A negative result on piecewise quadratic Lyapunov functions for decay rate analysis

In recent years, the advent of efficient methods for solving linear matrix inequality (LMI) problems has yielded a new interesting class of Lyapunov functions, the piecewise quadratic Lyapunov functions, subject to efficient computation. This class has shown promise, e.g. in the stability analysis of hybrid systems. In this paper, we describe one example that we have analyzed in order to investigate how LMI methods and piecewise quadratic Lyapunov functions can be used to iteratively synthesize a switched control law in order to obtain an improved decay rate. The example shows that piecewise quadratic Lyapunov functions are not sufficient to analyze the decay rate of a piecewise affine system, given that the partitioning used in the Lyapunov function is the same as in the piecewise affine system. As noted in other performance analysis examples, a finer partitioning is needed