Derive the closed loop transfer functions for the system

Question

Assignment: CONTROL SYSTEMS AND AUTOMATION

1. Choose a simple single control loop used on a process you are familiar with (it could be domestic or industrial).

(a) Explain why the control is necessary.
(b) Write a description of the control system.
(c) Produce an algorithm of the control system.
(d) Draw a block diagram of the control system.
(e) State the type(s) of signal used in the process.
(f) State whether the control is open or closed loop, feed forward or feedback.
(g) State and describe the sensor used for measuring the process variable to be controlled.

2. The curve in FIGURE 1 shows the response of a bare thermocouple which has been subjected to a step change in temperature from 50°C to 10°C. Assuming that the bare thermocouple behaves as a single transfer lag system, determine the mathematical relationship between the
temperature (T) and time (t) [i.e. determine the equation relating T to t].

FIG. 1

3. FIGURE 2 shows the behaviour of a control system which has been switched from manual to automatic mode. Comment on the conditions existing in the control loop which would have given rise to the response shown.

FIG. 2

4. FIGURE 3 shows an open loop system containing a distance velocity lag and a single transfer lag.

FIG. 3

If the system input x_i is subjected to a step disturbance from 2 units to 12 units, plot the response of x_o on a base of time. Determine graphically, and verify mathematically, the time taken for the output to change by 4 units.

5.

FIG. 4

Derive the closed loop transfer functions for the system shown in FIGURE 4 and show that for large values of G V₀/V_i the value of approaches unity.

6. FIGURE 5 shows an electrically heated oven and its associated control circuitry. The current, I, to the oven's heating element is fed from a
voltage-controlled power amplifier such that I = εK₁. A voltage, V_D, derived from a potentiometer, sets the desired oven temperature, T_D. The oven temperature is measured using a thermocouple that, for simplicity, is assumed to generate a constant emf of 10 µV per degree Celsius. The effect of the ambient temperature is ignored.

FIG. 5

(i) Represent the arrangement by a conventional control-system block diagram. Identify the following elements in the block diagram: input; error detector (comparator); controller; controlled element; detecting element and feedback loop.

(ii) Derive an expression for the transfer function of the system, in terms of the system parameters k₁, k₂, k_O and k_t.

(iii) Using the data given in TABLE A, calculate the oven temperature when the potentiometer is at its mid-point.

PARAMETER	VALUE
kt	10 µV/°C
k0	6.9 °C/A
k1	6 A/V
k2	2400

7. (a) Derive an expression for the closed loop transfer function θ₀/θ_i for the control system shown in FIGURE 6.

FIG. 6

(b) Show that the value of θ₀/θ_i is 0.8 when the following values of the various control loop elements are adopted:

K₁ = 10
K₂ = 2
K₃ = 5
K₄ = 0.4
K₅ = 0.9.