The Problem of Quality for Nonlinear Self-Regulating Systems with Quadratic Metric

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.