PID Control System Design and Automatic Tuning using MATLAB/Simulink. Liuping Wang
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      The following example illustrates the applications of the Padula and Visioli tuning rules.

Graph depicting Time on the horizontal axis, and two curves plotted with arrows pointing two Unit step responses.

       Table 1.8 PI controller parameters with reaction curve.

images images
Ziegler–Nichols 6.4 108
Cohen–Coon 6.5667 75.9231
Wang–Cluett 3.8867 127.2154
Image described by caption and surrounding text.

       Consider the same third order system with time delay used in Example 1.7. Find the PI and PID controller parameters using Padula and Visioli tuning rules and simulate their closed-loop step response.

      Solution. The parameters used in the tuning rules are , , and . To evaluate the PI controller performance, Table 1.4 is used to calculate the controller parameters. For , we have and . For , we have and .

       With sampling interval (s), the closed-loop step responses are compared in Figure 1.19, where the IP controller structure is used to reduce overshoot to the step reference signal. It is clearly seen that the closed-loop system with is stable; however, with it is not.

       Now, we evaluate the closed-loop performance for a PID controller with filter, where the filter time constant is chosen to be . Based on Table 1.5, the PID controller parameters are calculated as for , , , , and for , , , .

Image described by caption and surrounding text. Image described by caption and surrounding text.

       With the same sampling interval , and both proportional and derivative control on output only (IPD structure) where the derivative filter time constant is selected as , the closed-loop responses are simulated. Figure 1.20 compares the closed-loop responses. Both tuning rules lead to stable closed-loop control systems. It is seen that there are overshoots in both reference responses, which was caused by the quite large derivative gains.

      1.5.2 Fired Heater Control Example

      (1.58)equation

      and at the high fuel operating condition,

      (1.59)equation

      where the time constant is in minutes. Note that there are dramatically differences in time delay and the steady-state gain of the transfer function models.

       In this example, we will show how to use the tuning rules to find the PID controller parameters for the fired heater at the lower operating condition using the transfer function (1.58) and simulate the closed-loop response with a step reference signal using sampling interval (min) and a negative step disturbance entering at the half of the simulation time.

       The higher operating condition case is left as an exercise.

      Solution. Figure 1.21 shows the unit step response with the lines drawn to identify the time delay and time constant for a first order approximation. From the graph, the time delay is found as 9.54 min and the time constant min. With the steady-state gain equal to 3, the approximation using first order plus model leads to the following transfer function:

      (1.60)СКАЧАТЬ