Phase Limitations of Multipliers at Harmonics

We present a phase condition under which there is no suitable multiplier for a given continuous-time plant. The condition can be derived from either the duality approach or from the frequency interval approach. The condition has a simple graphical interpretation, can be tested in a numerically efficient manner and may be applied systematically. Numerical examples show significant improvement over existing results in the literature. The condition is used to demonstrate a third order system with delay that is a counterexample to the Kalman Conjecture.

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

[2]  K. Narendra,et al.  An off-axis circle criterion for stability of feedback systems with a monotonic nonlinearity , 1968 .

[3]  W. Heath,et al.  Multipliers for Nonlinearities With Monotone Bounds , 2020, IEEE Transactions on Automatic Control.

[4]  N. Barabanov,et al.  On the Kalman problem , 1988 .

[5]  Ulf Jönsson,et al.  Robustness Analysis of Uncertain and Nonlinear Systems , 1996 .

[6]  Michael G. Safonov,et al.  Computation of Zames-Falb multipliers revisited , 2010, 49th IEEE Conference on Decision and Control (CDC).

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

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

[9]  A. Rantzer.  Friction analysis based on integral quadratic constraints , 2001 .

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

[11]  Alexander Lanzon,et al.  LMI searches for anticausal and noncausal rational Zames-Falb multipliers , 2014, Syst. Control. Lett..

[12]  Carsten W. Scherer,et al.  Full‐block multipliers for repeated, slope‐restricted scalar nonlinearities , 2017 .

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

[14]  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).

[15]  Michael G. Safonov,et al.  All stability multipliers for repeated MIMO nonlinearities , 2005, Syst. Control. Lett..

[16]  Nikolay V. Kuznetsov,et al.  Hidden attractors in Dynamical Systems. From Hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits , 2013, Int. J. Bifurc. Chaos.

[17]  M. Safonov,et al.  Zames-Falb multipliers for MIMO nonlinearities , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[18]  Carsten W. Scherer,et al.  IQC‐synthesis with general dynamic multipliers , 2014 .

[19]  Ulf T. Jönsson,et al.  Duality in multiplier-based robustness analysis , 1999, IEEE Trans. Autom. Control..

[20]  Matthew C. Turner,et al.  L gain bounds for systems with sector bounded and slope-restricted nonlinearities , 2012 .

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

[22]  Sourav Patra,et al.  Absolute stability analysis for negative-imaginary systems , 2016, Autom..

[23]  Joaquín Carrasco,et al.  Comment on "Absolute stability analysis for negative-imaginary systems" [Automatica 67 (2016) 107-113] , 2017, Autom..

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

[25]  Anders Rantzer,et al.  Duality Bounds in Robustness Analysis , 1996 .

[26]  D. Altshuller.  Frequency Domain Criteria for Absolute Stability: A Delay-integral-quadratic Constraints Approach , 2012 .

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

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

[29]  Murti V. Salapaka,et al.  A generalized Zames-Falb multiplier , 2011, Proceedings of the 2011 American Control Conference.

[30]  U.T. Jonsson,et al.  A MATLAB toolbox for robustness analysis , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[31]  Matthew C. Turner,et al.  On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities , 2009, IEEE Transactions on Automatic Control.

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

[33]  Alexandre Megretski,et al.  New results for analysis of systems with repeated nonlinearities , 2001, Autom..

[34]  R. Fitts,et al.  Two counterexamples to Aizerman's conjecture , 1966 .

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

[36]  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.

[37]  Michael G. Safonov,et al.  All multipliers for repeated monotone nonlinearities , 2002, IEEE Trans. Autom. Control..

[38]  J. Wen,et al.  Robustness analysis of LTI systems with structured incrementally sector bounded nonlinearities , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[39]  Joaquín Carrasco,et al.  Kalman Conjecture for Resonant Second-Order Systems with Time Delay , 2018, 2018 IEEE Conference on Decision and Control (CDC).