Zames–Falb Multipliers for Invariance

This letter provides a comprehensive framework for local stability analysis of uncertain feedback interconnections within the integral quadratic constraints theory using general dynamic multipliers. It is shown how so-called hard and soft constraints can be effectively combined in order to (locally) capture the action of the uncertainty. This is illustrated for Zames–Falb multipliers, where it is proven that the subclasses of causal and anticausal multipliers can easily be factorized into hard constraints and thus individually be incorporated into the framework without introducing conservatism.

[1]  C. W. Scherer,et al.  Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness , 2005, SIAM J. Matrix Anal. Appl..

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

[3]  U. Jönsson.  Stability analysis with Popov multipliers and integral quadratic constraints , 1997 .

[4]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[5]  Stephen P. Boyd,et al.  Analysis of linear systems with saturation using convex optimization , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[6]  Matthew C. Turner,et al.  Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches , 2016, Eur. J. Control.

[7]  Carsten W. Scherer,et al.  A General Integral Quadratic Constraints Theorem with Applications to a Class of Sampled-Data Systems , 2016, SIAM J. Control. Optim..

[8]  Venkataramanan Balakrishnan,et al.  Lyapunov functionals in complex μ analysis , 2002, IEEE Trans. Autom. Control..

[9]  Matthew C. Turner,et al.  ℒ2 gain bounds for systems with slope-restricted nonlinearities , 2010, Proceedings of the 2010 American Control Conference.

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

[11]  Carsten W. Scherer,et al.  Stability analysis with integral quadratic constraints: A dissipativity based proof , 2013, 52nd IEEE Conference on Decision and Control.

[12]  Sophie Tarbouriech,et al.  Stability and Stabilization of Linear Systems with Saturating Actuators , 2011 .

[13]  Tingshu Hu,et al.  Control Systems with Actuator Saturation: Analysis and Design , 2001 .

[14]  Peter Seiler,et al.  Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints , 2015, IEEE Transactions on Automatic Control.

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

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

[17]  Carsten W. Scherer,et al.  Stability and performance analysis on Sobolev spaces , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[18]  Carsten W. Scherer,et al.  Invariance With Dynamic Multipliers , 2018, IEEE Transactions on Automatic Control.

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

[20]  Tingshu Hu,et al.  Control Systems with Actuator Saturation: Analysis and Design , 2001 .

[21]  Alexandre Megretski.  New IQC for quasi-concave nonlinearities , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

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

[23]  Tingshu Hu,et al.  Analysis of linear systems in the presence of actuator saturation and I-disturbances , 2004, Autom..