The stability analysis of systems with nonlinear feedback expressed by a quadratic program

We consider the stability of the feedback connection of a stable linear time invariant (LTI) plant with a static nonlinearity expressed by a certain class of quadratic program (QP). We establish quadratic constraints from the Karush-Kuhn-Tucker (KKT) conditions that may be used to construct a piecewise quadratic Lyapunov function via the S-procedure. The approach is based on existing results in the literature, but gives a more parsimonious linear matrix inequality (LMI) criterion. Our approach can be extended to model predictive control (MPC), and gives equivalent results to those in the literature but with a much lower dimension LMI criterion