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Unformatted text preview: a region of gain? order to make a secondorder approximation.
or a negativefeedback system? transient response. . . iPROBLEMS 1. For each of the root=loci shown in Figure P81, tell
whether or not the 'sketch can be a root lpcus. If the
sketch cannot be a root locus, explain why. Give all
reasons. [Section: 8.4] 1'01 if!)
srplane sAplane
5 3
' (a) (b)
ja)
s—plane
h ‘ 1 a
(4)
1'91 1'60
splane s—plane
5 6 '
(9) (f)
fa: if”
s—plane . s—plane
6 , I 5
(g) (it) Figure P8.1 417 Problems 9. How can you tell from the root locus that the natural frequency does not change over It}. How would you determine whether or not a root locus plot crossed the real axis?
11. Describe the conditions that must exist for all closedloop poles and zeros in 12. What rules for plotting the root locus are the same whether the system is a positive 1_3. Brieﬂy describe how the zeros of the open—loop system affect the root locus and the 2. Sketch the general shape of the root locus
for each of the openloop polezero
plots shown in Figure P82. [Section: 8.4] if” jtu
splane s—plane
a a
(a) (b)
1'60 jw
splane splane
U. 0' a
(e) (d)
1'01 jw
splane splane
a i a
(e) (D Figure P8.2 418' Chapter 8 Root Locus Techniques 3. Sketch the root locus for the unity feedback system
shown in Figure PS3 for the following transfer func
tions: [Section: 8.4] Figure P83
Figure P8.4
K (s + 2)(S + 6) '
a. GS) = _—S2 + 83 + 25 7. Sketch the root locus of the unity feedback system
K ( 2 + 4) shown .in Figure,P8.3, where
S
2 " (s+5)(5+6)
c. G 5 = M and ﬁnd the breakin and breakaway points.
() 2
3 [Section: 8.5]
K
d Gm = ‘—3—7 8. The characteristic polynomial of a feedback control
(3+1) (5+ ) system, which is the denominator of the closed
4_ Let loop transfer function, is given by 33 + 252 +
(20K + 7): + 100K. Sketch the root locus for this
K(5 + 3) system. [Section: 8.8]
G0") L" m 9. Figure P85 shows open~loop poles and zeros. There are two possibilities for the sketch of the root locus.
. . . ‘ Sketch each of the two possibilities. Be aware that
m Figure P83 [Section 8'5] only one can be the real locus for speciﬁc openloop a. Plot the root locus. pole and zero values. [Section: 8.4] b. Write an expression for the closedloop transfer
function at the point where the three closedloop
poles meet. jt’D 5. Let HKo+1f 0“) = m with K > 0 in Figure P853). [Section: 8.5] a. Find the range of K for closedloop stability.
b. Sketch the system’s root locus. Figure P85 c. Find the position of the closed—loop poles when
K = l and K = 2. 10. Plot the root locus for the unity feedback system shown in Figure P83, where
6. For the openloop polezero plot shown in Figure 2
P8.4, sketch the root locus and ﬁnd the break—in point. (KS) : W
[Section: 8.5] (5 "l" 3X3 — 3) For what range of K will the poles be in the right half
plane? [Section: 8.5] 11. For the unity feedback system shown in WW1“?
Figure P83, where K(s2 — 9)
G“) = W sketch the root locus and tell for what values of K the
system is stable and unstable. [Section: 8.5] 12. Sketch .the root locus for the unity feedback system
shown in Figure P83, where Ids? + 2) G“) = (we???) Give the values for all critical points of interest. Is the
system ever unstable? If so, for what range of K?
