Partial realization for singular systems in standard form Original Research Article

The partial realization problem under consideration consists in finding, for a given sequence s=(sk)0N−1 of blocks, matrices (A,E,B,C) of appropriate size such that si=CEN−1−iAiB and the identity matrix is a linear combination of A and E. We discuss the question whether there is always a realization of this form for which the state space dimension is equal to the maximal rank of the underlying Hankel matrices. We show that this question has an affirmative answer if the block size is less than or equal to 2 and some other cases but not in general. The paper strengthens results obtained by Manthey et al. [cf. W. Manthey, U. Helmke, D. Hinrichsen, in: U. Helmke et al. (Eds.), Operators, Systems, and Linear Algebra, Teubner, Stuttgart, 1997, pp. 138–156]. The main tools are the results of the authors obtained in connection with Vandermonde factorization of block Hankel matrices. Finally, an interpretation of the problem in periodic discrete-time systems is given.