Qualitative Theory of Hybrid Dynamical Systems

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Springer Science & Business Media, Mar 23, 2000 - Mathematics - 348 pages
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Hybrid 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.
 

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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
References
339
Index
347
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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.

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