Constrained Common Invariant Subspace and Its Application

The notion of constrained common invariant subspaces (CCISs) is proposed in this paper as a generalization of the well-known invariant subspace to study the structural properties of multiple matrices. Specifically, some necessary and sufficient conditions for the existence of a CCIS are established to provide a methodology to compute such a CCIS. Then, the properties of CCISs and their relation to common eigenvectors are revealed. The existence of common eigenvectors leads to the existence of CCIS, but not vice versa, so the established CCIS can reveal the structural properties of multiple matrices better than common eigenvectors can. The established CCIS is applied to the reducibility of Fornasini-Marchesini (F-M) state-space models, i.e., the necessary and sufficient conditions and the related algorithm for reducibility of F-M models are developed. Finally, a gain-scheduled state-feedback control is proposed for a rational parameter system to further demonstrate the superiority of the established CCIS.