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.