Convergence Rates of a Class of Uncertain Dynamic Systems

The problem of stabilization of uncertain systems plays a broad and fundamental role in robust control theory. The paper examines a boundedness theorem for a class of uncertain systems characterized as having a decreasing Lyapunov function in a ringlike region. It is a systematic study on stability that embraces both the transient and steady analysis, covering such aspects as the maximum overshoot of the system state, the stability region and the exponential convergence rate. The emphasis throughout is on deriving dominant time constants and explicit time expressions for a state to reach an invariant set. The central theorem provides a complete treatment of the time evolution of trajectories depending on the specific compact set of initial conditions. Toward this end, the comparison lemma along with a particular Riccati differential equation are essential and conclusive. The scope of questions addressed in the paper, the uniformity of their treatment, the novelty of the proposed theorem, and the obtained results make it very useful with respect to other works on the problem of robust nonlinear control.