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Unformatted text preview: Assignment #21 — Solutions — p.l
ECSE~2410 Signals & Systems  Fall 2006 Tue 12/5/06 {(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K at which the feedback
systems shown are just on the verge of instability. Sketch the Nyquist and Bode (straight line magnitude
and smooth phase) diagrams. I 1
(a) G8) 3 (3+1)3 S3 +3.92 +3S+l
Nyquist 1310i for general K: Bode plot for general K: I
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I Assignment #21 — Solutions w p.2
ECSE~2410 Signals & Systems  Fall 2006 Tue 12/5/06 M30). Using Nyquist and Bode piots and Bode approximations, ﬁnd gain constant K at which the feedback
systems shown are just on the verge of instabiiity. Sketch the Nyquist and Bode (straight line magnitude and smooth phase) diagrams. I 1
' ' (l G = W
(a) con/[mus (S) (3 +1)3 33 + 352 +35 +1 Assignment #21 ~ Solutions w p.3
ECSE—24EO Signais & Systems  Fall 2006 Tue 12/5/06 W
W 160). Using Nyquist and Bode piots and Bode approxﬁmations, ﬁnd gain constant K at which the feedback
systems Shown are just on the verge of instabiiity. Sketch the Nyquist and Bode (straighi ﬁne magnitude and smooth phase) diagrams.
I 1 (H03 m1": (3) Continued. G(5) = TheNyquisi plot for K =1 is:
nummﬁ]; " '
den= 3 3 I];
nyquist(num,den) Nyquist Diagram n ,7
L} 313:3“ Imaginary Axis o
0‘}
(.J
J;
('3
ix (3
Q :.
{\J C Q
(n
(‘3
(r wee
Real Axis To ﬁnd the frequency at which the Nyquist piot passed through the —1 point: ﬁnd the gain margin. Gm telis
us haw many times we can increase the gain before the plot goes through the 1 point. Executing the code, ﬂemwi: i E; dent‘il 3 3 I}; { gm, pmgweg eve—p] «emerginmumﬁem gives the values:
gm 8.001 1 pm 9130 weg “51.7.3222
we? $0 in this ease, we ear: increase the gain by 8.0011 (i.e., K = 8.001 1) before the system goes unstable.
(Cempare this '10 our approximate gain constant of ( K = 6.96) — not too bad! Assignment #2}. — Soiutions — 13.4
ECSE—2418 Signais & Systems  Fall 2006 Tue 12/5/06 1(3 0). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K at which the feedback
systems shown are just on the verge of instability. Sketch the Nyquist and Bede (straight Iine magnitude
and smooth phase) diagrams. u 1 1
(a) Continued. G(S)=(S+1)3 3m New Eet’s put this exact gain constant (K = 8.0011) into MATLAB to see if the [KG( j (0)1 =1I_W_hen
4G(ja)) m —7I: numﬂSﬂOi I};
denﬂ 1 3 3 E};
margin(num,den)
grid
Bode Diagram
(3m = 3.44e~0{}5 dB (at 1.73 radfsec) , Pm : 0.DO238 deg (at 1.73 rad/sec) Magnitude {68) Phase {deg} m" m' m‘ is Frequency (radlsec) Yes! The gain is just about zero dB when phase is «180 degrees. New above that 1 80 degrees of phase occurs at a phase crossover fi‘equeacy of 1 .73 rad/sec, While our
Bock: approximation gives us an approximate frequency of 1.91 rad/sec. Again, not bad! Assignment #2] — Solutions — 13.5
ECSE~2410 Signals & Systems  Fall 2006 Tue 12/5/06 1(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K at which the feedback
systems shown are just on the verge of instability. Sketch the Nyquist and Bode (straight line magnitude and smooth phase) diagrams.
