Technical Notes and Correspondence Stabilization of Time-Delay Systems Using

Abstract-For linear time-invariant systems with one or more noncom-mensnrate time delays, necessary and sufficient conditions are given for the existence of, a fiite-dimensional stabilizing feedback compensator. In particular, it is shown that a stabilizable time-delay system can always be stabilized using a finite-dimensional compensator. The problem of explicitly constructing finite-dimensional stabilizing compensators is also considered. I. STABILIZATION OF SYSTEMS WITH DELAYS In this note we consider the problem of stabilizing a linear time-invariant continuous-time system with q noncommensurate time delays h, , hZ,.-., h,. The systems we shall study are given by a state representation of the form dxo = (fldh,, dh2.. ' ' 9 dhq)x)(t) + (G(dhl, dh2, '. ' 9 dhq)lo(t) dt y( ( t)=(ff(db, , dbp " ' I dh8)x)(r)+(J(dh,. dhz, dhq)U)(f) (1.1) where the m-vector u(t) is the input at time t, the n-vector x(t) is the instantaneous state at time t, the p-vector y(r) is the output at time t, and matrices whose entries are polynomials in the delay operators dh, , ..., dhq with coefficients in the reds m. (Here (d&f) ( t) = At- rhi) for any positive integer r.) With the system ( l. l) , we shall associate the quadruple (F(z), G(z)