ps4 - Massachusetts Institute of Technology Department of...

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Unformatted text preview: Massachusetts Institute of Technology Department of Mechanical Engineering 2.160 Identification, Estimation, and Learning Spring 2006 Problem Set No. 4 Out: March 8, 2006 Due: March 15, 2006 Problem 1 Consider a transfer operator function: H ( q ) = 1 − 1.1 q − 1 + 0.3 q − 2 Compute H − 1 ( q ) as an explicit infinite expansion: ∞ H − 1 ( q ) = 1 + ∑ g k q − k . k = 1 Namely, obtain the coefficient g k in the above expression. [Hint: Use partial-fraction expansion.] Problem 2 Consider a linear time-invariant, single-output model given by y ( t ) = 1 + bq aq − 1 − 1 u ( t ) + (1 + aq − 1 )( 1 1 + cq − 1 ) e ( t ) where e ( t ) is an uncorrelated random variable, y ( t ) the output, and u ( t ) the deterministic input. Answer the following questions. a). Obtain the one-step-ahead predictor of y ( t ) . Write out predictor y ˆ( t ) as a function of u ( t − 1) , u ( t − 2), " , y ( t − 1), y ( t − 2), " . b). Assuming that parameter c is known: c = 0.5 , rewrite the predictor in linear regression form: y ˆ( t | θ ) = ϕ T ( t ) ⋅ θ + µ ( t ) , where θ is an unknown parameter vector,...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.

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ps4 - Massachusetts Institute of Technology Department of...

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