Asymptotic Stabilization of Single-Input Linear Systems Having Unknown but Bounded Parameters

This paper investigates the stabilization problem of single-input linear uncertain systems via constant state feedback control. The systems under consideration contain time-invariant uncertain parameters whose values are unknown but bounded in given compact sets and are thus called interval systems. The criterion for the asymptotic stability of the closed-loop system (obtained when a state feedback control is applied) is that all the eigenvalues of the resulting system matrix are in the strict left half of the complex plane. First, we show that to insure an interval system stabilizable, some entries of the system matrices must be sign invariant. More precisely, the number of the least-required sign-invariant entries in system matrices is equal to the system order. Then, we study the stabilizability of a set of interval systems called standard systems which contain sufficient numbers of sign-invariant entries in proper locations. After dividing all standard systems into some subsets by the uncertainty locations, we then derive necessary and sufficient conditions under which every system in a subset is stabilizable, regardless of its parameter varying bounds. The conditions show that all uncertain entries in system matrices should form a particular geometrical pattern called a "generalized antisymmnetric stepwise configuration". The result is further generalized for unstandard systems via linear transformation.