EE4218 / EE4216 讲解

Posted 2021-09-09 一吟流曲年殇

EE4218 / EE4216
Faculty of Science and Engineering
Department of Electronic and Computer
Engineering
End of Semester Assessment Paper
Module Code: EE4218 / EE4216
Module Title: Control 2
Semester: Spring 2018
Duration of Exam: 21
Grading Scheme: Final Exam : 80%
Coursework : 20%
Instructions to Candidates:
Answer any FOUR questions. All questions carry equal marks.
A University Standard Calculator may be used.
Module Code & Title: EE4218 / EE4216 Control 2 Page 2 of 6
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1. (a) Consider the system
   You may assume that β is a parameter that is allowed to vary in the range
   β ∈ [0.25, . . . , 1.5]
   i. Determine the natural frequency of oscillation ωn for this system. 2
   ii. Write a MATLAB script (.m file) that will illustrate the effect of variation
   in the parameter β in the region [0.25, . . . , 1.5] on a system step
   response.
   Your answer should briefly sketch the typical step responses that you
   would expect to observe. 7
   (b) This question concerns the locus of the roots of the characteristic equation
   for the closed loop system described by Figure 1.
   i. Determine equations for the asymptotes to the locus as K → ∞. 2
   ii. What are the departure angles for the locus as it leaves the complex
   open loop poles? 3
   iii. Assume that one feasible multiple root (accurate to 1 decimal place)
   exists for this locus at s = ?4.8 and that two infeasible multiple roots
   exist at s = ?1.8 ± ?1.3.
   Determine the other feasible multiple root location for this locus. 8
   iv. Sketch the root locus for this system. 3
   Figure 1:
   Module Code & Title: EE4218 / EE4216 Control 2 Page 3 of 6
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2. This question considers the D.C Motor
   System performance is to be improved using the following controller.
   K(s) = 50(s + 3)(s + 7)
   (a) Derive a bound on the Phase Margin of a system in terms of its worst case
   system sensitivity kSk∞. 6
   (b) Draw a system sensitivity plot, S(?ω), for this combination of plant and
   controller.
   Your answer should compute S(?ω) at
   ω = [0, 14, 23, 50,∞] rad\\s
   You may assume that kSk∞ = ks(?23)k. 8
   (c) Describe how bounds on Phase Margin, Steady state error, 5% settling
   time and Bandwidth can be read from the S(?ω) you have computed in
   Part (b) of this question. 6
   (d) Estimate the 5% settling time for this design to a step input demand.
   Briefly describe how you would use MATLAB to construct S(?ω) and to
   observe the step response for this particular design.
   Your answer should sketch an indicative step response for this system. 5
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3. This questions considers Nyquist based tuning for the following system :-
   2s + 8
   (s + 1)(s + 3)((s + 2)2 + 5)
   (a) Using the following frequency vectorω ∈ [0, 1.25, 3.3, . . . ,∞]
   draw a Nyquist diagram for this system. 10
   (b) Use your diagram to estimate the system phase and gain margins. 3
   (c) Using your Nyquist diagram, design a PID controller, K(s), for this system
   that is tuned for good low frequency performance. You may assume that
   the following tuning rules are appropriate :-
   KP = 0.4 × Ku
   where Ku is the limit cycle gain with the application of proportional only
   control and
   TI = 1.0 × Tu
   TD = 0.25 × Tu
   where Tu is the corresponding limit cycle period. 7
   (d) How would you improve the speed of response of your design using Phase
   Lead methods?
   Please note that an exact design is NOT required, it is sufficient to provide
   the design steps that you would take. 5
   Module Code & Title: EE4218 / EE4216 Control 2 Page 5 of 6
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4. This question considers the system shown in Figure 3. Denote the state variables
   for this system as x1 = VC(t) x2 = iL(t). The following describing equations
   for the system may be useful.
   Vin(t) = u(t) = L
   diL(t)dt + VC(t) ; Vout(t) = Vin(t) ?VC(t)
   VC(t)R1+ CdVC(t)
   dt = iL(t) + L
   R2diL(t)
   Figure 2:
   (a) Briefly list TWO reasons for a state space approach to Control Systems
   Design 2
   (b) For the case where
   L = 2H, R1 = 1?, R2 = 3?, C = 0.5F
   Complete the continuous A state matrix for this system and hence write
   the continuous state and output equation. 8
   (c) Design a State feedback controller that will place the closed loop poles for
   this system at ?2 ± 0.2?.
   Your answer should provide a diagram illustrating how this design can be
   implemented in practice. 8
   (d) Determine the feasibility of an Observer based controller design strategy
   for this system. 2
   (e) Describe one advantage and one disadvantage of such an Observer based
   strategy over a more traditional state feedback approach. 2
   (f) Explain how such an Observer based strategy might be implemented in
   practice 3
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5. This question concerns the following system that you have studied extensively
   during your Laboratory work this semester
   (a) What type of problem does this system model and why is it such a difficult
   problem to control? 3
   (b) Determine the values of Proportional gain K that will stabilise this system.
   Why is the proportional control that is suggested by Figure 3 unsuitable
   for this system? 8
   (c) Discuss the various approaches that you used to control this system over
   the course of the semester. For each different strategy that you consider
   your answer should describe
   i. How you implemented the design in MATLAB.
   ii. The frequency domain analysis tools that you used to assess the design.
   iii. The steps that you took to improve the time domain performance of
   your design. 14

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