Full exam for Model Identification and Data Analysis in the Computer Engineering degree programme at Politecnico di Milano. The document covers: 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
Full exam for Model Identification and Data Analysis in the Computer Engineering degree programme at Politecnico di Milano. The document covers: 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
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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…
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