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Weakly coprime factorization and state-feedback stabilization of discrete-time systems

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Abstract

The LQ-optimal state feedback of a finite-dimensional linear time-invariant system determines a coprime factorization NM −1 of the transfer function. We show that the same is true also for infinite-dimensional systems over arbitrary Hilbert spaces, in the sense that the factorization is weakly coprime, i.e., Nf, \({Mf \in \mathcal {H}^2 \Longrightarrow f \in \mathcal {H}^2}\) for every function f. The factorization need not be Bézout coprime. We prove that every proper quotient of two bounded holomorphic operator-valued functions can be presented as the quotient of two bounded holomorphic weakly coprime functions. This result was already known for matrix-valued functions with the classical definition gcd(N, M) = I, which we prove equivalent to our definition. We give necessary and sufficient conditions and further results for weak coprimeness and for Bézout coprimeness. We then establish a variant of the inner–outer factorization with the inner factor being “weakly left-invertible”. Most of our results hold also for continuous-time systems and many are new also in the scalar-valued case.

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  1. Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015, Helsinki, Finland

    Kalle M. Mikkola

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Mikkola, K.M. Weakly coprime factorization and state-feedback stabilization of discrete-time systems. Math. Control Signals Syst. 20, 321–350 (2008). https://doi.org/10.1007/s00498-008-0034-z

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