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_{(p)}^{-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_{(p)}^{-1}
and
H
is
Q[x]= x+ \Sigma_{n=p+1}^{\infty} Q_{n}[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_{(p)}^{-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_{(p)}^{-1}
is obtained. The
p
th-order inverse offers an approach to the study of the system inverse,
H^{-1}
, since, as
p \rightarrow \infty , K_{(p)}^{-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.