Exam November 2016, questions PDF

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ELEN90055 Control Systems Student Number:...................................................... The University of Melbourne Department of Electrical and Electronic Engineering Semester 2 Assessment, 2016 ELEN90055 Control Systems Reading Time: 15 minutes Writing Time: 3 hours This paper has 9 pages, including this page

Authorized Materials: • Melbourne School of Engineering approved electronic calculators may be used (with no data stored in memory). • This is an open book exam: books, lecture notes and worked examples are all allowed.

Instructions to Invigilators: • This examination paper is to be collected together with answer script books. • This exam paper should not be lodged with the Baillieu library.

Instructions to Students: • • • • • • • •

Students should attempt all questions. Students should read the instructions for each question carefully. Students should answer all questions in script books provided. Questions are NOT of equal value and marks are shown in parentheses. Students should show all calculations and mathematical manipulations fully. Students should submit this paper together with answer script books. Answers to each question should be started on a new page. Maximum possible mark is 70.

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Question 1 (11 Marks) Consider the spherical tank in Figure 1. The output of the system is the level of water in the tank h and the input is the volumetric flow Q1 which is supplied from another tank via a pump. Assuming that water is incompressible, that the pressure of air above water and at the outlet is equal to the atmospheric pressure, we write the differential equation for the change of volume of water in the tank: √ dV = Q1 − Q2 = Q1 − k h . dt Note that the volume of water in the tank V is related to the level of   water h via: V = V (h) = 6π h 3h(2R − h) + h2 .

Figure 1: Spherical tank.

• (2 Mark) Write a mathematical model of the system that relates the output h to the input Q1. • (2 Marks) Find all equilibria of the system.

• (3 Marks) Assuming R = 3m, k = 0.1m5/2/sec, linearize the system around the equilibrium for Q∗1 = 0.1m3/sec (this is the nominal value of the volumetric flow that the pump can supply). Write a transfer function for the linearized model. • (4 Marks) Suppose that the pump supplies volumetric flows that is constant but slightly different from its nominal value Q∗1 = 0.1m3/sec. Design a controller that can reject this type of disturbance. Please turn the page over. 2

Question 2 (10 Marks) Consider the Nyquist plot in Figure 2 of the following open-loop transfer function: Λ0(s) =

5s + 15 s3 − s2 + 11s − 51

Figure 2: Nyquist plot for Question 1.

• (2 marks) Using the Routh-Hurwitz criterion, determine the number of unstable poles of Λ0 (Hint: the number of sign changes in the first column of the Routh-Hurwitz array for the denominator of Λ0(s) is equal to the number of poles with strictly positive real part); • (3 marks) Using the Nyquist criterion, determine the stability of the closed-loop system. • (5 marks) Consider

KΛ0(s), K > 0 .

Give a detailed stability analysis of the closed-loop system for different values of K. Please turn the page over. 3

Question 3 (10 marks) Consider a plant transfer function G0(s) =

s+1 . s+5

Suppose that you have measured the output disturbance for this system and recorded it, see Figure 3.

Figure 3: Disturbance for Question 2.

• (2 marks) Using Figure 3, propose a model that approximately models this disturbance (i.e. find its generating polynomial Λd(s)). • (2 marks) Can a controller of the form C(s) = K, K > 0 achieve stability and output disturbance rejection? • (3 marks) Using root locus, investigate whether a controller of the form K , K>0 (1) C(s) = Λd(s) 4

can achieve both stability and output disturbance rejection for some values of K. If so, specify the values of K for which stability can be achieved. Provide detailed explanations of all steps you used to plot the root locus. • (3 marks) Using root locus, investigate whether the controller of the form (1) can achieve stability and output disturbance rejection for some values of K for the same disturbance but the following plant: s+5 G0(s) = s+1 If so, specify the values of K for which stability with disturbance rejection can be achieved. Please turn the page over.

