On the inversion of certain nonlinear systems

In this letter, 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); (n-1)]+f[x(n-1); y(n-1)], here g[], h[; ], and f[; ] are causal, discrete-time and nonlinear operators and the inverse function g-1[.] exists. The exact inverse of such systems is given by z(n) = g-1[fu(n)-f[z(n-1); u(n-1)]g=h[z(n - 1); u(n-1)]]. Similarly, when h[; ] = 1, the pthorder inverse is given by z(n) = g-1 p u(n)-f[z(n-1); u(n-1)]] where g-1 p [.] is the pth-order inverse of g[.].