Remarks on input-to-state stability of collocated systems with saturated feedback
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Remarks on input-to-state stability of collocated systems with saturated feedback Birgit Jacob1
· Felix L. Schwenninger2,3
· Lukas A. Vorberg1
Received: 3 January 2020 / Accepted: 24 August 2020 / Published online: 7 September 2020 © The Author(s) 2020
Abstract We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption on the linear system, we show ISS for the saturated one. We discuss the sharpness of the conditions in light of existing results in the literature. Keywords Input-to-state stability · Saturation · Collocated system · Semilinear system · Infinite-dimensional system
1 Introduction In this note we continue the study of the stability of systems of the form
B
x(t) ˙ = Ax(t) − Bσ B ∗ x(t) + d(t) , x(0) = x0 ,
(Σ S L D )
Birgit Jacob [email protected] Felix L. Schwenninger [email protected] Lukas A. Vorberg [email protected]
1
School of Mathematics and Natural Sciences, University of Wuppertal, IMACM, Gaußstraße 20, 42119 Wuppertal, Germany
2
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500,AE Enschede, The Netherlands
3
Department of Mathematics, Universityof Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
123
294
Mathematics of Control, Signals, and Systems (2020) 32:293–307
derived from the linear collocated open-loop system x(t) ˙ = Ax(t) + Bu(t), y(t) = B ∗ x(t). by the nonlinear feedback law u(t) = −σ (y(t) + d(t)). Here X and U are Hilbert spaces, A : D(A) ⊂ X → X is the generator of a strongly continuous contraction semigroup, and B is a bounded linear operator from U to X , i.e. B ∈ L(U , X ). The function σ : U → U is locally Lipschitz continuous and maximal monotone with σ (0) = 0. Of particular interest is the case in which σ is even linear in a neighbourhood of 0. The open-loop system is called collocated as the output operator B ∗ equals the adjoint of the input operator B. In the following, we are interested in stability with respect to both the initial value x0 , that is, internal stability, and the disturbance d, external stability. This is combined in the notion of input-to-state stability (ISS), which has recently been studied for infinite-dimensional systems e.g. in [7,9,19,20] and particularly for semilinear systems in [5,6,23], see also [18] for a survey. The effect of feedback laws acting (approximately) linearly only locally is known in the literature as saturation and first appeared in [24,25] in the context of stabilization of infinite-dimensional linear systems, see also [10]. There, internal stability of the closed-loop system was studied using nonlinear semigroup theory, a natural tool to establish existence and uniqueness of solutions for equations of the above type, see also the more recent works [11,15,16]. The simultaneous study of internal stability and the robustness with respect to additive disturbances in th
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