[Section: 8.5] 13. For each system shown in Figure P86, make an
accurate plot of the root locus and ﬁnd the following:
[Section: 8.5] a. The breakaway and break—in points
b. The range of K to keep the system stable c. The value of K that yields a stable system with
critically damped secondorder poles d. The value of K that yields a stable system with a
pair of secondorder poles that have a damping
ratio of 0.707 KG + 2)(s + l) (s — 2X5 — 1) System 1 K(s+2)(5+l)
(2—25+2) S 03 System 2
Figure P8.6 14. Sketch. the root locus and ﬁnd the range of K for
stability for the unity feedback system shown in s; 419 Problems Figure P8.3 for the following conditions: [Section:
8.5] _ [{(s2 + 1)
3. 0(3) (s _ ”(s + 2)(s + 3)
b. C(S) _ K(52 —23 + 2) 15‘. For the unity feedback system of Figure
P83, where ' _ K(s+3)
_ (s2 + 2)(s — 2)(s + 5) 0(5) sketch the root locus and ﬁnd the range of K such that
there' will be only two righthalf—plane poles for the
closedloop system. [Section: 8.5] 16. For the unity feedback system of Figure P83, where K 0(5) = s(s+ 6)(s + 9) plot the root locus and calibrate your plot for gain. Find
all the critical points, such as breakaways, asymptotes,
jagaxis crossing, and so forth. [Section: 8.5] 17. Given the unity feedback system of Figure P83,
make an accurate plot of the root locus for the following: _ K(32 — 2s + 2)
3. 6(3) _ m
b. Gm K(s —1)(s —— 2) = (s+1)(s+2) Calibrate the gain for at least four points for each
case. Also ﬁnd the breakaway points, the jmaxis
crossing, and the range of gain for stability for
each case. Find the angles of arrival for Part a.
[Section: 8.5] 18. Given the root locus shown in Figure P87, [Section: 8.5] a. Find the value of gain that will make the system
marginally stable. b. Find the value of gain for which the closedloop
transfer function will have a pole on the real axis
at 75. 420 Figure P8.7 19. Given the unity feedback system of Figure P83,
where K(s+ l)
s(s+2)(s+3)(s+4) do the following: [Section: 8.5] Gfs) = a. Sketch the root locus.
b. Find the asymptotes. c. Find the value of gain that will make the system
marginally stable. d. Find the value of gain for which the closedloop transfer function will have a pole on the real axis at
——0.5. 20. For the unity feedback system of Figure \lllleyplUS
P83, where  . " K (s + a!)
s(s + 3)(s + 6)
ﬁnd the values of o: and K that will yield a second— order closedloop pair of poles at —l :I: j 100.
[Section: 8.5] . 5
0(5) = Q°"trolsiilut'°" 21. For the unity feedback system of Figure P83, where
_ [{(s —1)(s — 2) C(S) s(s + 1) sketch the root locus and ﬁnd the following:
[Section: 8.5] a. The breakaway and breakin points
b. The jcuaxis crossing
c. The range of gain to keep the system stable Chapter 8 Root Locus Techniques d. The value of K to yield a stable system with second
order complex poles, with a damping ratio of 0.5 22. For the unity feedback system shown in Figure P83,
where _ K(s +10)(s + 20)
_ (s + 30)(s2 — 203 + 200) do the following: [Section: 8.7] (3(5) a. Sketch the root locus. b. Find the range of gain, K, that makes the system
stable. c. Find the value of K that yields a damping ratio of
0.707 for the system’s closedloop dominant poles. d. Find the value of K that yields closedloop criti
cally damped dominant poles. 23. For the system of Figure P88 (a), sketch utteyPtUg
the .root locus and ﬁnd the following: ‘
[Section: 8.7] v w)—
(s + l)(s + 2)(s + 3x; + 4) I
(b) Figure P8.8 a. Asymptotes
b. Breakaway points
c. The range of K for stability d. The value of K to yield a 0.7 damping ratio for the
dominant secondorder pair To improve stability, we desire the root locus to cross the
jayaxis at j5.5. To accomplish this, the openloop func
tion is cascaded with a zero, as shown in Figure P8.8(b). e. Find the value of a and sketch the new root locus. f. Repeat Part c for the new locus. ’ 9. Compare the results of Part c and Part f. What
improvement in transient response do you notice? .24. Sketch the root locus for the positiveqfeedback system shown in Figure P8.9. [Section: 8.9] Figure P8.9 25. Root loci are usually plotted for variations in the gain. Sometimes we are interested in the variation of the
closedloop poles as other parameters are changed.
