# zbMATH — the first resource for mathematics

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

##### MSC:
 9.3e+11 Estimation and detection in stochastic control theory