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Often a simple lead-lag filter is used for feedforward. 2 Controller design in the z-domain Let the pulse transfer function of the process, together with the sample and hold device, be G'(z)(see Fig. 5). Let H(z) be the desired pulse transfer function of the closed-loop system; then since y{z) = Gr(z)az)[v(z)-y(z)]. ) G'(z) 1 ~H(z) ^ ' the desired closed-loop performance will result. ) G'(z) can be derived from the simple models of the previous section by making use of the relation: which is useful since the term(l —e~sT)/s represents the sample and hold device which will be present in practical control loops.

1. This is a significant and important result and the implications are serious: a stable transfer function in S may discretise into an unstable transfer fuction in Z. Trapezoidal discretisation A great improvement in descretisation accuracy may be obtained by using trapezoidal discretisation. This can be implemented by replacing s in the transfer function to be discretise d by T (z + 1) Under trapezoidal discretisation, the imaginary axis of the S plane maps into the unit circle in the Z plane as shown in Fig.

This involves linearisation at every time step but otherwise the technique is the same as before. Note, however, that the amount of computation is very considerably increased. 32 Measurement for control The extended Kalman filter can be used to estimate both the state and the parameters of a process. The procedure is to designate the parameters as dummy states (the resulting product terms introduce non-linearities, hence the need for the extended version of the Kalman filter). 0 Introduction This chapter is concerned with the application of traditional 'off-the-shelf controllers.

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