Qualitative Theory of Hybrid Dynamical Systems

by Matveev, Alexey S.; Savkin, Andrey V.

• ISBN13: 9780817641412
• ISBN10: 0817641416
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Preface	ix
Introduction	1	(12)
Hybrid Dynamical Systems	1	(2)
Two Contrasting Examples of Discretely Controlled Continuous Variable Systems	3	(3)
The Main Goal of This Book	6	(1)
Organization of the Book	7	(3)
List of Notations	10	(3)
Qualitative Analysis of Some Simple Hybrid Dynamical Systems	13	(30)
Introduction	13	(2)
Differential Automata and Their Trajectories	15	(2)
Cyclic Linear Differential Automata	17	(6)
Qualitative Analysis of Cyclic Linear Differential Automata	23	(4)
Switched Server Systems with a Cyclic Switching Policy	27	(3)
Switched Server Systems with Several Limit Cycles	30	(3)
Qualitative Analysis of Closed Switched Server Systems	33	(5)
Essentially Non-Periodic Dynamics of Switched Arrival Systems	38	(5)
General Theory of Multivalued Differential Automata	43	(62)
Introduction	43	(1)
Multivalued Differential Automata	44	(23)
Basic assumptions and definitions	45	(4)
Illustrative examples	49	(4)
Invariant sets	53	(3)
A partial classification of points in the phase space	56	(3)
Deterministic and well-posed systems	59	(4)
The skeleton and the backstepping mapping	63	(1)
Asymptotically stable limit cycles	64	(3)
Decomposition of Well-Posed Differential Automata	67	(5)
Existence of Periodic Trajectories	72	(2)
Proofs of the Theorems and Lemmas from Section 3.2	74	(11)
Proof of Theorem 3.2.26	85	(12)
Proofs of the Theorems Sections 3.3 and 3.4	97	(8)
Proof of Theorem 3.4.3	102	(3)
Two-Dimensional Hybrid Dynamical Systems	105	(114)
Introduction	105	(2)
An Analog of the Poincare-Bendixon Theorem	107	(13)
Basic assumptions	107	(8)
A simple periodic dynamics	115	(1)
A criterion for a simple periodic dynamics	116	(4)
A Switched Arrival System with Three Buffers	120	(9)
A Switched Server System with Three Buffers	129	(26)
Proofs of the Statements from Section 4.2	155	(64)
Proofs of the lemmas from Section 4.2	155	(11)
Proofs of Theorem 4.2.10 and the remarks following it	166	(53)
Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: General Theory	219	(50)
Introduction	219	(2)
Basic Assumptions and Definitions	221	(7)
Multivalued differential automata with constant derivatives	221	(3)
Key assumptions	224	(4)
Criteria for Existence and Stability of Limit Cycles	228	(8)
A complement concerning deterministic systems	232	(4)
Proofs of the Lemmas from Section 5.2	236	(5)
Proofs of the Theorem and Lemmas from Section 5.3	241	(20)
Proofs of the Theorems and Lemmas from Subsection 5.3.1	261	(8)
Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: Examples	269	(36)
Introduction	269	(3)
Qualitative Analysis of a Switched Server System	272	(10)
Description of a switched server system	272	(1)
A cyclic control policy	273	(2)
The Clear-the-Largest-Buffer-Level Policy	275	(2)
Structural stability of a switched server system	277	(5)
A Switched Arrival System with Three Buffers	282	(4)
Qualitative Analysis of Switched Single Server Flow Networks	286	(19)
Single server flow networks	286	(3)
A cyclic control policy	289	(7)
A composed cyclic control policy	296	(2)
A combined control policy	298	(7)
Globally Periodic Behavior of Switched Single Server Flow Networks	305	(10)
Introduction	305	(1)
Description of Switched Single Server Flow Networks	306	(5)
Analysis of Switched Single Server Flow Networks	311	(4)
Regularizability of Switched Multiple Server Flow Networks	315	(16)
Introduction	315	(1)
Description of Switched Multiple Server Flow Networks	316	(5)
Regularizable Switched Multiple Server Flow Networks	321	(7)
Illustrative Example	328	(3)
Open Problems	331	(8)
Introduction	331	(1)
Switched Server Systems	332	(1)
Essentially Nonperiodic Multidimensional Switched Arrival Systems	333	(1)
Switched Server/Arrival Systems with Several Servers	333	(1)
A Generalized Processor Sharing Model	334	(2)
Stabilizability of Switched Multiple Server Flow Networks	336	(1)
Chaotic Switched Flow Networks	336	(1)
Existence and Global Stability of Limit Cycles in Nonlinear Differential Automata	337	(2)
References	339	(8)
Index	347