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 $$ar{C}(z) for the plant $$G(z) z^{-k} so that the closed loop transfer function $$ar{H}(z) equals $$H(z) z^{-k}.

Solving $$frac{ar{C} G z^{-k{1+ar{C}G z^{-k = z^{-k} frac{C G }{1 + C G}, we obtain $$ar{C} = frac{C}{1+CG(1-z^{-k})}. 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]