Stability analysis of hybrid integrator-gain systems using linear programming

Abstract

Analyzing the stability and performance of (possibly discontinuous) conewise linear systems, e.g., hybrid integrator-gain systems (HIGS), requires tools specifically designed for hybrid systems. For HIGS in particular, most results rely on constructing continuous piecewise quadratic (CPQ) Lyapunov functions by solving suitable sets of linear matrix inequalities. For the purpose of creating additional efficient methods for the analysis of HIGS-controlled systems and other conewise linear systems, a method for constructing continuous piecewise affine (CPA) Lyapunov functions with linear programming is proposed. We present the following main contributions: First, we extend the CPA method to the construction of input-to-state stable Lyapunov functions for (discontinuous) conewise linear systems by using Farkas’ lemma. Second, we show that the CPA method can also be used to calculate upper bounds on important performance metrics of conewise linear systems, specifically, the L1-gain and H1-norm. Third, a converse theorem is presented, which shows that, given an input-to-state stable conewise linear system and sufficiently refined CPA function, the linear conditions of the proposed method become necessary. The CPA method is demonstrated in a numerical example for stability and performance analysis of a HIGS-controlled system. In this example, the CPA method is compared to the CPQ method in terms of computational efficiency.