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# midterm - FEB-18-2011 08:29 P.001 ECE311 Dynamic Systems...

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Unformatted text preview: FEB-18-2011 08:29 P.001 ECE311 - Dynamic Systems and Control, Midterm Test Feb. 17, 2011, 7:30 pm — 9:00 pm First name: Family name: Student number: Note A one-sided aid sheet is permitted, but no calculator. To get part marks you must ShOW your work. There are 8 problems. Problem Max Mark Your Mark 1 2 2 4 3 4 4 3 5 5 6 5 7 4 8 4 total 31 FEB-18-2011 08:29 P.002 1. [2 marks] Suppose the Laplace transform F (s) of f (t) has poles at 0.01 :l: 103'. Sketch what the graph of f (t) could look like for t Z 0. S‘lnMS’oioiai ’Of‘ pNLVteth l0 s m Phil/Left View/L] mcr term] 2. [4 marks] Deﬁne these terms: an eigenvalue of the square matrix A; an equilibrium of the state equation 3': = f (:r, u). ‘ILILJQhV'Equi 6 complex VIMHJWV 7. fer wiu‘CL AX=7lY for \$0,140 /)(:fg‘ . 9 . . ,. '2, NHL; 6 MiiiiwriW/hi '6 caugiahf )‘oiui 50‘“ Oi] “w of] Werenfrrf rim/mm 77a] “7(0) eacft ] ' who], 3. [4 marks] Consider the following four transfer functions. The input is the signal cos(t) starting at time t = 0. For each of the transfer functions, state if the output is bounded. 2 s s — 1 2 G1(3) = 5— 13 612(3) 2 5+ 1’ 03(3) : 52+25’ G4(8) = 52 + 1 i‘ Mi Lowndﬂi wine 0, My 3 [4an wif]. We 5‘?0 1. Low/14%] ﬁnes G? {s sldfe l marl 1220‘»! Q.) LOW/\(fed Since Yb] Ufa-[S‘h'l Zia/e a reVgezng/ [Iva/Q on 1114 ”“9 WW7 e m. ‘ 4 r m] Lowmqihi 5?th stl flag a Veyéafeol WW 6! S“) . FEB-18-2011 08:29 P.003 1 (8 - 1)2 4. [3 marks] The Laplace transform of te‘ is . What is the inverse Laplace trans— 8 7 (s - 1)” 4 r) | TM answer {8‘ ) {1&4 019m eiIv-e of f2 , form of 5. [5 marks] We studied how to get a state model (A, B, C, D) from a transfer function G'(s). The relationship between the two is C(s) = 0(31 — A)“1B + D. Not every G(s) has a state model. For each of the following transfer functions ﬁnd a state model if it exists. If it does not exist, tell why. 2 —s 233 01(3) = 5+2’ 02(5) 2 e ’ G3(S) = 233 — 1‘ ‘(SWZAz/2353hC12‘036 E :. QQ L L ‘L “0 SNLEJQ MHJQ(\ mof rehma! (May I !/ G ‘ "'1‘”— : I + N ‘ “Shark ? l t 7.8"; “I 33" J7: U i 0 0 AV 0 d 1 , 55 O , O t it 0 FEB-18-2011 08:29 P.004 6. [5 marks] The following ﬁgure shows an electromagnet that provides a magnetic force to a metal cart. Think of the electromagnet as an RL circuit and assume the force on the cart is proportional to the current in the circuit. Derive the matrix A in a state model and mark its eigenvalues on the complex plane. +u_ y rem ~14 0 o. L Ill— 1 e ~-JV§+KIZ1 o O 1 baS L)‘JT4) .9 .E M 1'1 0 FEB-18-2011 08:30 P.005 7. [4 marks] Consider the system modeled by the coupled differential equations 371+yzﬂ1+1=ul .792 — 3/2 = 91112- Take the input vector, state vector, and output vector to be 91 u:|:u1:|, :13: :91 7y=|:y1:|. m 1/2 Then the nonlinear state equation has the form 53 = f(x, u), y = h(x,u). Find the Jacobian matrix of f with respect to x. \‘tltxz :qci': —')(1’¥~5 -" +Ul FEB-18-2011 08:30 P.006 8. [4 marks] Let 1 10 A: —2 —2 O . 0 0 0 Which of these is true as t tends to 00: e converges to 0; eAt converges but not to 0: eAt does not converge? Justify your answer. At V O V "' I! 0 ’l “t V o * 1" 1 3V3: 7. , V: o 1 2 . . r TAM? 00% C .hu/{y c;- [.1 LAL £9 Cl ‘ [Wilt/“Cf S0 0’06? (3,4 u UU‘,‘ [DUI 14h £24.21”me QPWOZCA [I 710 COWLPUJQ [y:_l4)~l 3140/ IDOL 31' Me Paley of!) {gall ﬁfe/meal, TOTAL P . 006 ...
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midterm - FEB-18-2011 08:29 P.001 ECE311 Dynamic Systems...

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