Smith predictor

Smith predictor is a type of predictive controllers that deals with pure time delay. The idea can be illustrated as follows.

Suppose the plant consists of $G(z)$ followed by a pure time delay $z^\{-k\}$. Suppose we only consider $G(z)$ and design a controller $C(z)$ with a closed loop transfer function $H(z)=frac\{C(z)\; G(z)\}\{1+C(z)G(z)\}$.

Our objective is to design a controller for the plant $G(z)\; z^\{-k\}$ so that the closed loop transfer function equals $H(z)\; z^\{-k\}$.

Solving , we obtain . The controller is implemented as shown in the following figure, where $G(z)$ has been changed to $hat\{G\}(z)$ to indicate that it is a model used by the controller.

A re-arrangement can be made as followsHere we can see that if the model used in the controller, $hat\{G\}(z)z^\{-k\}$, matches the plant $G(z)\; z^\{-k\}$ perfectly, then the two outer feedback loops cancel each other, and the controller generates the "correct" control action.

References

K. Warwick and D. Rees, "Industrial Digital Control Systems", IET, 1988. [http://books.google.com/books?id=4dURB2NTstAC&pg=PA100&dq=%22smith+predictor%22+inauthor:warwick&lr=&as_brr=0&ei=-a_OSIz6BJWKyQSOwJDjBA&sig=ACfU3U36uGEHj5Azv-prDeRtQ0SE51QkVg]