The State Feedback Regulator Problem for Regular Linear Systems

Abstract

This paper is about the state feedback regulator problem for infinite-dimensional linear systems. The plant, assumed to be an exponentially stable regular linear system, is driven by a linear (possibly infinite-dimensional) exosystem via a disturbance signal. The exosystem has its spectrum in the closed right half-plane and also generates the reference signal for the plant output. The regulator problem is to design a controller that, while guaranteeing the stability of the closed-loop system without the exosystem, drives the tracking error to zero. A particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller. Under suitable assumptions, we show that the latter problem is solvable if and only if a pair of algebraic equations, called the regulator equations, is solvable. We derive conditions, in terms of the transfer function of the plant and eigenvalues of the exosystem, for the solvability of the regulator equations. Three examples illustrating the theory are presented.Index Terms-Infinite-dimensional exosystem, regulator equations, state feedback controller, unbounded control and observation operators.

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

“…In this paper we also extend the characterization of the controllers achieving output regulation via the regulator equations and the internal model principle for regular linear systems in the situation where the operator B d is unbounded and the controller (3) is a regular linear system. These results generalize the ones in [30,35] where B d was assumed to be bounded and in [25] where feedforward control was studied. The controllers constructed in this paper are infinite-dimensional due to the full order observers and the internal models of the exosystem (2).…”

“…The output tracking and disturbance rejection for regular linear systems without the robustness requirement have been studied in [25,43]. In this paper we also extend the characterization of the controllers achieving output regulation via the regulator equations and the internal model principle for regular linear systems in the situation where the operator B d is unbounded and the controller (3) is a regular linear system.…”

“…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].…”

“…Output tracking and disturbance rejection of nonsmooth signals with high accuracy have applications in the control of motor and disk drive systems and in power electronics [10]. Output tracking of signals generated by an infinite-dimensional exosystem have been studied using state space methods in [20,17,28,25,30], and using frequency domain techniques in [45,34,46,22]. Robust tracking of nonsmooth periodic functions has also been studied extensively in repetitive control [18,44,41] where the control objective is to achieve precise tracking for a finite number of frequency components of y ref (·).…”

A backstepping approach to the output regulation of boundary controlled parabolic PDEs

“…Starting with the pioneering work in Pohjolainen (1982) solutions of the robust output regulation problem for infinite-dimensional systems with unbounded observation and control are developed for finite-dimensional signal models in Hämäläinen and Pohjolainen (2009), Rebarber and Weiss (2003). Recently, results for output regulation problems regarding this system class with infinite-dimensional signal models are given in Paunonen and Pohjolainen (2013), Natarajan, Gilliam, and Weiss (2014). An essential assumption for the solution of the robust output regulation problem is the measurability of the outputs to be controlled, which is required by the internal model principle (see e.g.…”