ELEN30012 Signals and Systems Questions for Assignment Three - semester 1 2021 May 14, 2021 This assignment has a total of 100 marks and counts for 5% of your assessment for ELEN30012. It must be done in groups of two. You can choose your own partner. Alternatively, you can ask Pasha or Trang to allocate a partner to you. Your partner for this assignment does not have to be the same person as for assignment 2. MATLAB is required for most of the questions. You need to make two submissions: 1) your Matlab file in ”Assignments” and 2) your report in pdf format on ”Gradescope”. Everyone in this class needs to be familiar with uploading on Gradescope. Therefore, if you are the one in your group without Gradescope experience then you should be the one to upload this assignment. In Gradescope you need to specify which questions are at which page. If you are not sure how to do this, please view the instructional video in the ELEN30012 Module ”How to use Gradescope”. 1 COVID-19 Model: z-Transforms In this section you will undertake some crude modeling of the spread of an infectious disease. The work of mathematical modellers using Signals and Systems around the world has become very important over the last year as it is used to inform political decisions in order to deal with the COVID-19 pandemic. Detailed and accurate modelling is required to produce trustworthy outcomes. This is not what you are going to do in this assignment. The model will be very simple but still it tries to capture some of the important features of the spread of an infectious disease. One of the main questions in this assignment is whether you will be able to see how curves are flattened by government action, even when using a simple model. We start our model on day n = 0, coinciding with 3 March 2020 as the time when the disease started to spread in Australia. For simplicity we assume that, on average, each infected person infects α other people and people are no longer infecting others after 2 days. Here α ∈ R+. The above scenario can be described by a system equation, namely y[n+ 2] = αy[n+ 1] + αy[n], for all n ≥ 0. (1) Here y[n] denotes the number of newly infected people on day n. For α = 1 this system generates a famous series of numbers referred to as a “Fibonacci Sequence”. Around 3 March 2020 Australia was still wide open to international travellers. Because of these open borders, the dynamics of the number of newly infected cases is given by y[n+ 2] = αy[n+ 1] + αy[n] + βv[n], for all n ≥ 0 (2) where v[n] is the number of infected international arrivals on day n and β ∈ R+. 1 We are interested in the situation where there is initially one mystery infected case in Australia, thus y[0] = 0 and y[1] = 1. Also, we assume that α = 3 and that the input signal v is the step signal u. Furthermore, there are 50,000 international arrivals per day in Australia and 0.01% of these carry the virus. Thus β = 5. 1. (10 marks) SIMULINK: Use two integer delay blocks, gain blocks, a constant block, and feedback loops to build the system model in equation (2) in Simulink. Run the model for 7 days to find the number of newly infected people on day n = 7, i.e., y[7]. 2. (a) (2 marks) Find y[2] and y[3]. (b) (5 marks) Use z-transforms in (2) to obtain an expression for Y (z)/z. Use the z-pair anu[n]←→ z z − a and the Left Time Shift theorem: if x[n]←→ X(z), and q > 0, then x[n+ q]←→ zqX(z)− x[0]zq − x[1]zq−1 − · · · − x[q − 1]z 3. (2 marks) MATLAB: Use the command residue to obtain a partial fraction expansion for Y (z)/z. 4. (5 marks) Use the partial fraction expansion you obtained from MATLAB to find an analytic expression for the system response y in the time domain. You may use the z-transform pair anu[n]←→ z z − a Hence find the value of y[7] and compare with your answer from Simulink. 