Additional dynamics in transformed time-delay systems

In studying the stability of time-delay systems, many published results use a transformation to transform a system with single time-delay to a system with distributed delay. In this article, the inherent limitations of such approaches are studied. Specifically, it is shown that such a transformation incurs additional dynamics which can be characterized by appropriate additional eigenvalues. The critical delay values when such additional eigenvalues cross the imaginary axis can be explicitly calculated. If the smallest of such delays is the smallest of the stability delay limit of the original system, then any stability criteria obtained using such transformation will be conservative. Some examples are also included.

[1]  R. Datko.  A procedure for determination of the exponential stability of certain differential-difference equations , 1978 .

[2]  Erik I. Verriest,et al.  Stability of Systems with Distributed Delays , 1995 .

[3]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[4]  Erik I. Verriest,et al.  Frequency domain robust stability criteria for linear delay systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[5]  M. Fu,et al.  Robust stability and stabilization of time-delay systems via integral quadratic constraint approach , 1998 .

[6]  C. D. Souza,et al.  Criteria for Robust Stability of Uncertain Linear Systems with Time-Varying State Delays , 1996 .

[7]  V. Kolmanovskii,et al.  On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems , 1999 .

[8]  C. Nett,et al.  A new method for computing delay margins for stability of linear delay systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[9]  K. Gu.  Discretized LMI set in the stability problem of linear uncertain time-delay systems , 1997 .

[10]  T. Su,et al.  Robust stability of delay dependence for linear uncertain systems , 1992 .

[11]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[12]  Jean-Michel Dion,et al.  Stability and robust stability of time-delay systems: A guided tour , 1998 .

[13]  J. Richard,et al.  Stability of perturbed systems with time-varying delays , 1997 .

[14]  Bor-Sen Chen,et al.  Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach , 1991 .

[15]  Bor-Sen Chen,et al.  Robust stability of uncertain time-delay systems , 1987 .

[16]  H. Antosiewicz,et al.  Differential Equations: Stability, Oscillations, Time Lags , 1967 .

[17]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[18]  Juing-Huei Su.  Further results on the robust stability of linear systems with a single time delay , 1994 .

[19]  K. Furuta,et al.  Memoryless stabilization of uncertain linear systems including time-varying state delays , 1992 .

[20]  Jean-Pierre Richard,et al.  Some Model Transformation for the Stability Study of Linear Systems With Delay , 1998 .

[21]  V. Kolmanovskii,et al.  Applied Theory of Functional Differential Equations , 1992 .