Discrete-time compensator

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

Doss, J. E., and C. F. Moore, The Discrete Analytical Predictor—A Generalized Dead-Time Compensation Technique, ISA Trans., 20 (4), 77 (1982). 

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Digital controllers of the Direct Synthesis type share yet one more characteristic namely, they contain time-delay compensation in the form of a Smith predictor (see Chapter 16). In Eq. 17-61, for Gc to be physically realizable, (Y/Y ) must also contain a term equivalent to which is z, where N = 0/Ar. In other words, if there is a term z in the open-loop discrete transfer function, the closed-loop process cannot respond before NAt or 0 units of time have passed. Using YIYsp)d of this form in Eq. 17-63 yields a Gc containing the mathematical equivalence of time-delay compensation, because the time delay has been eliminated from the characteristic equation. 

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