The logarithm law of adaptive tracking for multivariable systems (Q2725162)

The author considers the MIMO linear discrete-time stochastic system NEWLINE\[NEWLINEA(z)y_n = B(z)u_{n-1} + C(z)w_n,\quad n\geqslant 0,NEWLINE\]NEWLINE where \(y_n\), \(u_n,\) and \(w_n\) are the \(m\)-dimensional system output, input and random disturbance, respectively, \(y_n = u_n = w_n = 0\) for all \(n < 0\), and \(A(z)\), \(B(z)\), and \(C(z)\) are polynomials in the backward-shift operator \(z\); \(A(z)=I+A_1z+\dots+A_pz^p\), \(p\geqslant 0\), \(B(z) = B_1 + B_2z +\dots +B_qz^{q-1}\), \(q\geqslant 1\), \(C(z) =I+ C_1z + \dots+C_rz^r\), \(r\geqslant 0\) with known upper bounds \(p\), \(q\) and \(r\) for true orders and unknown coefficients \(A_i\), \(B_j\), and \(C_k\). The objective of the paper is to construct a control sequence \(\{u_n\}\) such that the cost function \(R_n=(1/n)\sum_{i=1}^n\|y_i-y_i^\ast-w_i\|^2\) is asymptotically minimized, where a reference sequence \(\{y_i^\ast\}\) is to be tracked. It is shown that \(\lim_{n\to\infty}(n/\log n) R_n\) exists and is finite. It depicts accurately the convergence rate of self-tuning regulators for multivariable systems.