For the system shown in Figure P810, sketch the root
locus as or is varied. [Section: 8.8] Figure 8.10
26. Given the unity feedback system shown in Figure
P83, where
K
0(3) = (5+ l)(s+2)(s+3) do the following problem parts by ﬁrst making a
secondorder approximation. After you are ﬁnished
with all of the parts, justify your secondorder approx
imation. [Section: 8.7] a. Sketch the root locus. r’ b. “Find K for 20% overshoot. c. For K found in Part b, what is the settling time, and
what is the peak time? d. Find the Ideations of higherorder poles for K
"found in Pan [3. e. Find the range of K for stability. 27. For/the unity feedback system shown in Figure P83, where Ids? 125 + 2) GQF=Q+2X&+®@+5M5+® do the following: [Section: 8.7] a. Sketch the root locus.
b. Findthe asymptotes. c. Find the range of gain, K, that makes‘the system
stable. 28. 29. 30. 31. 421 Problems d. Find the breakaway‘points. e. Find the value of K that yields a closedloop step
response with 25% overshoot. f. Find the location of higherorder closedloop poles
when the system. is operating with 25% overshoot. 9. Discuss the validity of your secondorder approx
imation. h. Use MATLAB to obtain the closed
Ioop step response to validate or
refute your secondorder
approximation. NthTLAR The unin feedback system shown in Figure 8.3, where Kn+mn+n
G“) = W— is to be designed for minimum damping ratio. Find the
following: [Section: 8.7] a. The value of K that will yield minimum damping
ratio b. The estimated percent overshoot for that case c. The estimated settling time and peak time for that
case d. The justiﬁcation of a secondorder approximation
(discuss) e. The expected steadystate error for a unit ramp
input for the case of minimum damping ratio For the unity feedback system shown in Figure P83,
where [((s + 2)
5(5 + 6) (s +10)
.ﬁnd K to yield closedloop complex poles with a
damping ratio of 0.55. Does your solution require a
justiﬁcation of a secondorder approximation? Explain.
::[Section: 8.7] For the unity feedback system shown in WWW?
Figure P8.3, where ‘ . G(s) G(s) = __ K(s + a)
_ 5(5 +1)(s +10)
tindt the value of or so that the system will have a settling tirrieofA seconds for large values of K. Sketch
the resulting' root locus. [Section: 8.8] For the unity feedback system, shown in Figure 8.3,
where K(s+6) Gts) — W 422 32. 33. 34.. 35. design K and a so that the dominant complex poles of
the closedloop function have a damping ratio of 0.45
and a natural frequency of 9/8 radfs. For‘the unity feedback system shown in Figure 8.3,
where K 0(5) = so +5)“ + 4)(s + 8) do the following: [Section: 8.7] a. Sketch the root locus.
b. Find the value of K that will yield a 10% overshoot. c. Locate all nondominant poles. What can you say
about the secondorder approximation that led to
your answer in Part 13‘? d. Find the range of K that yields a stable system. mATLAa Repeat Problem 32 using MATLAB.
Use one program to do the following: a. Display a root locus and pause. b. Draw a close—up ofthe root locus where the axes
go from —2 to O on the real axis and —2 to 2 on
the imaginary axrs. c. Overlay the 10% overshoot line on the closeup
root locus. d. Select interactively the point where the root
locus crosses the 10% overshoot line, and
respond with the gain at that point as well as
all of the.closedloop poles at that gain. e. Generate the step response at the gain for 10%
overshoot. For the unity feedback system shown in WWII!
Figure 8.3., where " _ K(sz+4s+5) .
701+m+SXs+ao+4) do“the following: [Section: 8.7] 0(5) a. Find the gain, K, to yield a 1second peak time if
one assumes a secondorder approximation. b. Check the accuracy of the second— onus
order approximationusing
MATLAB to simulate the system. For the unity feedback system shown in Figure P83,
where _1__ K(s+2)(s+3)
G“) "(s2 +2s+2J<s+—4' )(s+5)(s+6) Chapter 8 Root Locus Techniques 36. 37. 38. do the following: [Section: 8.7] a. Sketch the root locus. b. Find the jcoaxis crossing and the gain, K, at the
crossing. C. Find all breakaway and breakin points.
