Evaluating Tuning Parameters (4) - Answers

Part A

When the loop is open the following values are obtained from the Bode Diagrams:

Critical Frequency = 1.53
Gain Margin = 22.8 dB (13.8)

Note that there is no crossover frequency and hence no Phase Margin.

Part B

The gain of a PI controller is equal to GM/2.2.

GM = 13.8
Gain = 13.8/2.2 = 6.27

To obtain the reset time of a PI controller we have to convert the frequency to the correct units, take the reciprocal and then divide by 1.2.

CF = 1.53 / = 0.244
Pu = 4.11
Reset Time = 3.42

Part C

When the controller is added to the loop, using the above values for the controller parameters the following values are obtained from the Bode Diagrams:

Critical Frequency = 1.24
Gain Margin = 2.91 dB (1.40)
Crossover Frequency = 1.03
Phase Margin = 11

These values are too low. The phase margin especially should be greater. This is reflected in the response obtained when the above values are used in the controller model. As you can see the response does not reach steady state. The reason that this is so bad is because it is not a delay followed by first order lag model. The ZN closed loop method (which in effect we are using) only gives a starting point for the tuning parameters.

Now try fine tuning the controller parameters using the process response. A smoother response is obtained when you use a gain of 2.5 and a reset time of 4. Working backwards and using these in the Bode Diagram Applets results in the following values:

Critical Frequency = 1.28
Gain Margin = 10.9 dB (2.5)
Crossover Frequency = 0.583
Phase Margin = 46.8

As you can see these are closer to the limits specified at the start of the exercise.

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