25 06 2024 2I TS Mida II

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1 MODEL IDENTIFICATION AND DATA ANALYSIS 2nd module (MIDA2) Academic Year 2023/2024 – 25/06/2024 Surname Name Matr. Number Signature ................................ ............................... .....................………. .....................………. It is not allowed to consult books, notes, lecture notes etc. =============================================================================================================== 1a) Given the following impulse response: 𝜔(𝑡)= { 0 𝑡 < 𝑡0 1 3 (−3)2+𝑡 𝑡 ≥ 𝑡0 where 𝑡0 ≥ 0 Answer the following questions: a) Is the system asymptotically stable? For which values of 𝑡0 is it strictly proper? b) Set 𝑡0 = 2. Compute the transfer function associated with the system. What is the system order? a) For 𝑡 → ∞ |𝜔(𝑡)| → ∞, hence the system is NOT asymptotically stable (it is unstable). If 𝑡0 = 0 𝜔(0)= 3 → system is just proper If 𝑡0 > 0 𝜔(0)= 0 → system is strictly proper b) The transfer function is: 𝑊(𝑧)= ∑ 𝜔(𝑡)𝑧−𝑡 ∞ 𝑡=0 = ∑ 1 3 (−3)2+𝑡𝑧−𝑡 ∞ 𝑡=2 = 3 ∑(−3)𝑡𝑧−𝑡 ∞ 𝑡=2 = 3 ∑(−3)𝑘+2𝑧−𝑘−2 ∞ 𝑘=0 = 27𝑧−2 ∑(−3)𝑘𝑧−𝑘 ∞ 𝑘=0 = = 27𝑧−2 ∑(−3𝑧−1)𝑘 ∞ 𝑘=0 = 27𝑧−2 1 1 + 3𝑧−1 = 27 𝑧2 + 3𝑧 The system is of order 𝑛 = 2. 1b) In MATLAB, you are given a generic extended observability matrix 𝑂4 and a generic extended reachability matrix 𝑅4 for a system of order 𝑛 = 3 with 𝜔(0)= 3. Write the code to find the system matrices F, G, H and D. % O4 and R4 are provided F = inv(O4(1:3, :))*O4(2:4, :); G = R4(:, 1); H = O4(1, :); D = 3; Accuracy, clarity and organization of the answers will be evaluated 2 1c) Given the following impulse response coefficients: 𝜔(0)= 0 𝜔(1)= − 1 4 𝜔(2)= 0 𝜔(3)= 1 𝜔(4)= 0 𝜔(5)= −4 𝜔(6)= 0 Answer the following questions: a) Identify the system matrices using the 4SID method. What is the system order? b) Compute the eigenvalues from the identified…