On improving Popov's criterion for nonlinear feedback system stability

Abstract

For the L2-stability of a nonlinear single-input–single-output (SISO) feedback system, described by an integral equation and with the forward block transfer function and a first- and third-quadrant non-monotone nonlinearity in the feedback path, we derive an interesting generalization of the celebrated criterion of Popov [(1962). Absolute stability of nonlinear systems of automatic control. Automation and Remote Control, 22(8), 857–875]: , where is a constant. The generalization entails the addition of a general causal+anticausal O'Shea–Zames–Falb multiplier function whose time-domain -norm is constrained by certain characteristic parameters (CPs) of the nonlinearity obtained from certain novel algebraic inequalities. If the nonlinearity is monotone or belongs to any prescribed subclass of , its CPs are reduced, thereby relaxing the time-domain constraint on the multiplier. An important special feature of the new stability results is a partial bridging of the significant gap between the Popov criterion and the stability results that appeared post-Popov in the form of considering monotone and other subclasses of nonlinearities in exchange for weakening the restrictions on the phase angle behaviour of . Extensions to time-varying nonlinearities more general than those in the literature are also presented. Numerical examples are given to illustrate the theorems and to demonstrate their superiority over the existing literature.