.

Links with Controllability . . . . . . . . . . . . . . . . . . . . . 1697.2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.3 Kalman-Bucy Filter and Observer Modes Placement . . . . . . . . 1717.3.1 The Kalman-Bucy Filter . . . . . . . . . . . . . . . . . . . . . . 1737.3.2 Convergence of the Filter. Links with Observability. . . 178

7.4 Formulas in the Continuous-Time Case . . . . . . . . . . . . . . . . . 1797.4.1 Optimization in Finite Horizon . . . . . . . . . . . . . . . . . 1807.4.2 Optimization in Infinite Horizon.

Links with Controllability . . . . . . . . . . . . . . . . . . . . . 1817.4.3 Asymptotic Observer . . . . . . . . . . . . . . . . . . . . . . . . 184

7.5 Practical Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Part III Disturbance Rejection and Polynomial Approach

8 Polynomial Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.3 Results on Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . 196

8.3.1 Elementary Operations: Hermiteand Smith Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.3.2 Division and Bezout Identities . . . . . . . . . . . . . . . . . . 2008.4 Poles and Zeros. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.5 Equivalence Between Linear Differential Systems . . . . . . . . . . 2038.6 Observability and Controllability . . . . . . . . . . . . . . . . . . . . . . 205

8.6.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.6.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.7 From the State-Space Representation to the PolynomialController and Observable Forms. . . . . . . . . . . . . . . . . . . . . . 2128.7.1 From the State-Space Representation

to the Polynomial Observer Form . . . . . . . . . . . . . . . 2128.7.2 From the Polynomial Observer

Form to the Polynomial Controller Form . . . . . . . . . . 2138.8 Closed-Loop Transfer Functions from the Input

and the Disturbances to the Outputs . . . . . . . . . . . . . . . . . . . . 215

xiv Contents

8.9 Affine Parameterization of the Controller and ZerosPlacement with Fixed Poles. . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.10 The Inverted Pendulum Example . . . . . . . . . . . . . . . . . . . . . . 2188.10.1 Computation of the Polynomial Observer

and Controller Forms . . . . . . . . . . . . . . . . . . . . . . . . 2188.10.2 Computation of the Closed-Loop Transfer

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2198.10.3 Affine Parameterization of the Controller . . . . . . . . . . 2208.10.4 Placement of Regulation Zeros with Fixed Poles . . . . . 222

8.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Appendix A: The Discrete-Time Stationary Riccati Equation . . . . . . . 227

Appendix B: Laplace Transform and z-Transform . . . . . . . . . . . . . . . 233

Appendix C: Gaussian Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Appendix D: Bode Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

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