Infinite-dimensional Exosystems Research Articles

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

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.

Frequency Domain Robust Regulation of Signals Generated by an Infinite-Dimensional Exosystem

This paper deals with frequency domain robust regulation of signals generated by an infinite-dimensional exosystem. The problem is formulated and the stability types are chosen so that one can generalize the existing finite-dimensional theory to more general classes of infinite-dimensional systems and signals. The main results of this article are extensions of the internal model principle, of a necessary and sufficient solvability condition for the robust regulation problem, and of Davison's simple servo compensator for stable plants in the chosen algebraic framework.

The Internal Model Principle for Systems with Unbounded Control and Observation

In this paper the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation. As the main result, we present the internal model principle for linear infinite-dimensional systems with unbounded input and output operators. We do this for two different definitions of an internal model found in the literature, namely, the p-copy internal model and the $\mathcal{G}$-conditions. We also introduce a new way of defining an internal model for infinite-dimensional systems. The theoretic results are illustrated with an example where we consider robust output tracking for a one-dimensional heat equation with boundary control and pointwise measurements.

Sufficient and necessary conditions for the solvability of the state feedback regulation problem

In this paper, we discuss the state feedback output regulation problem (SFRP) for infinite-dimensional linear control systems with infinite-dimensional exosystems. Under the polynomial stabilizability assumption, sufficient and necessary conditions are given for the solvability of the SFRP. The solvability of this problem is characterized in terms of the solvability of a pair of linear regulator equations. An application of the solvability of the SFRP for polynomial stable SISO system is given. Copyright © 2013 John Wiley & Sons, Ltd.

Output Regulation for General Infinite-Dimensional Exosystems

In this paper we study output regulation of distributed parameter systems with infinite-dimensional exosystems. The purpose of the paper is to find simple and minimal conditions on the signal generator under which the solvability of the output regulation problem can be characterized by the solvability of the regulator equations. We also study the properties of the dynamic steady state of the closed-loop system and the uniqueness of the solution of Sylvester operator equations. The presented results have applications in robust regulation of infinite-dimensional systems.

Robust Regulation of Distributed Parameter Systems with Infinite-Dimensional Exosystems

In this paper a robust regulation problem for infinite-dimensional systems with infinite-dimensional exosystems is discussed. The input and output spaces are also allowed to be infinite-dimensional. A new definition of internal model in terms of the controller parameters is given. It is shown that there exists a feedback controller containing an internal model of the exosystem, which robustly regulates the class of signals generated by the exosystem and strongly or weakly stabilizes the closed-loop system. As far as the authors know, the results are new even for finite-dimensional systems with infinite-dimensional exosystems.

Robust Regulation of Infinite-Dimensional Systems with Infinite-Dimensional Exosystems

In this paper robust regulation problem for infinite-dimensional systems with infinite-dimensional exosystems is discussed. The exosystems considered in this paper have infinite number of eigenvalues on the imaginary axis and thus include periodic signals. It is shown that there exists a feedback controller which robustly regulates the class of signals generated by the exosystem and strongly stabilizes the closed-loop system. As far as the authors know, the result is new even for finite-dimensional systems.

A Simple Robust Controller for the Approximate Regulation of Almost Periodic Signals for Linear Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> By using a delay line as a signal generator, we design a simple robust controller for the approximate regulation (i.e., approximate asymptotic tracking and rejection) of almost periodic signals for linear systems. As opposed to the related existing results, our controller design does not require online parameter tuning. Besides the usual assumptions about stabilizability, detectability, and nonexistence of transmission zeros for the plant, the controller design algorithm of this paper only requires knowledge of the desired tracking accuracy <formula formulatype="inline"><tex>$\epsilon&gt;0$</tex></formula>. Based on this information alone, we can only design a robust controller achieving, in the presence of unknown trigonometric polynomial disturbances and additive perturbations to the plant and the controller, <formula formulatype="inline"><tex>$\lim\sup_{t\rightarrow\infty}\Vert y(t)-y_{ref}(t)\Vert≪\epsilon\Vert y_{ref}\Vert$</tex></formula>. Here, <formula formulatype="inline"><tex>$y$</tex></formula> is the output of the plant and <formula formulatype="inline"><tex>$y_{ref}$</tex></formula> is an arbitrary almost periodic reference signal taken from an infinite-dimensional generalized Sobolev space of almost periodic functions. In this paper, we also study robust output regulation in the limiting case where <formula formulatype="inline"><tex>$\epsilon\rightarrow 0$</tex></formula>. It turns out that, due to the general loss of exponential closed loop stability, asymptotic tracking/rejection can also be lost as <formula formulatype="inline"><tex>$\epsilon\rightarrow 0$</tex></formula>, unless the exogenous signals are smooth enough. This last result extends some recent theorems on (nonrobust) open loop output regulation for infinite-dimensional exosystems. The results of this paper are new and potentially useful for finite-dimensional plants, but the main results are also true for infinite-dimensional systems with bounded control and observation operators. </para>

Some Properties of Infinite-dimensional Systems Capable of Asymptotically Tracking Bounded Uniformly Continuous Signals

We study several properties of infinite-dimensional systems capable of asymptotically tracking (i.e., regulating) a given bounded uniformly continuous (BUC) reference signal yref using a feedforward controller. We show that the regulability of the signal can be characterized by the solvability of the regulator equations. Using this information we construct the so called regulable space \(\mathcal{R}_{y_{\rm ref}}\) associated to the reference signal yref. This space contains other reference signals which can also be regulated if we know that yref can be regulated. Using methods of harmonic analysis we obtain a complete description of the elements of \(\mathcal{R}_{y_{\rm ref}}\) whenever yref is almost periodic, and we show that ergodic vectors and isolated spectral points of yref yield nontrivial information about \(\mathcal{R}_{y_{\rm ref}}\) even if yref is not almost periodic. We also show that certain spectral points of the regulable reference signals—and their generators—cannot be zeros of the feedforward control system. These results generalize the well known results from finite-dimensional control theory; a crucial feature in our approach is the infinite-dimensionality of the exogenous signal generator. We conclude the article by demonstrating how infinite-dimensional exosystems can be used in practice.