1 1
t. G 3 m
(a) con mue (S) (S ~+~ 1)3 S3 + 352 + 35 +1 Now check to see if the Nyquist plot passes through the —1 point:
num:[8.0011}; denﬂl 3 3 1]; nyquist(num,den) axis([»1.5 1‘5 w1.5 1.51) Nyquist Diagram imaginary Axis 0 0 Real Axis Yes? Assignment #21 —— Solations — 13.6
ECSE~2410 Signals & Systems ~ Fall 2006 Tue 12/ 5/06 WWW—Wm
WWW 1(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constan’i K at which the feedback
systems shown are just on the verge of instability. Sketch the Nyquist and Bode (straight line magnitude
and smooth phase) diagrams. 1 0 mm
(b) (S) S(S+l)2 1 will not do the MATLAB plots for this problem, just the Bode aphroximations. Nyquist pie: for general K: w ; MW________ Ages} 3' ____l...._._l_____.....»———____—....—___ Assignment #21 —~ Solutions —— p.7
ECSE—2410 Signals & Systems — Fail 2006 Tue 12/ 5/06 WWWWm
WM 1(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K at which the feedback
systems shown are just on the verge of instabiiity. Sketch the Nyquist and Bode (straight line magnitude
and smooth phaSe) diagrams. 6453
3(5 +1) (C) G(S) "w" Let’s do MATLAB plots for this probkem, because it shows how to use time delays. Nyquist plot for generai K: W
£7” 5%,
7;”
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I 6 ._ 93.1; _______...,H,..________.'.... Assignment #21 — Solutions  [3.8
RISE2410 Signals & Systems ~ Fali 2006 Tue 12/5/06 1(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K at which the feedback
systems shown are just on the verge of instability. Sketch the Nyquist and Bode (straight line magnitude
and smooth phase) diagrams. 6753 5(5 +1) The Nyquist piot for K m1 is:
numxﬂ}; denzﬂ 1 O};
Gztfmumﬁenjiobeiay', 5)
nyqaist(num,don)
axis({~i0iO1010]) (c) 6(3) x Nyquist Diagram imaginary Axis Real Axis Assignment #21 »— Solutions — 9.9
ECSE—2410 Signals & Systems  Fail 2006 Tue 12/5/06 1(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K atwhgch the feeﬁback
systems shown are just on the verge of instability, Sketch the Nyquist and Bode (straight line magnitude
and smooth phase) diagrams. eS: 5(5 +1) (C) Continued. 0(5) = To ﬁnd the frequency at which the Nyquist plot passes through the »1 point,1et’s plot the Bode plot and use
the cursor to ﬁnd the frequency at which the phase is ~180 degrees; munxﬂ]; denxﬁ 1 G]; GetﬂnumﬁenjieDelay', 5); welogspacepLO) Bode(G,w) grid Bode Diagram System:G Ffequency (radfsec): 0.262
Magnitude {(18): 11,3 “I  Syetem: G
Frequency (rad/sec): 0262
Phase (deg): ~180 Magnitude (£18) us
x_.? s :; 1;! :1} Frequency {rad/sec) Compare this. exact value of 5on m 0.262 rad/sec to the approximate value found using Bode 7':
appfexiznationsg namely, cow = E = 0.2618. Not bad! A150 note that the exact (ﬁe, the MATLAB) magnitude at cow m 0.262 is 11.3 (113, While the Bode eppmxématien gave us a magnitude of 11.6 dB. Assignment #21 — Soiutions — p.10
BOSE—2410 Signals & Systems  Fall 2006 Tue 12/5/06 mum—WW— 1(30) [33133 NquSt and BOde Plots and 130% approximations, ﬁnd gain constant K at which the feedback
systems Shown are just on the verge of instability. Sketch the Nyquist and Bode (straight line magnitude
anti smooth phase) diagrams. W55 3(5 +1). (0) Continued. 6(5) = Another method of getting the magnitude at the phase crossover frequency:
Hume[l]; deiirtil 1 0}; Gmtf(num,den,‘io[)elay‘, 5); “13.262; {mag phase]:Bode(G,w) 'inagdBsz’tloglOGnag) The alphanumeric output is: mag : 3.6922 (absolute value, not dB)
phase : 43797390 magdB 2 21.3456 = 1 results in, Choosing K so that the QKG(%) Kw 3— 1 50.2708 mag W 36922 Recall the Bode approximation for the gain constant was K x 17:;— m 02613 Assignment #21 — Solutions — p.1 1 I
ECSE~2410 Signals & Systems  FaEl 2006 Tue 12/5/06 1(30). Using Nyquist and Bode plots and Bode approximations, ﬁnd gain constant K at which the feedback
Systems shown are just on the verge of instability. Sketch the Nyquist and Bode (straight line magnitude and smooth phase) diagrams. —5: (0) Continued. G(s) = Sb“). Plotting the Bode plot With K = 0.2708 so the feedback system is on the verge of' instability shows that the
