Trajectories bounds for nonlinear systems: a BMI approach

This paper deals with the analysis and synthesis of nonlinear systems, and considers the problems of computing extremal values of the trajectories over a given set of initial conditions as well as finding output controllers minimizing these extremal values under time-domain constraints. It is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one-parameter sequence of BMI optimizations. Moreover, a necessary and sufficient condition is proposed to establish the tightness of the found upper bound in spite of the conservatism introduced by the non-convexity of BMI optimizations and the chosen degree of the Lyapunov function and relaxing polynomials.