The Corduneanu-Popov Approach to the Stability of Nonlinear Time-Varying Systems

Abstract

Previous extensions of the stability criterion of Popov [1] for systems with a stable linear time-invariant plant $G(s)$ followed by a nonlinear time-varying gain $k(t)f( \cdot )$ entailed the definition of nonlinearity classes $[ f( \cdot ) \in N]$ and corresponding frequency domain multipliers $Z_N (s)$. Then, defining a measure of nonlinearity \[ F_{\min } \equiv \mathop {\min }\limits_x \{ \frac{{xf( x)}}{{\int_0^x {f( z )dz} }} \} \] stability is ensured if $Z(s) \in Z_N (s)$ exists, such that (i) $G(s)Z(s)$ is strictly positive real, (ii) $Z(s - \Lambda ) \in Z_N (s)$ and (iii)\[\frac{1}{k}\frac{{dk}}{{dt}}\leqq \Lambda F_{\min } \quad ( {\text{see [ 5 ]}}).\] In this paper, the point by point requirement of (iii) is replaced by an integral criterion. In many cases the constraints on $k(t)$ predicated by the integral inequality are substantially less strict than those of (iii) above. As (iii) is a special case of the integral relation, the results are never more strict. The development here will be limited to the two most general classes of nonlinearities, F (first and third quadrant nonlinearities, introduced by Lur’e and Postnikov [15] and considered by Popov [1]) and $FM$ (the subclass of F consisting of monotonically increasing functions); for more restricted classes the details are identical. For a complete exposition, see [13].