Stabilization of linear time-invariant interval systems via constant state feedback control

This paper investigates the stabilization problem of 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, the author shows 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, the author studies 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, the author then derives 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 antisymmetric stepwise configuration”. For an interval system satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example. The result is further generalized for nonstandard systems via linear transformation