Linear preservers of controllability and/or observability Original Research Article

Consider a linear control differential equations system image, y = Cx + Du, where x ε Cn, u ε Cm, y ε Cp, and A, B, C, D are matrices of appropriate sizes with entries in C. This system, or the matrix pair (A,B), or the matrix 4-tuple (A,B,C,D), is called controllable if rank(A – λI, B) = n for all λ ≠ 0. Let φ be a linear transformation on Cn × (n+m), the linear space of all matrix pairs (A,B). Then φ is said to preserve controllability if it maps controllable matrix pairs to controllable matrix pairs. We prove that φ preserves controllability if and only if φ(A,B) = β(SAS−1 + SBF, SBR) + f(A,B)(I,0) where β is a nonzero scalar, S,R are nonsingular, and f is a linear functional. Based on this result, we also find all linear mappings on the linear space of all matrix 4-tuples (A,B,C,D) which preserve controllability. Characterizations of linear preservers of observability—a concept dual to controllability—hence follow. Some variations of the above problems are also discussed.