Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Abstract

Abstract. We construct two error feedback controllers for robust output tracking and disturbance rejection of a regular linear system with nonsmooth reference and disturbance signals. We show that for sufficiently smooth signals the output converges to the reference at a rate that depends on the behaviour of the transfer function of the plant on the imaginary axis. In addition, we construct a controller that can be designed to achieve robustness with respect to a given class of uncertainties in the system, and present a novel controller structure for output tracking and disturbance rejection without the robustness requirement. We also generalize the internal model principle for regular linear systems with boundary disturbance and for controllers with unbounded input and output operators. The construction of controllers is illustrated with an example where we consider output tracking of a nonsmooth periodic reference signal for a two-dimensional heat equation with boundary control and observation, and with periodic disturbances on the boundary. IntroductionThe purpose of this paper is to construct controllers for robust output regulation of a regular linear system 1 [39, 40, 36]on an infinite-dimensional Banach space X. The main goal in the control problem is to achieve asymptotic convergence of the output y(t) to a given reference signal y ref (t) despite external disturbance signals w(t). In addition, it is required that the controller is robust in the sense that output tracking is achieved even under perturbations and uncertainties in the operators (A, B, B d , C, D) of the plant. The class of regular linear systems facilitates the study of robust output tracking and disturbance rejection for many important classes partial differential equations with boundary control and observation with corresponding unbounded operators B, B d and C [9,16,47,25]. In this paper we continue the work on designing robust controllers for regular linear systems begun recently in [26].The reference signal y ref (t) and the disturbance signals w(t) considered in the robust output regulation problem are assumed to be generated by an exosystem of the forṁ

Stability and Robust Regulation of Passive Linear Systems

“…The proof of Theorem 3.5 also improves the result [27,Thm. 27] The non-uniform decay rates for the semigroup follow from [7,8]…”

“…The controller constructions in this section are based on the internal model principle [16,28,29] which implies that a controller solves the robust output regulation problem provided that its dynamics contain a suitable number of copies of the frequencies {iω k } k∈I of the signals (4.1) and the closed-loop system is stable. If dim Y < ∞, then (A c , B c , C c , D c ) contains an internal model of the signals (4.1) if [27,Thm. 13]…”

“…13]. The results in [27] are presented for controllers with D c = 0, but they are applicable since D c ≥ 0 can be written as an output feedback for the system (2.1) without changing the properties of the closed-loop system. Moreover, the results are presented for an infinite set {iω k } k∈I , but they also apply trivially when I is finite.…”

“…In addition, the controller is required to be robust in the sense that it should achieve output tracking even if the parameters (A, B, C, D) experience small changes or contain uncertainties. This control problem has been studied actively in the literature for various classes of infinite-dimensional linear systems [42,24,17,31,21,18,28,37] including regular linear systems [40,9,29,41,26,27] and passive systems [31]. The robust output regulation problem can be solved with a dynamical error feedback controller of the forṁ…”

“…Moreover, our design methods allow considering nonsmooth periodic reference and disturbance signals with infinite numbers of frequencies. In earlier references, the robust output regulation of nonsmooth signals has only been achieved using an observer in the controller [18,27]. We solve this problem with two new robust controllers having the property that A c contains only the internal model of the reference and disturbance signals.…”