Genetic algorithm for simultaneous stabilization based on stability preserving maps

Simultaneous stabilization is an open problem in the design of robust controllers. Based on the stability preserving maps (SPM), genetic algorithms (GA) are applied to solving the problems of simultaneous stabilization of a set of SISO plants. The paper proposes several special kinds of matrices, which transform directly the space of the coefficients of a stable polynomial into a bounded domain without destroying the stability. The corresponding polynomial converted is suitable for processing in GAs. Hence, the simultaneous stabilization problem can be solved effectively by using GAs. Some illustrative examples are presented and related topics are discussed. Our examples show that while the simultaneous stabilization problem is generally NP-hard or rationally undecidable, in most cases, it can be tackled numerically by some optimization algorithms. Thus, simultaneously stabilizing controllers can be found accordingly.