# Implementing improper transfer functions with preview information available

I am trying to design a feedforward controller. It is actually an inverse of another transfer function, say $\mathcal{G}(s)$, which is a nonminimum phase transfer function. Since, I cannot take its inverse directly, I have to use a stable inversion technique available in control theory. One such way is to use the 'zero error phase tracking controller' or ZPETC. While we do not need to go into details of this technique, it is necessary to point out that the technique yields an improper transfer function, say $\mathcal{H}(s)$, without any delays.

Now, input to $\mathcal{H}(s)$ is a reference signal and I have future information available for this signal. In literature, it is said that using preview of the reference signal at future times, it is possible to implement an improper transfer function. But I do not know how? Basically what I need to know is how to implement an improper transfer function in MATLAB using this preview information available.

P.S. I know about the alternate way of implementing such systems by dividing them into proper and improper parts and then using a chain of differentiators for the improper part. But I do not know if this is a good option and yields a valid result. Any insights are welcome!

• Are you implementing it in discrete-time to add the preview information? – Carlos Massera Filho Nov 8 '16 at 21:32
• Sorry was off the grid for a few days... Yes, in discrete time – Zero Nov 16 '16 at 10:36

$$H(z) = \sum_{n=1}^{p>q} a_nz^n \bigg/ \sum_{m=1}^q b_mz^m.$$
Suppose $H = Y/U$, where $Y$ is the output and $U$ is the input. Then $$\sum_{m=0}^q b_mz^mY(z) = \sum_{n=0}^{p>q} a_nz^nU(z).$$ Recall the inverse $z$-transform is $$f[n] = Z^{-1}f(z) = \frac{1}{2\pi i}\oint_\Gamma f(z)z^{n-1}dz,$$ where $\Gamma$ is entirely contained in the radius of convergence of $f$. Clearly the inverse $z$-transform is linear. Therefore we have $$\sum_{m=0}^q b_m Y[m] = \sum_{n=1}^{p>q} a_nU[n],$$ which would be a problem for $n = q+1,\dots,p$ except that you assume you have this future information already available.