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On least squares estimation in continuous time linear stochastic systems. (English) Zbl 0781.93085
The authors consider the asymptotic limit of the family of estimations \(\{\alpha^*(T),\;T>0\}\) where \(\alpha^*(T)\) is the minimizer of the formal functional \[ \begin{split} L(T;\alpha)=\int^ 0_ 0\bigl\{(\dot X(t)- f(\alpha)x(t)-g(\alpha)U(t))'l(\dot X(t)\\ -f(\alpha)X(t)- g(\alpha)U(t))-\dot X'(t)l\dot X(t)\bigr\}dt\end{split} \] under the condition that \(X(t)\) together with the stochastic “control” \(U(t)\) satisfy the stochastic linear system \[ dX(t)= f(\alpha)X(t)+ g(\alpha)U(t)dt+ dB(t),\;X(0)=x, \] where \(B(\cdot)\) is an \(n\)-dimensional Wiener process, \(f(\alpha)\), \(g(\alpha)\) are matrix-valued polynomials and \(l\) is a weight matrix.
Simply speaking, under the conditions that \(U\) does not depend on \(X\), i.e. \(U\) is not feedback control and the empirical covariance function of \(U\) converges to a nonrandom limit as \(T\to\infty\), the authors establish the so-called consistency property, i.e. \(\alpha^*(T)\to \alpha\) in probability. Under linear feedback controls \(U(t)=kX(t)\), some stability conditions on \(X\) and \(U\), and an excitation condition on \(U\) and some unified conditions on the matrices \(k\), \(f\), \(g\), the authors establish the strong consistency, i.e. \(\alpha^*(T)\to \alpha\) almost surely as \(T\to\infty\). Remark that the consistency is the basis of the concepts of identification and self-tuning. This work is a continuation of the authors’ research started in 1988.

MSC:
93E10 Estimation and detection in stochastic control theory
93E12 Identification in stochastic control theory
60G35 Signal detection and filtering (aspects of stochastic processes)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62M09 Non-Markovian processes: estimation
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References:
[1] M. Boschková: Self-tuning control of stocharic systems in premence of drift. Kybernetika 24 (1988), 5, 347 - 362. · Zbl 0654.93040
[2] T. E. Duncan, B. Pasik-Duncan: Adaptive control of continous-time linear stochastic systems. Math. Control Signals Systems 9 (1990), 45 - 60. · Zbl 0682.93057
[3] P. Mandl T. E. Duncan, B. Pasik-Duncan: On the consistency of a least squares identification procedure. Kybernetika 24 (1988), 5, 340 - 346. · Zbl 0657.93072
[4] P. Mandl: Consistency of estimators in controlled systems. Stochastic Differential Systems (N. Christopeit, K. Helmes and M. Kohlmann, Springer-Verlag, Berlin - Heidelberg 1989, pp. 227 - 234. · Zbl 0689.93059
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