Approximate Tracking and Disturbance Rejection for Stable Infinite-Dimensional Systems Using Sampled-Data Low-Gain Control

Abstract

Abstract. In this paper we solve tracking and disturbance rejection problems for stable infinitedimensional systems using a simple low-gain controller suggested by the internal model principle. For stable discrete-time systems, it is shown that the application of a low-gain controller (depending on only one gain parameter) leads to a stable closed-loop system which asymptotically tracks reference signals r of the form r(k) = N j=1 λ k j r j , where r j ∈ C p and λ j ∈ C with |λ j | = 1 for j = 1, . . . , N. The closed-loop system also rejects disturbance signals which are asymptotically of this form. The discrete-time result is used to derive results on approximate tracking and disturbance rejection for a large class of infinite-dimensional sampled-data feedback systems, with reference signals which are finite sums of sinusoids, and disturbance signals which are asymptotic to finite sums of sinusoids. The results are given for both input-output systems and state-space systems.Key words. discrete-time systems, disturbance rejection, infinite-dimensional systems, internal model principle, low-gain control, sampled-data control, tracking AMS subject classifications. 93C25, 93C55, 93C57, 93C80, 93D15, 93D25 DOI. 10.1137/080716517 1. Introduction. The synthesis of low-gain integral controllers for uncertain stable continuous-time plants has received considerable attention in the last thirty years. Let G be a stable proper rational continuous-time transfer function matrix. The main existence result for robust low-gain integral control states that if all of the eigenvalues of G(0) have positive real parts, then there exists ε * > 0 such that for all ε ∈ (0, ε * ), the controller (ε/s)I stabilizes G. Moreover, the resulting closed-loop system asymptotically tracks arbitrary constant reference signals. This result has been proved by Davison [2] using state-space methods and Morari [11] using frequencydomain methods. This low-gain controller allows stabilization and tracking with very little information about the plant, and it is not based on system identification. The above regulator result has been extended to various classes of (abstract) infinitedimensional continuous-time systems: in [12] for exponentially stable parabolic systems, in [7] for systems in the Callier-Desoer algebra (CD-algebra), and in [9] for exponentially stable regular systems.In the case that the reference and disturbance signals are of the form

The State Feedback Regulator Problem for Regular Linear Systems

“…The finite-dimensional controller proposed in [42] solves the error feedback regulator problem. Sampled-data versions of this controller can be found in Ke, Logemann and Rebarber [31], [32]. In spite of the results in the cited papers, in this work we pursue the state feedback regulator problem (and will address the error feedback regulator problem in a future work) in the hope that, as in finite-dimensions, the state space approach developed here will be more suitable (in comparison to the approach in [23], [32], [42]) for a generalization addressing the regulator problem for non-linear infinite-dimensional plants.…”

A Sampled-Data Servomechanism for Stable Well-Posed Systems

“…We emphasize that the proof given here is new, with coprime factorizations playing a pivotal role and thereby providing an alternative approach to that developed in [3]. It is convenient to first state and prove the following lemma which will facilitate the proof of Proposition III.…”

Sampled-data output regulation of unstable well-posed infinite-dimensional systems with constant reference and disturbance signals

“…For regular systems, which is a subclass of well-posed systems, the authors of [7,31,33,51] have provided design methods of continuous-time controllers for robust output regulation. For stable wellposed systems with finite-dimensional exosystems, low-gain controllers suggested by the internal model principle have been constructed for the continuoustime setup in [29,40] and for the sampled-data setup in [18][19][20]28]. The difficulty of the problem we consider arises from the instability of well-posed systems.…”