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.

Answer: B

The set point is the normal desired value of the controlled variable.

In a physiological control system the set point is the reference value the system works to hold, such as a core temperature of about 37 degrees Celsius set by the hypothalamus. The comparator measures the variable against it and the effectors act to remove any difference.

The maximum output is a limit of the system, not its target. The error signal is the difference between the set point and the measured value, not the set point itself. Sensor sensitivity is a property of the receptor.

Answer: A

The axis is an example of a closed-loop (feedback) control system (printed in the paper as 'Closed-loop [feedback) control system').

The hypothalamus releases TRH, which drives the pituitary to release TSH, which drives the thyroid to make T3 and T4. Those hormones then feed back to inhibit TRH and TSH release. The output is measured and returned to the controller, which is negative feedback in a closed loop, and it keeps thyroid hormone levels steady.

An open-loop system has no such return path. Feed-forward control acts on a disturbance before it changes the output. Cascade control nests one controller inside another, which is not how this axis is usually described.

Answer: D

The item that is not a component is the disturbance generator.

A basic biological control system has a receptor (sensor) that detects the variable, a comparator or control centre that compares it with the set point, and an effector that acts to correct it. Body temperature control has skin and hypothalamic thermoreceptors, the hypothalamus as the integrator, and sweat glands and muscles as the effectors.

Disturbances, such as a cold environment, come from outside and are what the system resists. They are inputs to the system, not one of its parts.