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# midsol - Exam#1 EEL 5173(Spring 2005 Name \$114 Em SS Please...

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Unformatted text preview: Exam #1 EEL 5173 (Spring 2005) Name: \$114 Em; SS#: Please show all work for partial credit. (100 points total) Problem 1. (25 points) A mechanical translational system’s differential equation is given by d 2y(t) dy(t) M = ' t — B —— - K t . dtz f() db yo Express the system in the state space representation, considering y(t) to be the output and f(t) to be the input. golht‘m; Define slabs; ‘x,=ym X7. zyl’t) X| :XL _. _ k B J, {-9. ‘ o X : 0 l ,3 VB X+ A ‘4 i (M [74 M Problem 2. (25 points) Suppose x is a'column vector Then with respect to the standard basis (e1,e2,e3,e4) the representation is just x = [1 2 4 —1]T. Suppose we take as a basis the vectors 1 l p] a P2: : P3: , P4 0 0 0 _ 0 0 1 " 0 1 1 _ 1 ’ 1 1 1 1 What is the representation of x in the new basis? Wm; ' x: only. +01sz +01%, Wit 0w= Q3+qwz 2 Q‘1+Q3+Q\t=4 Q‘+0\1T0\3+Q+= v[ :> (11:9, @152, ,~ 05H /‘ Ow: Problem 3. (20 points) We denote a linear vector space consisting of polynomials of degree less than 3 with real coefﬁcients as R2[t]. Let a linear operator T=d/dt: R2[t]—> R2[t] to be the derivative map. Given b, = z‘2 + kt + 1,132 = t2 + 31‘ + 2,b3 = t2 + 22‘ +1 , ﬁnd the matrix representation of the operator in the {bi} basis. APPYB‘M/L l : Sohﬁm: Hm}, \Ndk (Nit ‘L’Le VFW 'Cv Mala/vol £013; Q‘:+Z r ﬁst, egtl Tﬁi : 2f :2 [69,1 €51 [(7).] 0. Tea: [ ‘—'[€“€L€5']l:::l l T€3= 0 46‘ 9&1“ => J91 = 0t! (tlrtﬂ) + altt‘ﬁlﬂ) + a; (9+ 2H!) = (Citraubm) {1+ (Q,+5q1+ZQ3)f +(0I,+2ql+q}) all 2 ‘: : i) T“ +05 t. 3) O” l 21> QIT-{Ei in. ’35} .1. (31+ 501+ZQ3R) Ch :‘(t’ .. Ql‘i'loll-l'qb :0 Ch):@ _ ! e2, ' OIL"? a)! LL+Q;'53 r -( ' \ (1’ T:BTB-f:{»l ~I .( O l \ .. “I f * I % O-LE ATFVOOKOL, 2 : WJD‘: (ZJC +' Z 011 L\+dlj)1+d5b5 : Col +9< “h L ' \ L 130* + (0((1’50111' 10(3).t+(0(|+20(1+0(5) :3 ‘4\+°{Z+°{5:O 00: ‘- oz»+ - :3 0(l My"? M§-l 0(2: ¢ \+ lo<1+ol3:. 01;: o Thy-2H5 = 3, 135+ Bun-+331); :— (_(;|+(lz+ Emil-P (P,+5P;+23;)f +(@,+ Qﬁzwss) :5 P1+EL+P550 r112, (Maw 2/331 :9 G” 3 (5(+2(§2+ ﬁg :3 652-4} Tb; :2icﬁrz = I. BMW/152+ (PM); ? VI: —0 Y2: 2 X32 —7_ Problem 4. (30 points) For the following system [EH—‘2 Eli] , 1). Obtain the state—transition matrix e’". 2). Find eigenvalues and eigenvectors of matrix A. 3). Find a transformation matrix to transform A to a diagonal matrix. 91,qu 1 I). @A‘alemu ‘— ciel (AI-A)=o ‘ :> )xf “l , Azrl. ﬁxbem , 3m=do+dn\ Q , 020mm) ‘ on: 8"“ .2t - LS9 Q ' Ole +1“ (—2) d0” 2€~JC go eA‘t : aol‘f‘d‘A Z lext- @1th O t M (b l & Y (Q l m 3). 2 ,Il ' MM. 6‘3”“ \$0 [—3], [A] A W N X : MG» 0AM aw W 1’42 .5 W 10'” who 938w ...
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midsol - Exam#1 EEL 5173(Spring 2005 Name \$114 Em SS Please...

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