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Unformatted text preview: 206 Chapter 4 Time Response PROBLEMS 1/ 1. Derive the output responses for all parts of Figure 4.7.
[Section: 4.4] 2. Find the output response, C(t), for each of wileyPLt/s
the systems shown in Figure P4.l. Also i
ﬁnd the time constant, rise time, and settling time for each case. [Sections:
4.2, 4.3] C .
antreal Solui'on hi— 5 C (5}
5+5 (:2) 1
3 20 C(s)
S+20
(b)
Figure P4.1 “ATLAE 3. Plot the step responses for Problem 2
using MATLAB. 4. Find the capacitor voltage in the network shown in
Figure P42 if the switch closes at t 2 0. Assume zero
initial conditions. Also ﬁnd the time constant, rise time, and settling time for the capacitor voltage.
[Sections: 4.2, 4.3] Figure P4.2 VS. Plot the step response for Problem 4 name
using MATLAB. From your plots, find
the time constant, rise time, and
settling time. it»??? For the system shown in Figure P43, (a) ﬁnd an E‘Jequation that relates settling time of the velocity of
the mass to M ; (b) ﬁnd an equation that relates rise time
of the velocity of the mass to M. [Sections: 4.2, 4.3] Figure P43 7. Plot the step response for Problem 6 MATLAB
using MATLAB. From your plots find:
the time constant, rise time, and
settling time. gUse M = 1 and M = 2. 8. For each of the transferfunctipgs shown below, ﬁn
the locations of the poles and zeros, plot them on th
s—plane, and then write an expression for the gener:
form of the step responserwithout solving for tk
inverse Laplace transform. State the nature of eac
response (overdamped, underdamped‘, and so on). @ Tm =s'—l22 7 5
“m” = (Ha—n+6?
_ 10(s + 7)
" T(5)“(5+10)(s + 20}
up, a 20
{if 1%) _ s2 + 6s +144
e. T(s) = 52:29
_ (s + 5)
f' Tts) _ (s+10)2 9. Use MATLAB to find the poles of. Mme
[Section: 4.2] 52+2$+2 TS =—+——«——————
“ 54+6s3‘+452+7s+2 10. Find rthe transfer function and poles of the syster
represented in state space here: [Section: 4.10} 3 —4 1 I
X: ——3 2 0 x+ 3 u(t)
5 7 9 7 0
y=[2 8 —3]§; x(0)= 0
0 11. Repeat Problem 10 using MATLAB.
[Section: 4.10]. 12. Write the general form of the capacitor=
voltage for the electrical network
shown in Figure P4.4. [Section: 4.4]. R1=10k9 v(r) = 11(1) e C: lOpF Figure P4.4 13. Use MATLAB to plot the capacitor
voltage in Problem 12. [Section: 4.4]. “ATLAE 14. Solve for x(t) in the system shown in Figure P45 if
f (I) is a unit step. [Section: 4.4]. M: 1 kg K,=5N/m fv= l Nslm
ﬂ!)=u(r)N fit) Figure P4.5 (,3 15. The system shown in Figure P4.6 has a unit step input.
Find the output response as a function of time.
Assume the sySterii is underda'mped. Notice that
the result will be Eq. (4.28). [Section: 4.6]. Ru) wﬁ co) sz+2 gm "5+ to}
Figure P4.6 16. Derive the relationship for damping ratio as a function
of percent overshoot, Eq. (4.39). [Section: 4.6]. 1?. Calculate the exact response of each system of Prob
lem 8 using Laplace transform techniques, and com pare the results to those obtained in that problem.
[Sections: 4.3, 4.4]. l8: Find the damping ratio and natural frequency for each
secondorder system of Problem 8 and show that the
'value of the damping ratio conforms to the type of
response (underdamped, overdarnped, and so on)
predicted in that problem. [Section: 4.5]. (1:; Problems A system has a dampingratio of 0.5, a
natural frequency of 100 rad/s, and a
dc gain of 1. Find the response of the
system to a unit step input. [Section: 4.6]. l=0“th ’ a a)!!! a 16 ‘ _. T =———
5 m ﬁ+n+m
A. / 0.04 Kb! T“)? s2 +0.02s+ 0.04 C. T(S) = 1.05 x 107 52+.1.6 x103s+1.05 x107 207
wieyPLUs solutiﬂﬂs /‘
(26) For each of the secondorder systems that follow, ﬁnd
‘4 5, Tp, Tr, and %0S. [Section: 4.6]. 21. Repeat Problem 20 using MATLAB. Have the com 22. 23. F. ’\ puter program estimate the given spe
cifications and plot the step responses.
