Robustness analysis of LTI systems with structured incrementally sector bounded nonlinearities

This paper addresses the stability analysis of a negative feedback interconnection of a multivariable linear time-invariant system and a structured time-invariant incrementally sector bounded nonlinearity. The classic Zames-Falb multiplier (1968) is extended to the multivariable case and is approximated arbitrarily closely by linear matrix inequalities. The problem of finding the multiplier that provides the largest stability bound then becomes a convex optimization problem over state space parameters. The method is also applied to symmetric incrementally sector bounded structured nonlinearities and provides an upper bound for the generalized structured singular value. Numerical examples are provided to demonstrate the effectiveness of this method.

[1]  G. Zames.  On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity , 1966 .

[2]  C. P. Neuman,et al.  Stability of continuous time dynamical systems with m-feedback nonlinearities. , 1967 .

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

[4]  G. Zames.  Stability of systems with sector nonlinearities: A comparison of various inequalities , 1968 .

[5]  Brian D. O. Anderson,et al.  Algebraic Structure of Generalized Positive Real Matrices , 1968 .

[6]  K. Narendra,et al.  Frequency Domain Criteria for Absolute Stability , 1973 .

[7]  John C. Doyle.  Analysis of Feedback Systems with Structured Uncertainty , 1982 .

[8]  Michael G. Safonov,et al.  Computer-aided stability analysis renders Papov criterion obsolete , 1987 .

[9]  Andrew Packard,et al.  A General Statement of Structured Singular Value Concepts , 1988, 1988 American Control Conference.

[10]  The Optimal Multiplier Method for Nonlinear Robustness Analysis , 1990, 1990 American Control Conference.

[11]  A. Tits,et al.  Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics , 1991 .

[12]  Peter M. Young,et al.  Robustness with parametric and dynamic uncertainty , 1993 .

[13]  Jose C. Geromel,et al.  A convex approach to the absolute stability problem , 1994, IEEE Trans. Autom. Control..

[14]  Dennis S. Bernstein,et al.  Extensions of mixed-µ bounds to monotonic and odd monotonic nonlinearities using absolute stability theory† , 1994 .

[15]  A. Packard,et al.  Linear matrix inequalities in analysis with multipliers , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[16]  M. Safonov,et al.  Real/complex multivariable stability margin computation via generalized Popov multiplier-LMI approach , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[17]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.