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1 MODEL IDENTIFICATION AND DATA ANALYSIS (INF-MAT-ELT), A.Y. 2018/2019 Prof. Sergio M. Savaresi – Prof. Sergio Bittanti – September 2nd, 2019 Surname Name University ID Number Signature ................................ ............................... .....................……… ………… .. ....................……… …………….. =========================================================================================================== - Write the solutions (including procedures and intermediate steps) in the blank areas (use the back of the page, if needed) - Additional papers will not be considered. - Clarity, order and precision are strongly considered for the final evaluation. =========================================================================================================== 1. Consider the stationary process generated as: . Put the model in canonical form and determine the optimal predictor 𝑦𝑦�(𝑡𝑡+ 𝑘𝑘/𝑡𝑡) for k=1, 2, 3, … Draw the diagram of the prediction error variance as a function of the prediction horizon k. We rewrite the process equation as follows 𝑦𝑦(𝑡𝑡) = 1 (−5) 1 − 5𝑧𝑧−1 1 − 1 5 𝑧𝑧−1 �1 + 1 4 𝑧𝑧−1��1 − 1 4 𝑧𝑧−1�(−5)𝑒𝑒(𝑡𝑡) = �1 − 1 16 𝑧𝑧−2�𝑒𝑒̃(𝑡𝑡) where 𝑒𝑒̃(𝑡𝑡) ∼ 𝑊𝑊𝑊𝑊(0,25). The one above is the required canonical form. Using the long division, we compute that 𝑦𝑦�(𝑡𝑡|𝑡𝑡− 1) = 𝑦𝑦�(𝑡𝑡|𝑡𝑡− 2) = − 1 16 1 − 1 16 𝑧𝑧−2 𝑦𝑦(𝑡𝑡− 2) 𝑦𝑦�(𝑡𝑡|𝑡𝑡− 𝜏𝜏) = 0 for 𝜏𝜏≥ 3 Correspondingly, denoting 𝜀𝜀(𝑡𝑡|𝑡𝑡− 𝜏𝜏) = 𝑦𝑦(𝑡𝑡) − 𝑦𝑦�(𝑡𝑡− 𝜏𝜏), we compute that 𝜀𝜀(𝑡𝑡|𝑡𝑡− 1) = 𝜀𝜀(𝑡𝑡|𝑡𝑡− 2) = 𝑒𝑒̃(𝑡𝑡) 𝜀𝜀(𝑡𝑡|𝑡𝑡− 𝜏𝜏) = 𝑦𝑦(𝑡𝑡) for 𝜏𝜏≥ 3 Therefore 𝑣𝑣𝑣𝑣𝑣𝑣�𝜀𝜀(𝑡𝑡|𝑡𝑡− 1)� = 𝑣𝑣𝑣𝑣𝑣𝑣(𝜀𝜀(𝑡𝑡|𝑡𝑡− 2)) = 25 𝑣𝑣𝑣𝑣𝑣𝑣�𝜀𝜀(𝑡𝑡|𝑡𝑡− 𝜏𝜏)� = �1 + � 1 16� 2 � ⋅ 25 for 𝜏𝜏≥ 3 The corresponding plot is 2 2. Assume that data are available, generated according to the system 𝒮𝒮: 𝑦𝑦(𝑡𝑡) = 𝑒𝑒(𝑡𝑡) + 3𝑒𝑒(𝑡𝑡− 1), 𝑒𝑒∼ 𝑊𝑊𝑊𝑊(0,1) For…