# ps5 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Mechanical Engineering 2.160 Identification, Estimation, and Learning Spring 2006 Problem Set No. 5 Out: March 15, 2006 Due: March 22, 2006 Problem 1 Consider a linear stable system, as shown below. Variable u ( t ) is input, y ( t ) is output, and e ( t ) is uncorrelated white noise with zero mean values. Answer the following questions. 1 1 1 + aq ) ( t u ) ( t y + + 1 bq + 1 + dq 1 1 + cq 1 e ( t ) Figure Block diagram a). Obtain a one-step-ahead predictor for the output: y ˆ( t | t 1) . b). Prove that the prediction error y ( t ) y ˆ( t | t 1) is equivalent to the white noise e ( t ) , if the parameters involved in the system are exactly known. Problem 2 Consider the continuous-time, Laguerre series expansion of the following two transfer functions: 1

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G 1 ( s ) = ( s + 1) 2 , G 1 1 2 ( s ) = ( s + 1)( s + 2) a). The Laguerre series expansion is associated with the following transformation of variables: s + a z = s a Change the variable of the above transfer function G 1 ( s ) from s to z , and obtain the new transfer function G 1 ( z ) . Find the poles of G 1 ( z ) , when the parameter a is 1; a = 1 . b). Setting the parameter
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ps5 - Massachusetts Institute of Technology Department of...

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