d'Andréa-Novel / De Lara Control Theory for Engineers

A Primer
2013
ISBN: 978-3-642-34324-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 261 Seiten

Control Theory is at the heart of information and communication technologies of complex systems. It can contribute to meeting the energy and environmental challenges we are facing. The textbook is organized in the way an engineer classically proceeds to solve a control problem, that is, elaboration of a mathematical model capturing the process behavior, analysis of this model and design of a control to achieve the desired objectives. It is divided into three Parts. The first part of the text addresses modeling aspects through state space and input-output representations. The notion of the internal state of a system (for example mechanical, thermal or electrical), as well as its description using a finite number of variables, is also emphasized. The second part is devoted to the stability analysis of an equilibrium point. The authors present classical tools for stability analysis, such as linearization techniques and Lyapunov functions. Central to Control Theory are the notions of feedback and of closed-loop, and the third part of the textbook describes the linear control synthesis in a continuous and discrete-time framework and also in a probabilistic context. Quadratic optimization and Kalman filtering are presented, as well as the polynomial representation, a convenient approach to reject perturbations on the system without making the control law more complex. Throughout the text, different examples are developed, both in the chapters and in the exercises.

Autoren/Hrsg.

D'Andréa-Novel, Brigitte

De Lara, Michel

Inhaltsverzeichnis

1;Foreword;5
2;Preface;9
3;Contents;10
4;Part I Modelling, Dynamical Systems and Input-Output Representation;15
5;1 Basics in Dynamical System Modelling;16
5.1;1.1 Introduction;16
5.2;1.2 Balance Equations and Phenomenological Laws;16
5.2.1;1.2.1 Balance Equations;17
5.2.2;1.2.2 Phenomenological Laws;18
5.3;1.3 Basic Laws and Principles of Physics;19
5.3.1;1.3.1 Conservation of Mass;20
5.3.2;1.3.2 Principles of Thermodynamics;20
5.3.3;1.3.3 Point Mechanics;21
5.3.4;1.3.4 Electromagnetism Equations;21
5.4;1.4 Applications in Solid Mechanics, Fluid Mechanics and Electricity;22
5.4.1;1.4.1 Solid Mechanics;22
5.4.2;1.4.2 Fluid Mechanics;27
5.4.3;1.4.3 Elementary Models of Electrical Circuits;29
5.5;1.5 Conclusion;29
6;2 Finite Dimensional State-Space Models;30
6.1;2.1 Introduction;30
6.2;2.2 Definitions of State-Space Models;30
6.3;2.3 Examples of Modelling;34
6.3.1;2.3.1 The Inverted Pendulum;34
6.3.2;2.3.2 A Model of Wheel on a Plane;36
6.3.3;2.3.3 An Aircraft Model;39
6.3.4;2.3.4 Vibrations of a Beam;41
6.3.5;2.3.5 An RLC Electrical Circuit;42
6.3.6;2.3.6 An Electrical Motor;43
6.3.7;2.3.7 Chemical Kinetics;44
6.3.8;2.3.8 Growth of an Age-Structured Population;46
6.3.9;2.3.9 A Bioreactor;47
6.4;2.4 Dynamical Systems;48
6.5;2.5 Linear Dynamical Systems;52
6.6;2.6 Exercises;55
7;3 Input-Output Representation;58
7.1;3.1 Introduction;58
7.2;3.2 Input-Output Representation;59
7.2.1;3.2.1 Definitions and Properties;59
7.2.2;3.2.2 Characteristic Responses and Transfer Matrices;60
7.3;3.3 Single-Input Single-Output l.c.s. Systems;63
7.4;3.4 Stability and Poles: Routh's Criteria;65
7.5;3.5 Zeros of a Transfer Function;66
7.6;3.6 Controller Synthesis: The PID Compensator;68
7.6.1;3.6.1 First-Order Open-Loop System;70
7.6.2;3.6.2 Open-Loop Second-Order System;70
7.7;3.7 Graphical Methods: Gain and Phase Margins---Stability-Precision Dilemma;70
7.8;3.8 Lead and Lag Phase Compensators;76
7.9;3.9 Exercises;78
8;Part II Stabilization by State-Space Approach;81
9;4 Stability of an Equilibrium Point;82
9.1;4.1 Introduction;82
9.2;4.2 Stability and Asymptotic Stability of an Equilibrium Point;82
9.3;4.3 The Case of Linear Dynamical Systems;84
9.4;4.4 Stability Classification of the Zero Equilibrium for Linear Systems in the Plane;86
9.5;4.5 Tangent Linear System and Stability;91
9.6;4.6 Lyapunov Functions and Stability;95
9.7;4.7 Sketch of Stabilization by Linear State Feedback;100
9.8;4.8 Exercises;104
10;5 Continuous-Time Linear Dynamical Systems;107
10.1;5.1 Introduction;107
10.2;5.2 Definitions and Examples;108
10.3;5.3 Stability of Controlled Systems;110
10.4;5.4 Controllability. Regulator;111
