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Solution Assignment #8 True or False: (1) True (2) True (3) True (4) True (5) False 6.) b %Skill check 6 p=3.25*[1/6 1]; q=[1/24 11/24 1 0]; sys=tf(p,q); [mag,phase,w]=bode(sys); [Gm,Pm,Wcg,Wcp]=margin(mag,phase,w) Script Run Gm = 5.2850e+03 Pm = 54.8907 (Phase margin) Wcg = 262.3778 Wcp = 2.5586 (Cross over frequency) 7.) a p=[1 0.2]; q=[1/8 11/8 19/4 5]; sys=tf(p,q); bode(sys) The gain margin is ∞ so the system is stable
Figure Skill check 7
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Mag
nitu
de (d
B)
10 2 10 1 100 101 102180
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Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
8.) d
! ! =9 !
(!" + 1)( !" ! + 3!" + 9) =9 ![ 9− 4!! − !(12! − !!)]
9− 4!! − !(12! − !!)
At limiting K, imaginary of L(s)=0 => ω=0 or ! = 2 3 rad/s
!"#$ ! ! = −1 =>9 ! 9− 4!! − ! 12! − !!
9− 4!! − ! 12! − !!!!! !
= −1 => − 351 !1521
= −1 => ! = 4.33 9.) a %Skill check 9 p=4.3*9*[0.2 1]; q=[1 4 12 9]; sys=tf(p,q); [mag,phase,w]=bode(sys); [Gm,Pm,Wcg,Wcp]=margin(mag,phase,w) Script run Gm = 5.1077 Pm = 28.1321 Wcg = 7.1834 Wcp = 3.7548 10.) d %Skill check 10 p=[1 1]; q=[4 1 0 0]; sys=tf(p,q); [mag,phase,w]=bode(sys); [Gm,Pm,Wcg,Wcp]=margin(mag,phase,w) Script run Gm = 1.3693e-05 Pm = -35.7368 Wcg = 0.0037 Wcp = 0.6537
The phase margin is negative so the system is unstable 11.) b !!" = 180+ !"#!!
!4 − 180 => ! = 3.3564 !"#/!
20 log ! + 20log ( !" + 4 )+20log( !! )=0 (eq11) using ! = 3.3564 !"#/! in (eq11) we get K=2.15 12.) a Replace !!!.!! !" !!.!!!!
!.!!!! and repeat the procedure of question #11
14.) c p=[-0.3 1]; q=[3/50 1/2 1 0]; sys=tf(p,q); bode(sys) grid on
15.) a %Skill check 15 p=[1 4]; q=[1 6 5 0]; sys=tf(p,q); [mag,phase,w]=bode(sys); [Gm,Pm,Wcg,Wcp]=margin(mag,phase,w)
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Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
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40System: sysFrequency (rad/sec): 1.37Magnitude (dB): 3.05
Mag
nitu
de (d
B)
Gm = 7.2196e+04 Pm = 58.1073 Wcg = 268.7613 Wcp = 0.6678 Word Match (in or, top to bottom): f, e, k, b, j, a, i, d, h, c, g E9.16
The phase approximation is
E9.24 Using the Nyquist criterion, we have P=1 and N=0 which implies Z=N+P=1. Hence the system has root in the right half-plane. E9.25 p=[11.7]; q=[1/200 3/20 1 0]; sys=tf(p,q); bode(sys) grid on p=[11.7]; q=[1/200 3/20 1 0]; sys=tf(p,q); bode(sys) grid on Using the bode plot of the loop transfer function