Stability Properties of Coupled Impedance Passive LTI Systems

Abstract

Abstract-We study the stability of the feedback interconnection of two impedance passive linear time-invariant systems, of which one is finite-dimensional. The closed-loop system is well known to be impedance passive, but no stability properties follow from this alone. We are interested in two main issues: (1) the strong stability of the operator semigroup associated with the closed-loop system, (2) the input-output stability (meaning transfer function in H ∞ ) of the closed-loop system. Our results are illustrated with the system obtained from the non-uniform SCOLE (NASA Spacecraft Control Laboratory Experiment) model representing a vertical beam clamped at the bottom, with a rigid body having a large mass on top, connected with a trolley mounted on top of the rigid body, via a spring and a damper. Such an arrangement called a tuned mass damper (TMD), is used to stabilize tall buildings. We show that the SCOLE-TMD system is strongly stable on the energy state space and that the system is input-output stable from the horizontal force input to the horizontal velocity output.

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

“…Finally, we would like to highlight Theorem 6.16 which shows that positive realness of a transfer function H is equivalent to H being stabilized by every stable transfer function K such that − K is strongly positive real. Establishing alternative sufficient conditions for the stability of the feedback interconnection of two transfer functions H and K in H 0 (L(U )) in terms of positive-real type properties of H and − K seems an interesting problem for future work (see also [56,Theorem 4.2] which has some overlap with Theorem 6.16). We feel that the theory developed in the current paper is likely to be useful in this context.…”

“…Other sufficient conditions for the stability of the feedback interconnection of H and K in terms of positive-real properties of H and − K have appeared in the literature, and we only refer here to the recent result [56,Theorem 4.2].…”

The regulation problem for the one-dimensional Schrödinger equation via the backstepping approach

“…We introduce a special class of well-posed systems, following the terminology in [29], [31], [32], [34] and many other papers. We do this because our systems (1.1) and (1.2) fall into this category (as we shall see), and we will use tools developed for such systems.…”

“…Let G be a function defined on some domain in C that contains a right half-plane, with values in a normed space. Following [34], we say that G is strictly proper if lim Re s → ∞ G(s) = 0, uniformly with respect to Im s.…”

Stability and Robust Regulation of Passive Linear Systems

“…In this paper we introduce new results for studying polynomial and the more general non-uniform stability for coupled passive abstract linear systems (1.1) and (1.2). Strong and exponential closed-loop stabilities of infinite-dimensional systems have been studied in the literature for passive one-dimensional boundary control systems [36,30], coupled systems with collocated inputs and outputs [15], and passive systems coupled with finite-dimensional systems [44]. Polynomial stability of coupled systems has been studied extensively in the context of coupled linear partial differential equations [3,1,6,2], and for abstract hyperbolic-parabolic systems [20].…”