Chapter 10 - Closed Loop and PID Control amplitude) by the width (the time interval between samples), and then summing the rectangular slices, as shown in Figure 10-15. Doing so creates an approximation of the integral called the numerical integral or discrete integral. (It should be noted that in Figure 10-15 the integral of a sine wave over one complete cycle, or any number of complete cycles, is zero because the algebraic sum of the positive slices and negative slices is zero.) As we will see, in a PID control system, the integrate function is used to minimize the offset. Since it will reset the system so that the SP and PV are equal, the integrate function is usually called the reset.
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-1 -2 Figure 10-15 - Discrete Integral of a Sine Wave The error incurred in taking a numerical integral when compared to the true (calculus) integral depends on the sampling rate. A slower sampling rate results in fewer samples and larger error, while a faster sampling rate results in more samples and smaller error. When used in a PID control system, this type of numerical integrator has an inherent problem. Since the integrator keeps a running sum of area slices, it will retain sums beginning with the time the system is switched on. If the control system is switched on and the machine itself is not running, the constant error can cause extremely high values to accumulate in the integral before the machine is started. This phenomenon is called reset wind-up or integral wind-up. If allowed to accumulate, these high integral values can cause unpredictable responses at the instant the machine is switched on that can damage the machine and injure personnel. To prevent reset windup, when the CV reaches a predetermined limit, the integral function is no longer calculated. This will keep the value of the CV within the upper and lower limits, both of which are specified by the PLC programmer. In some systems, these CV limits are called saturation, or output (CV) min and output (CV) max, or batch unit high limit and batch unit preload.
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