Stability Analysis of Inverse Volterra Series

Abstract

Among various nonlinear control methods, the one based on the Volterra series expansion is a promising approach for chemical process control. Almost all compensator design methods based on Volterra series system models utilize the inverse or some type of pseudo-inverse of the models. It is well known that this inverse is usually stable only for a limited amplitude of input signals, and this limited range is not understood quantitatively. Traditional input-output stability analysis methods cannot be used to analyze such an input amplitude dependent stability problem. Under the assumption of the open-loop system being strictly causal, Local Small Gain Theorem (LSGT) is first developed in the paper, which states a sufficient condition for the stability of the closed-loop nonlinear system. Using the new theorem, not only can one determined the local stability of the closed-loop system but also obtain a bound on the external input signal which guarantees BIBO stability. Then, this theorem is used to analyze the stability problem of inverse Volterra series. It so happens that for the Volterra series models an approximation of the local system gain can be easily obtained. By solving a simple single-variable optimization problem, a bound on the external input signal can be obtained, which guarantees the stability of the inverse Volterra series. Both mathematical analysis and simulation results are presented.