CONTROL SYSTEMS AND AUTOMATION

Question 1. A 5 to 20 bar reverse acting proportional pressure controller has an output of 4 to 20 mA. The set point is 11 bar. Determine:

(a) the measured value pressure which gives an output of 15 mA when the proportional band setting of the controller is 40%

(b) the proportional band setting which will give an output of 8 mA when the measured value is 14 bar and the desired value is 11 bar.

Question 2. FIGURE 1(a) shows a flow control system whose output Q_o is regulated by using a proportional controller to control Q₁. (Note that Q₂ is not controlled). The control system can be represented by the block diagram shown in FIGURE 1(b).

(a) Derive a relationship between Q_o and Q₂, B, C and DV, where C is the gain of the controller.

(b) If Q₂ is 3000 m³h^-1 when the bias B is 1000 m³h^-1, the PB setting of the controller is 40% and the desired value of the controller is 4000 m³ h^-1, determine the resultant value of Q_o.

(c) If Q₂ now changes to 2500 m³h^-1, determine the new bias figure required to ensure that Q_o is maintained at its original flow rate value.

Question 3. (a) With the aid of a sketch explain how proportional action is produced in a pneumatic controller whose output is 0.2 to 1.0 bar. Assume that the controller is direct acting.

(b) Show, mathematically, that the output is dependent on the difference between the measured and desired values.

(c) With the aid of a well annotated sketch describe the construction and operation of a P + I + D controller having a pneumatic output.

Question 4. (a) FIGURE 2 shows an electronic 'black box' whose output is ten times the difference between its two input signals. Show how the 'black box' could be realised using just two operational amplifiers and five resistors. Give the relative values of the resistors.

(b) With the aid of circuit diagrams show diagrammatically, and prove mathematically, how op-amps are utilized to produce:

(i) integral action
(ii) derivative action.

(c) Show how generation of the above actions are combined with proportional action generation to produce a three term electronic output controller.

Question 5. The proportional control system of FIGURE 3(a) has an input, θ₁, of 10 units. The uncontrolled input, θ₂, has a value of 50 units, prior to a step change down to 40 units. The result of this disturbance upon the output, θo, is shown in FIGURE 3(b).

(a) Calculate the change in offset in the output produced by the step change.

(b) Draw a modified block diagram to show how the offset could be minimised by the inclusion of another control action. Also, show by means of a sketch how the modification might be expected to affect the output response.

(c) Show, by drawing a modified block diagram, how the magnitude of the disturbance could be minimised by the inclusion of a third type of control action.

Question 6. FIGURE 4(a) shows a flow control system that is controlled by a P + I controller. The control objective is to maintain a constant flow rate, Q_o, for varying values of input flow rate, Q₂. The system can be represented by the block diagram shown in FIGURE 4(b).

Assume that the following initial conditions apply.

Show that following a permanent step disturbance in Q₂ from 1000 m³ h^-1 to 1200 m³ h^-1, the resulting offset is eliminated.

You should continue your calculations until Q_o is within 4 m³h^-1 of its final value for two successive calculations.* Derive any formulae used.

Question 7. (a) FIGURE 5 shows the input and output waveforms for a proportional plus integral controller. State:

(i) the controller's proportional gain

(ii) the controller's integral action time.

(b) FIGURE 6 shows a proportional plus derivative controller that has a proportional band of 20% and a derivative action time of 0.1 minutes. Construct the shape of the output waveform for the triangular input waveform shown, if the input rises and falls at the rate of 4 units per minute.

Question 8. (a) FIGURE 7 shows the closed-loop response of a plant to a step input when the proportional only gain was set to 4. Use the 'Quarter Amplitude Response Method' to estimate the required settings of a P + I + D controller.

(b) If the same plant was 'tuned' using the 'Ultimate Cycle Method', estimate the P + I + D controller settings if a proportional only gain of 6 was required to produce steady oscillations.