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is expected to produce a slower closed-loop response in comparison to that with . It is worthwhile mentioning from Tables (1.4) and (1.5) that both PI and PID controller parameters are invalid for the time delay because as , the proportional control gain .

A frequency response analysis for the Padula and Visioli tuning rules based control system will be presented in Chapter 2 where an example (see Example 2.4) will be given to show the sensitivity functions and their Nyquist diagrams.

      1.4.3 Wang and Cluett Tuning Rules

In Wang and Cluett (2000), a first order plus delay model was used to derive several tuning rules for PID controllers. The rules were calculated using a frequency response analysis based on the ratio of the time constant to the time delay , which is defined as . A desired closed-loop time constant is chosen as a scale of the time delay . Among them, which works quite well in applications, is a particular choice with the desired closed-loop time constant being equal to the time delay . For this choice, the PID controller parameters are presented in Table 1.6.

Table 1.6 Wang–Cluett tuning rules with reaction curve (

).

P
PI
PID

      1.4.4 Food for Thought

      1 In the IMC-PID controller tuning rules, if a faster closed-loop response is desired, would you increase or decrease the desired closed-loop time constant ?

      2 For the tuning rules derived by Padula and Visioli, are there any closed-loop performance parameters chosen by the user?

      3 Intuitively, would you increase the proportional controller gain if the PID controlled system is unstable when using the Padula and Visioli's tuning rules?

      4 Would you increase the integral time constant if the PID controlled system is oscillatory when using the Padula and Visioli's tuning rules?

1.5 Examples for Evaluations of the Tuning Rules

      Several examples are presented in this section for evaluation of the tuning rules that are based on the first order plus delay model.

      1.5.1 Examples for Evaluating the Tuning Rules

      The first example is based on a first order plus delay plant and the second example is based a high order plant so that an approximation is made during the graphic procedure.

Example 1.6

       The unit step response of a continuous time transfer function model

      (1.55)

       is shown in Figure 1.15. Instead of using the first order plus delay model directly, we will find the PI controller parameters using the values of , , and delay .

      Solution. A Simulink simulator is built to collect the step response testing data and to produce a figure for the step response. On this figure, a line is drawn to reflect the maximum slope of the reaction curve; there are two arrows marking the points of interest. Using MATLAB command ginput(2), with a click on the bottom point, we find the coordinates ; and with a click on the top point, we find .

       From the readings of the two points, we find that

      (1.56)

Figure 1.15 Unit step response (Example 1.6)

       where is one since a unit step signal is used as the input. The time delay , and the parameter With these parameters, we calculate the PI controller parameters using the reaction curve based methods (see Tables 1.2, 1.3, and 1.6). The PI controller parameters are summarized in Table 1.7. Their closed-loop step responses are compared in Figure 1.16.

      In

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