Henri Bourlès 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.

• Preface ixList of Notations xiiiChapter 1. Representation of Systems: A Historical Overview 11.1. Transfer functions and matrices 11.1.1. Transfer functions 11.1.2. Transfer matrices 31.1.3. The discrete-time case 61.2. State-space representation 71.2.1. Continuous-time state-space systems 71.2.2. Discrete-time state-space systems 71.2.3. Controllability and observability 81.2.4. Poles of as state-space system 111.2.5. Stability of linear time-invariant systems 131.3. "Geometric" approach 141.3.1. Formalism of the geometric approach 141.3.2. Reachable and non-observable subspaces 151.3.3. State-feedback controls, observers 161.3.4. Canonical Kalmande composition, stabilizability and detectability 171.4. Polynomial matrix description 211.4.1. PBH test (Hautus criterion) 211.4.2. Rosenbrock representation 231.5. The behavioral approach 261.5.1. Controllability without control variables 261.5.2. Observability in the behavioral approach 301.6. Module of a system 321.6.1. Using modules in control theory 321.6.2. The Fliessian approach 331.6.3. Characterization in terms of modules of controllability and observability 341.7. The formalism of algebraic analysis 351.7.1. Nature of algebraic analysis 351.7.2. Oberst's contribution and its consequences 35Chapter2. Linear Systems: Concepts and General Results 432.1. Control systems 432.1.1. The formalism of control systems 432.1.2. Transfer matrix: general case 452.1.3. State-space representation of a control system 462.2. Strict equivalence of Rosenbrock systems 492.2.1. Admissible Rosenbrock representations 492.2.2. Strict equivalence 512.3. Controllability, observability and their duality: the algebraic point of view 552.3.1. Algebraic controllability 552.3.2. Algebraic controllability of state-space systems (continuous time) 572.3.3. Algebraic controllability of state-space systems (discrete-time) 632.3.4. Algebraic duality 662.3.5. Algebraic observability and the algebraic duality principle 742.4. Reachability, observability and their duality: Kalmanian point of view 772.4.1. Complete controllability of a state-space system (continuous time) 782.4.2. Complete reachability and controllability of a state-space system (discrete-time) 832.4.3. Observability of state-space systems (continuous-time) 862.4.4. Complete observability and constructability of state-space systems (discrete-time) 91Chapter 3. Poles and Zeros of Linear Systems, Interconnectedness and Stabilization 973.1. Poles and zeros of continuous or discrete invariant linear systems 973.1.1. System poles, transmission poles and zeros 973.1.2. Input-output-decoupling zeros and hidden modes 1093.1.3. Relations between poles, transmission poles and hidden modes 1123.1.4. Invariant zeros 1133.1.5. Dynamic interpretation of different poles and zeros 1183.2. Poles and zeros of interconnected systems 1193.2.1. Diagram of a control system 1193.2.2. System interconnection 1203.2.3. Series interconnection 1213.2.4. Parallel interconnection 1263.2.5.Feedback interconnection 1293.2.6. Youla-Kucera parametrization of stabilizing controllers 135Chapter 4. Systems with Differential Equations and Difference Equations 1454.1. Systems governed by functional differential equations 1454.1.1. Functional differential equation of retarded type 1464.1.2. Functional differential equations of neutral type 1474.1.3. Case of infinite delays 1494.1.4. Linear functional differential equations 1514.1.5. Stability of functional differential equations 1564.2. Time-in variant linear systems with lumped delays 1594.2.1. Definition and simplified framing 1594.2.2. Commensurability or incommensurability of delays 1624.2.3. Case of commensurable delays 1624.2.4. Homological questions 1654.3. Time-invariant linear systems with distributed delays 1724.3.1. RingH: case of non-commensurable delays 1724.3.2. RingsH and H0: case of commensurable delays 1804.3.3. Controllability and observability of H-systems 1814.3.4. Stability of H-systems 1824.3.5. Poles and zeros of H-systems 185Appendix. The Mathematics of the Theory of Systems 187References 267Index 279

Henri Bourlès 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.

