29 01 2024 E TS Mida II

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1 MODEL IDENTIFICATION AND DATA ANALYSIS 2nd module (MIDA2) Academic Year 2022/2023 – 29/01/2024 Surname Name Matr. Number Signature ................................ ............................... .....................………. .....................………. It is not allowed to consult books, notes, lecture notes etc. =============================================================================================================== 1. Given the following impulse response coefficients: 𝜔𝜔(0) = 0 𝜔𝜔(1) = 3 𝜔𝜔(2) = 0 𝜔𝜔(3) = 1 3 𝜔𝜔(4) = 0 𝜔𝜔(5) = 1 27 𝜔𝜔(6) = 0 answer the following questions: a. Identify the system matrices using the 4SID method. What is the system order? b. Check the observability and reachability of the model identified in (a). c. Compute the transfer function of the model identified in (a). d. Provide the analytical expression of the impulse response 𝜔𝜔(𝑡𝑡) for the identified system. e. What happens to the matrices 𝐹𝐹, 𝐺𝐺, 𝐻𝐻and 𝐷𝐷 identified with 4SID when 𝜔𝜔(0) = 1 ? SOLUTION a) 4SID method: 1) STEP 1: identify the system order. 𝐻𝐻1 = [3] 𝐻𝐻2 = �3 0 0 1/3 � 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝐻𝐻2) = 2 𝐻𝐻3 = � 3 0 1/3 0 1/3 0 1/3 0 1/ 27 � 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝐻𝐻3) = 2 ⇒ system order 𝑟𝑟= 2 2) S TEP 2: factorize 𝐻𝐻3 𝑅𝑅3 = �3 0 1/3 0 1/3 0 � 𝑟𝑟𝑟𝑟𝑎𝑎 𝑂𝑂3 = � 1 0 0 1 1/9 0 � 3) S TEP 3: estimate the system matrices 𝐻𝐻� = 𝑂𝑂3(1, ∶) = [1 0 ] 𝐺𝐺� = 𝑅𝑅3(: , 1) = �3 0� 𝐹𝐹� = 𝑂𝑂 3(1: 2, :)−1𝑂𝑂3(2: 3,∶) = �1 0 0 1 � −1 � 0 1 1/9 0 �= � 0 1 1/9 0 � 𝐷𝐷� = 𝜔𝜔(0) = 0 b) By definition: 𝑅𝑅= [𝐺𝐺� 𝐹𝐹�∙ 𝐺𝐺�] = �3 0 0 1/3 � 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑅𝑅) = 2 ⇒ reachable 𝑂𝑂= � 𝐻𝐻� 𝐻𝐻� ∙ 𝐹𝐹��= �1 0 0 1 � 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑂𝑂) = 2 ⇒ observable Since the model is identified from data, it must be both reachable and observable. c) The transfer function is defined as 𝑊𝑊(𝑧𝑧) = 𝐻𝐻��𝑧𝑧𝑧𝑧− 𝐹𝐹�� −1 𝐺𝐺�+ 𝐷𝐷� Accuracy, clarity and organization of the answers will…