Abstract
The concept of fixed modes plays an important role in stabilization and pole assignment for linear time-invariant systems. Several characterizations of fixed modes have been obtained.
In this paper, a concept of degrees of fixed modes is introduced for linear time-invariant control systems with constrained control structures, typically including decentralized control systems. The degree of a fixed mode corresponds to its geometric multiplicity, and is defined by the minimal dimension of all eigen spaces of feedback system matrices for the fixed mode. It coincides with minimal multiplicity of the fixed mode as the transmission zero for regular subsystems. Moreover, in decentralized control systems, the degree has an equivalent expression related to the algebraic characterization of fixed modes by partitions of stations.
Moreover, a concept of minimal effective subsystems corresponding to each fixed mode is introduced by considering the Jordan canonical form of the linear system. It is proved that at least one minimal effective subsystem exists for every fixed mode. This fact is important in applications such as elimination of fixed modes.
