Estimation and control of periodic piecewise dynamic systems

Abstract

Recently, periodic piecewise dynamic systems have received increasing attention. Some preliminary studies have been conducted, mostly focusing on stability, performance analysis, and several control problems. However, many problems remain open. To enrich the research topic, this thesis focuses on some estimation problems and control problems under the effects of uncertainties, disturbances, and faults.

In terms of estimation, by considering the effects of exogenous disturbance inputs and unexpected fault inputs, three types of estimation problems for different periodic piecewise dynamic scenarios have been investigated: (1) Peak-to-peak filtering is performed for periodic piecewise polytopic systems to estimate the system state and the performance signal. Under peak-bounded disturbances, the attention is mainly focused on designing periodic filters that ensure the filtering error systems to be robustly asymptotically stable and satisfy a peak-to-peak disturbance attenuation level. Two design methods based on parameter-dependent and parameter-independent Lyapunov matrix functions are utilized to establish sufficient conditions. A tractable algorithm is proposed to compute the periodic filter parameters governed by non-convex conditions; (2) Under peak-bounded disturbances, the reachable set estimation for a type of periodic piecewise time-varying systems is studied. Based on a generally reachable set estimation lemma for periodic piecewise systems with bounded disturbances, an efficient criterion of reachable set estimation is obtained. By adopting the Lyapunov function based on interval segmentation, the conditions can contribute to less conservative results in optimizing the bounding region of the reachable set; (3) The fault estimation problem is addressed for periodic piecewise Takagi-Sugeno (T-S) fuzzy systems affected by both faults and disturbances. By developing the fuzzy-dependent fault estimation observer with periodic time-varying matrix parameters, sufficient condition is established to guarantee the asymptotic stability and $L_2$-$L_{\infty}$ (energy-to-peak) performance of the error dynamic systems. The observer gains are obtained with conservatism comparison via solving LMIs, which are derived by utilizing the techniques of dealing with matrix polynomials.

In terms of control, the stabilization and observer-based control are developed: (1) Stabilization problem is investigated for periodic piecewise systems with dwell time uncertainty. Considering the uncertainty in dwell time for the first time, a novel mixed-mode time-varying Lyapunov function is constructed. The periodic time-scheduled state-feedback controller is designed via a tractable iterative algorithm to stabilize the periodic piecewise system affected by dwell time uncertainty; (2) Observer-based control for periodic piecewise time-varying systems is addressed to enclose the output reachable set with a bounding region. By decoupling the time-varying observer and controller gains, sufficient condition is developed to design the observer-based controller. The conservatism reduction is achieved by utilizing the relaxed quadratic matrix polynomial definiteness results. (421 words)