On minimax approach to non-parametric adaptive control (Q2722622)

\textit{A. S. Nemirovskij} and \textit{Ya. Z. Tsypkin} [Autom. Remote Control 45, 1589-1600 (1984); translation from Avtom. Telemekh. 1984, No. 12, 64-77 (1984; Zbl 0573.93035)] proposed the lower information bound for the performance of adaptive control algorithms for linear sampled-data control systems. Here, the adaptive stabilization problem is considered for a functional autoregressive model. Consider a plant described by the nonlinear difference equations \(y_t=-g(y_{t-1})+u_{t-1}+e_t,t=1,2,\dots\), where \(y_t\) and \(u_t\) are the output and the control, \(e_t\) are i.i.d. Gaussian random variables with \(Ee_t=0,Ee_t^2=\sigma^2<\infty\), and the function \(g\) belongs to a certain a priori specified class \(\mathcal F\) of measurable functions. Let \({\mathcal F}={\mathcal F}(s,L)\) be a class of bounded functions satisfying the Hölder condition with the constant \(L<\infty\) and with the degree of smoothness \(s>0\). Denote by \(\mathcal U\) the set of all control strategies \({\mathcal U}=\{u_t(\cdot)|t\geq 0\}\).NEWLINENEWLINENEWLINEThe authors [Autom. Remote Coltrol 60, No. 3, Pt. 2, 445-457 (1999);translation from Avtom. Telemekh. 1999, No. 3, 180-196 (1999)] proposed the lower information bound for an arbitrary control strategy: NEWLINE\[NEWLINE \lim\inf_{n\to\infty}n^{2s/(2s+1)}\sup_{g\in{\mathcal F}(s,L)}n^{-1} \sum_{t=1}^nE_{g,U}(y_t-e_t)^2\geq C(s)b(s,L),NEWLINE\]NEWLINE where \(C(s)>0\) and \(b(s,L)\) are some constants.NEWLINENEWLINENEWLINEIn the present paper the authors propose the upper bound: NEWLINE\[NEWLINE \lim\inf_{n\to\infty}\left({n\over{\log n}}\right)^{2s/(2s+1)} \sup_{g\in{\mathcal F}(s,L)}n^{-1} \sum_{t=1}^nE_g,\;{\mathcal U}(y_t^2-\sigma^2)\leq C(s) (L\sigma^{3s})^{2/(2s+1)}(1+O(1)).NEWLINE\]NEWLINE A ``quasi-optimal'' adaptive control algorithm, which attains the minimax rate of convergence up to a logarithmic factor, is proposed.