26 07 2024 E TS Mida II

Pagina 1

1 MODEL IDENTIFICATION AND DATA ANALYSIS 2nd module (MIDA2) Academic Year 2023/2024 – 26/07/2024 Surname Name Matr. Number Signature ................................ ............................... .....................………. .....................………. It is not allowed to consult books, notes, lecture notes etc. =============================================================================================================== 1a) A system is subject to the following input: 𝑢𝑢(𝑡𝑡) = �3 𝑖𝑖𝑖𝑖 𝑡𝑡= 2 0 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 The response of the system is: 𝑦𝑦(0) = 0 𝑦𝑦(1) = 0 𝑦𝑦(2) = 6 𝑦𝑦(3) = 0 𝑦𝑦(4) = 3 8 𝑦𝑦(5) = 0 𝑦𝑦(6) = 3 32 𝑦𝑦(7) = 0 a) Identify the system order. b) Identify the system matrices using the 4SID method. c) Provide the MATLAB code to compute the eigenvalues of matrix 𝐹𝐹 found in question (b),then write the system in (b) as a discrete-time state-space MATLAB object. a) We need to use the Hankel matrix. Note that 𝑖𝑖𝑖𝑖𝑖𝑖𝑢𝑢 𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) = 1 3 𝑢𝑢(𝑡𝑡+ 2) ⇒ 𝜔𝜔(𝑡𝑡) = 1 3 𝑦𝑦(𝑡𝑡+ 2) 𝜔𝜔(0) = 2 𝜔𝜔(1) = 0 𝜔𝜔(2) = 1 8 𝜔𝜔(3) = 0 𝜔𝜔(4) = 1 32 𝜔𝜔(5) = 0 We can now proceed with the method: 𝐻𝐻1 = [𝜔𝜔(1)] = 0 ⇒ 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝐻𝐻1) = 0 𝐻𝐻2 = � 0 1/8 1/8 0 � ⇒ 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝐻𝐻2) = 2 𝐻𝐻3 = � 0 1/8 0 1/8 0 1/ 32 0 1/ 32 0 � ⇒ 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝐻𝐻3) = 2 ⇒ system order 𝑟𝑟= 2 b ) We must factorize 𝐻𝐻3 into 𝑅𝑅3 and 𝑂𝑂3: 𝑅𝑅3 = 𝐻𝐻3(1: 2, :) = � 0 1/8 0 1/8 0 1/ 32� 𝑂𝑂3 = �𝐼𝐼2 ⋯� = � 1 0 0 1 1/4 0 � Therefore: 𝐺𝐺� = 𝑅𝑅3(: ,1) = � 0 1/8� 𝐻𝐻� = 𝑂𝑂3(1, : ) = [1 0 ] 𝐹𝐹� = 𝑂𝑂3(1: 2, :)−1 ⋅ 𝑂𝑂3(2: 3, :) = � 0 1 1/4 0 � 𝐷𝐷� = 𝜔𝜔(0) = 2 c) MATLAB code: F = [0 1; 1/4 0]; G = [0; 1/8]; H = [1 0]; D = 2; eig(F) sys = ss(F, G, H, D, -1); Accuracy, clarity and organization of the answers will be evaluated 2 1b) Consider the system whose input/output behavior is described by: 𝑦𝑦(𝑡𝑡) = −3𝑧𝑧−2(1 − 3𝑧𝑧−1) 1 − 7 2� 𝑧𝑧−1 + 3 2� 𝑧𝑧−2 𝑢𝑢(𝑡𝑡) + 2…