This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Problems 497 4. What kind of compensation improves transient response?
5. What kind of compensation improves both steadystate error and transient response? 6. Cascade compensation to improve the steadystate error is based upon what pole—
zero placement of the compensator? Also, state the reasons for this placement. 7. Cascade compensation to improve the transient response is based upon what pole
zero placement of the compensator? Also, state the reasons for this placement. 8. What difference on the S—plane is noted between using a PD controller or using a
lead network to improve the transient response? .9. In order to speed up a system without changing the percent overshoot, 'where must
the compensated system’s poles on the s—plane be located in comparison to the
uncompensated system’s poles? 10. Why is there more improvement in steadystate error if a PI controller is used
instead of a lag network? 11: When compensating for steadystate error, what effect is sometimes noted in the
transient response? . 12. A lag compensator with the zero 25 times as far from the imaginary axis as the
compensator pole will yield approximately how much improvement in steadystate
error? 13. If the zero of a feedback compensator is at —3 and a closedloop system pole is at
3.001, can you say there will be polezero cancellation? Why? 14. Name two advantages of feedback compensation. um PROBLEMS 1. Design a PI controller to drive the step \tiiieYPlUS
response error to zero for the unity feed
back system shown in Figure P9.1“, where . 5
K (Emmi suutm" b. Use MATLAB to simulate your design
for K = 1. Show both the input
ramp and the output response on
the same plot. G(s) = ————.—
(S + ”(.5 + 3) (S + .10) . 3‘. The unity feedback system shown in Figure P9.1 with
The system operates With a damping ratio of 0.5.
Compare the speciﬁcations of the uncompensated and GU) = K
compensated systems. [Section: 9.2] (s + l)(s + 3)(s + 5) is operating with 10% overshoot. [Section: 9.2] a. What is thewvalue of the appropriate static error constant?
Figure P9.1 b. Find the transfer function of a lag network so that
the appropriate static error constant equals 4 with
2. Consider the unity feedback system shown in out appreciably changing the dominant P0168 0f
Figure P9.1, where theuntmmpensated system.
K c. Use MATLAB or any other computer “ATM“
G(s) = ——m program to simulate the system to
5“ + 2)“ ‘l' 5) see the effect of your compensator. a. Design a :PI controller to drive the ramp response
error to zero for any K that yields stability. 4. Repeat Problem 3 for G(s) = ————.
[Section: 9.2] [Section: 9.2] 5“ + 3X5 + 5) K 498 5. Consider the unity feedback system shown in Figure
P9.1 with G(s) = K (S + 2)(s + 4)(s + 6) a. Design a compensator that will yield KP = 20
without appreciably changing the dominant pole
location that yields a 10% overshoot for the
uncompensated system. [Section: 9.2] b. Use MATLAB or any other computer
program to simulate the uncom
pensated and compensated systems. c. Use MATLAB or any other computer
program to determine how much
time it takes the slow response of
the lag compensator to bring the output
to within 2% of its final compensated value. mATlAs “Ml—As 6. The unity feedback system shown in
Figure P9.1 with K(s+6) Gﬁy=o+2xa+no+s) is operating with a dominant—pole damping ratio of
0.707. Design a PD controller so that the settling time
is reduced by a factor of 2. Compare the transient and”
steadystate performance of the uncompensated: and
“compgrﬂted systems. Describe any problems with
year design. [Section: 9.3], I “ATLAB 7. Redo Problem 6 using MATLAB in the
following way: a. MATLAB will generate the root locus for the
uncompensated system along with the 0.707
damping ratio line. You will interactively select
the operating pdint. MATLAB will then inform
you of the coordinates of the operating point,
the gain at the operating point, as well as the
estimated 0/005, T5, T,,, 1;, can, and K,J repre
sented by a secondorder approximation at
the operating point. b. MATLAB will display the step response of the
uncompensated system. c. Without further input, MATLAB will. calculate
the compensated design point and wiil then ask
you to input a value for the PD compensator zero
from the keyboard. MATLAB will respond with a
plot of the root locus showing the compensated
design point. MATLAB will then allow you to Chapter 9 Design via Root Locus i keep changing the PD compensator value from
the: keyboard 'until a root locus is plotted that
goes through the design point. d. For the compensated system, MATLAB will
inform you of the coordinates of the operating
point, the gain at the operating point, as well as
the estimated %05, T5, Tp, t, (on, and KD repre
sented by a secondorder approximation at the
operating point. e. MATLAB will then display the step response of
the compensated system. 8. Design a PD controller for the system shown in Figure
P92 to reduce the settling time by a factor of 4
while continuing to operate the system with 20%
overshoot. Compare your performance to that ob
tained in Example 9.7. L
s(s + 5)(s + 15) rFigure F92 9. Considerthe unity feedback system shown in Figure .
