A Nonlinear Integral Equation from the Theory of Servomechanisms

A general class of nonlinear servomechanisms is described by the integral equation x(t) = sit) hit - u)F(x(u)) du, - o c / 00 , (1) where s( ·) is an input signal, A-( ·) is an impulse response function, and F(') is a nonlinear function. The equation (1) represents the system diagram of Fig. 1, wit h F( ·) as above, and with K( ·) the transfer function corresponding to /»'(·)· We assume that F( ·) satisfies the uniform Lipschitz condition | F(x) - F(y) | £ 0 x - y , and that F( 0) = 0. A classical method for studying nonlinear servomechanisms like that of Fig. 1 is to specify exactly the nonlinear element F( ·), to assume t h a t the response k( ·) is the Green's function of a differential operator of low order, and to use some sort of phase-plane analysis. This method has two theoretical disadvantages: it lacks generality, and, when applied, it tends to give more information than is needed; thus it provides detailed knowledge about a restricted class of cases. In this paper we shall use a method that has the opposite characteristics: it provides a small amount of highly relevant information about a 1309