Extension of minimum variance estimation for systems with unknown inputs.

*(English)*Zbl 1036.93058The paper deals with optimal unbiased filtering for stochastic discrete time-varying systems with unknown inputs. Minimum variance unbiased state estimators are considered, and the solutions are given in the time domain. The general case is investigated where the unknown inputs affect both the model and the outputs. In this case, the major problem is that the state estimation error is correlated with the model and the measurement noises. The unbiasedness of this error leads to the minimization of the variance of the estimation error under algebraic constraints. The authors parametrize all solutions of the algebraic constraints by two parameter matrices. The first matrix plays the role of a gain matrix and the second one is used to guarantee rank conditions. In the general case where the estimation error is correlated with the model and measurement noises, only a suboptimal filter can be obtained. Necessary and sufficient conditions for the existence of this filter are presented. The conditions for the estimation error to be uncorrelated with the system noises are given; these conditions lead to two filters. The first one is an estimator, and the second one is a predictor. All the obtained results are expressed in terms of the initial system matrices. In the time-invariant case, the conditions for the stability of the filters are derived. When the measurements are free of unknown inputs, these conditions become necessary and sufficient.

Reviewer: Yuliya S. Mishura (Kyïv)

##### MSC:

93E10 | Estimation and detection in stochastic control theory |

93C55 | Discrete-time control/observation systems |

93E11 | Filtering in stochastic control theory |

##### Keywords:

minimum variance filter; unknown inputs; stability; convergence; linear time-varying systems; noise-estimation error correlation; unbiased estimators
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\textit{M. Darouach} et al., Automatica 39, No. 5, 867--876 (2003; Zbl 1036.93058)

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