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 seq...

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Bibliographic Details
Published in:Linear algebra and its applications 2017-04, Vol.518, p.97-143
Main Authors: Baragaña, I., Puerta, F., Zaballa, I.
Format: Article
Language:English
Subjects:
Block-Hankel matrices
Brunovsky/Kronecker indices
Control algorithms
Controllability
Differential manifold
Geometry
Integer programming
Integers
Linear algebra
Manifolds (mathematics)
Markov analysis
Markov parameters
Markov processes
Matrix
Minimal realizations
MISO (control systems)
MISO systems
Observability
Observability (systems)
Sequences
SIMO systems
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.12.032