Mathematical Modelling of Systems Questions and Answers

Answer: D

The closed loop example is the refrigerator.

A refrigerator's thermostat senses the temperature inside the cabinet and compares it with the setting. When it is too warm, the compressor is switched on; when the set temperature is reached, it is switched off. The output (temperature) is measured and fed back to control the input, which is the defining feature of a closed loop system.

An ordinary room fan, a simple air cooler and a basic laboratory soldering iron run at whatever setting the user chooses without measuring their effect, so they are open loop systems (unless fitted with a thermostat).

Answer: C

The keyed disadvantage is that electronic control is susceptible to fire risk.

Compared with pneumatic control, which uses only compressed air and is intrinsically safe, electronic control systems carry electrical energy into the plant. In hazardous areas, such as refineries and chemical plants with flammable gases, a spark from electrical equipment can ignite the atmosphere, so electronic instruments need explosion-proof or intrinsically safe designs.

Electronic controllers draw very little current, are generally cheaper and more flexible than pneumatic ones, and are not inherently affected by the seasons, so the other options are not their characteristic disadvantage.

Answer: A

The correct code is (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii).

A gear train acts as a rotational transformer: the load angle is the motor angle times the gear ratio, \(\theta_{L} = n\theta_{m}\) (ii). A tachometer produces a voltage proportional to shaft speed, \(V_{2}(s) = K_{t}\,\omega(s)\) (i). A dc amplifier is modelled as a gain with a single time constant, \(K_{a}/(s\tau + 1)\) (iv). An accelerometer is a mass-spring-damper, giving the second-order ratio with \(-s^{2}\) in the numerator (iii).

The other codes give the tachometer the first-order amplifier model or match the gear train to the second-order accelerometer equation.

Answer: D

Backlash can cause low level oscillations.

Backlash is the play between meshing gears, a non-linearity in which the output does not move while the input reverses through the dead zone. In an otherwise stable control system it can produce a limit cycle, a sustained oscillation of small amplitude around the desired position, because the controller keeps over-correcting across the gap.

Backlash does not make a system overdamped, and it does not specifically cause poor stability at reduced gain; it tends to cause hunting at higher gains. 'Underdamping' describes a linear system's response, not the limit cycle that backlash produces.

Answer: A

The open loop system is the field controlled D.C. motor (A), as the University's key states.

In field control the speed is set by changing the field current, for example with a rheostat, and nothing measures the speed and feeds it back to correct the setting. The output has no effect on the input, which is the definition of an open loop system.

The Ward-Leonard system and the metadyne are generator-fed drive arrangements that are normally described with a speed or voltage feedback loop. This is the textbook convention behind the key: a Ward-Leonard drive can also be run without feedback, so the choice is one of usual description and not of principle. A stroboscope is a measuring instrument, not a control system.