On the inversion of certain nonlinear systems

We present some theorems for the exact inversion and the pth-order inversion of a wide class of causal, discrete-time, nonlinear systems. The nonlinear systems we consider are described by the input-output relationship y(n)=g[x(n)]h[x(n-1). y(n-1)]+f[x(n-1). y(n-1)], where g[/spl middot/], h[/spl middot/. /spl middot/], and f[/spl middot/. /spl middot/] are causal, discrete-time and nonlinear operators and the inverse function g/sup -1/[/spl middot/] exists. The exact inverse of such systems is given by z(n)=g/sup -1/[{u(n)-f[z(n-1). u(n-1)]}/h[z(n-1). u(n-1)]]. Similarly, when h[/spl middot/. /spl middot/]=1, the pth-order inverse is given by z(n)=g/sub p//sup -1/[u(n)-f[z(n-1), u(n-1)]] where g/sub p//sup -1/[/spl middot/] is the pth-order inverse of g[/spl middot/].