# exam2_sol - Exam#2 EEL 5173(Spring 2005 i Name Bill Lang SS...

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Unformatted text preview: Exam #2 ; EEL 5173 (Spring 2005) i Name: Bill) Lang; SS#: Please show all work for partial credit. (100 points total) Problem 1. (25 points) Consider a mechanical system shown in the ﬁgure: k1 n k2 k1 u2__. 1 Its differential equation is given by d2 t ml (:1; ) +(kl +k2)y1(t)‘k2.V2 =u1 d2 t ' m2 y2()—k2y,(t)+(k,+k2)y2=u2 dt2 Express the system in the state space representation, considering y1(t) and y2(t) to be the output and u1(t) and u2(t) to be the input (note that this is a two-input two-output system). 9W Let x: 2' Wm y; n] y' U1 Y1 / I YA” VL’ [kl 2- \ X‘:X2,XL:—k;’ yll’é/l‘l—i’l—i ' - - KL kn‘kz stx‘t, Xt- m V. - m Vanni . O ' O O 00 >0 K+t< Xl 40 \;._i 0 £ 0 ml u “H “M 00 k0 0 o I W O _l<I‘H<L 0 0J— Problem 2. (25 points) Suppose x is a column vector 4:]— Then with respect to the standard basis (e1,ez,e3) the representation is just x = [7 2 4]T. Suppose we take as a basis the vectors p1 ii- p: {it a i What is the representation of x in the new basis? Gown; X=Q1T, +04% M3 7’5 I kt ‘ 2 :Q, Z +% f +a3[g 4 3 '4 8 37> (JUL-M @2318 Q l / ;:‘7 901 (m NWﬂKu ofy [Hit in: M} ii ,16 \2 /" I Problem 3. (25 points) For the following system 561 0 O 0 xl 562 = l 0 2 x2 3'63 0 1 1 x3 ﬁnd its eigenvalues, eigenvectors, and then diagonalize it. 3 b Show v. 0 0 O A = 1 0 2 0 1 1 Its characteristic polynomial is A 0 0 A(A) = det(AI —- A) = det —1 A —2 0 —-1 A — 1 = A[A(A — 1) — 2] = (A — 2)(A‘+ 1)A Thus A has eigenvalues 2, —1, and 0. The eignevector associated with A = 2 is any nonzero solution of i —2 0 0 1 (A—21)q1= 1 —2 2 ‘q1=0 o 1 —1 Thus q1 = [O 1 l]’ is an eigenvector associated with A = 2. Note that the eigenvector is not unique, [0 a a]’ for any nonzero real (1 can also be chosen as an eigenvector. The eigenvector associated with A = ——1 is any nonzero solution of 1 0 0 (A - (—1)I)q2 = 1 1 2 (12 = 0 0 1 2 which yields q; = [0 -— 2 l]’. Similarly, the eigenvector associated with A = 0 can be computed as q3 = [2 1 —— 1]’. Thus the representation of A with respect to {q1, q2, q3} is 2 o 0 A: 0 —1 0 0 0 0 It is a diagonal matrix with eigenvalues on the diagonal. This matrix can also be obtained by computing A=Q‘1AQ with 0 o 2 Q=[q1q2q3]= 1 —2 1 (3,55,; Problem 4. (25 points) For the following system EH? illilHil“ QOLAQWK Cmdvnllalm‘lan‘ha moth“)? “I F: [e m : [I‘ 4] WM) :{ is it controllable? No’t ' ...
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