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Question 4 (Total 11 marks) Consider the block diagram in Figure 4 of a wheelchair velocity control system designed for people that are paralysed from head down. The idea is that the person’s head movement is converted into a signal that powers the motor driving the wheelchair. The head movement encodes the error between the desired velocity (in the person’s brain) and the actual velocity that the person observes. Bode diagrams for the sensor and for the wheelchair with motor are given in Figure 5 at the end of exam paper.

Figure 4: Block diagram for the wheelchair control system.

(a) (3 marks) Supposing that the amplifier gain is K = 1, check if the closed loop system is stable and determine the phase and gain margins. (b) (4 marks) Adjust the amplifier gain so that the phase margin is at least 30◦ . What is the gain margin in this case? (c) (4 marks) Is it possible to adjust the amplifier gain so that you achieve steady state error smaller than 0.01 and phase margin of at least 30◦ ? If so, find the range of K for which this is possible (provide detailed calculations). Please turn the page over.

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Short answer questions [Total mark 28] Question 5 (4 marks) Suppose that you have found that the characteristic polynomial of a closed-loop system takes the following form: s3 + (kp + 3)s2 + 4kp s + 15 where kp > 0 is a controller parameter that you can tune. Explain how you can use the Routh-Hurwitz criterion to check if the settling time (5% of the final value) is smaller than 2sec (you do not need to complete all calculations - just explain the steps you would use). Question 6 (3 marks) Suppose that a controller with a transfer function C(s) was designed so that it achieves tracking of step signals and rejects ramp input disturbances for a given plant. What can you say about the stability of this controller when considered as system given by its transfer function C(s). Please explain your answer in detail. Question 7 (4 marks) If you want a given closed-loop system to track sinusoidal references of the form r(t) = A sin(ωt), ω ∈ [0, 1]rad/sec, explain in detail a design procedure for an appropriate controller that achieves this design specification. Question 8 (4 marks) Suppose that you performed an experiment on a stable system in which the sensor gives you the following measurement Z Z t

m(t) =

t

e(t)dt =

0

0

(r (t) − y(t))dt ,

where r(t) is a unit step input and y(t) is the step response of the plant. Suppose that your measurements show that m(10) < 0 . What can you conclude about the number of poles of your plant? Question 9 (3 marks) Suppose that you designed two controllers for a given plant and they both yield a stable closed-loop system. The 7

first controller gives you rise time tr = 0.5sec and settling time ts = 10sec. The second controller gives you the rise time tr = 1sec and settling time ts = 4sec. Which controller gives you a faster response for the closed-loop system? Question 10 (4 marks) Suppose that you have found that your plant has two stable poles and one stable zero and your boss told you that you have the budget of $500 to buy an actuator and, moreover, it is essential that the closed-loop system does not have any undershoot in its step response. You need to choose between three actuators that have the following transfer functions: C1(s) =

s−1 1 s − 0.5 . ; C (s) = ; C (s) = 2 3 s2 + 2s + 4 s3 + s2 + s + 1 s2 + s + 2

The first actuator costs $350, the second actuator costs $450 and the third actuator costs $550. Which actuator would you recommend to your boss for purchase? Question 11 (3 marks) Suppose that the impulse response of a system contains the term t2. What is the minimum order of this system? Question 12 (3 marks) Which of the following combinations may not give stability of the closed-loop system in a unit feedback structure? 1. Stable plant and stable controller. 2. Unstable plant and stable controller. 3. Stable plant and unstable controller. 4. Unstable plant and unstable controller. Please elaborate on your answer in detail (you can give examples to back up your answer). End of the examination paper. 8

9 Figure 5: Bode diagrams for the wheelchair control system.

Library Course Work Collections

Author/s: Electrical and Electronic Engineering Title: Control Systems, 2016 Semester 2, ELEN90055 Date: 2016 Persistent Link: http://hdl.handle.net/11343/127717...