• Preface ixList of Notations xiiiChapter 1. Representation of Systems: A Historical Overview 11.1. Transfer functions and matrices 11.1.1. Transfer functions 11.1.2. Transfer matrices 31.1.3. The discrete-time case 61.2. State-space representation 71.2.1. Continuous-time state-space systems 71.2.2. Discrete-time state-space systems 71.2.3. Controllability and observability 81.2.4. Poles of as state-space system 111.2.5. Stability of linear time-invariant systems 131.3. "Geometric" approach 141.3.1. Formalism of the geometric approach 141.3.2. Reachable and non-observable subspaces 151.3.3. State-feedback controls, observers 161.3.4. Canonical Kalmande composition, stabilizability and detectability 171.4. Polynomial matrix description 211.4.1. PBH test (Hautus criterion) 211.4.2. Rosenbrock representation 231.5. The behavioral approach 261.5.1. Controllability without control variables 261.5.2. Observability in the behavioral approach 301.6. Module of a system 321.6.1. Using modules in control theory 321.6.2. The Fliessian approach 331.6.3. Characterization in terms of modules of controllability and observability 341.7. The formalism of algebraic analysis 351.7.1. Nature of algebraic analysis 351.7.2. Oberst's contribution and its consequences 35Chapter2. Linear Systems: Concepts and General Results 432.1. Control systems 432.1.1. The formalism of control systems 432.1.2. Transfer matrix: general case 452.1.3. State-space representation of a control system 462.2. Strict equivalence of Rosenbrock systems 492.2.1. Admissible Rosenbrock representations 492.2.2. Strict equivalence 512.3. Controllability, observability and their duality: the algebraic point of view 552.3.1. Algebraic controllability 552.3.2. Algebraic controllability of state-space systems (continuous time) 572.3.3. Algebraic controllability of state-space systems (discrete-time) 632.3.4. Algebraic duality 662.3.5. Algebraic observability and the algebraic duality principle 742.4. Reachability, observability and their duality: Kalmanian point of view 772.4.1. Complete controllability of a state-space system (continuous time) 782.4.2. Complete reachability and controllability of a state-space system (discrete-time) 832.4.3. Observability of state-space systems (continuous-time) 862.4.4. Complete observability and constructability of state-space systems (discrete-time) 91Chapter 3. Poles and Zeros of Linear Systems, Interconnectedness and Stabilization 973.1. Poles and zeros of continuous or discrete invariant linear systems 973.1.1. System poles, transmission poles and zeros 973.1.2. Input-output-decoupling zeros and hidden modes 1093.1.3. Relations between poles, transmission poles and hidden modes 1123.1.4. Invariant zeros 1133.1.5. Dynamic interpretation of different poles and zeros 1183.2. Poles and zeros of interconnected systems 1193.2.1. Diagram of a control system 1193.2.2. System interconnection 1203.2.3. Series interconnection 1213.2.4. Parallel interconnection 1263.2.5.Feedback interconnection 1293.2.6. Youla-Kucera parametrization of stabilizing controllers 135Chapter 4. Systems with Differential Equations and Difference Equations 1454.1. Systems governed by functional differential equations 1454.1.1. Functional differential equation of retarded type 1464.1.2. Functional differential equations of neutral type 1474.1.3. Case of infinite delays 1494.1.4. Linear functional differential equations 1514.1.5. Stability of functional differential equations 1564.2. Time-in variant linear systems with lumped delays 1594.2.1. Definition and simplified framing 1594.2.2. Commensurability or incommensurability of delays 1624.2.3. Case of commensurable delays 1624.2.4. Homological questions 1654.3. Time-invariant linear systems with distributed delays 1724.3.1. RingH: case of non-commensurable delays 1724.3.2. RingsH and H0: case of commensurable delays 1804.3.3. Controllability and observability of H-systems 1814.3.4. Stability of H-systems 1824.3.5. Poles and zeros of H-systems 185Appendix. The Mathematics of the Theory of Systems 187References 267Index 279

Henri Bourlès 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.

