Frequency Response Exercise

Evaluating Controller Parameters (2)

Introduction

The aim of this exercise is to obtain controller parameters for the process given using the bode diagram. The second process being controlled has the following process parameters:

Delaytime = 1.5
Lag Time = 3.5
Lag Gain = 0.8

This is the same as in process 2 in the Comparing Controller Tuning Methods Experiment. In that exercise the following values were obtained:

Tuning Method	Controller Gain	Controller Reset
ZN Open Loop	2.625	5
Ciancone	1.375	3.6
Lopez	2.625	3.5
ZN Closed Loop	2.27	4.92
Damped Osciallation	2.48	4.75
Relay-Based	2.0	4.92

It is now possible to try a different technique to compare values.

Exercise

There are a number of steps in this procedure. The aim is to obtain controller parameters which give

A gain margin of between 6dB and 8dB (2 - 2.5)
A phase margin of between 45 and 60 degrees (not less than 30)

First let us remind ourselves of some definitions.

The Critical Frequency is the frequency at which the Phase Angle has a value of -180.
The Gain Margin is the modulus of the Amplitude Ratio at the Critical Frequency.
The Crossover Frequency is the frequency when the Amplitude Ratio is 0.
The Phase Margin is equal to (180 + Phase Angle) at the crossover frequency.

Now it is possible to use the applets below to answer the following questions.

Part A

Consider the open loop process i.e. the controller gain and reset are both zero. Evaluate

The Critical Frequency
The Gain Margin
The Crossover Frequency
The Phase Margin

Part B

From the gain margin and critical frequency determine the controller parameters. (Remember for this step that the units for AR have to be converted from decibels back into their actual values and that the critical frequency is in radians/time unit and this has to be converted to time units by dividing by and taking the reciprocal)

Part C

Now try the process with the above controller parameters. Again evaluate

The Critical Frequency
The Gain Margin
The Crossover Frequency
The Phase Margin

Are the gain margin and phase margin within the limits required? If not evalute controller parameters that are and discuss your results.

Bode Diagram Applets

Below are two applets showing the amplitude ratio and phase angle with respect to time for this process. Note that it is possible to change the values of the controller parameters and click on the graphs to read the coordinates.

Please use these applets to work through the above exercises.

Process and Controller Applet

The applet below shows the response of the process when run with control. Simply input the required controller parameters remembering to press return. Again it is possible to click on the graph to read what the coordinates are.

Answers

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