zbMATH — the first resource for mathematics

On the consistency of a least squares identification procedure. (English) Zbl 0657.93072
The paper considers the estimation of the parameter $$\alpha$$ in the equation $dX_ t=f(\alpha)X_ tdt+U_ tdt+dW_ t.$ If $$\alpha^*_ T$$ is the least squares estimate of $$\alpha$$ based on $$\sigma \{X_ t,U_ t$$; $$t\leq T\}$$ then heuristically $$\alpha^*_ T$$ is the minimizer of the quadratic function $\int^{T}_{0}(\dot X_ t-f(\alpha)X_ t-U_ t)'\Lambda (\dot X_ t-f(\alpha)X_ t-U_ t)dt$ for some nonnegative definite symmetric matrix $$\Lambda$$. If $$\alpha_ 0$$ is the true value of $$\alpha$$ it is shown that $$\lim_{T\to \infty} \alpha^*_ T=\alpha_ 0$$ under certain conditions.
Reviewer: R.Elliott

MSC:
 93E12 Identification in stochastic control theory 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93E10 Estimation and detection in stochastic control theory 93E03 Stochastic systems in control theory (general) 93C40 Adaptive control/observation systems
Full Text:
References:
 [1] M. Boschková: Self-tuning control of stochastic linear systems in presence of drift. Kybernetika 24 (1988), 5, 347-362. · Zbl 0654.93040 [2] T. E. Duncan, B. Pasik-Duncan: Adaptive control of continuous time linear systems. Preprint. University of Kansas 1986. · Zbl 0595.93055 [3] P. R. Kumar, P. Varaiya: Stochastic Systems: Estimation, Identification, and Adaptive Control. Prentice Hall, Engelwood Cliffs 1986. · Zbl 0706.93057 [4] P. Mandl: On evaluating the performance of self-tuning regulators. Proc. of 2nd Intemat. Symp. on Numerical Analysis, Prague 1987. B. E. Teubner, Leipzig. To appear. [5] B. Pasik-Duncan: On Adaptive Control. Central School of Planning and Statistics Publishers, Warsaw 1986. In Polish. · Zbl 0595.93055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.