Disturbance feed-forward on the controller

According to Figure 11.1 the disturbance will feed via the transfer function to the controller, which will compensate the influence of the disturbance.

Figure 11.1: Block diagram of the feed-forward on the controller

From this diagram the controlled variable directly follows as

With some manipulations one obtains from this

which gives

where for brevity the argument is omitted. From the transfer functions of Eq. (11.3) one can see that the characteristic equation is

with regard to disturbance behaviour and

with regard to the reference behaviour. The disturbance will be fully compensated if

from which the required transfer function for the feed-forward element is

This approach can only be realised by a controller if the pole excess of is not larger than that of . Otherwise a total compensation is not possible. Moreover, the polynomial must be Hurwitzian.

For the frequent case that the disturbance and control behaviour are equal, i.e.  the case of , the transfer function of the feed-forward elements is

As the total compensation of a disturbance in a plant with P behaviour is only possible by a controller with I behaviour, the transfer function of the feed-forward element, according to Eq. (11.8), should thus show ideal D behaviour. If there is a PI controller in the loop, the feed-forward element must be designed as a  element.

Often the feed-forward element cannot be realised as ideally designed according to Eqs. (11.7) or (11.8), because , besides pure I behaviour, normally contains delay elements. Also in these cases a  element is recommended.