10.4.1;5.4.1 Controllability;111
10.4.2;5.4.2 Systems Equivalence. Controllable Canonical Form;114
10.4.3;5.4.3 Regulator;117
10.5;5.5 Observability. Observer;118
10.6;5.6 Observer-Regulator Synthesis. The Separation Principle;124
10.7;5.7 Links with the Input-Output Representation;126
10.7.1;5.7.1 Impulse Response and Transfer Matrix;126
10.7.2;5.7.2 From Input-Output Representation to State-Space Representation;128
10.7.3;5.7.3 Stability and Poles;129
10.8;5.8 Local Stabilization of a Nonlinear Dynamical System by Linear Feedback;130
10.9;5.9 Tracking Reference Trajectories;132
10.9.1;5.9.1 Stabilization of an Equilibrium Point of a Linear Dynamical System;132
10.9.2;5.9.2 Stabilization of a Slowly Varying Trajectory;133
10.9.3;5.9.3 Stabilization of Any State Trajectory;135
10.10;5.10 Practical Set Up. Stability-Precision Dilemma;135
10.10.1;5.10.1 Steps for the Elaboration of a Control Law;135
10.10.2;5.10.2 Sensitivity to Model Parameter Uncertainty: Precision;137
10.10.3;5.10.3 Sensitivity to Input Delay: Stability;139
10.11;5.11 Exercises;140
11;6 Discrete-Time Linear Dynamical Systems;142
11.1;6.1 Introduction;142
11.2;6.2 Exact Discretization of a Continuous-Time Linear Dynamical System;143
11.3;6.3 Stability of Discrete-Time Classical Dynamical Systems;146
11.3.1;6.3.1 Stability of an Equilibrium Point;146
11.3.2;6.3.2 Case of Discrete-Time Linear Dynamical Systems;148
11.4;6.4 Stability of Controlled Discrete-Time Linear Dynamical Systems;152
11.5;6.5 Controllability. Regulator;153
11.6;6.6 Observability. Observer;155
11.7;6.7 Observer-Regulator Synthesis. Separation Principle;157
11.8;6.8 Choice of the Sampling Period;159
11.9;6.9 Links with the Input-Output Representation;160
11.9.1;6.9.1 Impulse Response, Transfer Matrix and Realization;160
11.9.2;6.9.2 Stability and Poles. Jury Criterion;162
11.9.3;6.9.3 Zeros of a Discrete-Time Scalar l.c.s. System;163
11.9.4;6.9.4 Relation Between an l.c.s. System in Continuous-Time and the Exact Discretized;164
11.10;6.10 Local Stabilization of a Nonlinear Dynamical System;168
11.11;6.11 Practical Set Up;172
11.12;6.12 Exercises;172
12;7 Quadratic Optimization and Linear Filtering;174
12.1;7.1 Introduction;174
12.2;7.2 Quadratic Optimization and Controller Modes Placement;175
12.2.1;7.2.1 Optimization in Finite Horizon;175
12.2.2;7.2.2 Optimization in Infinite Horizon. Links with Controllability;178
12.2.3;7.2.3 Implementation;180
12.3;7.3 Kalman-Bucy Filter and Observer Modes Placement;180
12.3.1;7.3.1 The Kalman-Bucy Filter;182
12.3.2;7.3.2 Convergence of the Filter. Links with Observability;187
12.4;7.4 Formulas in the Continuous-Time Case;188
12.4.1;7.4.1 Optimization in Finite Horizon;189
12.4.2;7.4.2 Optimization in Infinite Horizon. Links with Controllability;190
12.4.3;7.4.3 Asymptotic Observer;193
12.5;7.5 Practical Set up;193
12.6;7.6 Exercises;193
13;Part III Disturbance Rejection and Polynomial Approach;197
14;8 Polynomial Representation;198
14.1;8.1 Introduction;198
14.2;8.2 Definitions;201
14.3;8.3 Results on Polynomial Matrices;203
14.3.1;8.3.1 Elementary Operations: Hermite and Smith Matrices;204
14.3.2;8.3.2 Division and Bezout Identities;207
14.4;8.4 Poles and Zeros. Stability;208
14.5;8.5 Equivalence Between Linear Differential Systems;210
14.6;8.6 Observability and Controllability;212
14.6.1;8.6.1 Controllability;212
14.6.2;8.6.2 Observability;216
14.7;8.7 From the State-Space Representation;219
14.7.1;8.7.1 From the State-Space Representation to the Polynomial Observer Form;219
14.7.2;8.7.2 From the Polynomial Observer form to the Polynomial Controller Form;220
14.8;8.8 Closed-Loop Transfer Functions from the Input and the Disturbances to the Outputs;222
14.9;8.9 Affine Parameterization of the Controller and Zeros Placement with Fixed Poles;224
14.10;8.10 The Inverted Pendulum Example;225
14.10.1;8.10.1 Computation of the Polynomial Observer and Controller Forms;225
14.10.2;8.10.2 Computation of the Closed-Loop Transfer Functions;226
14.10.3;8.10.3 Affine Parameterization of the Controller;227
14.10.4;8.10.4 Placement of Regulation Zeros with Fixed Poles;229
14.11;8.11 Exercises;231
15;Appendix AThe Discrete-Time Stationary Riccati Equation;234
16;Appendix BLaplace Transform and z-Transform;240
17;Appendix CGaussian Vectors;245
18;Appendix DBode Diagrams;250
19;References;254
20;Index;257