Exponential stabilization of well-posed systems by colocated feedback

Curtain, Ruth F.

doi:10.1137/040610489

Cited by 48 publications

(65 citation statements)

References 44 publications

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“…More details on stabilizability and detectability of regular linear systems can be found in [32,38]. For techniques for choosing K and L, see for instance [33,42,12].…”

Section: The System the Controller And The Closed-loop Systemmentioning

confidence: 99%

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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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“…More details on stabilizability and detectability of regular linear systems can be found in [32,38]. For techniques for choosing K and L, see for instance [33,42,12].…”

Section: The System the Controller And The Closed-loop Systemmentioning

confidence: 99%

“…It should be noted that ( P L (iω k )K 1k ) k∈Z ∈ ℓ 2 (C) implies that (P L (iω k )K 1k ) −1 → ∞ as |k| → ∞, and thus the condition (12) requires that Eφ k and F φ k decay sufficiently fast as |k| → ∞.…”

Section: The New Controller Structurementioning

confidence: 99%

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Paunonen¹

2017

SIAM J. Control Optim.

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“…The following example (adapted from [15,Example 5.6]) shows that Corollary 7.4 is in general not valid for unbounded B, that is, in the system node context, dissipativity of A together with the conditions C = B * and D H = 0 is not sufficient for the positive realness of H.…”

Section: ])mentioning

confidence: 99%

Transfer functions of infinite-dimensional systems: positive realness and stabilization

Guiver

Logemann

Opmeer

2017

Math. Control Signals Syst.

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We consider a general class of operator-valued irrational positive-real functions with an emphasis on their frequency-domain properties and the relation with stabilization by output feedback. Such functions arise naturally as the transfer functions of numerous infinite-dimensional control systems, including examples specified by PDEs. Our results include characterizations of positive realness in terms of imaginary axis conditions, as well as characterizations in terms of stabilizing output feedback, where both static and dynamic output feedback are considered. In particular, it is shown that stabilizability by all static output feedback operators belonging to a sector can be characterized in terms of a natural positive-real condition and, furthermore, we derive a characterization of positive realness in terms of a mixture of imaginary axis and stabilization conditions. Finally, we introduce concepts of strict and strong positive realness, prove results which relate these notions and analyse the relationship between the strong positive realness property and stabilization by feedback. The theory is illustrated by examples, some arising from controlled and observed partial differential equations.

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“…For the case of skew-adjoint A and bounded C see our Proposition 3.7. Curtain and Weiss [10] contains many further references on this topic and it gives rather general sufficient conditions on (A, B) for −B * to be a stabilizing state feedback operator. Ammari and Tucsnak [2] considered the case of second order systems without damping.…”

Section: Background On Admissibility Observability and Observersmentioning

confidence: 99%

Recovering the initial state of an infinite-dimensional system using observers

2010

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. Recovering the initial state of an infinitedimensional system using observers. Automatica, Elsevier, 2010, 46 (10) Abstract: Let A be the generator of a strongly continuous semigroup T on the Hilbert space X, and let C be a linear operator from D(A) to another Hilbert space Y (possibly unbounded with respect to X, not necessarily admissible). We consider the problem of estimating the initial state z 0 ∈ D(A) (with respect to the norm of X) from the output function y(t) = CT t z 0 , given for all t in a bounded interval [0, τ ]. We introduce the concepts of estimatability and backward estimatability for (A, C) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z 0 from y. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell's principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.

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Exponential stabilization of well-posed systems by colocated feedback

Cited by 48 publications

References 44 publications

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Transfer functions of infinite-dimensional systems: positive realness and stabilization

Recovering the initial state of an infinite-dimensional system using observers

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