Robust Control

Abstract

The problems of robuststabilityand robust stabilization of linear state-space systems with parameter uncertainties have been extensively studied in the pastd ecades [25, 209]. Manyr esults on these topics have been proposed. Among the differenta pproaches that deal with these problems, the methods based on the concepts of quadratic stabilityand quadratic stabilizabilityhave become popular. An uncertain system is quadratically stable if there exists a fixed Lyapunovf unction to infer the stabilityo ft he uncertain system, while an uncertain system is quadratically stabilizable if there exists afeedbackcontroller sucht hat the closed-loop system is quadratically stable. Manyr esults on quadratic stability and quadraticstabilizabilityhave been reported in both the continuousa nd discrete contexts; see, e.g., [8, 57,1 82, 210], and ther eferences therein. However, in the context of uncertain singulars ystems,i th as been shown that the problems of robust stability and robusts tabilization are more complicated than those for state-space systems. Specifically,t he robust stabilityp roblem for singulars ystemsr equires considering not only stability robustness, but also regularityand non-impulsiveness (for continuous singular systems) or causality( for discrete singular systems) [48, 49,1 12, 177, 183], while thel atter twoa re intrinsic properties of state-space systems. Similarly, the robuststabilization problem foruncertain singular systems must consider not only stabilization but also regularization and impulse elimination, while the latter twoi ssues do not arise in the state-space case.In this chapter,w ec onsider thep roblemso fr obust stability analysis and robusts tabilization foru ncertain singular systems. The concepts of generalized quadratic stabilitya nd generalized quadratic stabilizabilityf or both continuous-and discrete-time singular systemsa re proposed. It is shown that generalized quadratic stabilitya nd generalized quadratic stabilizabilityi mply

Nonfragile passivity and passification of nonlinear singular networked control systems with randomly occurring controller gain fluctuation

“…Therefore, by employing the similar methods in [36][37][38], where a matrix G satisfying EG 5 0 and rank G5n2r is introduced, and defining P5XE T 1GW T , X > 0, the following theorem is further provided.…”

Optimal control of large-scale singular linear systems via hierarchical strategy

“…Mathematically speaking, a singular system model is formulated as a set of coupled differential and algebraic equations, which include information on the static as well as dynamic constraints of a real plant; such systems are called singular systems (Xu and Lam, 2006). As singular systems constitute an important class of systems of both theoretical and practical significance, they have been the subject of extensive research during past decades (Balasubramaniam, 2006; Li and Lin, 2004; Verghese et al, 1981; Vlasenko and Perestyuk, 2005; Xu and Lam, 2006).…”

“…Mathematically speaking, a singular system model is formulated as a set of coupled differential and algebraic equations, which include information on the static as well as dynamic constraints of a real plant; such systems are called singular systems (Xu and Lam, 2006). As singular systems constitute an important class of systems of both theoretical and practical significance, they have been the subject of extensive research during past decades (Balasubramaniam, 2006; Li and Lin, 2004; Verghese et al, 1981; Vlasenko and Perestyuk, 2005; Xu and Lam, 2006). Singular systems, such as robotics (McClamroch, 1986), power systems (Xu and Lam, 2006), networks (Shi et al, 2016), economical systems (Luenberger and Arbel, 1997), biological systems (Zhang et al, 2012), and highly interconnected large-scale systems (Gui et al, 2006), have been studied in the past decades due to the fact that singular models describe physical systems more directly and generally than regular state-space ones (Xu and Lam, 2006).…”

Input–output finite time stability of fractional order linear systems with 0 < α < 1

“…The problem of the IO-FTS for fractional order systems is challenging. On the one hand, this is because of the complexity of the fractional order calculus equation and the fact that integer order algorithms (Chen, 2014;Wang et al, 2013; Xu and Lam, 2006) cannot be directly applied to the fractional order system. On the other hand, if we want to provide the condition of the IO-FTS for fractional order systems by employing the Lyapunov function, we have to find a Lyapunov candidate function in the fractional order.…”