On the Boundedness of the Set of Stabilizing Controllers

This paper shows that the set of rational, strictly proper stabilizing controllers for single input single output (SISO) linear time invariant (LTI) plants will form a bounded (can even be empty) set in the controller parameter space if and only if the order of the stabilizing controller can not be reduced any further; if the set of proper stabilizing controllers of order tau is not empty and the set of strictly proper controllers of order r is bounded, then tau is the minimal order of stabilization. The paper also extends this result to characterize the set of controllers that guarantee some prespecified performance specifications. In particular, it is shown here that the minimal order of a controller that guarantees specified performance is I iff (1) there is a controller of order I guaranteeing the specified performance and (2) the set of strictly proper stabilizing controllers of order I and guaranteeing the performance is bounded. Moreover, if the order of the controller is increased, the set of higher order controllers which satisfies the specified performance, will necessarily be unbounded. This characterization is provided for performance specifications, such as gain margin and robust stability, which can be posed as the simultaneous stabilization of a family of real polynomials. Other performance specifications, such as phase margin and H_infin norm, can be reduced to the problem of determining a set of stabilizing controllers that renders a family of complex polynomials Hurwitz. The characterization of the set of controllers for the stabilization of complex polynomials is provided and is used to show the boundedness properties for the set of controllers that guarantee a given phase margin or an upper bound on the H_infin norm.