On canonical forms for linear constant systems

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Abstract
Let ( F, 0, H) be a reachable system. We introduce the notion of a nice selection and that of a minimal nice selection of column vectors of the reachability matrix ( G FG.). These notions lead to a unified treatment and a parametrization of all the # and all the t> canonical forms of ( F, 0, H), which are sometimes called first and second control canonical forms. This global approach shows that the # and the ♭ canonical forms are local coordinates of the (quasi-projeetive) algebraic variety of all reachable systems of fixed dimension (modulo basis change in the state space). More-over, the canonical forms of ( F, G, H) we mentioned above, are in bijective correspondence with the so-called # and [and ♭matrix fraction representations of its transfer function. Finally, we investigate a canonical form for (proper rational) matrix fractions, which turns out to be a ♭ matrix fraction representation.

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