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Introduction to the Modern Theory of Dynamical Systems

This paper studies a number of quantized feedback design problems for linear systems. We consider the case where quantizers are static (memoryless). The common aim of these design problems is to stabilize the given system or to achieve certain performance with the coarsest quantization density. Our main discovery is that the classical sector bound approach is nonconservative for studying these design problems. Consequently, we are able to convert many quantized feedback design problems to well-known robust control problems with sector bound uncertainties. In particular, we derive the coarsest quantization densities for stabilization for multiple-input-multiple-output systems in both state feedback and output feedback cases; and we also derive conditions for quantized feedback control for quadratic cost and H/sub /spl infin// performances.

The sector bound approach to quantized feedback control | NOVA. The University of Newcastle's Digital Repository

Introduction to lie algebras and representation theory

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Feedback systems: Input-output properties

We study the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to counteract the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization.

Feedback Stabilization of Switched Linear Systems With Unknown Disturbances Under Data-Rate Constraints

A Lyapunov-based small-gain theorem for interconnected switched systems

The input-to-state stability (ISS) of a nonlinear switched system is investigated in the scenario where there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signal neither switches too frequently nor activates non-ISS subsystems for too long, a hybrid ISS Lyapunov function can be constructed to guarantee ISS of the switched system. With the constraints on the switching signal being modeled by a novel auxiliary timer, a hybrid system is defined so that the solutions to the two systems are correspondent. After the construction and verification of an ISS Lyapunov function, ISS of all complete solutions to the hybrid system, and therefore all solutions to the switched system, is conveniently proved.

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Input-to-state stability for switched systems with unstable subsystems: A hybrid Lyapunov construction

Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems

This paper introduces a notion of topological entropy for switched systems, formulated using the minimal number of initial states needed to approximate all initial states within a finite precision. We show that it can be equivalently defined using the maximal number of initial states separable within a finite precision, and introduce switching-related quantities such as the active time of each mode, which prove to be useful in calculating the topological entropy of switched linear systems. For general switched linear systems, we show that the topological entropy is independent of the set of initial states, and establish upper and lower bounds using the active-time-weighted averages of the norms and traces of system matrices in individual modes, respectively. For switched linear systems with scalar-valued state or simultaneously diagonalizable matrices, we derive formulae for the topological entropy using active-time-weighted averages of eigenvalues, which can be extended to the case with simultaneously triangularizable matrices to obtain an upper bound. In these three cases with special matrix structure, we also provide more general but more conservative upper bounds for the topological entropy.

Guosong Yang

Papers

Feedback Stabilization of Switched Linear Systems With Unknown Disturbances Under Data-Rate Constraints

A Lyapunov-based small-gain theorem for interconnected switched systems

Input-to-state stability for switched systems with unstable subsystems: A hybrid Lyapunov construction

Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems

On Topological Entropy of Switched Linear Systems with Diagonal, Triangular, and General Matrices