2.6 Problems (Solved and Unsolved for Modeling of Systems) – 3 hours 1. Derive a differential equation relating the input and output for each of the systems shown in figure. (Bolton)

Figure

2. A closed-loop negative feedback system for the control of the height of liquid in a tank by pumping liquid from a reservoir tank can be considered to be a system with a differential amplifier having a transfer function of 5, its output operating a pump with a transfer function 5/(s + 1). The coupled system of tanks has a transfer function, relating height in the tank to the output from the pump, of 3/(s + 1)(s + 2). The feedback sensor of the height level in the tank has a transfer function of 0.1. Determine the overall transfer function of the system, relating the input voltage signal to the system to the height of liquid in the tank. (Bolton) 3. A closed-loop negative feedback system to be used for controlling the position of a load has a differential amplifier with transfer function K~ operating a motor with transfer function 1/(sL + R). The output of the motor operates a gear system with gear ratio N and this, in turn, operates a screw with transfer function 1/s to give the resulting displacement. The position sensor is a potentiometer and this gives a feedback voltage related to the position of the load by the transfer function K:. Derive the transfer function for the system as a whole, relating the input voltage to the system to the displacement output. (Bolton) 4. Develop a model for the hydraulic system shown in Figure 2.18 where there is a liquid entering a container at one rate q~ and leaving through a valve at another rate q2. (Bolton)

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5. Define hydraulic resistance, capacitance and inertance. 6. Determine a model for the temperature of a room (Figure 2.16) containing a heater which supplies heat at the rate q~ and the room loses heat at the rate q2. (Bolton)

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7. Develop a model for the simple thermal system of a thermometer at temperature T being used to measure the temperature of a liquid when it suddenly changes to the higher temperature of TL (Bolton)

From Burns 8.

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From Ogata:

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Consider the liquid level system shown in Figure 3.60. Assuming that

sec/02.03 3mQandmH == and the cross sectional area of the tank is equal to 5 m2, obtain the time constant of the system at the operating point ( QH , ). Assume that the flow through the valve is turbulent.

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Figure. Liquid Level Tank

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A thermocouple has a time constant of 2 sec. A thermal well has a time constant of 30 secs. When the thermocouple if inserted into the well, this temperature measuring device can be considered a two capacitance system. Determine the time constants of the combined thermocouple thermal well system. Assume that the weight of the thermocouple is 8g and the weight of the thermal well is 40g. Assume also that the specific heats of the thermocouple and thermal well are the same. References: 1. Advanced Control Engineering, Ronald Burns, First Edition, Butterworth –

Heinemann, 2001. 2. Modern Control Engineering, K. Ogata, Third Edition, Prentice Hall of India. 3. Control Systems, Bolton

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