gain is zero (18 (close!) when the phase is —180° : numﬁ2’708}; denr:[1 1 0};
Grtf(num,den,*ioDelay', S);
Wﬁlogspaiee(w i 90)
Bode(G,w) grid Bode Diagram 14\  ,___ System: 6
Frequency (rad/sec): 0.26
Magnitude (dB): 0.0636
l___ System: G _ Frequency (rad/sec}: 0.262
Phase {6629): 480 Magnitude {68} Phase {deg} .2552 Frequency {radisec} Assignment #21 — Solutions — p.12
BOSE2410 Signals & Systems ~ Pal} 2006 Tue 12/5/06 1(30). Using Nyquist and Bode plots and Bode approximations, find gain constant K at which the feedback
systems shown are just on the verge of instability. Sketch the Nyquist and Bode (straight ﬁne magnitude
and smooth phase) diagrams. est s(s+1)' (C) Continued. 0(5) = Plotting the Nyquist plot with K 2 0.2708 so the feedback system is on the verge of instability should
result in a piot passing through the 1 point: We get the same information from the Nyquist plot, where the range of frequencies is restricted to
10"1 g a) $10”: numﬂLZ’fQS}; den:{ 1 1 O};
GztfmumoenjioDelay', S);
wwiogspaeebl ,0) N yquist(G,w) Nyquist Diagram Emaginary Axis “333 he 0,4 :3 5,: {:12 Real Axis Assignment #21 — Solutions m 13.13
ECSE2410 Signals & Systems  FaH 2006 Tue 12/5/06 2(20). Using the feedback block diagram shown
400 X09)
where KG(3) M. [1 +§][1+ 1%][1+E%] 3 (a) Sketch both the Bode straight line magnitude and the smooth phase diagg'ams. (b) Calcuiate the approximate gain and phase margins using Bode approximatgons, (0) Sketch the Nyquist diagram. ((3) Use MATLAB 10 Pic“ the exact BOde diagr 31118, the Nyqnist plot (scale so the diagram around the —1
point is dearly Visible), and using the Margin command calculate the exact gain and phase margins. Y(s) Assignment #21 —— Solutions — p.14
ECSE2410 Signals & Systems — Fall 2006 Tue 12/5/06 2(20). Using the feedback block diagram shown
400 X09 [Hi—ltnatH—sl (a) Sketch both the Bode straight line magnitude and the smooth phase diagrams. (b) Calculate the approximate gain and phase margins using Bode approximations. (C) Sketch the Nyquist diagram. H ' (d) Use MATLAB to plot the exact Bode diagrams, the Nyquist plot (scale so the diagram around the 1
point is dearly Visibie), and using the Margin command calculate the exact gain and phase margins. —————> Y(s)
where KG(s) :2 (b) Gain and phase margins don’t make much sense if the system is unstabiet Sorry! Assignment #21 «— Solutions — p.15
ECSE~2410 Signais & Systems  F311 2006 Tue 12/ 5/06 2(20). Using the feedback block diagram shown
400 Xe)
where KG(S) = (d) Continued. Use MATLAB to plot the exact Bode diagrams, the Nyquist piot (scale so the diagram
around the 1 point is Clearly Visibie), ancf using the Margin command calcaiate the exact gain and phase
margins. Y(s) Note negative signs in Gm and Pm! Makes no sense! gode Diagram
Gm = 40.3 GB (at 33.3 rad/sec) , Pm x 49.6 deg (at 58.3 radlsec) Magnitude {dB} Fhase (deg) Frequency (rad/sec} numeriﬁlﬂeiOOSE;
rootseil ti 0 —100];
denimpeEyUeeis};
margmmumgden) Assignment #21 — Solutions —~ p.1 6 ECSE~2410 Signals & Systems « Fall 2006 Tue 12/5/06 2(20). Using the feedback block diagram shown
, 400 X(s)
where KG(S) m WWW ((1) Continued. Use MATLAB to plot the exact Bode diagrams, the Nyquistpiot (scale so the diagram around the —} point is Greatly Visible), and using the Margin command calculate the exact gain and phase
margins. ' ' Y(S) Nyquist Diagram 4:»
r') émagmary Axis u in 5 ’ ma 1% ~1ij Real Axis x’su:w={4.0c«¥005};
rcoisisiwi w i 0 ~11 (3%};
ricnmpoiyirmis);
My; iliSfﬁRlIﬁngn) Assignment #21 — Solutions — p.17
BCSE—2410 Signals & Systems  Fall 2006 Tue 12/5/06 2(20), Using the feedback block diagram shown
400 X(s)
where KG(s) .— (saw (6) Continued. Use MATLAB to plot the exact Bode diagrams, the Nyquist plot (scale so the diagram
around the —1 point is clearly visible), and using the Margin commané calculate the exact gain and phase
margins, ' Y(s) Getting closer to the «1 point in the Nyquist diagram. The ~1 point is encirc§ed so the system is unstable.
