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Some connections between statistics and control theory. (English) Zbl 0642.93066
Mathematical statistics and probability theory, Proc. 6th Pannonian Symp., Bad Tatzmannsdorf/Austria 1986, Vol. B, 155-168 (1987).
[For the entire collection see Zbl 0625.00018.]
This paper deals with the identification of linear quadratic control systems by least squares methods under varying supplementary assumptions. Asymptotically efficient statistical estimates of the cost are studied, esp. conditions such as lim[(l/T). \(E| X_ T|\) 2)], \(T\to +\infty\), and the asymptotic distribution of the time spent by the cost function above its true value for the true model parameters. The arcsine distribution for this time is lim \(P(B_ T\leq y)=(2/\pi)\) arcsin(\(\sqrt{y}).\)
The first example is for a linear control problem with one linear unknown parameter in the state equation. This is extended to self-tuning control laws with Wiener processes and quadratic costs. A theorem gives the limit distribution of the cost function of such a self-tuning control in terms of an explicit Wiener process (two-dimensional) in some cases and of a normal process in others.
Reviewer: L.F.Pau

MSC:
93E20 Optimal stochastic control
93C05 Linear systems in control theory
93E12 Identification in stochastic control theory
60J60 Diffusion processes
62F12 Asymptotic properties of parametric estimators
93E10 Estimation and detection in stochastic control theory