On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

This paper considers the robust stability of a discrete-time Lurye system consisting of the feedback interconnection between a linear system and a bounded and monotone nonlinearity. It has been conjectured that the existence of a suitable linear time-invariant (LTI) O’Shea-Zames-Falb multiplier is not only sufficient but also necessary. Roughly speaking, a successful proof of the conjecture would require: (a) a conic parameterization of a set of multipliers that describes exactly the set of nonlinearities, (b) a lossless S-procedure to show that the non-existence of a multiplier implies that the Lurye system is not uniformly robustly stable over the set of nonlinearities, and (c) the existence of a multiplier in the set of multipliers used in (a) implies the existence of an LTI multiplier. We investigate these three steps, showing the current bottlenecks for proving this conjecture. In addition, we provide an extension of the class of multipliers which may be used to disprove the conjecture.

[1]  Randy A. Freeman,et al.  Noncausal Zames-Falb Multipliers for Tighter Estimates of Exponential Convergence Rates , 2018, 2018 Annual American Control Conference (ACC).

[2]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

[3]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[4]  J. Isbell.  Birkhoff’s problem 111 , 1955 .

[5]  C. Scherer,et al.  Absolute stability analysis of discrete time feedback interconnections , 2017 .

[6]  Shuai Wang,et al.  Convex Searches for Discrete-Time Zames–Falb Multipliers , 2018, IEEE Transactions on Automatic Control.

[7]  M. Dahleh,et al.  Control of Uncertain Systems: A Linear Programming Approach , 1995 .

[8]  William P. Heath,et al.  Phase Limitations of Zames–Falb Multipliers , 2017, IEEE Transactions on Automatic Control.

[9]  W. Heath,et al.  Duality Bounds for Discrete-Time Zames–Falb Multipliers , 2020, IEEE Transactions on Automatic Control.

[10]  Matthew C. Turner,et al.  Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches , 2015, 2015 European Control Conference (ECC).

[11]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[12]  T. Georgiou,et al.  Robustness analysis of nonlinear feedback systems: an input-output approach , 1997, IEEE Trans. Autom. Control..

[13]  P. Seiler,et al.  Zames–Falb multipliers for convergence rate: motivating example and convex searches , 2020, International Journal of Control.

[14]  Ulrich Eggers,et al.  Introduction To Infinite Dimensional Linear Systems Theory , 2016 .

[15]  Manuel de la Sen,et al.  Second-order counterexamples to the discrete-time Kalman conjecture , 2015, Autom..

[16]  J. Willems,et al.  Some new rearrangement inequalities having application in stability analysis , 1968 .

[17]  M. Zwaan.  An introduction to hilbert space , 1990 .

[18]  Discrete‐time systems with slope restricted nonlinearities: Zames–Falb multiplier analysis using external positivity , 2021, International Journal of Robust and Nonlinear Control.

[19]  R. O'Shea,et al.  A frequency-time domain stability criterion for sampled-data systems , 1967, IEEE Transactions on Automatic Control.

[20]  Benjamin Recht,et al.  Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints , 2014, SIAM J. Optim..

[21]  R. O'Shea.  An improved frequency time domain stability criterion for autonomous continuous systems , 1966, IEEE Transactions on Automatic Control.

[22]  Simon Michalowsky,et al.  Robust and structure exploiting optimisation algorithms: an integral quadratic constraint approach , 2019, Int. J. Control.

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

[24]  Peter Seiler,et al.  Construction of Periodic Counterexamples to the Discrete-Time Kalman Conjecture , 2020, IEEE Control Systems Letters.

[25]  Ulf Jönsson,et al.  Integral Quadratic Constraints , 2010 .

[26]  Sei Zhen Khong,et al.  On the Necessity and Sufficiency of the Zames-Falb Multipliers , 2021, Autom..

[27]  A. Megretski.  Combining L1 and L2 methods in the robust stability and performance analysis of nonlinear systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[28]  U. Jonsson,et al.  Stability analysis of systems with nonlinearities , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[29]  Alexander Lanzon,et al.  Equivalence between classes of multipliers for slope-restricted nonlinearities , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[30]  R. Saeks,et al.  The analysis of feedback systems , 1972 .