A note on quadratic performance of a class of switched uncertain nonlinear systems

We address quadratic performance, specifically quadratic stabilization and L2 gain for a class of switched systems consisting of uncertain nonlinear subsystems. In each subsystem, there exist norm-bounded uncertainty and constant vector in the linear part, and the vector field includes a nonlinear term with a quadratic constraint. Under the assumption that a convex combination of subsystems' constant vectors reaches the origin, we define a new convex combination of subsystems (CCS) without a constant vector, which incorporates the system matrices, uncertainties and nonlinear terms in a unified manner. If the CCS is quadratically stable (QS) at the origin, we propose a state-dependent switching law under which the switched uncertain nonlinear system (SUNS) is also QS at the origin. A detailed discussion is also presented on the upper bound of the system state when the convex combination of the subsystems' constant vectors is not zero. Furthermore, a parallel discussion and results are established for the quadratic L2 gain of the SUNS.