P9.l with__[Section: 9.3] ' K
Q+W a. Find the location of the dominant poles to yield a
1.6 second settling time and an overshoot of 25%. G(s)'= b. If a compensator with a zero at —I is used to
achieve the conditions of Part a, what must the
angular contribution of the compensator pole be? c. Find the location of the compensator pole. d. Find the gainrequired to meet the requirements
stated in Part a. e. Find the location of other closedloop poles for the
compensated system. f. Discuss the validity of your secondorder approx
imation. 9. Use MATLAB or any other computer
program to simulate the compen
sated system to check your design. 10. The unity feedback system shown in Figure. P9.1 with amt; i1. 12. is to be designed for a settling time of 1.667 seconds
and a 16.3% overshoot. If the compensator zero is
placed at —.1, do the following: [Section: 9.3] a. Find the coordinates of the dominant poles.
b. Find the compensator pole. c. Find the system gain. d. Find the location of all nondorninant poles. e. Estimate the accuracy of your secondorder
approximation. f. Evaluate the steady—state error characteristics. 9. Use MATLAB or any other computer mama
program to simulate the system and
evaluate the actual transient res
ponse characteristics for a step input. Given the unity feedback system of Figure P9. 1 , with
K (5 l 6) 1
(s + 2)(s + 4)(s + 7)(s + 8) do the following: [Section: 9.3] G(s) = a. Sketch the root locus. b. Find the coordinates of the dominant poles for
which i; = 0.8. c. Find the gain for whichc = 0.8. d? If the system is to be cascadecompensated so that
T5 = 1 second and i; = 0.8. ﬁnd the compensator
poleif the compensator zero is at —4.5. e. Discuss the validity of "your second—order approx
iman'on. f. Use MATLAB or any other computer
program to simulate the compen
sated and uricompensatedzsystems
and compare the results to those expected. MATLAB “ATLAS
Redo Problem 1 1 using MATLAB in the
following way: a. MATLAB 'will ' generate the root locus for the
uncompensated system along with the 0.8 damp
ing~ratio line. You will interactively select the oper
ating point. MATLAB will then inform you of the
coordinates of the operating point, the gain at the
operating point, as well as the estimated %05, T5,
Tp, g, a)", ande represented by a secondorder
approximation at the operating point. 13. 499 Problems b. MATLAB will display the step response of the
uncompensated system. c. Without further input, MATLAB will calculate
the compensated design point and will then ask
you to input a value for the lead compensator
pole from the keyboard. MATLAB will respond
with a plot of the root locus showing the com
pensated design point. MATLAB will then allow
you to keep changing the lead compensator pole
value from the keyboard until a root locus is
plotted that goes through the design point. d. For the compensatedssystem, MATLAB will in—
form you of the coordinates of the operating
point, the gain at the operating point, as well as
the estimated %05, T5, Tp, ;, can, and Kp repre
sented by a second—order approximation at the
operating point. e. MATLAB will thendisplayvthe step response of
the compensated system. f. Change the compensator's zero location a few
times and collect data on the compensated sys
tem to see if any other choices of compensator
zero yield advantages over the original design. Consider the unity feedback System of Figure 139.1 with
_ K
— s(s + 20)(s + 40) The system is operating at 20% overshoot. Design a
compensator to decrease the settling time by a factor
of 2 without affecting the percent overshoot and do the
following: [Section: 9.3] G(S) a. Evaluate the uncompensated system’s dominant
poles, gain, and settling time. b. Evaluate the compensated system’s dominant
poles and settling time. c. Evaluate the compensator’s pole and zero. Find the
required gain. d. Use MATLAB or any other computer
program to simulate the compen— sated and uncompensated systems'
step response. ManAs 14. The unity feedback system shown in Figure P9.l with K G(s) : (s +15)(52 l— 63 +13) is operating with 30% overshoot. [Section: 9.3] 500 a. Find the transfer function of a cascade compen
sator, the system gain, and the dominant pole location that will cut the settling time in half if the compensator zero is at —7. b. Find other poles and zeros and discuss your sec
ondorder approximation. c. Use MATLAB or any other computer
program to simulate both the un—
compensated and compensated sys
tems to see the effect of your compensator. “ATLAE 15. For the unity feedback system of Figure P9.1 with
K 5(5 ~l—1)(s2 +10: +26)
do the following: [Section: 9.3] G(s) = a. Find the settling time for the system if it is operat
ing with 15% overshoot. b. Find the zero of a compensator and the gain, K, so
that the settling time is 7 seconds. Assume that the
pole of the compensator is locatedat —15. c. Use MATLAB or any other computer
program to simulate the system's
step response to test the compensator. 16. A unity feedback control system has the
following forward transfer function:
[Section: 9.3] G(s) = K 52(5 + 4)(s +12)
a. Design a lead compensator to yield a closedloop step response with 20.5% overshoot and a settling
time of 3 seconds. Be sure to specify the value of K. b. Is your second—order approximation valid? c. Use MATLAB or any other computer “Am”
program to simulate and compare the
transientresponseofthecompensated
system to the predicted transient response. 17. For the unity feedback,system of Figure P9.1, with _ K
Gts) — mm the damping ratio for the dominant poles is to be 0.4,
and the settling time is to be 0.5 second. [Section: 9.3] a. Find the coordinates of.the dominant poles. b. Find the location of the compensator zero if the
compensator pole is at —15. Chapter 9 Design via Root Locus c. Find the required system gain. d. Compare the performance of the uncompensated
and compensated systems. e. Use MATLAB or any other computer
program to simulate the system to
check your design. Redesign if
necessary. “MlAs 18. Consider the unity feedback system of Figure P9.1,
with
K G“) = m a. Show that the system cannot operate with a settling .
