Extension of Popov absolute stability criterion to non-autonomous systems with delays

This paper extends in a simple way the classical absolute stability Popov criterion to multivariable systems with delays and with time-varying memoryless non-linearities subject to sector conditions. The proposed sufficient conditions are expressed in the frequency domain, a form well-suited for robustness issues, and lead to simple graphical interpretations for scalar systems. Apart from the usual conditions, the results assume basically a generalized sector condition on the derivative of the non-linearities with respect to time. Results for local and global stability are given, the latter concerning in particular the linear time-varying ones. For rational transfers, the frequency conditions are equivalent to some easy-tocheck linear matrix inequalities:this leads to a tractable method of numerical resolution by rational approximation of the transfer. As an illustration, a numerical example is provided.

[1]  C. Desoer.  Frequency domain criteria for absolute stability , 1975, Proceedings of the IEEE.

[2]  P. Borne,et al.  Matrix approaches to the absolute stability of time-varying Lurie-Postnikov systems , 1979 .

[3]  F. R. Gantmakher.  Absolute stability of nonlinear control systems , 1968 .

[4]  R. Curtain,et al.  L ∞ approximation and nuclearity of delay systems , 1988 .

[5]  P. Bliman,et al.  Absolute stability of nonautonomous delay systems: delay-dependent and delay-independent criteria , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[6]  Z. Rekasius,et al.  A stability criterion for feedback systems containing a single time-varying nonlinear element , 1965 .

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

[8]  C. Corduneanu,et al.  ABSOLUTE STABILITY OF SOME INTEGRO-DIFFERENTIAL SYSTEMS , 1972 .

[9]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[10]  A. Halanay,et al.  APPROXIMATIONS OF DELAYS BY ORDINARY DIFFERENTIAL EQUATIONS , 1981 .

[11]  A. Rantzer.  On the Kalman-Yakubovich-Popov lemma , 1996 .

[12]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[13]  Liapunov Functions for Feedback Systems Containing a Single Time-varying Non-linearity† , 1968 .

[14]  Kestutis Pyragas.  Control of chaos via extended delay feedback , 1995 .

[15]  J. Walker.  Stability of Feedback Systems Involving Time-delays and a Time-varying Non-linearity , 1967 .

[16]  T. Başar.  Absolute Stability of Nonlinear Systems of Automatic Control , 2001 .