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HW9_Solns

# HW9_Solns - Wโ โ1 K W9 9 6>:52 Hโ<> 3โ K/9...

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Unformatted text preview: \ Wโ โ1 K W9 9 ( 6:: > :52 : Hโ(<>) 3โ K /9 +~j~~mem M We.ยป (mag F/laiim 11.22. The closed-loop system function is given by __ H(s) Qโ) โ 1 + G(s)H(s)' (a) Rom the given H(s) and 0(5), we obtain 1 1 Q(5)=W=(s+2)2' Taking the inverse Laplace transform, we get q(t) = teโ2tu(t). (b) From the given H(s) and G(s), we obtain 3+1 5+1 Q(S)=W:(s+2)2' The paxtial fraction expansion of Q(s) is 1 1 Q(s)=s+2~(s+2)2' Taking the inverse Laplace transform, we get q(t) = eโzt'u.(t) โ teโ2tu(t). (c) From the given H(s) and 0(3), we obtain 1/2 ~53 a a = โโโ-' I: a at โ;โ 31ft, ms) 1+ (1 MW ,3 41:33 )K Taking the inverse Laplace transform, we get \$Qr โ43 W) = l 5, 6(t โ k/3). {the} 41:23 ๏ฌ๏ฌfโ " L-n I? Imag Axis I _. Sf Imag Axis @ Imag Axis I O 01 | N I 01 I 01 O 01 -2 โ1 0 1 2 L O _. N ReaI Axis Real Axis I .s O _.. Real Axis Real Axis K>O Imag Axis L o โA |\) . i'u A I N Real Axis Imag Axis O -1 0 Real Axis 2 .2 1 รฉ o: 0 โ E - -1 -2 -3 -2 โ1 0 Real Axis Figure \$11.24 2 2 5 1 0 .5 .m .. .m x O 3 M M A 7 a 0| 0| K e a a R .H H 2 _ a o 1 _ J. 210444420444210444504 m_x< mmE_ m_x< mmE. m_x< mmE. m_x< mmE. 2 2 K <0 ' 0 Real Axis 0 Real Axis 0 Real AXIS I โ1 โ1 cw 2 1 0 4 9__9__ 9... w_x< mmE. w_x< mmE. m_x< mmE. m_x< mac: ) \l e \I ) {L .U tau ๏ฌux Figure \$11.24 -' 5+2. _ 6โ+2.ยข.1โโFโ 44โ2.. โ- 11โ;4+4-โ+=โi1ยข<ยข2-= )โ , 9+2. 62' + Cz+1โฌ)9+ MEL) ...
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HW9_Solns - Wโ โ1 K W9 9 6>:52 Hโ<> 3โ K/9...

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