Introduction to Mathematical Systems Theory. A Behavioral Approach

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Cover Title page Preface How to Use tbis Book 1 Dynamical Systems 1.1 Introduction 1.2 Models 1.2.1 The universum and the behavior 1.2.2 Behavioral equations 1.2.3 Latent variables 1.3 Dynamical Systems 1.3.1 The basic concept 1.3.2 Latent variables in dynamical systems 1.4 Linearity and Time-Invariance 1.5 Dynamical Behavioral Equations 1.6 Recapitulation 1.7 Notes and References 1.8 Exercises 2 Systems Defined by Linear Differential Equations 2.1 Introduction 2.2 Notation 2.3 Constant-Coefficient Differential Equations 2.3.1 Linear constant-coefficient differential equations 2.3.2 Weak solutions of differential equations 2.4 Behaviors Defined by Differential Equations 2.4.1 Topological properties of the behavior 2.4.2 Linearity and time-invariance 2.5 The Calculus of Equations 2.5.1 Polynomial rings and polynomial matrices 2.5.2 Equivalent representations 2.5.3 Elementary row operations and unimodular polynomial matrices 2.5.4 The Bezout identity 2.5.5 Left and right unimodular transformations 2.5.6 Minimal and full row rank representations 2.6 Recapitulation 2.7 Notes and References 2.8 Exercises 2.8.1 Analytical problems 2.8.2 Algebraic problems 3 Time Domain Description of Linear Systems 3.1 Introduction 3.2 Autonomous Systems 3.2.1 The scalar case 3.2.2 The multivariable case 3.3 Systems in Input/Output Form 3.4 Systems Defined by an Input/Output Map 3.5 Relation Between Differential Systems and Convolution Systems 3.6 When Are Two Representations Equivalent? 3.7 Recapitulation 3.8 Notes and References 3.9 Exercises 4 State Space Models 4.1 Introduction 4.2 Differential Systems with Latent Variables 4.3 State Space Models 4.4 Input/State/Output Models 4.5 The Behavior of i/s/o Models 4.5.1 The zero input case 4.5.2 The nonzero input case: The variation of the constants formula 4.5.3 The input/state/output behavior 4.5.4 How to calculate e^{At}? 4.5.4.1 Via the Jordan form 4.5.4.2 Using the theory of autonomous behaviors 4.5.4.3 Using the partial fraction expansion of (l-A)^{-1} 4.6 State Space Transformations 4.7 Linearization of Nonlinear i/s/o Systems 4.8 Recapitulation 4.9 Notes and References 4.10 Exercises 5 Controllability and Observability 5.1 Introduction 5.2 Controllability 5.2.1 Controllability of input/state/output systems 5.2.1.1 Controllability of i/s systems 5.2.1.2 Controllability of i/s/o systems 5.2.2 Stabilizability 5.3 Observability 5.3.1 Observability of i/s/o systems 5.3.2 Detectability 5.4 The Kalman Decomposition 5.5 Polynomial Tests for Controllability and Observability 5.6 Recapitulation 5.7 Notes and References 5.8 Exercises 6 Elimination of Latent Variables and State Space Representations 6.1 Introduction 6.2 Elimination of Latent Variables 6.2.1 Modeling from first principles 6.2.2 Elimination procedure 6.2.3 Elimination of latent variables in interconnections 6.3 Elimination of State Variables 6.4 From i/o to i/s/o Model 6.4.1 The observer canonical form 6.4.2 The controller canonical form 6.5 Canonical Forms and Minimal State Space Representations 6.5.1 Canonical forms 6.5.2 Equivalent state representations 6.5.3 Minimal state space representations 6.6 Image Representations 6.7 Recapitulation 6.8 Notes and References 6.9 Exercises 7 Stability Theory 7.1 Introduction 7.2 Stability of Autonomous Systems 7.3 The Routh-Hurwitz Conditions 7.3.1 The Routh test 7.3.2 The Hurwitz test 7.4 The Lyapunov Equation 7.5 Stability by Linearization 7.6 Input/Output Stability 7.7 Recapitulation 7.8 Notes and References 7.9 Exercises 8 Time- and Frequency-Domain Characteristics of Linear Time-Invariant Systems 8.1 Introduction 8.2 The Transfer Function and the Frequency Response 8.2.1 Convolution systems 8.2.2 Differential systems 8.2.3 The transfer function represents the controllable part of the behavior 8.2.4 The transfer function of interconnected systems 8.3 Time-Domain Characteristics 8.4 Frequency-Domain Response Characteristics 8.4.1 The Bode plot 8.4.2 The Nyquist plot 8.5 First- and Second-Order Systems 8.5.1 First-order systems 8.5.2 Second-order systems 8.6 Rational Transfer Functions 8.6.1 Pole/zero diagram 8.6.2 The transfer function of i/s/o representations 8.6.3 The Bode plot of rational transfer functions 8.7 Recapitulation 8.8 Notes and References 8.9 Exercises 9 Pole Placement by State Feedback 9.1 Open Loop and Feedback Control 9.2 Linear State Feedback 9.3 The Pole Placement Problem 9.4 Proof of the Pole Placement Theorem 9.4.1 System similarity and pole placement 9.4.2 Controllability is necessary for pole placement 9.4.3 Pole placement for controllable single-input systems 9.4.4 Pole placement for controllable multi-input systems 9.5 Algorithms for Pole Placement 9.6 Stabilization 9.7 Stabi1ization of Nonlinear Systems 9.8 Recapitulation 9.9 Notes and References 9.10 Exercises 10 Observers and Dynamic Compensators 10.1 Introduction 10.2 State Observers 10.3 Pole Placement in Observers 10.4 Unobservable Systems 10.5 Feedback Compensators 10.6 Reduced Order Observers and Compensators 10.7 Stabilization of Nonlinear Systems 10.8 Control in a Behavioral Setting 10.8.1 Motivation 10.8.2 Control as interconnection 10.8.3 Pole placement 10.8.4 An algorithm for pole placement 10.9 Recapitulation 10.10 Notes and References 10.11 Exercises A Simulation Exercises A.1 Stabilization of a Cart A.2 Temperature Control of a Container A.3 Autonomous Dynamics of Coupled Masses A.4 Satellite Dynamics A.4.1 Motivation A.4.2 Mathematical mode1ing A.4.3 Equilibrium Analysis A.4.4 Linearization A.4.5 Analysis of the model A.4.6 Simulation A.5 Dynamics of a Motorbike A.6 Stabilization of a Double Pendulum A.6.1 Modeling A.6.2 Linearization A.6.3 Analysis A.6.4 Stabilization A.7 Notes and References B Background Material B.1 Polynomial Matrices B.2 Partial Fraction Expansion B.3 Fourier and Laplace Transforms B.3.1 Fourier transform B.3.2 Laplace transform B.4 Notes and References B.5 Exercises Notation References Index