Exponential convergence bounds using integral quadratic constraints

The theory of integral quadratic constraints (IQCs) allows verification of stability and gain-bound properties of systems containing nonlinear or uncertain elements. Gain bounds often imply exponential stability, but it can be challenging to compute useful numerical bounds on the exponential decay rate. In this work, we present a modification of the classical IQC results of Megretski and Rantzer [13] that leads to a tractable computational procedure for finding exponential rate certificates. We demonstrate the effectiveness of our method via a numerical example.

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

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

[3]  M. Corless,et al.  Bounded controllers for robust exponential convergence , 1993 .

[4]  Stephen P. Boyd,et al.  Method of centers for minimizing generalized eigenvalues , 1993, Linear Algebra and its Applications.

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

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

[7]  A. Rantzer,et al.  System Analysis via Integral Quadratic Constraints. Part II , 1997 .

[8]  U. Jonsson.  A nonlinear Popov criterion , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

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

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

[11]  Keiji Konishi,et al.  Robust stability of Lure systems with time-varying uncertainties: a linear matrix inequality approach , 1999, Int. J. Syst. Sci..

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

[13]  Adrian Wills,et al.  Zames-Falb Multipliers for Quadratic Programming , 2007, IEEE Transactions on Automatic Control.

[14]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[15]  P. Olver.  Nonlinear Systems , 2013 .

[16]  Manfred Morari,et al.  Embedded Online Optimization for Model Predictive Control at Megahertz Rates , 2013, IEEE Transactions on Automatic Control.

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

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