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<p>Preface ix</p> <p>List of Notations xiii</p> <p><b>Chapter 1. Representation of Systems: A Historical Overview 1</b></p> <p>1.1. Transfer functions and matrices 1</p> <p>1.1.1. Transfer functions 1</p> <p>1.1.2. Transfer matrices 3</p> <p>1.1.3. The discrete-time case 6</p> <p>1.2. State-space representation 7</p> <p>1.2.1. Continuous-time state-space systems 7</p> <p>1.2.2. Discrete-time state-space systems 7</p> <p>1.2.3. Controllability and observability 8</p> <p>1.2.4. Poles of as state-space system 11</p> <p>1.2.5. Stability of linear time-invariant systems 13</p> <p>1.3. "Geometric" approach 14</p> <p>1.3.1. Formalism of the geometric approach 14</p> <p>1.3.2. Reachable and non-observable subspaces 15</p> <p>1.3.3. State-feedback controls, observers 16</p> <p>1.3.4. Canonical Kalmande composition, stabilizability and detectability 17</p> <p>1.4. Polynomial matrix description 21</p> <p>1.4.1. PBH test (Hautus criterion) 21</p> <p>1.4.2. Rosenbrock representation 23</p> <p>1.5. The behavioral approach 26</p> <p>1.5.1. Controllability without control variables 26</p> <p>1.5.2. Observability in the behavioral approach 30</p> <p>1.6. Module of a system 32</p> <p>1.6.1. Using modules in control theory 32</p> <p>1.6.2. The Fliessian approach 33</p> <p>1.6.3. Characterization in terms of modules of controllability and observability 34</p> <p>1.7. The formalism of algebraic analysis 35</p> <p>1.7.1. Nature of algebraic analysis 35</p> <p>1.7.2. Oberst's contribution and its consequences 35</p> <p><b>Chapter2. Linear Systems: Concepts and General Results 43</b></p> <p>2.1. Control systems 43</p> <p>2.1.1. The formalism of control systems 43</p> <p>2.1.2. Transfer matrix: general case 45</p> <p>2.1.3. State-space representation of a control system 46</p> <p>2.2. Strict equivalence of Rosenbrock systems 49</p> <p>2.2.1. Admissible Rosenbrock representations 49</p> <p>2.2.2. Strict equivalence 51</p> <p>2.3. Controllability, observability and their duality: the algebraic point of view 55</p> <p>2.3.1. Algebraic controllability 55</p> <p>2.3.2. Algebraic controllability of state-space systems (continuous time) 57</p> <p>2.3.3. Algebraic controllability of state-space systems (discrete-time) 63</p> <p>2.3.4. Algebraic duality 66</p> <p>2.3.5. Algebraic observability and the algebraic duality principle 74</p> <p>2.4. Reachability, observability and their duality: Kalmanian point of view 77</p> <p>2.4.1. Complete controllability of a state-space system (continuous time) 78</p> <p>2.4.2. Complete reachability and controllability of a state-space system (discrete-time) 83</p> <p>2.4.3. Observability of state-space systems (continuous-time) 86</p> <p>2.4.4. Complete observability and constructability of state-space systems (discrete-time) 91</p> <p><b>Chapter 3. Poles and Zeros of Linear Systems, Interconnectedness and Stabilization 97</b></p> <p>3.1. Poles and zeros of continuous or discrete invariant linear systems 97</p> <p>3.1.1. System poles, transmission poles and zeros 97</p> <p>3.1.2. Input-output-decoupling zeros and hidden modes 109</p> <p>3.1.3. Relations between poles, transmission poles and hidden modes 112</p> <p>3.1.4. Invariant zeros 113</p> <p>3.1.5. Dynamic interpretation of different poles and zeros 118</p> <p>3.2. Poles and zeros of interconnected systems 119</p> <p>3.2.1. Diagram of a control system 119</p> <p>3.2.2. System interconnection 120</p> <p>3.2.3. Series interconnection 121</p> <p>3.2.4. Parallel interconnection 126</p> <p>3.2.5.Feedback interconnection 129</p> <p>3.2.6. Youla-Kucera parametrization of stabilizing controllers 135</p> <p><b>Chapter 4. Systems with Differential Equations and Difference Equations 145</b></p> <p>4.1. Systems governed by functional differential equations 145</p> <p>4.1.1. Functional differential equation of retarded type 146</p> <p>4.1.2. Functional differential equations of neutral type 147</p> <p>4.1.3. Case of infinite delays 149</p> <p>4.1.4. Linear functional differential equations 151</p> <p>4.1.5. Stability of functional differential equations 156</p> <p>4.2. Time-in variant linear systems with lumped delays 159</p> <p>4.2.1. Definition and simplified framing 159</p> <p>4.2.2. Commensurability or incommensurability of delays 162</p> <p>4.2.3. Case of commensurable delays 162</p> <p>4.2.4. Homological questions 165</p> <p>4.3. Time-invariant linear systems with distributed delays 172</p> <p>4.3.1. RingH: case of non-commensurable delays 172</p> <p>4.3.2. RingsH and H0: case of commensurable delays 180</p> <p>4.3.3. Controllability and observability of H-systems 181</p> <p>4.3.4. Stability of H-systems 182</p> <p>4.3.5. Poles and zeros of H-systems 185</p> <p>Appendix. The Mathematics of the Theory of Systems 187</p> <p>References 267</p> <p>Index 279</p>

<p><b>Henri Bourlès</b> is Professor Emeritus at the Conservatoire National des Arts et Métiers and a researcher at the SATIE Laboratory (CNAM/ENS Paris-Saclay) in France. He has been teaching automatic control in engineering schools and universities for more than thirty years. His research work focuses on systems theory.</p>

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