On the Dimensions of Controllability Subspaces: A Characterization via Polynomial Matrices and Kronecker Invariants

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Abstract
Summary:Transfer function models used for early stages of design are large dimension models containing all possible physical inputs, outputs. Such models may be badly conditioned and possibly degenerate. The problem considered here is the selection of maximal cardinality subsets of the physical input, output sets, such as the resulting model is nondegenerate and satisfies additional properties such as controllability and observability and avoids the existence of high order infinite zeros. This problem is part of the early design task of selecting well-conditioned progenitor models on which successive design has to be carried out. The conditions for different type of degeneracy are investigated and this leads to necessary and sufficient conditions required to guarantee nondegeneracy. The sufficient conditions for nondegeneracy also lead to models with no infinite zeros. Furthermore, additional conditions are derived which guarantee controllability and observability of the resulting model. The results are then used to develop a selection procedure for natural subsets of inputs and outputs, which guarantee transfer function and input, output nondegeneracy, as well as controllability and observability of the resulting system. A parameterisation of solutions that satisfy the above requirements is given