Discrete time digital fixed parameter feedback controllers

Discrete time controllers will not normally be stand-alone units but will be simulated within the software of a digital computer. The capacity of the computer can be used if necessary to produce more complex forms of feedback control than those provided by the standard algorithms of the classical fixed parameter controller. 

The Position Form of the Discrete Time Control Algorithm 

The general relationship for the classical PID analog controller is given by (Section 7.2.2)  

The discrete time equivalent to this can be expressed in finite difference form as  

Equation 7.237 is termed the position form of the algorithm as it gives the actual value (or position) of the controller output signal. In this form, at the nth sampling instant, the PI algorithm saves only the current value of the error e and the sum 

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