Quadratic stabilizability of linear systems with structural independent time-varying uncertainties

The author investigates the problem of designing a linear state feedback control to stabilize a class of single-input uncertain linear dynamical systems. The systems under consideration contain time-varying uncertain parameters whose values are unknown but bounded in given compact sets. The method used to establish asymptotical stability of the closed-loop system (obtained when the feedback control is applied) involves the use of a quadratic Lyapunov function. The author first shows that to ensure a stabilizable system some entries of the system matrices must be sign invariant. He then derives necessary and sufficient conditions under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties. The conditions show that all uncertainties can only enter the system matrices in such a way as to form a particular geometrical pattern called an antisymmetric stepwise configuration. For the systems satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example