Concise stability conditions for systems with static nonlinear feedback expressed by a quadratic program

We consider the stability of the feedback connection of a strictly proper linear time invariant (LTI) stable system with a static nonlinearity expressed by a convex quadratic program (QP). From the Karush-KuhnTucker (KKT) conditions for the QP, we establish quadratic constraints that may be used with a quadratic Lyapunov function to construct a stability criterion via the S-procedure. The approach is based on existing results in the literature, but gives a more parsimonious linear matrix inequality (LMI) criterion and is much easier to implement. Our approach ∗G. Li is with Automatic Control Laboratory, Department of Mechanical Engineering, The University of Bristol, Queens Building, University Walk, Bristol, BS8 1TR, UK. Email: [email protected] †W. P. Heath and B. Lennox are with Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, PO Box 88, Sackville Street, Manchester M60 1QD, UK. Email: [email protected] [email protected]