5. (8 marks) SIMULINK: Now run the model while implementing the following government actions: • On day n1 = 9 (starting from 0:00 AM), the government recommends social distancing, so that α = 0.5. • On day n2 = 19, the government imposes lockdown, closing universities and schools, so that α = 0.2. • On day n3 = 27, a larger percentage of the international arrivals carry the virus, namely 10%, resulting in β = 500. • On day n4 = 38 the government closes the borders, so that β = 0 for n ≥ n4. (a) (2 marks) Plot the output y[n] as a function of time n. Comment on its shape—- where can you see a flattening of the curve? (b) (2 marks) Let n5 > n4 be the smallest integer such that on day n5 the number of newly infected people is less than 10. What is the value of n5? (c) (2 marks) Suppose that the government eases some of the restrictions on day n5 + 1 so that α = 0.6. Plot the output y[n] as a function of time n for n4 ≤ n ≤ n5 + 20. (d) (2 marks) For which values of α ∈ R+ will there be a flattening of the curve for n > n5? 6. (2 marks) Suppose that people who got the new variant of the coronavirus are infectious for 4 days rather than 2 days. How should the system model of equation (1) be changed? 2 2 COVID-19 model: zero input response via DT state representation Consider again the infectious disease model of the previous section, given by equation (2). Again assume that α = 3 and β = 5. 7. (5 marks) Introduce state variables for the system given by equation (2) and write down the state equations in controller canonical form. 8. (5 marks) Introduce a state vector x and suitable matrices (A,B,C,D) to write down a state representation for the system. 9. (5 marks) Give an expression for the zero-input response yzi[n] in terms of A, C and the state-transition matrix of the system, assuming initial conditions y[0] = 0 and y[1] = 1 and input v[n] = 0. 10. (4 marks) MATLAB Use stem to plot yzi[n] for 10 days. What is the value of yzi[7]? 11. (2 marks) SIMULINK: Simulate the system response and show the result yzi[7] displayed in SIMULINK, i.e., in scope, to verify your answer for Question 10. 12. (5 marks) MATLAB Use dlsim to plot yzi[n] of the system for 10 days. Compare the value of yzi[7] with your answer to Question 10. What is the system response as n goes to infinity? 3 COVID-19 model: step response via DT state represen- tation Consider again the infectious disease model of the previous section, given by equation (2). Again assume that α = 3 and β = 5. 13. (5 marks) Return to the controller canonical form state representation (A,B,C) that you obtained in the previous section. Without MATLAB, calculate the eigenvalues and eigen- vectors of the matrix A. 14. (5 marks) MATLAB Use [V,L] = eig(A) to find eigenvalues and eigenvectors of A. If the eigenvectors you have calculated are different from MATLAB, explain the reason. 15. (5 marks) MATLAB Obtain a coordinate transformation x¯ = Px that will give an equiva- lent diagonal state representation (A¯, B¯, C¯) for the system representing number of infected people. Use your diagonal state representation (A¯, B¯, C¯) for Questions 16 to 20. 16. (5 marks) Assume that there are initially no cases in Australia and that there is a constant influx of infected international arrivals. This means v[n] = u[n], and y[0] = y[1] = 0. Use your equivalent diagonal state representation (A¯, B¯, C¯) to write down a summation formula for yzs[n], for n ≥ 1, the step response with zero initial conditions, in terms of A¯, B¯, and C¯. (Note: You are not required to simplify this summation). 3 17. (5 marks) MATLAB Use symsum to symbolically evaluate the summation in Question 16. You will need to introduce symbolic variables. Simplify your answer using vpa(ans,3). Using the formula, calculate yzs[7]. 18. (5 marks) MATLAB Use dlsim to plot the system response to a step input with these initial conditions. What is yzs[7]? Compare your answer obtained via simulation with the theory (Question 17) 19. (5 marks) Now as in Part 1, suppose there is already one infected case in Australia, which means y[0] = 0 and y[1] = 1 for equation (2). Find x0 and x¯0, and then write down a summation formula for the system overall response y[n], for n ≥ 1, with a step input and these initial conditions, in terms of A¯, B¯, C¯ and x¯0. Evaluate y[7] using the obtained formula. How does your answer relate to the yzs[7] and yzi[7]? Explain why. 20. (5 marks) MATLAB Use dlsim to plot the system overall response y[n] for 10 days. What is y[7]? Compare your answer with the answer obtained in the Question 19. 4
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