مرکز منطقه ای اطلاع رساني علوم و فناوري - The Problem of Quality for Nonlinear Self-Regulating Systems with Quadratic Metric

Abstract :

The systems considered in this paper are characterized by differential equations of the form $dot{x}_{k} = \\sum _{\\alpha } b_{k\\alpha } x_{\\alpha } + f_{k}(\\alpha _1,\\cdots ,x_n, t) (k= 1, \\cdots , n)$ which are defined over a region N of Euclidean space $E_n$ with metric $R^2 = \\sum _{i=1} x_i^2$ , and with $t$ ranging over some interval $T$ . The $f_k (k+1,\\cdots , n)$ are assumed to be such that 1) $f(0,\\cdots ,0,t) \\equiv 0$ and 2) there exist positive constants $L_k$ such that $|f_k(x_1,\\cdots ,x_n,t) | \\leq L_k R$ for all points in $N$ and all $t$ in $T$ . The problem of quality means the problem of determining the values of $m$ adjustable parameters $p_i,\\cdots ,p_m$ in the $b_{k\\alpha }$ , and $f_k$ in such a way as to result in a rapid return to equilibrium of the representative point in phase-space subject to a limitation on the amount of overshoot. This problem is formulated in precise terms, and a method of solution for it is indicated. As an illustration, the method is applied to a problem in regulation which was formulated by Bulgakov.