(I. Find angles of departure from the complex poles. e. Find the gain, K, to yield a damping ratio of 0.3 . " for the closedloop dominant poles. Repeat Parts. a through c and e of Froblem 35 for
[Section'. 8.7] K(s+ 8)
s(s +2) (.5 + 4)(s + 6) For the unity feedback system shown in Figure P83,
where Gfs) '2 K
(sr'wi 3)(s2 + 45 + 5) do the following: [Section: 8.7] 0(3) Z a. Find the location of the closed loop dominant polesll
the system is operating with 15% overshoot. b. Find the gain for Part a.
c. Find‘all other closedloop poles. d. Evaluate the accuracy of your secondorder
approximation. For the system shown in Figure P811, do the follow
ing: [Section‘ 8.7] Figure P8.11 a. Sketch the root locus. .b. Find the jagaxis crossing and the gain, K. at the crossing. C. Find the realaxis breakaway to twodecimalplace
accuracy. d. Find angles of arrival to the complex zeros. 9. Find the closedloop zeros. ;. f. Find the gain, K, for a closedloop step response
with 30% overshoot. ‘9. Discuss the validity of your second—order approx
imation. 39. Sketch the root locus for the system of Figure P8.12
and ﬁnd the following: [Section: 8.7] _K—_.
5(5 + 3)(s + 7)(s + 8) (s + 30)
(s2 + 20s + 200) Figure P8.12 _i a. The range ofi gain to yield stability ' b. The value of gain that will yield a damping ratio of
0.707 for the system’s dominant poles i c. The value of gain that will yield closedloop poles
1 that are critically damped
l “ATLAB 40; Repeat Problem 39 using MATLAB. The
program will do the following in one
program: a. Display a root locus and pause. b. Display a closeup of the root locus where the
axes go from —2 to 2 on the real axis and —2 to 2
on the imaginary axis. c. Overlay the 0.707 damping ratio line on th
close—up root locus. ‘ d. Allow you to select interactively the point where l the root locus crosses the 0.707 damping ratio line, and respond by displaying the gain at that point as well as all of the closedloop poles at that gain. The program will then allow you to select interactively the imaginaryaxis crossing and respond with a display of the gain at that point as well as all of the closed—loop poles at that gain. Finally, the program will repeat the evaluation for
critically damped dominant closed—loop poles. e. Generate the step response atthe gain for 0.707
damping ratio. {t_.u.:_:_i:muma .. .. _. ti; .\ Problems 6 423 41. Given the unity feedback system shown «Wilts
in Figure P83, where 9 ﬂ K(s + z) . (
6(3) — s2(s +20) 501M do the following: [Section: 8.7]" a. If .2 = 6, ﬁnd K so that the damped frequency
of oscillation of the transient response is 10 rad/s. b. For the system of Part a, what static error constant
(ﬁnite) can be speciﬁed? What is its value? c. The system is to be redesigned by changing the
values of z and K. If the new speciﬁcations are
%OS = 4.32% and Ts = 0.4 s,ﬁndthe new values
of z and K. 42. Given the unity feedback system shown in Figure
P83, where K
(s +1)(s + 3)(s + 6)2
ﬁnd the following: [Section: 8.7] C(s) : a. The value of gain, K, that will yield a settling time
of 4 seconds 5b. The value of gain, K, that will yield a critically
damped system 43. Let
K(s — l)
(s + 2)(s + 3) in Figure P83. [Section: 8.7]. G(s) = a. Find the range of K for closedloop stability.
b. Plot the root locus for K > 0.
c. Plot the root locus for K < 0. d. Assuming a step input, what value of K will result
in the smallest attainable settling time? e. Calculate the system’s ess for a unit step input
assuming the value of K obtained in Part (I. f. Make an apprOXimate hand sketch of the unit step
response of the system if K has the value obtained
in Part (1. 44. Given the unity feedback system shown in Figure
P83, where am K = s(s+ l)(s+5) 424 Chapter 8 Root Locus Techniques evaiuate the pole sensitivity or the closedloop system 45. Figure P8. 130:) shows a robot equipped to perform if the second—order, underdamped closedloop poles arc welding. A similar device can be conﬁguredasa
are set for [Section: 8.10] sixdegreesoffreedo‘m industrial robot that can I
transfer objects according to a desired program
a' C: 0591 Assume the block diagram of the swing motionf
b E = 0456 system shown in Figure P8.13(b). If K: 64,510, ; .
