Fig. 27.6. The terms involved in the PID control algorithms. The error, e(t), is the error curve evaluated at time t; the integral term, ie(t)dt, is the (shadowed) area under the error curve; and the derivative term, de(t)/dt, is the slope of the error curve at time t

The user can specify the relative contribution of each term in Eq. (27.2) by assigning appropriate values to the parameters Kc (proportional gain), (integral time), and rd (derivative time). This is called tuning and many times is not an easy or intuitive task. Control textbooks describe many tuning procedures of varying degrees of difficulty. In this chapter we will present the relay technique, which is simple to follow, and we will apply it to the heat exchanger model. This technique, which was proposed by Astrom and Hagglund (1984), uses the process response under On/Off control and the well-known Ziegler and Nichols (1942) tuning rules.

First, the period and amplitude of the process measurement oscillation must be obtained, as seen in Fig. 27.8, which is a magnification of Fig. 27.4.

Then, a parameter called ultimate gain, Kcu, must be computed using the following formula:

where u2 and u1 are the On/Off controller values and A is the amplitude of the process output oscillations. From Fig. 27.8 we can see that A = 0.15°C, and in Fig 27.4 the value of (u2 - u1) is 0.05 GPM, therefore, the ultimate gain is Kcu = 0.42 GPM °C-1. We also need to get from the figure the ultimate period, Pu.

Ziegler and Nichols (Z-N) tuning rules provide a reasonably good performance for proportional (P), proportional/integral (PI), and proportional/integral/derivative (PID) controllers. We will illustrate their performance below using the heat exchanger model.

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