ECE-520: Discrete-Time Control Systems Homework 2 Due: Tuesday December12 in class 1) For this problem, assume 12,,xaepx Axbg⎡⎤fh⎡⎤⎡== =⎢⎥⎤⎢⎥⎢⎥⎣⎦⎣⎣⎦⎦ and show the following: a) for () ,Tdffxpx pdx== b) for () ,Tdffxxpdx==p c) for () ,TdffxAx Adx== d) for () ,TdffxAxdx==A e) for () , ( )TTdffxxAx AAdx==+x 2) The error vector ebetween observation vector dand the estimate of the input ˆxisˆedAx=−. We want to weight the errors by a symmetric matrixR. Findˆxto minimize ReeT. (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 . db) Using your least squares estimate of the parameters, estimate the value of . (2.5)yc) Suppose we believe the first three measurements are twice a reliable as the last three. Determine a reasonable weighted least squares estimate of parameters and . c d4) Assume we expect a process to follow the following equation: ( )xxeβγε= Assume we measure ()xγat various locationsx: x 0.0 0.1 0.4 2.0 4.0 ()xγ 2.45 2.38 2.30 1.40 0.70 a) Determine a least squares fit to the parameters ε and β. (Hint: Try logarithms.) b) Estimate (3.0)γ 5) Assume we have an experimental process we are modeling, and, based on sound physical principles, we believe the relationship between xand to be y()yxxβα⎛⎞=⎜⎟⎝⎠ 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 α andβ. (Hint: You cannot solve for αdirectly. Let log( )wβα=w, solve for and β, and then infer α.) b) Estimate . (3.5)yPreparation for Lab 2 (to be done individually, No Maple) 6) Consider the following model of the two degree of freedom system we will be using in lab 2. ccmmkkk123121 2F(t)x (t)1x (t)2 a) Draw free body diagrams for each mass and show that the equations of motion can be written as 11 11 1 2 1 2 222 22 2 3 2 21()()mx cx k k x F kxmx cx k k x kx+++ = ++++ =   b) Defining , , , and 11qx=21qx=3qx=2 24qx=, show that we get the following state equations 12 1 21 11112 213 34 4232222201000010001000kk c kqqmmmqqFmqqqqkkkcmmm⎡⎤⎡⎤⎢⎥⎛ ⎞ ⎛⎞ ⎛⎞+⎡⎤ ⎡⎤⎢⎥⎢⎥−−⎛⎞⎜ ⎟ ⎜⎟ ⎜⎟⎢⎥ ⎢⎥⎢⎥⎢⎥⎝ ⎠ ⎝⎠ ⎝⎠⎜⎟⎢⎥ ⎢⎥⎢⎥⎢⎥=+⎝⎠⎢⎥ ⎢⎥⎢⎥⎢⎥⎢⎥ ⎢⎥⎢⎥⎢⎥⎛⎞ ⎛ ⎞⎛⎞+⎣⎦ ⎣⎦⎢⎥⎢⎥−−⎜⎟ ⎜ ⎟⎜⎟⎣⎦⎢⎥⎝⎠ ⎝ ⎠⎝⎠⎣⎦ In order to get theAand Bmatrices for the state variable model, we need to determine all of the quantities in the above matrices. The matrix will be determined by what we want the output of the system to be. C c) If we want the output to be the position of the first cart, what should Cbe? If we want the output to be the position of the second cart what should be? Cd) Now we will rewrite the equations from part (a) as 22111111 21122222222 12122kxxx xmmkxxx xmζω ωζω ω++=+++=  F We will get our initial estimates of 1ζ, 1ω, 2ζ, and 2ω using the log-decrement method (assuming only one cart is free to move at a time). Assuming we have measured these parameters, show how 2,1A, 2,2A, 4,3A, and 4,4Acan be determined. e) By taking the Laplace transforms of the equations from part (d), show that we get the following transfer function 21222222 2211 1 2 2 212()()(2 )(2 )kmmXskFssss smmζω ω ζω ω⎛⎞⎜⎟⎝⎠=+++ +− f) It is more convenient to write this as 2122222()() ( 2 )( 2 )aa a bb bkmmXsFs s s s s2ζωωζωω⎛⎞⎜⎟⎝⎠=++++ By equating powers of in the denominator of the transfer function from part (e) and this expression you should be able to write down four equations. The equations corresponding to the coefficients of , , and do not seem to give us any new information, but they will be used to get consistent estimates of s3s2ss1ζ and 1ω. The equation for the coefficient of will give us a new relationship for 0s2212kmmin terms of the parameters we will be measuring. g) We will actually be fitting the frequency response data to the following transfer function 222222()2211()(1)(abaa bbXs KFsss ssζζωω ωω=1)++++ What is in terms of the parameters of part (f)? 2Kh) Using the transfer function in (f) and the Laplace transform of the second equation in part (d), show that the transfer function between the input and the position of the first cart is given as ()2222 21122212()() ( 2 )( 2 )aa a bb bssXs mFs s s s sζω ω2ζωωζωω++=++++ i) This equation is more convenient to write in the form 22122212222211()2211()(1)(abaa bbKs sXsFsss ssζωωζζωω ωω⎛⎞++⎜⎟⎝⎠=1)++++ What is in terms of the quantities given in part (h)? 1K j) Verify that 2224,1 221kKAmKω== k) Verify that 22 221222,314,1abkAmAωωωω−== l) Verify that222214,11abKBmAωω==. Note that this term contains all of the scaling and unit