time of 2/3 second and a percent overshoot of 1.5%
with a simple gain adjustment. b. Design a lead compensator so that the system '
meets the transient response characteristicsoi
Part a. Specify the compensator's pole, zero,
and the required gain. 19. Given the unity feedback system of Figure P9.1
with
K 0(3) = W Find the transfer function of a laglead compensator i .
that will yield a settling time 0.5 second shorter than .
that of the uncompensated system, with a damping
ratio of 0.5, and improve the steadystate error bya
factor of 30. The compensator zero is at ;5. Also, ﬁnd
the compensated system’s gain. Justify any second
order approximations or verify the design through
simulation. [Section_: 9.4] 20. Redo Problem 19 using a lag—leradrcom— “TL“
pensator and MATLAB in the following way: a. MATLAB will generate the root locus for the” uncompensated system along with the 0.5,
dampingratio line. You will interactively select
the operating point. MATLAB will then proceed
to inform you of the coordinates of the operat
ing point, the gain at the operating point, aswell
as the estimated %05, T5, Tp, E, can, and KR
represented by a secondorder approximation,
at the operating point. b. MATLAB will display the step response of the
uncompensated system. 21. c. Without further input, MATLAB will calculate
the compensated design point and will then ask
you to.input a.value for the lead compensator
pole from the keyboard. MATLAB will respond
with a plot of the root locus showing the com
pensated design point. MATLAB will then allow
you to keep changing the lead compensator pole
value from the keyboarduntil a root locus is
plotted that goes through the design point. d. For the compensated system, MATLAB will
"inform you of the coordinates of the operating
point, the gain at the operating point, as well as
the estimated %05, T5, TD, (1, run, and'Kp repre
sented by a secondorder approximation at the
operating point. e. MATLAB will then display the step response of
the compensated _System. f. Change the compensator‘s zero location a few
times and collect data on the compensated system
to see if any other choices of the compensator zero
yield advantages over the original design. 9. Using the steadystate error of the uncompen
sated system, add a lag compensator to yield an
improvement of 30 times over the uncOmpen
sated system’s steadystate error, with minimal
effect on the designed transient response. Have
MATLAB plot the step response. Try several
values for the lag compensator's pole and see
the effect on the step response. Given theuncompensated unity feedback system of
Figure P9.1, with _K__
s(s +1)(s —l— 3) do the following: [Section: 9.4] Gfs) = a. Design a compensator to yield the following spe
ciﬁcations: settling time = 2.86 seconds; percent
overshoot = 4.32%; the steadystate error is to be
improved by a factor of 2 over the uncompensated
system. b. Compare the transient and steady—state error spe
ciﬁcations of the uncompensated and compen
sated systems. c. Compare the gains of the uncompensated and
compensated systems. d. Discuss the validity of your secondorder approx
imation. 501E Problems e. Use MATL‘AB or any other computer
program to simulate the uncompen—
sated and compensated systems and
verify the specifications. 22. For the unity feedback system given in WWWUs
Figure P9.1 with '  K 6(3) : s(s+ 5)(s—l— 11) do the following: [Sectiom 9.4] a. Find the gain, K, for the uncompensated system to
operate with 30% overshoot. b. Find the peak time and K, for the uncompensated
system. c. Design a laglead compensator to decrease the
peak time by a factor of 2, decrease the percent
overshoot by a factor of 2, and improve the steady
state error by a factor of 30. Specify all poles, zeros,
and gains. 23. The unity feedback system shown in Figure P9.] with . K
G“) = (52 +4s+ 8)(s+ 10) is to be designed to meet the following speciﬁcations: Overshoot'. Less than 25%
Settling time: Less than 1 second
Kp = 10 Do the following: [Section: 9.4] 3. Evaluate the performance of the uncompensated
system operating at 10% overshoot. b. Design a passive compensator to meet the desired
speciﬁcations. 1:. Use MATLAB to simulate the comp
ensated system. Compare the
response with the desired specifica
tions. “ATLAB 24. Consider the unity feedback system in Figure P9.1,
