The solution of a transcendental problem and its applications in simultaneous stabilization problems

A complete solution for a special transcendental problem in system theory is given. One of the interpretations of this transcendental problem is the following simultaneous stabilization problem: under what conditions can two linear plants be simultaneously stabilized by a single compensator having no real poles in the closed right half of the complex plane? These conditions are completely general and therefore encompass unstable non-minimum-phase plants as well. It is shown that the simultaneous stabilizability of two given plants via such a restricted compensator depends solely on the locations of the real zeros of the two plants and their difference plant in the closed right half of the complex plane. For two plants satisfying the stabilizability conditions, a computational design procedure for constructing a desired compensator is provided and illustrated. A computationally checkable necessary condition for simultaneous stabilization of three plants is also given