ENG5314 Control Systems Analysis and Design M 2020
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Control Systems Analysis and Design M (ENG5314)
2020
SECTION A
Attempt BOTH questions
Refer to the closed-loop system shown in Figure Q1(a)
Derive the closed-loop equations relating the plant output y to the signals r, d, and n. Discuss how should the feedback system be designed in order to respond appropriately to each of these signals and what the associated limitations are. [5]
Figure Q1(a)
(b) |
Derive an expression linking the vector margin Sm of the closed-loop system to the peak magnitude of the sensitivity function SO . Derive an equivalent expression linking the complementary vector margin Tm of the closed-loop system to the peak magnitude of the complementary sensitivity function TO . Briefly discuss the relevance of these expression in relation to stability robustness of the feedback system. [5] |
(c) Consider the linear state space-system
Ẋ (t) = AX(t) + Bu(t) X (0)= X0
y(t) = CX(t) + Du(t)
Show how a linear transfer function G(S) can be derived. Explain which element of the linear state space representation cannot be represented in the
linear transfer function G(S). [5]
(d) In your own words, explain what is meant by controllability in the context of
state feedback control. Describe a test for controllability of a state space system. [5]
Q2 (a) Show that the sensitivity So is equivalent to the relative changes in Towhich
result from changes in the plant Po , i.e. show that So = − . Explain in your
own words what the significance of this result is. [5]
(b) |
Explain why it is highly undesirable to have low-frequency measurement noise in a feedback system, and discuss why high-frequency measurement noise is relatively unimportant. [5] |
(c) Explain what is meant by observer canonical form of a state-space system. Given the transfer function:
G(s) =
derive the state-space description in observer canonical form. [5]
(d) Explain in your own words what is meant by the Separation Theorem in the context of state-estimator feedback control. [5]
SECTION B
Attempt ONE question
Q3 |
(a)
(b) |
Consider a feedback control system with loop gain Lo (S). Discuss design targets of the closed loop system in terms of the sensitivity function So (S) and the complementary sensitivity function To (S). How can these design targets be translated into requirements for the frequency response of Lo ? [6] Consider a PID controller. (i) State the control law in the time domain and in the Laplace domain. [3] (ii) Derive the transfer function of the PID controller in terms an overall controller gain K, a time-constant associated with the integral term, TI , and a time constant associated with the derivative term, TD . What are the poles and zeros of C(S)? [4] (iii) Sketch the Bode frequency response of a PID controller with K = 100, TI = 1 and TD = 0.05. Clearly mark the corner frequencies and the corresponding asymptotes of the magnitude and phase components of the frequency response. [4] (iv) Describe how the PID controller can be extended to make it realisable. Based in the numerical values in Q3(b)(iii), choose a suitable value for the extra component and explain your choice. Amend the Bode plot of the PID controller accordingly. [3] |
Q4 |
(a) |
Explain in your own words what is meant by state estimator feedback control. Use a block diagram to illustrate your explanations and mark the elements which form the compensator. Discuss the reasons for using a state estimator. [5] |
(b) |
For the structure described in (a), describe in detail the state estimator (observer). Derive the equations for the state estimation error (t) = X (t) − (t) and discuss its behaviour. [5] |
(c) Consider the plant
[Ẋ(Ẋ)2(1)] = [ ] [X(X)2(1)] + [23(2)] u
y = [1 0] [X1 ]
Derive the observer gain vector L such that the closed loop observer poles are
located at - 100 and - 110. [5]
(d) Describe a test for observability of a state space system. Show whether the following system is observable:
⌊Ẋ(Ẋ)2(1)⌋ = [ ] [X(X)2(1)] + ⌊2(5)⌋ u
y = [ −2 1] [X(X)2(1)]
[5]
2022-08-01