Control of linear systems with fixed noncommensurate point delays

Abstract

The First solution is given to the fundamental open problem of stabilizability and detectability (necessary sufficient conditions for internal stabilization by feedback) of retarded and a large class of neutral delay-differential systems with several fixed, noncommensurate point delays, using causal compensators (observers and state-feedback or dynamic output feedback), which are also the same type of neutral or retarded delay-differential systems with fixed point delays only. Our results are rank conditions on the system matrices [ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zI - F:G</tex> ] and [ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zIF^{T}:H^{T}</tex> ] evaluated at points in the complex plane and are the weakest possible generally applicable sufficient such rank conditions for stabilization of neutral systems in the light of what is known on the stability of such systems. These conditions are necessary for most practical purposes. The class of systems we consider includes all retarded delay-differential systems with noncommensurate, fixed point delays. In the case of retarded systems, these rank conditions are necessary and sufficient conditions for stabilization via compensators which are causal retarded delay-differential systems with fixed point delays only. These constitute the first full solution of these previously unsolved problems of stabilizability and detectability (which, together, are necessary and sufficient conditions for internal stabilization by feedback) for delay-differential systems even in the retarded single fixed point delay case. An application of our results to a problem of practical importance in control of linear systems with no delays provides a stabilization criterion interesting in itself.