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Control 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. Upload

your 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 questions

and 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 the

chat function in zoom to avoid distracting others. You are not allowed to communicate

or 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. The questions 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 differential

equation

y“ ? y步

y

+ 6y步 ? 6 ln |y| = u步? u,

where 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‘ = 0 and y‘ =

1. Express your answer as a differential equation involving the incremental

variables 汛y, 汛u and their time-derivatives.

(c) Find the simplified transfer function G(s) for your linearised model as a ratio

of 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 define

BIBO stability).

(d) Suppose u‘ and 汛u are both = 0 for all t ≡ 0?. It is observed that when the

incremental initial condition 汛y(0

?) is not exactly zero, the output y(t) does

not approach the equilibrium value y‘ = 1 as t ↙ ﹢. Explain briefly why this

happens even though G(s) is BIBO stable.

1

2. (3 + 2 + 5 + 3 = 13 marks) Consider a plant with transfer function

G(s) =

1

s2 + 4

You wish to use a controller C(s) = q(s + 1) + k/s to stabilise this plant in a unity

feedback 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 the

reference signal to the plant output. Express your answer as a ratio of two

polynomials.

(b) Assuming closed-loop stability, show that this controller achieves a steady-state

error of zero when the reference is a step function.

(c) The characteristic equation is

s3 + 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 would

improve 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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