c. Which of the two previous cases has more desir— make a secondorder approximation and estimate 3 able sensitivity? the following (Hardy, 1967): ‘ Load '
actuator
Input ' Rem
position + __K__ pOSIuon
_ 3‘2 + 75 + 1220
Network Pressure Position feedback (I?) Figure P8.13 a. Robot equipped to perform arc welding; I). block diagram for swing motion system a. Damping ratio b. Percent overshoot
c. Natural frequency
d. Settling time a. Peak time What can you say about your original secondorder
approximation? i 46. During ascent the automatic steering program aboard
the space shuttle provides the interface between the
lowrate processing of guidance (commands) and the
highrate processing of ﬂight control (steering in
response to the commands). The function performed
is basically that of smoothing. A simpliﬁed represen
tation of a maneuver smoother linearized for coplanar
maneuvers is shown in Figure P8.14. Here 6CB(S) is
the commanded body angle as calculated by gui
dance, and 633(5) is the desired body angle sent to
ﬂight control after smoothing.3 Using the methods of
Section 8.8, do the following: ‘ a. Sketch a root locus where the roots vary as a
function of K3. b. Locate the closedloop zeros. c. Repeat Parts 3 and b for a root locus sketched as a
function of K2. Figure P8.14 Block diagram of smoother 47. Repeat Problem 3 but sketch your root loci for
negative values of K. [Section: 8.9] 48. Large structures in space, such as the space station,
have to be stabilized against unwanted vibration. One
niethod is to use an active vibration absorber to
control‘ the structure, as shown in Figure P8.15(a)
(Brunei; 1992). Assuming that all values except the JSource: Rockwell International. vibration
absorber Output structure
acceleration C (5) Input force / Active vibration absorber (b) Figure P8.15 3. Active vibration absorber (©1992 AIAA);
b. control system block diagram. mass ofthe active vibration absorben areknown and
are equalto unity, do the following: a. Obtain 6(5) and Ho) = H1(s)H2(5) in the block
diagram representation of the system of Figure
8.115(1)), which shows that the active vibration
absorber acts as a feedback element to control
the structure. (Hint: Think of K5 and Dc as pro
ducing inputs to the structure.) b. Find the steadystate position of the structure for a
force disturbance input. c. Sketch the root locus for the system as a function of
active vibration absorber mass, MC. 49. Figure P8.16 shows the block diagram of the closed
loop control of the linearized magnetic levitation Figure P8.16 Linearized magnetic levitation system block
diagram 426 50. 51. system described in Chapter 2, Problem 58. (Galvao,
2003). Assuming/1 = 1300 and n = 860, draw the root locus
and ﬁnd the range of K for closed—loop stability when: 3. 0(3) = K; F)
K(s + 200)
5+ 1000 The simpliﬁed transfer function model from steering
angle 6(5) to tilt angle rp(s) in a bicycle is given by b. 0(5) '= () +V
gas aVS Z
G's):m=ESZﬂ.g. In this model it represents the vertical distance from
the center of mass to the ﬂoor, so it can be ureadily
veriﬁed that the model is openloop unstable. (Astrb‘m,
2005). Assume that for a speciﬁc bicycle, a = 0.6 m,
b = 1.5 m, h = 0.8 m, andg = 9.8 m/sec. In orderto
stabilize the bicycle, it is assumed that the bicycle is
placed in the closedloop conﬁguration shown in
Figure P83 and that the only available control vari
able is V, the rear wheel velocity. a. Find the range of V for closed—loop stability. b. Explain why the methods presented in this chapter
cannot be used to obtain the root locus. . MATLM
c. Use MATLAB to obtain the system's
rootlocps. A technique to control the steering of a vehicle that
follows a line located in the middle of a lane is to
deﬁne a lookahead point and measure vehicle devia
tions with respect to the point. A linearized model for
such a vehicle is b1K
V all 012 —biK T V .
i; _ sz r
11:1 — 6121 (122 52K d “if
Yg 0 1 0 0 Y3. 1 0 U 0 where V = vehicle’s lateral velocity, r = vehicle‘s yaw
velocity, 1,0 = vehicle’s yaw position, and Ya 2 the
yaxis coordinate of the vehicle’s center of gravity. K
is a parame...
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