Advanced Control Systems PDF

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THE ROYAL MELBOURNE INSTITUTE OF TECHNOLOGY UNIVERSITY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

June 2006

EEET 2100/1368 ADVANCED CONTROL SYSTEMS

TWO (2) HOURS

INTERNAL EXAMINATION CITY CAMPUS • Total number of questions: 6 • Total marks: 100 • Answer ALL questions • Questions are not worth equal marks • University calculators are allowed • Examination is closed book • 10 minutes reading time allowed

Examiner: A/Professor Liuping Wang Discipline: Electrical Energy and Control Systems

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1.

EEET2100/1368 ADVANCED CONTROL SYSTEMS

(20 marks) • Write down the transfer function of an ideal PID controller in terms of proportional gain, integration time and derivative time. (3 marks) • Given a plant transfer function G(s), sketch the block diagram for – PI control system; (5 marks) – PID control system with derivative implementation using output signal and a derivative filter (6 marks); Mark setpoint signal, input disturbance and output disturbance on both diagrams. 1 , find • Given a first order plant transfer function model G(s) = 2s+1 proportional gain Kc and integration time τI , where the desired closedloop polynomial is chosen to be s2 + 2 × 0.707 × 3s + 9 (6 marks).

2.

(15 marks) • Given a loop transfer function G(s)C (s), give expressions for sensitivity function and complementary sensitivity function. (5 marks) • Explain why it is possible to use feedback control to simultaneously reject low frequency disturbance and attenuate high frequency measurement noise? (10 marks)

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EEET2100/1368 ADVANCED CONTROL SYSTEMS

3. (15 marks) Figures 1 and 2 (see pages 5 and 6) show two Nyquist plots of open-loop transfer function G(s)C(s) =

Kc (2s + 1) s(0.1s + 1)(10s − 1)

where Kc is positive. • From the Nyquist stability criteria, which plot corresponds to a stable closed-loop system? Explain your answer. (5 marks) • Illustrate its gain margin and phase margin on the Figure that indicates a stable closed-loop system. (5 marks) • How much gain variation can the system tolerate before it becomes unstable? (5 marks) (Detach pages 5 and 6, and submit them together with the examination paper.) 4. (15 marks) Given a state space model below with three inputs and three outputs and 5 state variables x( ˙ t) = Ax(t) + Bu(t); y(t) = Cx(t) • write down the state feedback control equation with controller K and the observer equation with observer gain J. (10 marks) • Draw the block diagram of observer. (5 marks)

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EEET2100/1368 ADVANCED CONTROL SYSTEMS

5. (15 marks) Check controllability and observability of the following state space model:     

x˙ 1 x˙ 2 x˙ 3 x˙ 4

    

   

= 

y(t) =

h

0 1 3w2 0 0 0 0 −2w

0 0 x1  0 2w    x2  0 1   x3 x4 0 0 





i 0 0 1 0   

x1 x2 x3 x4





    +  

0 1 0 0



   u(t) 

(1)

    

(2)

where w 6= 0. 6.

(20 marks) • Suppose that the output disturbance is a sinusoidal signal of frequency √ 6 (rad/sec) and the plant is described by the transfer function G(s) =

s+4 . (s − 1)(s + 2)

– Design a pole-assignment controller to minimize the effect of the disturbance. Three of the closed-loop poles are chosen to be −4, and the rest of the closed-loop poles are chosen to be −2. (15 marks) – Will the output of the closed-loop system follow a sinusoidal setpoint signal of the same frequency with zero steady-state error? Explain your answer by using sensitivity function analysis (5 marks).

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EEET2100/1368 ADVANCED CONTROL SYSTEMS

4 w=0.1 3.5

3

2.5

Imaginary

2

1.5

1

0.5

w=1000

0

−0.5

−1 −3

−2.5

−2

−1.5

−1 Real

−0.5

0

0.5

Figure 1: Nyquist plot A for question number 3

1

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EEET2100/1368 ADVANCED CONTROL SYSTEMS

4

3

2

Imaginary

1 w=0.1 0 w=1000 −1

−2

−3

−4

−16

−14

−12

−10

−8 Real

−6

−4

−2

Figure 2: Nyquist plot B for question number 3

0...