Partial stability preserving maps and stabilization

This paper deals with the stabilization of systems using low-order controllers. We introduce the concept of a partial stability preserving map and show that stabilization with a low-order controller is equivalent to the existence of such a matrix map. This provides a different characterization of stabilization and allows for the development of tests and design techniques. We then combine this concept with the frequency parameterization of stable polynomials and show that stabilization is equivalent to the existence of common zeros of a finite number of multiaffine expressions in the space of ordered frequencies. We suggest an optimization technique for solving this problem that takes advantage of the special structure present.