Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

This paper considers robust stability and robust performance analysis for discrete-time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time-invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur’e-Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H2 and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.

[1]  J. Pearson,et al.  On the absolute stability of sampled-data systems: The "Indirect control" case , 1964 .

[2]  D. S. Bernstein,et al.  Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their application to robust stability , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[3]  A. Rantzer,et al.  System analysis via integral quadratic constraints , 1997, IEEE Trans. Autom. Control..

[4]  A. Megretski,et al.  Integral quadratic constraints for monotonic and slope restricted diagonal operators , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[5]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[6]  A. I. Lurʹe.  Some non-linear problems in the theory of automatic control , 1957 .

[7]  P. Falb,et al.  Stability Conditions for Systems with Monotone and Slope-Restricted Nonlinearities , 1968 .

[8]  Sang Woo Kim,et al.  A Revisited Tsypkin Criterion for Discrete-Time Nonlinear Lur'e Systems with Monotonic Sector-Restrictions , 1998, Autom..

[9]  J. Pearson.  A note on the existence of Liapunov functions for a class of discrete-time systems , 1963 .

[10]  D. Siljak.  Algebraic criteria for positive realness relative to the unit circle , 1973 .

[11]  J. E. Gibson,et al.  On the Asymptotic Stability of a Class of Saturating Sampled-Data Systems , 1964, IEEE Transactions on Applications and Industry.

[12]  G. Zames,et al.  On the stability of systems with monotone and odd monotone nonlinearities , 1967, IEEE Transactions on Automatic Control.

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

[14]  Brian D. O. Anderson,et al.  Discrete positive-real fu nctions and their applications to system stability , 1969 .

[15]  D. Bernstein,et al.  Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete-time theory , 1993 .

[16]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[17]  Wassim M. Haddad,et al.  A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach , 1996, IEEE Trans. Autom. Control..

[18]  E. I. Jury,et al.  On the absolute stability of nonlinear sample-data systems , 1964 .

[19]  A. Stoorvogel.  The robust H2 control problem: a worst-case design , 1993, IEEE Trans. Autom. Control..

[20]  E. I. Jury,et al.  On the stability of a certain class of nonlinear sampled-data systems , 1964 .

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

[22]  Dennis S. Bernstein,et al.  Parameter-Dependent Lyapunov Functions and the Discrete-Time Popov Criterion for Robust Analysis and Synthesis , 1992 .