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Stochastic Systems with Inputs

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Linear Stochastic Systems

Part of the book series: Series in Contemporary Mathematics ((SCMA,volume 1))

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Abstract

In this chapter we study the stochastic realization problem with inputs. Our aim is to provide procedures for constructing state space models for a stationary process y driven by an exogenous observable input signal u, also modeled as a stationary process.

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Notes

  1. 1.

    Here the explicit computation of the update \(\Omega _{\nu -1}^{\dag }\hat{Y }_{[t+1,T]}\) can be avoided using instead a “shift-invariance” argument on the estimated observability matrix, see Sect. 13.3.

  2. 2.

    “Complementary”, since it is the orthogonal complement of U [t, T] in the data space \(Z_{[t_{0},t)} \vee U_{[t,T]}\). The notation \(U_{[t,T]}^{\perp }\) is not completely consistent since the ambient space of the complement varies with t.

  3. 3.

    Recall that strict stability of the predictor is always required for prediction error methods [212].

  4. 4.

    Here the system order n is also assumed to be known. Of course any consistent order estimation procedure used in subspace identification would serve to the purpose. Order estimation is performed in most subspace identification algorithms by a (weighted) SVD truncation step which was discuss in the previous subsection.

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Authors and Affiliations

  1. Departments of Automation and Mathematics, Shanghai Jiaotong University, Shanghai, China

    Anders Lindquist

  2. Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden

    Anders Lindquist

  3. Department of Information Engineering, University of Padova, Padova, Italy

    Giorgio Picci

Authors
  1. Anders Lindquist
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  2. Giorgio Picci
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Lindquist, A., Picci, G. (2015). Stochastic Systems with Inputs. In: Linear Stochastic Systems. Series in Contemporary Mathematics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45750-4_17

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