An iterative system inversion technique (Q2708071)

A generic discrete-time single-input-single-output system described by the following difference equation NEWLINE\[NEWLINE y(k+1) = F[y(k), \dots , y(k-m), u(k-d+1), \dots , u(k-d-n+1)] NEWLINE\]NEWLINE is under consideration. The inverse of \(F\), in case it is realizable and stable, is NEWLINE\[NEWLINE u(k-d+1) = F^{-1}[y_d (k+1), y(k), \dots , y(k-m), u(k-d), \dots , u(k-d-n+1)]. NEWLINE\]NEWLINE The proposed solution to the inversion problem is based on the following iteration NEWLINE\[NEWLINE u(k-d+1)_{i+1} = u(k-d+1)_i + \alpha e(k+1)_i, NEWLINE\]NEWLINE where \(u(k-d+1)\) is the control signal at time \(k-d+1, e(k+1)_i = y_d(k+1) - y(k+1)_i\) is the error signal at time \(k+1\) caused by \(u(k-d+1)_i\), and the subscript \(i\) is the iteration index. Computer simulations show the feasibility of the inversion technique which gives good results using practical amounts of memory.