On the geometry of realizable Markov parameters by SIMO and MISO systems

Let p, m, n, d be positive integers and let Ln(d) denote the set of sequences L=(L1,…,Ln) of p×m real or complex matrices which are realizable by systems of minimal order d. It was shown in [5,14] that Ln(d) can be endowed with a structure of differentiable manifold when p=m=1; that is, when the sequences are realizable by Single Input/Single Output (SISO) systems. In this paper a similar result is obtained for more general sequences. Specifically, we will consider the set Ln(r_,s_) of sequences L which are realizable by systems of minimal order d and having r_ and s_ as Brunovsky indices of controllability and observability, respectively. It is shown in this paper that when one of the two collections of indices r_ or s_ is constant, then Ln(r_,s_) can be provided with a structure of differentiable manifold and a formula of its dimension is given. The special cases r_=(1,…,1) or s_=(1,…,1) correspond to sequences realizable, respectively, by Single Input/Multi Output (SIMO) or Multi Input/Single Output (MISO) systems.