ENG5314 Control Systems M 2019
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Control Systems M (ENG5314)
2019
SECTION A
Q1 |
(a) |
Draw the structure of an open-loop control system and of a two-degree-of- freedom feedback control system. Show the different signals, including disturbances, and explain their physical meaning. Explain which additional components are required in the feedback control structure. [5] |
(b) Show that for an open-loop structure the ideal compensator should cancel the
plant dynamics, and state the drawbacks of this approach. Demonstrate that a feedback control structure can provide approximate plant inversion and show therefore that high-gain closed-loop control implicitly inverts plant dynamics. [5]
(c) What is observability in the context of state feedback control? Describe a test for observability of a state space system. [5]
(d) What is meant by state estimator feedback control? Use a block diagram to illustrate your explanations and mark the elements which form the compensator. Explain what is meant by the Separation Theorem in the context.
[5]
Q2 |
(a) |
Assume that the nominal plant Po (s) changes to some other value P(s). If this change causes the Nyquist plot of the loop gain L(s) to move substantially closer to the - 1 point, explain how this may affect the closed-loop sensitivity function S(s) and the closed-loop complementary sensitivity function T(s), and how this in turn may adversely affect the various closed-loop performance targets. [5] |
(b) |
Describe the design goals which one attempts to achieve when designing closed-loop feedback systems, and describe factors which limit the extent to which these goals can be achieved. [5] |
(c) Explain the concept of “internal stability” and relate this to BIBO stability, using a block-diagram similar to that which you derived in Q1(a). Why is it important to consider internal stability in a feedback control system? [5]
(d) Consider the state space system [ẋ(ẋ)2(1)] = [ ] [x(x)2(1)]+ [0(2)] u
1
Show whether this system is controllable. Is it also stabilisable? [5]
SECTION B
Q3 (a)
Explain why ‘peaking’ in the sensitivity function S0 and the complementary sensitivity function T0 should be avoided. The explanation should be based on performing the following analysis.
i. Derive the closed-loop equations of a feedback control system in terms of So and To , considering the output responses to the reference, disturbance and noise. Discuss how peaking would affect the system’s response to these signals. [4]
ii. Show that the complementary vector margin for the inverse loop gain is equal to the inverse of the peak value of |T0 |. By a symmetry argument, briefly describe the effect of |S0 | on the vector margin for the loop gain. Based on this, explain what a strongly oscillatory response of a closed- loop system means for stability-robustness. [5]
iii. Show that T0 is equal to the sensitivity of S0 to changes in the plant P0 . By a symmetry argument, briefly describe the effect of S0 on the sensitivity of T0 to plant changes. [5]
(b) Refer to a closed loop control structure. Assume that the compensator C(s) is
chosen in such a way that the nominal loop gain Lo (s) gives a stable closed- loop system. Assume also that the nominal loop gain Lo (s) is perturbed to the actual loop gain L(s), i.e. Lo (s) → L(s) (or, equivalently, that the nominal inverse loop gain is perturbed as 1/Lo (s) → 1/L(s)). Derive a sufficient condition for closed-loop stability which combines two tests, one involving |So |
and the other involving |To |. What is the name of this criterion? [6]
Consider the linear state space system
ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
with x(0) = x0 .
(i) Show how a linear transfer function G(s) can be derived. [5]
(ii) |
What are the requirements in terms of the properties of the matrices / vectors A, B, C and D for which the system is stable? What are the corresponding requirements for the transfer function G(s)? [4] |
(b) Consider a state space system. What is controller design by pole assignment
and how can it be used to design a state feedback controller? What is the relevance of the controller canonicalform and of controllability in this context? [6]
(c) Design a state feedback controller K for the state space system
「x1 ] 「−8 −1]「x1 ] 「1]
| | = | | | |+ | | u
Lx2 」
using the pole placement method in such a way that the closed system has an overshoot of Mp = 1% and a rise time of tT = 0. 1 seconds. You can use the following equations to derive the natural frequency and the damping of the desired closed loop system:
n(o) 吴
毛 = − ; 0 三 毛< 1
[5]
2022-08-01