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# Empirical tuning rules according to Ziegler and Nichols

Many industrial processes show step responses with pure aperiodic behaviour according to Figure 8.6. This S-shape curve is characteristic of many high-order systems and such plant transfer functions may be approximated by the mathematical model

 (8.15)

which contains a 1st-order delay element and a dead time. Figure 8.6 shows the approximation by a element.

Here the step response is characterised by constructing the tangent at the turning point with the following three values: ( gain of the plant), (rise time) and (delay time). Then a rough approximation according to Eq. (8.15) is to set and .

For a plant of the type described above a lot of tuning rules for standard controllers have been developed. These have been mostly developed empirically from simulation studies. The most famous empirical tuning rules are those of Ziegler and Nichols. These tuning rules have been derived to provide step responses for the closed loop, where the response shows a decrease of the amplitude of approx. 25% per period. For the application of these rules according to Ziegler and Nichols two different approaches can be used:

a)
Method of the stability margin(I): Here, the following steps are used:
1. The controller is switched to pure P action.
2. The gain of the P controller is continuously increased until the closed loop shows permanent oscillations. The value of the gain at this state is denoted as the critical controller gain .
3. The length of period (critical period) of the oscillations is measured.
4. From and one determines the controller tuning values , and using the formulas given in Table 8.1.

b)
Method of the step response (II): In the case of an industrial plant it is often not possible, suitable or allowed to drive the plant into permanent oscillations for determining and . Measuring the step response of the plant does not generally cause difficulties. Therefore, in many cases the second form of the Ziegler-Nichols approach is more expedient. The rules are based directly on the slope of the tangent at the turning point and on the delay of the step response. One has to observe that the measurement of the step response needs only to be taken at the turning point T, as the slope of the tangent already describes the ratio . Using the measured data and as well as the formula given in Table 8.1 the controller tuning parameters can be determined by simple calculations.

Demonstration Example 8.1   A virtual experiment using PID control for tracking

Demonstration Example 8.2   A virtual experiment using PID control for high-precision positioning

DYNAST study example 8.1   PI control of a PT1Tt plant

DYNAST study example 8.2   PID control of a PT1Tt plant

DYNAST study example 8.3   Disturbance response for PI control of a PT1Tt plant

Next: Design of controllers using Up: PID control and associated Previous: Advantages and disadvantages of   Contents
Christian Schmid 2005-05-09