Analysis and synthesis of dynamic systems with periodic piecewise characteristics

Abstract

This thesis is concerned with the analysis and synthesis problems for three types of continuous-time systems with periodic piecewise characteristics. In terms of analysis, several issues related to stability and L2 performance are addressed for periodic piecewise systems with different characterisations. Focuses are given on three aspects: a) For a class of periodic piecewise linear systems, the finite-time stability and L2-gain characterisation are investigated by developing a basic time segment-based piecewise Lyapunov-like function. Considering uncertainties in such kind of systems, sufficient conditions are presented to ensure the exponential stability and a guaranteed upper bound of the time-weighted quadratic cost function via Lyapunov functions in time-varying forms; b) For periodic piecewise time-delay systems, delay-dependent sufficient conditions of exponential stability and weighted L2-gain are obtained based on introducing a periodic matrix function into the Lyapunov-Krasovskii functional. Moreover, an improved Lyapunov-Krasovskii functional is proposed for asymptotic stability, without requiring all the parameter matrices to be positive definite; c) For a class of periodic piecewise time-varying systems constituted by subsystems in time-interpolative forms, a general sufficient stability criterion is given based on a Lyapunov function involving periodically time-varying matrix functions. In terms of synthesis, some periodic control schemes are established based on the corresponding analysis results. The specific studies on controller synthesis are listed as follows: a) For periodic piecewise linear systems, finite-time stabilisation and H∞ control schemes are designed with time-varying controller gains, which are shown to be effective in guaranteeing finite-time stability and finite-time boundedness with less conservative performance, respectively. For uncertain periodic piecewise linear systems, it is shown that the proposed robust time-weighted guaranteed cost controller can be solved via iterative convex optimisation, and achieve faster response with the increase of weighting factor; b) For periodic piecewise time-delay systems, the H∞ control problem is addressed, and a guaranteed cost control scheme is established under an iterative multi-objective optimisation framework. It is illustrated that both the periodic matrix function and relaxed constraints can achieve less conservative upper bounds of H2 guaranteed cost and H∞ disturbance attenuation performance index; c) For periodic piecewise time-varying systems, an H∞ tracking controller is developed by taking advantage of the negative definiteness property of a type of matrix polynomials. It is demonstrated that the proposed scheme not only guarantees the closed-loop stability and state tracking, but also provides an alternative approach to solve the periodically time-varying controller gains in a straightforward way.