Controllability and observability of linear differential behaviors

Two central definitions in systems theory are controllability and observability. For state-space systems, controllability is defined as the possibility of transferring the state from any initial to any terminal value, while observability is defined as the possibility of deducing the state from an observed output. Recently (J.W. Polderman et al., 1998), a more general definition has been put forward that generalizes these notions to more general model classes. A number of conditions have been derived that allow one to deduce the controllability of a system described by the system of differential equations R₀w+R₁dw/dt+...+R_Ld^L w/dt^L=0, with R_i∈R^p×q and observability for R₀w₁+R₁dw₁/dt+...+R_L d^Lw₁/dt^L=M₀w₂ +M₁dw₂/dt+...+M_Nd^Nw ₂/dt^N, with R_i∈R^p×q and M_i∈R^p×l. However, the conditions of Polderman et al. are not explicit in the coefficient matrices (R₀, ...R_L) or (M₀, ...M_N ). J. Hoffmann (1994) derived conditions of such an explicit nature but, in the controllability case for example, they involve checking the rank of a matrix of dimension Lpq×L(p+1)q. This paper derives conditions in terms of (R₀, ...R_L) or (M ₀, ...M_N). The conditions involve explicit rank tests and are presented as a recursive algorithm in MATLAB-style pseudocode. The algorithm generalizes the familiar (B, AB, ..., A^n-1B) or (C^T, A^TC^T, ...(A ^T)^n-1C^T) rank test for state-space systems and the Euclidean algorithm for checking co-primeness of two polynomials