Simultaneous stabilization of multivariable linear systems.

The simultaneous stabilization of a collection of systems has received considerable attention over a number of years. The practical motivation for a solution to the simultaneous stabilization problem (SSP) stems from the stability requirements of multimode systems in practical engineering. For example, a real plant may be subjected to several modes due to the failure of sensors and nonlinear systems are often represented by a set of linear models for design purposes. To examine these problems, it is necessary to establish a simultaneous stabilization theory. This dissertation considers the problem of simultaneously stabilizing a set of linear multivariable time-invariant systems. Three methodologies are presented. The first method is based on finding new approaches to solving the strong stabilization problem (i.e. stabilization by a stable controller) which can then be used in the SSP of two plants. New sufficient conditions and algorithms are derived for the solution to this problem. The second method utilizes robust stability theory applied to a "central" plant obtained from a given set of plants. A generalized two-block L-optimization problem is formulated and solved to find the central plant. The third method utilizes the parametrization of all stabilizing controllers. Sufficient conditions for the existence of a solution are derived and in the case of two plants a formula is derived for finding a simultaneously stabilizing controller. The work advances the theory of the SSP (and the Strong Stabilization Problem) by introducing and investigating several new approaches, and deriving new sufficient conditions. The work is less successful in deriving practical algorithms for the SSP except in the second method where a reliable algorithm is given for finding a central plant on which existing robust stabilization methods can be applied. This method is illustrated by its application to helicopter control.