STABILIZATIONS OF CONTINUOUS TIME SYSTEMS BY FIRST ORDER CONTROLLERS

Abstract In this paper we consider the problem of stabiliz-ing a given but arbitrary linear time invariant con-tinuous time system with transfer function , bya first order feedback controller   .The complete set of stabilizing controllers is deter-mined in the controller parameter space  !#" !#$&% ;this includes an answer to the existence question ofwhether '( is “first order stabilizable” or not. Theset is shown to be computable explicitly, for fix ed #$ by solving linear equations and the three dimen-sional set is recovered by sweeping over the scalarparameter #$ . This result is applicable to a) the si-multaneous stabilization problem and b) the robuststabilization problem of a continuum of plants. Thelatter is illustrated by applying it to the stabiliza-tion of an interval family of transfer functions '( which reduces to the stabilization of the Kharitonovvertex plants. In each case the solution is facilitatedby the fact that linear equations are involved in thesolution so that the intersection of sets can be foundby adding more equations. Illustrative examples areincluded. They demonstrate that the shape of the sta-bilizing set in the controller parameter space is quitedifferent and much more complicated compared tothat of PID controllers despite the fact that both are“three term controllers”. It is remarkable that despitethis complicated topology the set can be unravelledvia “linear computations”. Extensions and applica-tions to design are discussed.