Feedback and Feedforward Output Regulation of Bounded Uniformly Continuous Signals for Infinite‐Dimensional Systems

Immonen, Eero; Pohjolainen, Seppo

doi:10.1137/050623000

Cited by 51 publications

(45 citation statements)

References 15 publications

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“…Indeed, the straightforward approach is to use a full state observer, which of course is infinite-dimensional. This is the approach taken in several references, for example, in Byrnes, Lauko, Gilliam and Shubov [4] or in Immonen and Pohjolainen [28]. For plants that are already stable, the real challenge (that we shall address in our follow-up paper) is to design a finite-dimensional error feedback controller.…”

Section: Introductionmentioning

confidence: 97%

“…Regulator theory for infinite-dimensional linear systems with bounded control and observation operators has been significantly advanced by a group of researchers at Tampere University of Technology (Finland) who have developed a sophisticated theory of infinite-dimensional exosystems, see for instance [24], [26]- [28], [39], [40]. The state feedback regulator problem for exponentially stabilizable linear plants driven by infinite-dimensional exosystems generating periodic signals was addressed in [27].…”

Section: Introductionmentioning

confidence: 99%

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The State Feedback Regulator Problem for Regular Linear Systems

Natarajan

Gilliam

Weiss

2014

IEEE Trans. Automat. Contr.

129

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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.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

The State Feedback Regulator Problem for Regular Linear Systems

Natarajan

Gilliam

Weiss

2014

IEEE Trans. Automat. Contr.

129

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“…In particular, if g(ω) = M 0 ω α for some α, M 0 > 0, then M −1 log (ct) ∼ (t/ log t) 1/α [3, Ex. 1.7], and thus (20) and (21) In the case where the operator B d and the operator L in the controller are bounded, the convergence rate (21) is achieved for all v 0 ∈ D(S) and x e0 ∈ D(A e ), since in this case we have B e ∈ L(W, X e ) and A −1 e B e v 0 ∈ D(A e ). The norm on the right-hand side of (21) can then be estimated by…”

Section: If We In Particular Choosementioning

confidence: 99%

“…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 (·).…”

Section: Introductionmentioning

confidence: 99%

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Paunonen¹

2017

SIAM J. Control Optim.

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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

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“…For time-invariant systems and for τ -periodic reference and disturbance signals y ref (·) and w dist (·) it is possible to solve the output regulation problem by solving the regulator equations associated to an infinite-dimensional autonomous exosystem [14,Thm. 3.1].…”

Section: Construction Of the Controllersmentioning

confidence: 99%

Robust Output Regulation for Continuous-Time Periodic Systems

Paunonen

2017

IEEE Trans. Automat. Contr.

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Abstract. We consider controller design for robust output tracking and disturbance rejection for continuous-time periodic linear systems with periodic reference and disturbance signals. As our main results we present four different controllers: A feedforward control law and a discrete-time dynamic error feedback controller for output tracking and disturbance rejection, a robust discrete-time feedback controller, and finally a discrete-time feedback controller that achieves approximate robust output tracking and disturbance rejection. The presented constructions are also new for time-invariant finite and infinite-dimensional systems. The results are illustrated with two examples: A periodically time-dependent system of harmonic oscillators and a nonautonomous two-dimensional heat equation with boundary disturbance. IntroductionIn this paper we study the output regulation problem for an exponentially stable τ -periodic system of the forṁwith state space X, input space U 0 , and output space Y 0 . The main goal in our control problem is to design a control law in such a way that the output y(t) ∈ Y 0 converges asymptotically to a τ -periodic reference signal y ref (·) despite the external τ -periodic reference signal w dist (·). In the robust output regulation problem we in addition require that the same controller achieves the output tracking even for perturbed parameters (Ã(t),B(t),B d (t),C(t), D(t)) of the system (1). Throughout the paper we consider systems (1) on a Banach or Hilbert space X. This class of systems includes a wide range of nonautonomous partial differential equations, delay equations, and infinite systems of ordinary differential equations. However, the presented results are also new and directly applicable for finite-dimensional periodic systems on X = C n and X = R n . Finally, our results offer three new controllers for output regulation of linear time-invariant systems with nonsmooth periodic reference and disturbance signals.2010 Mathematics Subject Classification. 93C05, 93B52 (93B28).

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Feedback and Feedforward Output Regulation of Bounded Uniformly Continuous Signals for Infinite‐Dimensional Systems

Cited by 51 publications

References 15 publications

The State Feedback Regulator Problem for Regular Linear Systems

The State Feedback Regulator Problem for Regular Linear Systems

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Robust Output Regulation for Continuous-Time Periodic Systems

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