ECE486 Homework 1 Solution

1. Do not start writing until you are instructed to do so.
2. Do not continue to write when you are told to stop.
3. You are not allowed to communicate with one another during the quiz.
5. Answer in the answer-sheet and submit both question- and answer-sheets before the end of the quiz.
6. Write your name and student number clearly in all the sheets.
7. Answer all questions. There are 4 questions with sub-questions.

Question 1 (12 Points)
a) Consider a water tank of cross-sectional area A with volume of water flowing in at a rate of 𝑞𝑖𝑛 and volume flowing out at a rate of 𝑞𝑜𝑢𝑡. Taking gravitational constant to be g,
𝑔

Figure 1
Show that the dynamics of the system can be expressed an ordinary differential equation:
𝑔
𝐴𝑦̇(𝑡)+ 𝑦(𝑡)=𝑞𝑖𝑛
𝑅 (3 Points)
** Hint the rate of change of water volume in the tank is the same as the net volume flow
b) How are the system input 𝑞𝑖𝑛(𝑡) and the output 𝑦(𝑡) related via the convolution operation? (1 Points)
c) If at t=0, the height y(0)=5 and 𝑞𝑖𝑛 = 0 (the inlet to the tank is switched off), obtain the system response i.e. the variation of height, h(t) over time assuming A=1. (3 Points)
d) Write down τ, the time constant (i.e. time taken for the tank to drain to a height of 1/e of its initial height) in terms of e, A, R and g if 𝑞𝑖𝑛=0. (2 Points)
e) Using Laplace transformation, find the transfer function that relates the input and output in the s-domain assuming zero-initial condition. (3 Points) Question 2 (6 Points)
A dynamic system with no input is governed by the equation:
𝑥̈ = 0.5(𝑥2 − 1)𝑥̇ + 1.5𝑥
a. Choosing state variables (𝑥1 𝑥2) = (𝑥 𝑥̇), write down its non-linear state-space model for
the system. (2 Points) b. Derive the linearized state-space model at the equilibrium point. (4 Points)


Question 3 (12 Points)
A dynamic system can be represented by the following block diagram:

Figure 2

a) Show that the transfer function of the system could be expressed in
𝑌 𝜔𝑛2
𝐻(𝑠) =𝑈 = 𝑠2 +2𝜁𝜔𝑛𝑠+𝜔𝑛2
Also, write down the value for 𝜔𝑛 and 𝜁. (4 Points)

b) Use Routh-Hurwitz stability analysis to check if system in (a) is stable. (4 Points)
c) A specification on 5% settling time 𝑡𝑠 < 0.5 is required for the 2nd order system
i) Sketch the region of pole locations on the complex plane to meet the spec.
ii) Explain whether the system in (a) meets the requirement. (4 Points)



𝜔𝑛2 𝜎2 𝜔𝑑2
𝐻(𝑠)=𝑠 2+2𝜁𝜔𝑛𝑠+𝜔𝑛2 = 𝜔𝑑

Question 4 (10 Points)
a) State the key purpose of incorporating integral control in a PID controller (1 Points)
b) The sensor of a control system is subject to a lot of noise from the working environment which term in the PID control is likely to worsen the effect of noise. Explain. (1 Points)

c) A control system is implemented as represented by the block diagram.

Figure 3
Fill in the blocks (i)-(iii)
(3 Points)
iv) Write down the closed-loop transfer function. (5 Points)