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Unformatted text preview: FEB182011 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 onesided 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 FEB182011 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 . . ,.
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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
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(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 eiIve 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
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it 0 FEB182011 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.
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M 1'1 0 FEB182011 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 FEB182011 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
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 Spring '11
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