An approach to simultaneous system design. I - Semialgebraic geometric methods

This paper introduces semialgebraic parameterization as an approach to analyze simultaneous stabilization and pole placement problems. Rational families of plants of a given McMillan degree, that are simultaneously stabilizable by a fixed family of compensators, are parameterized. For a discrete family of plants, the parameterization problem reduces to the simultaneous stabilization or the pole placement problem of a r-tuple of multi input multi output plants by a nonswitching compensator. It is shown that by removing a semialgebraic subset of a proper algebraic set, the 'space of plants' can be decomposed into components that are either simultaneously stabilizable or simultaneously unstabilizable. Under special cases, explicit parameterization of the semialgebraic set is obtained. Finally a necessary condition for the simultaneous stabilization of single input or single output plants is obtained.