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# ELEC ENG 7082 Principles of Control Systems Exam Paper

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ELEC ENG 7082 Principles of Control Systems

Sample Examination Paper

Total Duration: 120 mins Total Marks: 50 marks Instructions for Candidates • This examination is Open book . • Answer all questions. Permitted Materials • Graphing Calculator • Hand -written Notes • Lecture Notes • Printed Notes • Scientific Calculator • Text Books as Prescribe d

Question 1 [10 marks] A pendulum feedback control system is shown in Figure 1 . (a) When 𝐷(𝑠) is a PI controller, determine whether it will yield a stable closed -loop system. (4 marks)dd (b) When 𝐷(𝑠) is a PID controller, determine whether it can stabilise the system. If so, what constraints must the PID parameters 𝐾𝑃, 𝐾𝐼, and 𝐾𝐷 satisfy? (6 marks)

Figure 1

Question 2 [10 marks] The response 𝑦(𝑡) of a linear system is described by the differential equation 𝑑3𝑦(𝑡) 𝑑𝑡3+5𝑑2𝑦(𝑡) 𝑑𝑡2+11𝑑𝑦(𝑡) 𝑑𝑡+15 𝑦(𝑡)=𝑑𝑥(𝑡) 𝑑𝑡−𝑥(𝑡) where 𝑥(𝑡) denotes the input to the system. (a) Assume that 𝑦(𝑡) and all its time derivatives are zero at 𝑡=0, what is the transfer function of the system? (3 marks) (b) Given that there is a system pole at 𝑠=−3, find the other poles. (2 marks) (c) Find the time domain response, 𝑦(𝑡), of the system to a unit step input. (5 marks)

Question 3 [10 marks] The vertical motion of a hot air balloon is described by the following equation { 𝑑𝜓 𝑑𝑡=30−𝜓+0.05𝑞 𝑑2ℎ 𝑑𝑡2=20−960 𝜓−𝑑ℎ 𝑑𝑡 where 𝜓 is the temperature of the hot air, ℎ is the altitude of the balloon, and 𝑞 is the rate at which heat energy is supplied to the hot air. (a) For a constant altitude ℎ0, find the equilibrium temperature 𝜓0 and the equilibrium rate of heat input 𝑞0. (5 marks) (b) Derive a state model which is linearised abou t the equilibrium point (𝜓0, ℎ0, 𝑞0). Use 𝑥1= 𝜓−𝜓0; 𝑥2=ℎ−ℎ0; 𝑥3=𝑑ℎ 𝑑𝑡 and 𝑢=𝑞−𝑞0 in your model. Let the output 𝑦 be 𝑥1. (5 m arks)

Question 4 [8 marks] Given a unity -feedback closed -loop system shown in Figure 2a. The Bode plot of its forward - loop transfer function 𝐺𝑅𝐺𝑝 is as shown in Figure 2b. (a) How much can the gain of the controller or plant be increased without making the closed loop system unstable? (4 marks) (b) How much additional phase shift can be introduced without making the closed loop system unstable? (4 marks)

Question 5 [12 marks] The position of a hard drive head is described by the state space model. 𝑥̇= [−0.5010]𝑥+𝑢, and 𝑦 =𝑥 (a) Let 𝐿 be the state feedback gain vector, and 𝑙𝑟 be the scalar gain for the reference input 𝑟. Determine the values for 𝐿 and 𝑙𝑟 of the state feedback control law 𝑢 = −𝐿𝑥 + 𝑙𝑟𝑟 such that the poles of the closed loop system are placed at 𝑠=−4±4𝑗 and result in static gain = 1 from the reference to the output . (4 marks) (b) The state observer for estimating 𝑥 is 𝑥̂̇= 𝐴𝑥̂+𝐵𝑢+𝐾(𝑦−𝐶𝑥̂), where 𝐾 is the observer gain vector. Determine 𝐾, and explain how you choose the pole location of the observer with respect to that of the state feedback controller. (4 marks) (c) Introduce 𝑥𝑒= [𝑥𝑥̃] where 𝑥̃=𝑥−𝑥̂. (i) Form the state model in terms of 𝑥𝑒. (ii) Evaluate the characteristic polynomial of the closed -loop system det( 𝑠𝐼− 𝐴𝑒)=0. (4 marks)

END OF SAMPLE EXAMINATION PAPER

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