Theory of pth-order inverses of nonlinear systems

The concept and theory of the p th-order inverse of a nonlinear system es developed in this article. The p th-order inverse, K (₏) ^-1, of a system H is defined as a system for which the Volterra series of the system Q formed by the tandem connection of K (₏) ^-1 and H is Qx= x+ ₍=₏+₁^ Q₍x so that the 2nd through the p th-order Volterra operators of Q are zero. The necessary and sufficient conditions for the existence of K (₏) ^-1 are determined. It is shown that the p th-order pre-inverse of a system H is identical to its p th-order post-inverse. In addition, a synthesis of K (₏) ^-1 is obtained. The p th-order inverse offers an approach to the study of the system inverse, H^-1, since, as p, K (₏) ^-1 becomes the Volterra series of H^-1. This approach is discussed and some applications with regard to nonlinear differential equations and nonlinear feedback systems are presented.