04 09 2024 E TS Mida II

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1 MODEL IDENTIFICATION AND DATA ANALYSIS 2nd module (MIDA2) Academic Year 2023/2024 – 04/09/2024 Surname Name Matr. Number Signature ................................ ............................... .....................………. .....................………. It is not allowed to consult books, notes, lecture notes etc. =============================================================================================================== 1a) Consider the system whose input/output behavior is described by: 𝑦(𝑡) = 𝑧−2(1 − 3 2⁄ 𝑧−1) 1 − 2𝑧−1 + 3 4⁄ 𝑧−2 𝑢(𝑡) answer the following questions: a) Is the system asymptotically stable? Is it strictly proper? b) Identify the system matrices from the transfer function. c) Provide the analytical expression 𝜔(𝑡) as a function of time for the system. a) 𝑊(𝑧) = 𝑧−2(1−3 2⁄ 𝑧−1) 1−2𝑧−1+3 4⁄ 𝑧−2 = 𝑧−3 2⁄ 𝑧3−2𝑧2+3 4⁄ 𝑧 = (𝑧−3 2⁄ ) 𝑧(𝑧−3 2⁄ )(𝑧−1 2⁄ ) Since there is a pole in 𝑧𝑝 = 3/2, the system is unstable. The order of the denominator is strictly greater than the order of the numerator, so the system is strictly proper. b) The system is of order 𝑛 = 3. Using the controllable canonical form: 𝑊(𝑧) = 0𝑧2 + 𝑧 − 3 2⁄ 𝑧3 − 2𝑧2 + 3 4⁄ 𝑧 + 0 𝐻̂ = [− 3 2⁄ 1 0] 𝐺̂ = [ 0 0 1 ] 𝐹̂ = [ 0 1 0 0 0 1 0 − 3 4⁄ 2 ] 𝐷̂ = 0 c) We will use the geometric series formula. We will cancel out the unstable pole since we are only looking at the IO response of the system: 𝑊(𝑧) = (𝑧 − 3 2⁄ ) 𝑧(𝑧 − 3 2⁄ )(𝑧 − 1 2⁄ ) = 𝑧−2 1 1 − 1 2 𝑧−1 = 𝑧−2 ∑ (1 2 𝑧−1) 𝑡+∞ 𝑡=0 = ∑ (1 2) 𝑡 𝑧−(𝑡+2) +∞ 𝑡=0 = = ∑ (1 2) 𝑘−2+∞ 𝑘=2 𝑧−𝑘 𝜔(𝑡) = { 0 for 𝑡 < 2 (1 2) 𝑡−2 for 𝑡 ≥ 2 Accuracy, clarity and organization of the answers will be evaluated 2 1b) Given the following impulse response: 𝜔(𝑡) = { 0 𝑖𝑓 𝑡 𝑜𝑑𝑑 (− 1 3) 𝑡 2+1 𝑖𝑓 𝑡 𝑒𝑣𝑒𝑛 Answer the following questions: a) Compute the first 5 impulse response…