Structural Invariants of Linear Multivariable Systems

This article identifies some of the structural properties of the matrix triple $(C,A,B)$ which remain invariant under various transformation groups. The paper begins with a brief account of a recent result which states that the controllable space of $(A,B)$ can be decomposed into a direct sum of singly-generated controllability subspaces, the dimension of each subspace being determined by one of the controllability indices of $(A,B)$. In certain instances the component subspaces of this decomposition can be chosen so that their C-images also decompose system output space in a special way; matrix triples $(C,A,B)$ for which such a decomposition is possible are called prime. If C̄ is an appropriately defined group of system-coordinate and state-feedback transformations acting on prime triples, then the controllability indices of $(A,B)$ determine a complete orbital invariant under C̄. By imbedding C̄ in a richer transformation group C^ *, which also includes nonphysically realizable “output-injection” transformations, it is shown that the C^ *-orbit of any matrix triple $(C,A,B)$ is uniquely characterized by three lists of positive integers and a list of monic polynomials called the transmission polynomials of $(C,A,B)$. These lists determine a C^ *-canonical form for $(C,A,B)$.