with _K__ (s + 2) (s + 4) The system is operated with 4.32% overshoot. In
order to improve the steadystate error, KP is to be C(s) = 502 Chapter 9 Design via. Root Locus increased by at least a factor of S.~A lag‘ compensator
of the form (5 +0.5)
(3 + 0.1) is mine used. [Section: 9.4] GAS) = a. Find the gain required for both the compensated
and the uncompensated systems. b. Find the value of K p for both the compensated and
the uncompensated systems. c. Estimate the percent overshoot and settling time
for both the compensated and the uncompensated
systems. d. Discuss the validity of the secondorder approx
imation used for your results in Part c. e. Use MATLAB or any other computer
program to simulate the step
response for the uncompensated
and compensated systems. What do you notice
about the compensated system's response? f. Design a lead compensator that will correct the
objection you notice in Part e. “ATLAs ’2\5.For the unity feedback system in Figure P91, with \/ 02(3) (5+1) @mQA design a P’ID controller that will yield a peak time of
1.047 seconds and a damping ratio of 0.8, with zero 29
error for a step input. [Section: 9.4] 26. For the unity feedback system in Figure P9.l, with
K \{t‘ileyPtUs (s + 4)(3 + 6)(s +10)
do the following: 0(5) = f: ‘ . 5
ontrial smutla" a. Design a controller that will yield no more than
25% overshoot and no more than a 2 second
settling time for a step input and zero steadystate
error for step and ramp inputs. “ATLAS, b. Use MATLAB and verify your design. 27. Redo Problem 26 using MATLAB in the
following way: 28. . For the unity feedback system in Figure P9,] with} 30: a. MATLAB will ask .tor the desired percent over
shoot,.settling time, and PI compensator zero. b. MATLAB will design‘the PD controller's zero. c. MATLAB will display the root locus of the PID .
compensated system with the desired percent :
overshoot line. d. The user will interactively select the intersection
of the root locus and the desired percent over
shoot line. e. MATLAB will display the gain and transient
response characteristics of the PID compen
sated system.  1 f. MATLAB will display the step response of the .
FIBcompensated system. 1 g. “MATIAB will display the ramp reSponse of the f
FIBcompensated system. If the system of Figure P93 operates with a damping _
ratio of 0.517 forthe dominant secondorder poles. ‘
ﬁnd the location of all closedrloop poles and zeros. l (s + 2)
Figure P93 K
s(s+ 2)(s+4)(S+ 6) do the following: [Section: 9.5] GlSJ= a. Design rate feedback to yield a step response with
no more than 15% overshoot and no more than3
seconds settling time. Use Approach 1. b. Use MATLAB and simulate your
compensated system. MATLAs Given the system of Figure P94: :[Section: 9.5] a. Design the value of K1, as well as a in the feedbaclr . :
path of the minor loop, to yield a settling time all :
second with 5% overshoot for the step] response  b.. Design the value of K to yield a majorloop .
response with 10% overshoot for a step input. “AWAR c. Use MATLAB many other computer
program tolrr‘simulate the step
response toTthe'entire closedloop
system. d. Add a PI 'cornpensator to reduce the
majorbop steadystate error to
zerogand simulate the step response
using MATLAB or any other com
puter program. “ATLAB Figure P9.4 31. Identify and realize the following controllers with
operational ampliﬁers. [Section: 9.6] sl—0.01 s
b.s+2 a. 32. Identify and realize the following com—
pensators with passive networks.
[Section: 9.6] a s+0.1
5+0.01 19
5+2 V O
b. Lt/ v
5+5 t”l c s + 0.1 s + 1 ' 5+0.01 s+'10
E33. Repeat Problem 32 using operational ampliﬁers.
[Section: 9.6] Design Problems .34. The room temperature of an 11 m2 room is to be
controlled by varying the power of an indoor radiator.
For this speciﬁc room the openloop transfer function
from radiator power, Q(s), to temperature, T(s), is
(Thomas, 2005) , To) , (1 x 10—6)s2 + (1.314 x10’9)s+{2.66 x 10“) G _ . _—___m
(r) 9(5) s3 + 0.0016332 + (5.272 x 107); + (3.533 x 10~11) 503 Problems The system is assumed to be in. the closedloop
conﬁguration shown in Figure P9.1. a. For a unit step input, calculate the steadystate
error of the system. b. Try using the procedure of Section 9.2 to design a
PT controller to obtain z...
View
Full Document
 Fall '08
 ani

Click to edit the document details