Guaranteed Asymptotic Stability for Some Linear Systems With Bounded Uncertainties

We consider a class of linear dynamical systems in which the system and input matrices, as well as the input, are uncertain. The nominal system is time-invariant, while the uncertainties are assumed to be measurable functions of time whose values may range in given compact sets. Utilizing solely the knowledge of the sets from which uncertain quantities take their values, we derive a state feedback controller that guarantees global uniform asymptotic (Lyapunov) stability of the zero state in the presence of admissible uncertainties. The controller is nonlinear, namely componentwise switching; however, its construction requires only the solution of a linear matrix equation. Unlike linear feedback, this nonlinear controller assures asymptotic stability for any admissible realization of the system; this is illustrated by means of a simple example.