Controllability and Observability in Linear Time-Variable Networks With Arbitrary Symmetry Groups

This paper presents a unified treatment of linear time-variable networks displaying arbitrary geometrical symmetries by incorporating group theory into an analysis scheme. Symmetric networks have their elements arranged so that certain permutations of the network edges result in a configuration which is identical with the original. These permutations lead to a group of monomial matrices which are shown to commute with the network A-matrix and the state transition matrix of the normal form equation. The representation theory of groups facilitates the study of those network properties which are determined solely by symmetry. By using group theory, a simple arithmetic condition is derived which, when satisfied, implies that the network is noncontrollable or nonobservable because of symmetry alone. The results allow the determination by inspection of linear combinations of the original state variables which result in noncontrollable variables. It is shown that networks displaying axial point group symmetry are generally only weakly controllable.