## Qualitative Theory of Hybrid Dynamical SystemsHybrid dynamical systems, both continuous and discrete dynamics and variables, have attracted considerable interest recently. This emerging area is found at the interface of control theory and computer engineering, focusing on the analogue and digital aspects of systems and devices. They are essential for advances in modern digital- controller technology. "Qualitative Theory of Hybrid Dynamical Systems" provides a thorough development and systematic presentation of the foundations and framework for hybrid dynamical systems. The presentation offers an accessible, but precise, development of the mathematical models, conditions for existence of limit cycles, and criteria of their stability. The book largely concentrates on the case of discretely controlled continuous-time systems and their relevance for modeling aspects of flexible manufacturing systems and dynamically routed queuing networks. Features and topics: *differential automata*development and use of the concept "cyclic linear differential automata" (CLDA)*switched single-server flow networks coverage*application to specific models of manufacturing systems and queuing networks*select collection of open problems for the subject*self-contained presentation of topics, with the necessary background This new book is an excellent resource for the study and analysis of hybrid dynamical systems used in systems and control engineering. Researchers, postgraduates and professionals in control engineering and computer engineering will find the book an up-to-date development of the relevant new concepts and tools. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Introduction | 1 |

12 Two Contrasting Examples of Discretely Controlled Continuous Variable Systems | 3 |

13 The Main Goal of This Book | 6 |

14 Organization of the Book | 7 |

15 List of Notations | 10 |

Qualitative Analysis of Some Simple Hybrid Dynamical Systems | 13 |

22 Differential Automata and Their Trajectories | 15 |

23 Cyclic Linear Differential Automata | 17 |

452 Proof of Theorem 4210 and the remarks following it | 166 |

Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives General Theory | 219 |

52 Basic Assumptions and Definitions | 221 |

522 Key assumptions | 224 |

53 Criteria for Existence and Stability of Limit Cycles | 228 |

531 A complement concerning deterministic systems | 232 |

54 Proofs of the Lemmas from Section 52 | 236 |

55 Proofs of the Theorems and Lemmas from Section 53 | 241 |

24 Qualitative Analysis of Cyclic Linear Differential Automata | 23 |

25 Switched Server Systems with a Cyclic Switching Policy | 27 |

26 Switched Server Systems with Several Limit Cycles | 30 |

27 Qualitative Analysis of Closed Switched Server Systems | 33 |

28 Essentially NonPeriodic Dynamics of Switched Arrival Systems | 38 |

General Theory of Multivalued Differential Automata | 43 |

32 Multivalued Differential Automata | 44 |

321 Basic assumptions and definitions | 45 |

322 Illustrative examples | 49 |

323 Invariant sets | 53 |

324 A partial classification of points in the phase space | 56 |

325 Deterministic and wellposed systems | 59 |

326 The skeleton and the backstepping mapping | 63 |

327 Asymptotically stable limit cycles | 64 |

33 Decomposition of WellPosed Differential Automata | 67 |

34 Existence of Periodic Trajectories | 72 |

35 Proofs of the Theorems and Lemmas from Section 32 | 74 |

36 Proof of Theorem 3226 | 85 |

37 Proofs of the Theorems from Sections 33 and 34 | 97 |

371 Proof of Theorem 343 | 102 |

TwoDimensional Hybrid Dynamical Systems | 105 |

42 An Analog of the PoincareBendixon Theorem | 107 |

422 A simple periodic dynamics | 115 |

423 A criterion for a simple periodic dynamics | 116 |

43 A Switched Arrival System with Three Buffers | 120 |

44 A Switched Server System with Three Buffers | 129 |

45 Proofs of the Statements from Section 42 | 155 |

56 Proofs of the Theorem and Lemmas from Subsection 531 | 261 |

Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives Examples | 269 |

62 Qualitative Analysis of a Switched Server System | 272 |

622 A cyclic control policy | 273 |

623 The CleartheLargestBufferLevel Policy | 275 |

624 Structural stability of a switched server system | 277 |

63 A Switched Arrival System with Three Buffers | 282 |

64 Qualitative Analysis of Switched Single Server Flow Networks | 286 |

642 A cyclic control policy | 289 |

643 A composed cyclic control policy | 296 |

644 A combined control policy | 298 |

Globally Periodic Behavior of Switched Single Server Flow Networks | 305 |

72 Description of Switched Single Server Flow Networks | 306 |

73 Analysis of Switched Single Server Flow Networks | 311 |

Regularizability of Switched Multiple Server Flow Networks | 315 |

82 Description of Switched Multiple Server Flow Networks | 316 |

83 Regularizable Switched Multiple Server Flow Networks | 321 |

84 Illustrative Example | 328 |

Open Problems | 331 |

92 Switched Server Systems | 332 |

93 Essentially Nonperiodic Multidimensional Switched Arrival Systems | 333 |

95 A Generalized Processor Sharing Model | 334 |

96 Stabilizability of Switched Multiple Server Flow Networks | 336 |

98 Existence and Global Stability of Limit Cycles in Nonlinear Differential Automata | 337 |

339 | |

347 | |

### Other editions - View all

Qualitative Theory of Hybrid Dynamical Systems Alexey S. Matveev,Andrey V. Savkin Limited preview - 2012 |

Qualitative Theory of Hybrid Dynamical Systems Alexey S. Matveev,Andrey V. Savkin No preview available - 2012 |

Qualitative Theory of Hybrid Dynamical Systems Alexey S. Matveev,Andrey V. Savkin No preview available - 2000 |

### Common terms and phrases

arrival rates asymptotically stable automata Brouwer fixed-point theorem Cauchy problem chapter CLDA closed-loop system completes the proof connected component Consider contains converges countable set defined Definition Denote discrete path edge empty exists finite number flexible manufacturing systems Furthermore globally periodic graph Q hybrid dynamical systems hybrid systems implies initial condition intersection interval invariant domain Lebesgue measure limit cycle limsup multiple server flow multivalued differential automaton multivalued function nodes Notation ordinary differential equations periodic trajectory point a,p point of uncertainty positive switching point Proof of Lemma proof of Theorem qualitative analysis relation Section server flow network server switches single server flow solution subsection Suppose that Assumptions switched arrival system switched multiple server switched server system switched single server switching policy switching time sequence symbolic dynamics symbolic range three buffers trajectory lying vector violation

### Popular passages

Page 341 - Hybrid system modeling and autonomous control systems, In RL Grossman, A. Nerode, AP Ravn, and H. Rischel (Ed.). Hybrid Systems, Lecture Notes in Computer Science, Vol. 736, Springer- Verlag, New York. Back, A., J. Guckenheimer, and M. Myers (1993). A dynamical simulation facility for hybrid systems, In RL Grossman, A.

Page 343 - In Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996.

Page 345 - Stable distributed real-time scheduling of flexible manufacturing/assembly/disassembly systems," IEEE Transactions on Automatic Control, vol.

Page 342 - Scheduling jobs with simple precedence constraints on parallel machines", IEEE Control Systems Magazine, Vol.

Page 341 - A. Bemporad and M. Morari. Control of systems integrating logic, dynamics, and constraints.

Page 343 - Lemmon, JA Stiver, and PJ Antsaklis. Event identification and intelligent hybrid control. In RL Grossman, A. Nerode, AP Ravn, and H. Rishel, editors, Hybrid Systems.

Page 345 - AV Savkin and RJ Evans. A new approach to robust control of hybrid systems over infinite time.

Page 343 - C. Horn and PJ Ramadge. A topological analysis of a family of dynamical systems with nonstandard chaotic and periodic behavior. International Journal of Control, 67(6):979-1020, 1997.