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**Stochastic multivariable tracking: A polynomial equation approach.**
*(English)*
Zbl 0531.93069

The paper presents a general solution to the following stochastic tracking problem. Given the following input output equation for the plant \(y(s)=R(s)u(s)+S(s)w(s)\) and the reference \(r(s)=\bar s(s)\bar w(s),\) where R, S and \(\bar S\) are rational matrices (not necessarily proper) and w and \(\bar w\) independent stationary white noises with specified covariances (not necessarily positive definite), find the optimal controller of the following form \(u(s)=-P(s)(y(s)+v(s))+Q(s)(r(s)+\bar v(s))\) (where v and \(\bar v\) are independent measurement noises), such that a weighted sum of steady state covariances of the input and the tracking error is minimized, leaving the closed-loop system asymptotically stable. The design procedure allows to obtain also necessary and sufficient condition for the existence of the solution and it is based on polynomial matrices techniques. Such techniques allows to handle the problem in such a generality, but the optimal P and Q are not restricted to be proper, so that in general the optimal controller is not realizable. The procedure generalizes a construction made by the same author in the SISO case [IEEE Trans. Autom. Control AC-27, 468-470 (1982; Zbl 0488.93066)] and essentially follows the steps of the paper by V. Kučera [ibid. AC-25, 913-919 (1980; Zbl 0447.93031)] about regulation problems. Three spectral factorization and the solution of two linear equations in polynomial matrices are required. Moreover, the author shows that the feedback part P is independent of the reference, whose influence is restricted only to the feedforward part Q.

Reviewer: M.Piccioni

### MSC:

93E20 | Optimal stochastic control |

12E12 | Equations in general fields |

93B50 | Synthesis problems |

93C35 | Multivariable systems, multidimensional control systems |

12D05 | Polynomials in real and complex fields: factorization |

### Keywords:

tracking systems; polynomial matrices; spectral factorization; feedback part; feedforward part
Full Text:
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### References:

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