Balanced Realizations as Minimum Sensitivity Structures in Linear Systems

Recently, balanced realizations in linear systems have been proposed from the viewpoint of controllability and observability measure of the state-space. They have been extended to various linear systems such as time-variant systems and a class of 2-dimensional systems, and have been applied to model reduction. However, basic properties of balanced realizations have not been investigated so far. On the other hand, system sensitivities are important measures to compare realization structures. This paper shows that balanced realizations in discrete-time linear systems are derived as minimum sensitivity structures. The statistical sensitivity of linear state-space systems is proposed and represented by the controllability and observability Grammians. It is shown that the statistical sensitivity depends on realization structures. The statistical sensitivity is minimized, and minimum sensitivity stuctures are obtained via equivalent transformations. The class of minimum sensitivity structures includes balanced realizations as a special case.