Robust stability, constrained stabilization, and observer-based controller designs for time delay systems

Abstract

Listen

Translate article

This thesis initially presents the problem of stability pertaining to delay systems and provides useful results on stability analysis using linear matrix inequalities (LMIs). The connection between recently published results on stability robustness in the domain of time delays and robust stability with respect to structured perturbation of system matrices is investigated. Consequently, computable stability radius formulas for multi-delay systems using Rekasius substitution are derived. Using the concepts of kernel and offspring curves, which describe the complete portrait of possible imaginary characteristic roots of the nominal system, the fundamental stability region is defined among a set of disjoint stability regions for two independent time-delays. This makes it possible to give a necessary and sufficient condition for stability robustness of two time-delay systems with structured uncertainties in terms of two distinct LMIs. We also provide an explicit formula for stability radius of a special class of time-delay systems and generalize the result for multi-delay case. Important results on stabilization and robust stabilization using state feedback and dynamic output feedback controller for both cases of delay-independent and delay-dependent are provided. These results are essential in the development of constrained stabilization, which is explored in later chapter. Subsequently, design of observer-based controller for robust stabilization of time-delay systems is explored. We present design methods for Proportional Observer (PO) and Proportional-Integral Observer (PIO) and compare their robust performance. By using Lyapunov-based stability condition for time-delay systems, we establish the stability and convergence of the observer. A method for PIO design is proposed to attenuate the disturbance to a pre-specified level while estimating the state of the delay system. The method guarantees the stability of the observer and minimizes the H∞ norm between the disturbance and the estimated error. All designs require solving certain modified algebraic Riccati equations. Due to the fact that attenuation is not the only objective for the designer, it is also shown that PIO has the capability of making simultaneous estimation of states and unknown constant disturbance, which can reliably be used in robust fault detection. Finally, the class of positive systems for continuous-time delay systems, defined as Metzlerian delay systems, is considered. The stability and robust stability of this class of systems are analyzed and direct formula for robust stability radius is derived. The stabilization and robust stabilization of Metzlerian delay systems are also explored and design methods are provided. A by-product of the results is the design procedures of state feedback and dynamic output feedback controllers for general linear time-delay systems such that the closed-loop system is constrained to be Metzlerian delay stable. The thesis includes several illustrative examples to support the theoretical results.