Clicking the cursor on the plot gives the values shown. Nyquist Diagram
System: sys
‘13 Real: 323
:5 lmag: 0000638
g Frequency (rad/sec}: 35.8
§ ................................. , . .r ..................................... u + Real Axis ﬂame";{aﬁeﬁﬁﬁE;
tests1:01 ~ E 0  l 00];
Liﬁﬂ‘3p(§i}’(1“0£ﬂ$);
Nyquistmuquen)
axisQ—S 0.5 —l.5 15]) Assignment #21 — Solutions — p.18
BUSH—2410 Signals & Systems  Fall 2006 Tue 12/5/06 W
3(20). A boat is being steered automaticalfy by the following means: An underwater transmitter on the
boat sends out an acoustical signal to a reference point, the return from which continuonsiy indicates the
actual heading. By comparison with an onboard program of desired heading an error signal is obtained
which causes the rudder to be deflected an amount proportional to instantaneous error. The time lag in the
rudderactuating mechanism is negligible. For a step displacement of the rudder the boat exponentially
approaches a constant rate of turn with a time constant of ﬁve seconds. The velocity of sound in water is
roughly 1 mite/sec. Speciﬁcations demand that the entire steering Eoop have a gain crossover frequency of
at least 2 rad/sec. (Le, the frequency where the Bode magnitude plot crossed the zero dB axis.) Find the maximum distance from the reference point at which the system is just statute. Note. Use Bode plots and Bode approximations. If yo don’t understand nautical terms like heading, look it up on the web. _//ﬂ‘ Tag, ee mid same 5 2a rang” ﬁnes)
T‘ﬂ {r _zgtm Assignment #21 w Solutions  p.19
ECSE—2410 Signais & Systems  Fall 2006 Tue 12/5/06 W
3(20). A boat is being steered automatically by the following means: An underwater transmitter on the
boat sends out an acoustical signai to a reference point, the return from which continuously indicates the
actual heading. By comparison with an onboard program of desired heading an error signal is obtained
which causes the rudder to be deﬂected an amount proportional to instantaneous error. The time lag in the
rudderuactuating mechanism is negligible. For a step displacement of the rudder the boat exponentially
approaches a constant rate of turn with a time constant of five seconds. The velocity of sound in water is
mugth 1 mile/sec. Speciﬁcations demand that the entire steering loop have a gain crossover frequency of
at ieast 2 radfsec. (i.e.,_ the frequency where the Bode magnitude plot crossed the zero dB axis.) Find the maximum distance from the reference point at which the system is gust stable. I Nete. Use Bode plots and Bode approximations. if you don’t understand nautical terms iike heading, look it up on the web. sitii §.<§.~i:§~) Assignment #2} —~ Soiutiens — p.20
ECSE—2410 Signals & Systems — Fali 2006 Tue 12/5/06 3(20), Continued, Specifications demand that the entire steering 1001:) have a gain crossover frequency of
at keast 2 radfsee. (i.e., the frequency where the Bode magnitude plot crossed the zero dB axis.) Find the maximum distance from the reference point at which the system is just Stabie. Note. Use Bode
plots and Bede approximations. If you don’t understand nautical} terms like heading, look it up on the web.  . Zeﬁ
KAQ‘wEWJW a K .
‘5 WO+% as wreak? 9:? m, gﬁﬁf‘dhﬁm ham/a ere .
up!” as?“ saw‘0‘“ “‘5'
52.. Assignment #21 ~ Solutions —— p.21 ECSE24IO Signals & Systems ~ F31} 2006 Tue 1 2/5/06 4(10). Find the z~transf0rm, X(z), 0f the following repeating sequences: Assume x[n]mo, n<0. (a) xa[n]m{9_,1,0,1,0,1,..‘} (b) xb[n]z{;,1,0,1,1,0,;,1,0,...} 6‘1) Eéi%i$ _g°i+%'3+ %“S%H . T: gui+€a Ehl+£g+€“im>
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QM ?c>{£”gw CMWMQW? 5:? &§€g L"% {inﬁrm a 3 Assignment #21 — Solutions —~ 13.22 ECSB~2410 Signals & Systems  Fall 2006 Tue 12/5/06 5(20). Find the closed form analytic expressions for the inverse z—transfonns of the following. Use tables
and properties. Assume x[n] = 0 for n < 0. 1 “i in
(a)Xa<z a“? r: (a?) a 3 mi Kai—a1: 67%) MM j/
(“Tia 1%54) “6%)? ; _H__M__
M3  _ ~ Z : m imam” (Iowan—1M w gm: ..— (I?) vim 2}& X z z g
(C) J) (lmgzﬂlf we“?
n H} w Wa—
W ha KL Cinpg a ) WWW
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mm 9 3 2. gem (a)
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