Qualitative Theory of Hybrid Dynamical Systems

ISBN-13: 9781461271147 / Angielski / Miękka / 2012 / 348 str.

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

Opis

The emerging area of hybrid dynamical systems lies at the interface of control theory and computer science, i.e., analogue 'and' digital aspects of systems. This new monograph presents state-of-the-art concepts, methods and tools for analyzing and describing hybrid dynamical systems.

Recenzje

"This study investigates the qualitative behaviour of hybrid systems.... This evolution is described by the coupling of the dynamics of an ODE and that of an automaton. Once the continuous state reaches a prescribed set S, the state of the automaton switches to another state and another ODE is activated. The main theme is the characterization of periodic behaviour of such systems and, when it is obtained, the study of its stability.... This mongraph is an exploration, and its merit is to convince the reader of the difficulty and richness of the topic under investigation." -Zentralblatt Math

Spis treści

1 Introduction.- 1.1 Hybrid Dynamical Systems.- 1.2 Two Contrasting Examples of Discretely Controlled Continuous Variable Systems.- 1.3 The Main Goal of This Book.- 1.4 Organization of the Book.- 1.5 List of Notations.- 2 Qualitative Analysis of Some Simple Hybrid Dynamical Systems.- 2.1 Introduction.- 2.2 Differential Automata and Their Trajectories.- 2.3 Cyclic Linear Differential Automata.- 2.4 Qualitative Analysis of Cyclic Linear Differential Automata.- 2.5 Switched Server Systems with a Cyclic Switching Policy.- 2.6 Switched Server Systems with Several Limit Cycles.- 2.7 Qualitative Analysis of Closed Switched Server Systems.- 2.8 Essentially Non-Periodic Dynamics of Switched Arrival Systems.- 3 General Theory of Multivalued Differential Automata.- 3.1 Introduction.- 3.2 Multivalued Differential Automata.- 3.2.1 Basic assumptions and definitions.- 3.2.2 Illustrative examples.- 3.2.3 Invariant sets.- 3.2.4 A partial classification of points in the phase space.- 3.2.5 Deterministic and well-posed systems.- 3.2.6 The skeleton and the backstepping mapping.- 3.2.7 Asymptotically stable limit cycles.- 3.3 Decomposition of Well-Posed Differential Automata.- 3.4 Existence of Periodic Trajectories.- 3.5 Proofs of the Theorems and Lemmas from Section 3.2.- 3.6 Proof of Theorem 3.2.26.- 3.7 Proofs of the Theorems from Sections 3.3 and 3.4.- 3.7.1 Proof of Theorem 3.4.3.- 4 Two-Dimensional Hybrid Dynamical Systems.- 4.1 Introduction.- 4.2 An Analog of the Poincaré-Bendixon Theorem.- 4.2.1 Basic assumptions.- 4.2.2 A simple periodic dynamics.- 4.2.3 A criterion for a simple periodic dynamics.- 4.3 A Switched Arrival System with Three Buffers.- 4.4 A Switched Server System with Three Buffers.- 4.5 Proofs of the Statements from Section 4.2.- 4.5.1 Proofs of the lemmas from Section 4.2.- 4.5.2 Proof of Theorem 4.2.10 and the remarks following it.- 5 Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: General Theory.- 5.1 Introduction.- 5.2 Basic Assumptions and Definitions.- 5.2.1 Multivalued differential automata with constant derivatives.- 5.2.2 Key assumptions.- 5.3 Criteria for Existence and Stability of Limit Cycles.- 5.3.1 A complement concerning deterministic systems.- 5.4 Proofs of the Lemmas from Section 5.2.- 5.5 Proofs of the Theorems and Lemmas from Section 5.3..- 5.6 Proofs of the Theorem and Lemmas from Subsection 5.3.1.- 6 Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: Examples.- 6.1 Introduction.- 6.2 Qualitative Analysis of a Switched Server System.- 6.2.1 Description of a switched server system.- 6.2.2 A cyclic control policy.- 6.2.3 The Clear-the-Largest-Buffer-Level Policy.- 6.2.4 Structural stability of a switched server system.- 6.3 A Switched Arrival System with Three Buffers.- 6.4 Qualitative Analysis of Switched Single Server Flow Networks.- 6.4.1 Single server flow networks.- 6.4.2 A cyclic control policy.- 6.4.3 A composed cyclic control policy.- 6.4.4 A combined control policy.- 7 Globally Periodic Behavior of Switched Single Server Flow Networks.- 7.1 Introduction.- 7.2 Description of Switched Single Server Flow Networks.- 7.3 Analysis of Switched Single Server Flow Networks.- 8 Regularizability of Switched Multiple Server Flow Networks.- 8.1 Introduction 315 8.2 Description of Switched Multiple Server Flow Networks.- 8.3 Regularizable Switched Multiple Server Flow Networks.- 8.4 Illustrative Example.- 9 Open Problems.- 9.1 Introduction.- 9.2 Switched Server Systems.- 9.3 Essentially Nonperiodic Multidimensional Switched Arrival Systems.- 9.4 Switched Server/Arrival Systems with Several Servers.- 9.5 A Generalized Processor Sharing Model.- 9.6 Stabilizability of Switched Multiple Server Flow Networks.- 9.7 Chaotic Switched Flow Networks.- 9.8 Existence and Global Stability of Limit Cycles in Nonlinear Differential Automata.- References.

O autorze

The emerging area of hybrid dynamical systems lies at the interface
of control theory and computer science, i.e., analogue 'and' digital
aspects of systems.
A new monograph presenting state-of-the-art concepts, methods, and
tools for analyzing and describing hybrid dynamical systems.
Control engineers, system engineers and robotics researchers and
professionals will find the book a valuable resource for new
developments, applications, and research in the field.