ELEN90055控制系统

ELEN90055Control Systems

Midsemester Test
Semester 2, 2021

Instructions

This test consists of 2 questions, with marks as indicated, summing to 27. You have one(1) hour to complete this test, including reading, writing, scanning and uploading. Uploadyour answers through Gradescope by 1.05pm Melbourne time. Aim to finish writing by
12.50pm, so you have time for scanning and upload. For full marks, complete all questionsand show your working.This test is ＆open book＊ and you may refer to any subject materials. The lecturer will
be available on the usual zoom channel until 1.05pm if you need clarification - use thechat function in zoom to avoid distracting others. You are not allowed to communicateor collaborate with, or seek/provide assistance from/to, any other persons, from the start
time of the test until the late submission time is over. Thequestions are randomised.Late submissions are permitted until 1.30pm Melbourne time but a penalty may apply.Submissions after that time will not be accepted. In case of potential technical issues,download a copy of this question sheet onto your computer or device as soon as you start.If you have technical issues, take a screenshot of the error message.

1. (2 + 5 + 4 + 3 = 14 marks) Consider a system given by the time-domain differentialequationwhere y 6= 0 is the output and u is the input. Let (y‘, u‘) denote an equilibrium point.
   (a) Find an expression for y‘ in terms of u‘.
   (b) Find the linearised (i.e. incremental) model of this system if u‘
2. Express your answer as a differential equation involving the incrementalvariables 汛y, 汛u and their time-derivatives.
   (c) Find the simplified transfer function G(s) for your linearised model as a ratioof two coprime polynomials. Plot any poles and zeros (using x＊s and o＊s respec-tively) in the complex plane, and check that G(s) is BIBO stable (do not defineBIBO stability).
   (d) Suppose u‘ and 汛u are both = 0 for all t ≡ 0?. It is observed that when theincremental initial condition 汛y(0?) is not exactly zero, the output y(t) doesnot approach the equilibrium value y‘ = 1 as t ↙ ﹢. Explain briefly why thishappens even though G(s) is BIBO stable.
3. (3 + 2 + 5 + 3 = 13 marks) Consider a plant with transfer function
   G(s) =1s2 + 4You wish to use a controller C(s) = q(s + 1) + k/s to stabilise this plant in a unityfeedback loop, where k and q are real parameters to be chosen.
   (a) Find a simplified expression for the closed-loop transfer function H(s) from thereference signal to the plant output. Express your answer as a ratio of twopolynomials.
   (b) Assuming closed-loop stability, show that this controller achieves a steady-stateerror of zero when the reference is a step function.
   (c) The characteristic equation iss3 + qs2 + (4 + q)s+ k = 0.
   Find conditions on q and k for the closed-loop system to be stable.
   (d) Referring to suitable transfer functions, explain briefly why making q large wouldimprove how well the output follows the reference, but also increases the sensi-tivity of the output to high-frequency measurement noise.
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