Digital compensator types

In a continuous system, a differentiation of the error signal e can be represented as 

Taking Laplace transforms with zero initial conditions 

Digital PID controller. From equation (4.92), a continuous PID controller can be written as 

Tustin s Rule Tustin s rule, also called the bilinear transformation, gives a better approximation to integration since it is based on a trapizoidal rather than a rectangular area. Tustin s rule approximates the Laplace transform to 

Inserting this value of. v into the denominator of equation (7.98), still yields a digital PID controller of the form shown in equation (7.100) where 

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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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