• Preface ixList of Notations xiiiChapter 1. Representation of Systems: A Historical Overview 11.1. Transfer functions and matrices 11.1.1. Transfer functions 11.1.2. Transfer matrices 31.1.3. The discrete-time case 61.2. State-space representation 71.2.1. Continuous-time state-space systems 71.2.2. Discrete-time state-space systems 71.2.3. Controllability and observability 81.2.4. Poles of as state-space system 111.2.5. Stability of linear time-invariant systems 131.3. "Geometric" approach 141.3.1. Formalism of the geometric approach 141.3.2. Reachable and non-observable subspaces 151.3.3. State-feedback controls, observers 161.3.4. Canonical Kalmande composition, stabilizability and detectability 171.4. Polynomial matrix description 211.4.1. PBH test (Hautus criterion) 211.4.2. Rosenbrock representation 231.5. The behavioral approach 261.5.1. Controllability without control variables 261.5.2. Observability in the behavioral approach 301.6. Module of a system 321.6.1. Using modules in control theory 321.6.2. The Fliessian approach 331.6.3. Characterization in terms of modules of controllability and observability 341.7. The formalism of algebraic analysis 351.7.1. Nature of algebraic analysis 351.7.2. Oberst's contribution and its consequences 35Chapter2. Linear Systems: Concepts and General Results 432.1. Control systems 432.1.1. The formalism of control systems 432.1.2. Transfer matrix: general case 452.1.3. State-space representation of a control system 462.2. Strict equivalence of Rosenbrock systems 492.2.1. Admissible Rosenbrock representations 492.2.2. Strict equivalence 512.3. Controllability, observability and their duality: the algebraic point of view 552.3.1. Algebraic controllability 552.3.2. Algebraic controllability of state-space systems (continuous time) 572.3.3. Algebraic controllability of state-space systems (discrete-time) 632.3.4. Algebraic duality 662.3.5. Algebraic observability and the algebraic duality principle 742.4. Reachability, observability and their duality: Kalmanian point of view 772.4.1. Complete controllability of a state-space system (continuous time) 782.4.2. Complete reachability and controllability of a state-space system (discrete-time) 832.4.3. Observability of state-space systems (continuous-time) 862.4.4. Complete observability and constructability of state-space systems (discrete-time) 91Chapter 3. Poles and Zeros of Linear Systems, Interconnectedness and Stabilization 973.1. Poles and zeros of continuous or discrete invariant linear systems 973.1.1. System poles, transmission poles and zeros 973.1.2. Input-output-decoupling zeros and hidden modes 1093.1.3. Relations between poles, transmission poles and hidden modes 1123.1.4. Invariant zeros 1133.1.5. Dynamic interpretation of different poles and zeros 1183.2. Poles and zeros of interconnected systems 1193.2.1. Diagram of a control system 1193.2.2. System interconnection 1203.2.3. Series interconnection 1213.2.4. Parallel interconnection 1263.2.5.Feedback interconnection 1293.2.6. Youla-Kucera parametrization of stabilizing controllers 135Chapter 4. Systems with Differential Equations and Difference Equations 1454.1. Systems governed by functional differential equations 1454.1.1. Functional differential equation of retarded type 1464.1.2. Functional differential equations of neutral type 1474.1.3. Case of infinite delays 1494.1.4. Linear functional differential equations 1514.1.5. Stability of functional differential equations 1564.2. Time-in variant linear systems with lumped delays 1594.2.1. Definition and simplified framing 1594.2.2. Commensurability or incommensurability of delays 1624.2.3. Case of commensurable delays 1624.2.4. Homological questions 1654.3. Time-invariant linear systems with distributed delays 1724.3.1. RingH: case of non-commensurable delays 1724.3.2. RingsH and H0: case of commensurable delays 1804.3.3. Controllability and observability of H-systems 1814.3.4. Stability of H-systems 1824.3.5. Poles and zeros of H-systems 185Appendix. The Mathematics of the Theory of Systems 187References 267Index 279