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Consistency of estimators in controlled systems. (English) Zbl 0689.93059
Stochastic differential systems, Proc. 4th Conf., Bad Honnef/FRG 1988, Lect. Notes Control Inf. Sci. 126, 227-234 (1989).
[For the entire collection see Zbl 0679.00015.]
Linear controlled systems satisfying \[ (*)\quad dX_ t=f(\alpha)X_ tdt+gU_ tdt+dW_ t,\quad t\geq 0, \] are dealt with. \(f(\alpha)\) is a matrix depending linearly on \(\alpha \in R^ m\). \(\{W_ t\}\) denotes a Wiener process with incremental variance matrix \(h\neq 0\). \(\{U_ t\}\) is the control signals. \(\alpha\) is a parameter unknown to the controller. A persistent excitation condition for (*) is introduced. It implies the strong consistency of the least squares estimators, whenever \[ U_ t=K_ tX_ t,\quad | K_ t| \leq const.,\quad t\geq 0, \] \(\{K_ t\}\) nonanticipative. A procedure for numerical verification of the excitation condition is developed.
Reviewer: P.Mandl

93E10 Estimation and detection in stochastic control theory