Smith predictor

The Smith predictor (invented by O. J. M. Smith in 1957) is a type of predictive controller for systems with pure time delay. The idea can be illustrated as follows. The Smith predictor (invented by O. J. M. Smith in 1957) is a type of predictive controller for systems with pure time delay. The idea can be illustrated as follows. Suppose the plant consists of G ( z ) {displaystyle G(z)} followed by a pure time delay z − k {displaystyle z^{-k}} . z {displaystyle z} refers to the Z-transform of the transfer function relating the inputs and outputs of the plant G {displaystyle G} . As a first step, suppose we only consider G ( z ) {displaystyle G(z)} (the plant without a delay) and design a controller C ( z ) {displaystyle C(z)} with a closed-loop transfer function H ( z ) = C ( z ) G ( z ) 1 + C ( z ) G ( z ) {displaystyle H(z)={frac {C(z)G(z)}{1+C(z)G(z)}}} that we consider satisfactory. Next, our objective is to design a controller C ¯ ( z ) {displaystyle {ar {C}}(z)} for the plant G ( z ) z − k {displaystyle G(z)z^{-k}} so that the closed loop transfer function H ¯ ( z ) {displaystyle {ar {H}}(z)} equals H ( z ) z − k {displaystyle H(z)z^{-k}} . Solving C ¯ G z − k 1 + C ¯ G z − k = z − k C G 1 + C G {displaystyle {frac {{ar {C}}Gz^{-k}}{1+{ar {C}}Gz^{-k}}}=z^{-k}{frac {CG}{1+CG}}} ,we obtain C ¯ = C 1 + C G ( 1 − z − k ) {displaystyle {ar {C}}={frac {C}{1+CG(1-z^{-k})}}} . The controller is implemented as shown in the following figure, where G ( z ) {displaystyle G(z)} has been changed to G ^ ( z ) {displaystyle {hat {G}}(z)} to indicate that it is a model used by the controller.