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