Minimal-order inverses for linear systems with zero and arbitrary initial states†

The problem of constructing minimal-order inverses for linear, time-invariant systems with zero as well as arbitrary initial states is considered, and an algorithm which uses only simple matrix operations is presented for constructing these inverses. It is shown that, for a system described by the 4-tuple (A, B, C, D), a minimal-order inverse has dynamical order equal to the dimension of the maximal output-nulling invariant subspace of the state space of the system ; and in the case of a system with arbitrary initial states, this is identically zero.