The effect of small time-delays on the closed-loop stability of boundary control systems

It has been observed that for many stable feedback systems, the introduction of arbitrarily small time-delays into the loop causes instability. In this paper we present a systematic treatment of this phenomenon for a large class of boundary control systems which allows for in-span control. Our approach is based on a combination of input-output methods and modal analysis. We give a number of sufficient conditions for robustness/nonrobustness of closed-loop input-output stability with respect to delays. Our framework includes a large class of ill-posed systems, i.e., systems whose open-loop transfer function is unbounded on any right half-plane. We then analyze the relationship between the poles of the transfer function and the exponential modes of the underlying boundary-value problem to derive internal stability properties from external ones.

[1]  Andrew Packard,et al.  The complex structured singular value , 1993, Autom..

[2]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[3]  Goong Chen,et al.  Exponential stability analysis of Xa long chain of coupled vibrating strings with dissipative linkage , 1989 .

[4]  M. Omizo,et al.  Modeling , 1983, Encyclopedic Dictionary of Archaeology.

[5]  Yves Nievergelt,et al.  Complex analysis in one variable , 1985 .

[6]  R. Datko,et al.  Two examples of ill-posedness with respect to small time delays in stabilized elastic systems , 1993, IEEE Trans. Autom. Control..

[7]  Yuncheng You,et al.  Some second-order vibrating systems cannot tolerate small time delays in their damping , 1991 .

[8]  Hartmut Logemann,et al.  Conditions for Robustness and Nonrobustness of theStability of Feedback Systems with Respect to Small Delays inthe Feedback Loop , 1996 .

[9]  W. Desch,et al.  Destabilization due to delay in one dimensional feedback , 1989 .

[10]  Tryphon T. Georgiou,et al.  Graphs, causality, and stabilizability: Linear, shift-invariant systems on ℒ2[0, ∞) , 1993, Math. Control. Signals Syst..

[11]  L. Markus,et al.  Stabilization of a hybrid system of elasticity by feedback boundary damping , 1988 .

[12]  J. Lagnese,et al.  An example of the effect of time delays in boundary feedback stabilization of wave equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[13]  R. Rebarber.  Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach , 1995 .

[14]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .

[15]  J. F. Barman,et al.  L 2 -stability and L 2 -instability of linear time-invariant distributed feedback systems perturbed by a small delay in the loop , 1973 .

[16]  R. Datko,et al.  Two questions concerning the boundary control of certain elastic systems , 1991 .

[17]  L. Pandolfi.  The pole and zero structure of a class of linear systems , 1983 .

[18]  Michael P. Polis,et al.  An example on the effect of time delays in boundary feedback stabilization of wave equations , 1986 .

[19]  Stuart Townley,et al.  The effect of small delays in the feedback loop on the stability of neutral systems , 1996 .

[20]  A. Krall,et al.  Modeling stabilization and control of serially connected beams , 1987 .

[21]  R. Datko.  Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks , 1988 .

[22]  Hartmut Logemann,et al.  PDEs with Distributed Control and Delay in the Loop: Transfer Function Poles, Exponential Modes and Robustness of Stability , 1998, Eur. J. Control.

[23]  Kenneth B. Hannsgen,et al.  Effectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity , 1988 .