A Generalization of the Popov Criterion

A criterion for the stability of control systems which contain an arbitrary jinik number oj memoryless nonlinearities is considered. The criterion is such that the degree oj stability may be specijied, and such that for the case when the absolute stability of a control system having one memoryless nonlinearity is considered, it reduces to the original Popov criterion. Introduction The formulation and proof of absolute stability criteria applicable to systems having many memoryless nonlinearities is given in (1) and (2). The significant contribution of (1) is the demonstration of the existence of a frequency response stability criterion for multiple nonlinearityy systems. The contribution of (2) is that means of constructing system Lyapunov functions are given. Also, the convenient application of the network theory concept of a positive real function to the control theory concept of a minimal realization of a transfer function (3) to proving the system stability criteria is indicated. However, consideration of the degree of stability of such systems is not given except in the single nonlinearity case (4). This paper gives an application of the theorem of (3) to establish a stability criterion for control systems containing an arbitrary finite number of memoryless nonlinearities that is more general than either of those considered in (1) and (2), and states clearly the conditions for which it maybe applied. The criterion is such that the degree of stability may be specified. When the absolute stability of a control system having one memoryless nonlinearity is considered, the criterion reduces to that given originally by Popov (5). Stability Criterion The nonlinear systems considered are those where the system is of the form (or may be arranged to be of the form) of that shown in Fig. 1. Lyapunov stability is considered, hence, the inputs are not indicated, The matrix ?$’(s) is an n X n matrix of stable rational transfer functions, assumed to be such that w(m) = o. (1) The nonlinearities ~i(~i) (i = 1,2, 0.., n) are assumed to satisfy the conditions O < ~i(~i)~i < kiyiz for ki>O(i=l,2, ”.., n). (2)