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Degrees of MEng, BEng, MSc and BSc in Engineering

Control M (ENG5022)

Monday 12th December 2022

13:30 - 15:30

Attempt ALL questions in Section A and ONE question from Section B and ONE question from Section C.

TOTAL MARKS AVAILABLE

SECTION A

Attempt BOTH questions

(a)

Explain benefits and drawbacks of using a closed-loop control strategy compared to an open-loop setup. You should consider characteristics of each structure with respect to plant disturbances, plant changes, stability, and complexity. [5]

(b) In your own words, summarise the design goals which one attempts to achieve when designing closed-loop feedback systems. Describe factors which limit the extent to which these goals can be achieved. [5]

[2(1)t(t)] = [ − 1(0).3 − 0(14).7] [x2(x1)t(t)] + [0(1)] u(t)

y = [0 0.6] [x2(x1)t(t)]

[x2(x1)0(0)] = [1(0)]

Explain how the transfer function can be derived and obtain the transfer

function for this system. Which element of the state space system cannot be represented in the transfer function form? [5]

(d) 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) List three advantages and one disadvantage of using a digital controller, as opposed to an analogue one. [4]

(b) Derive the expression to find the discrete transfer function of a generic continuous plant G(s) preceded by a zero-order hold. [7]

(c) By using appropriate sketches and formulas, discuss how the Tustin rule maps the poles, and show whether it preserves the stability of a continuous transfer function, when transformed into discrete domain. [5]

(d) Starting from the condition for BIBO stability in the time domain for hk:

Derive a condition for the same using the poles of its transfer function. [4]

SECTION B

Attempt ONE question

Q3 Consider a plant P(s) = .

(a) Sketch the Bode frequency response of P(s), clearly marking the corner frequencies and the corresponding asymptotes of the magnitude and phase components. [4]

(b) Derive the transfer function of an ideal PID controller in terms of an overall controller gain K, a time-constant associated with the integral term, TI , and a time constant associated with the derivative term, TD , assuming that TD ≪ TI . What are the poles and zeros of C(s)? [5]

(c) For a PID controller with K = 10, TI = 0. 1 and TD = 0.01, sketch the Bode plot of the frequency response. Clearly mark the corner frequencies and the corresponding asymptotes of the magnitude and phase components of the frequency response. [3]

(d) Apply the PID controller from (c) to the plant P(s). Sketch the Bode frequency response of the loop gain L(s) and determine the cross-over frequency. What is the bandwidth of the closed loop system? [4]

(e) Explain why the ideal PID controller from (b) is not realisable, and describe how it can be extended to make it realisable. Based on the numerical values in (c), choose a suitable value for the extra component and explain your choice.[2]

(f) Amend the Bode plot of the PID controller with the extra component from (e). How does this affect the Bode plot of the loop gain? [2]

(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. [3]

(b) For the structure described in (a), describe in detail the state estimator

(observer). Derive the equations for the state estimation error ̃(x)(t) = x(t) − ̂(x)(t) and discuss its behaviour. [5]

(c) Explain in your own words what is meant by the Separation Theorem in the context of state-estimator feedback control. [4]

(d) Consider the following plant

[2(1)t(t)] = [ 29(−)44(.)3(8) 0(1)] [x(x)2(1)t(t)] + [204] u(t)

y = [1 0] [x2(x1)t(t)]

A state feedback controller has been designed for this system using pole- placement which results in has an overshoot of Mp = 10% and arise time of tr = 0.2 seconds.

(i) Calculate the location of the closed loop poles relating to the state feedback. You can use the following equations to derive the natural frequency and the damping of the desired closed loop system:

幼n ≅ t(1)T(.8) ξ = − √π2 ln(M)PMP)2 ; 0 ≤ ξ < 1 [4]

(ii) Calculate the gain vector L of a state estimator for the controller derived in (i). Choose the observer poles in such away that their time constants are both 10 times faster than the poles obtained for the state feedback controller. [4]

SECTION C

Attempt ONE question