10 02 2025 E TS Mida II

Pagina 1

1 MODEL IDENTIFICATION AND DATA ANALYSIS 2nd module (MIDA2) Academic Year 2023/2024 – 10/02/2025 Surname Name Matr. Number Signature ................................ ............................... .....................………. .....................………. It is not allowed to consult books, notes, lecture notes etc. =============================================================================================================== 1a) Given the following impulse response coefficients: 𝜔(0) = 1 𝜔(1) = − 1 4 𝜔(2) = − 1 16 𝜔(3) = − 1 64 𝜔(4) = − 1 256 answer the following questions: a) Identify the system matrices using the 4SID method. What is the system order? b) Compute the transfer function from the identified matrices. Is the system strictly proper? Is it asymptotically stable? c) In MATLAB, you are given a generic (noisy) dataset containing the input u_vec and output y_vec timeseries signals of a system of the third order. The sampling frequency of the vectors is 10 Hz. Write the MATLAB code to identify the system using the 4SID method. Display the identified F, G, H, D matrices. a) 4SID method: 1) STEP 1: identify the system order. 𝐻1 = [−1/4] 𝑟𝑎𝑛𝑘(𝐻1) = 1 𝐻2 = [ −1/4 −1/16 −1/16 −1/64] 𝑟𝑎𝑛𝑘(𝐻2) = 1 ⇒ system order 𝑛 = 1 2) STEP 2: factorize 𝐻2 𝑅2 = [−1/4 −1/16] 𝑎𝑛𝑑 𝑂3 = [ 1 1/4] 3) STEP 3: estimate the system matrices 𝐻̂ = 𝑂2(1, ∶) = [1] 𝐺̂ = 𝑅2(: , 1) = [−1/4] 𝐹̂ = 𝑂2(1, : )−1𝑂2(2, ∶) = [1]−1[1/4] = [1/4] 𝐷̂ = 𝜔(0) = 1 b) The transfer function is defined as 𝑊(𝑧) = 𝐻̂(𝑧𝐼 − 𝐹̂) −1 𝐺̂ + 𝐷̂ 𝑊(𝑧) = [1][𝑧 − 1/4]−1[−1/4] + 1 = − 1 4 𝑧 − 1 4 + 1 = 𝑧 − 1 2 𝑧 − 1 4 The system is just proper and it is asymptotically stable (since all poles are inside the unit circle). c) data = iddata(u_vec, y_vec, 1/10); sys = n4sid(data, 3); disp(sys.A); disp(sys.B); disp(sys.C); disp(sys.D);…