1 ECE-520: Discrete-Time Control Systems Homework 4 Due: Thursday September 30 in class 1) For this problem, assume 12,,xa e fp x Axb g h and show the following: a) for( ) ,T Tdff x p x pdx b) for ( ) ,TTdff x x p pdx c) for( ) ,dff x Ax Adx d) for ( ) ,T Tdff x A x Adx e) for ( ) , ( )TTTdff x x Ax x A Adx 2) The error vector ebetween observation vector dand the estimate of the input ˆxisˆe d Ax. We want to weight the errors by a symmetric matrixR. Findˆxto minimize ReTe. (This is a weighted least squares.) 3) Assume we expect a process to follow the equation1()ytct d t, and we have measurements: t 1.0 2.0 3.0 4.0 5.0 6.0 ()yt 0.30 0.21 0.14 0.12 0.11 0.09 a) Determine a least squares estimate of the parameters cand d. b) Using your least squares estimate of the parameters, estimate the value of (2.5)y. c) Assume we want to use a weighted least squares with r = diag([ 2 2 2 1 1 1]). Determine the weighted least squares estimate of parameters cand d.2 4) Assume we expect a process to follow the following equation:()xxe Assume we measure ()xat various locationsx: x 0.0 0.1 0.4 2.0 4.0 ()x 2.45 2.38 2.30 1.40 0.70 a) Determine a least squares fit to the parameters and . (Hint: Try logarithms.) b) Estimate (3.0) 5) Assume we have an experimental process we are modeling, and, based on sound physical principles, we believe the relationship between xand yto be ()yxx and we have the following measurements: x 1.0 2.0 3.0 4.0 ()yx 8.0 1.0 0.1 0.1 a) Find a least squares estimate for and. (Hint: You cannot solve for directly. Let log( )w, solve for w and , and then infer .) b) Estimate
View Full Document
Unlocking...