Estimate the rise time from the/plots. [Section: 4.6]. Use MATLAB'S LTl Viewer and obtain
settling time, peak time, rise time, and percent overshoot for each of the
systems in Problem 20. [Section: 4.6]. “ATM; Gut Too; For each pair of secondorder system speciﬁcations
that follow, ﬁnd the location of the secondorder pair
of poles. [Section: 4.6]. a. %OS= 12%; s = 0.6 second b. %OS = 10%; Tp = 5 seconds =c. Ts = 7 seconds; Tp = 3 seconds Find the transfer function of a second
order system that yields a 12.3% over
shoot and a settling time of 1 second. [Section: 4.6] $25! For the system shown in Figure P4.7, do the follow ing: {Section: 4.6] Figura\l"4.7
a. Find the transfer function C(s) = X (s) / F (S).
b. Find 5*, a)", %0S, Th1}, and T,. 26. For the system shown in Figure P4.8, a step torque is
applied at 61(1). Find i5 208 Chapter 4 Time Response a. The transfer function, 6(5) = 62(s)/T(s). b. The percent overshoot, settling time, and peak
time for 62 (t): [Section: 4.6]
71:) 910) a 0 l N—m—s/rad 920) Response Figure P4.8 27. Derive the unit step response for each transfer func
tion in Example 4.8. [Section: 4.7]. 28. Find the percent overshoot, settling time, rise time,
and peak time for _ 14.145
‘ (:2 +1.204s + 2.329)(s + 5) 30. T(s) [Section: 4.7] 29. For each of the unit step responses shown
in Figure P49, ﬁnd the transfer function
of the system. [Sections: 4.3, 4.6]. \itimyPLI/s 31. 0.1 0.15 Time (seconds) 10 15
Time (seconds) . (5) WC; Figure P43 (continued) For the following response functions, determine if
polezero cancellation can be approximated. If it can,
ﬁnd percent overshoot, settling time, rise time, and
peak time. [Section: 4.8]. (5+3)
s(s+ 2)(s2 + 35+10)
_ (s+2.5)
_ 5(5 + 2)(s2 + 4; + 20)
(s+2.l)
s(s+ 2)(s2 +S+ 5) _ (5+ 2.01)
d' C(S) — s(s + 2)(s2 ll SS i 20) a. C(s) =
b. C(s) c. C(s) = Using MATLAB, plot the time response
of Problem 30(a) and from the plot
determine percent overshoot, settling
time, rise time, and peaktime. [Section: 4.8] . Find peak time, settling time, and percent overshoot 2 3
Time (seconds) (13) M so Figure P43 (ﬁgure continues) for only those responses below that can be approxi
mated as secondorder responses. [Section: 4.8]. a. C(t) = 0.003500  0.0015243'4’
' 0.0019769‘3’ cos(22.16t)
._ 0.0005427e3‘ sin(22.16t) b. c0) = 0.05100 — 0.007353e8'
— 0.0076476—Eur 003(81) — 0.01309e6r sin(8t) " c. C(f) = 0.009804 — 0.00018572‘51"
— 0.0099909'2‘ cos(9.796t) — 000194252! sin(9.796t) 33. 35. 36. . Use MATLAB'S Simuiink to obtain the d. c0) = 0.007000 — 0.001667510’
— 0.008667e2f cos(9.951t) — 0.0008040e2r sin(9.951t) For each of the following transfer functions with
zeros, ﬁnd the component parts of the unit step
response: (1) the derivative of the response without
a zero and (2) the response without a zero, scaled to
the negative of the zero value. Also, ﬁnd and plot the
total response. Describe any nonminimumphase behavior. [Section: 4.8].
s + 2
G = ———
(S) 52 + 35 + 36
s — 2
b. G 2 ——
(S) 52 + 33 + 36 gmulink step response of a system, 1 6(5) =52+3s+10 under the following conditions: [Section: 4.9] a. The system is linear and driven by an amplifier
whose gain is 10. b. An amplifier whose gain is 10 drives the system.
The amplifier saturates at i025 volts. Describe
the effect of the saturation on the system’s
output. 1:. An amplifier whose gain is 10 drives the system.
The amplifier saturates at 3:025 volts. The sys
tem drives a 1 :1 gear train that has backlash. The
deadband width of the backlash is 0.02 rad.
Describe the effect of saturation and backlash
on the system's output. A system is represented by the state and gtﬁre Space
output equations that follow. Without
solving the state equation, ﬁnd the
poles of the system. [Section: 4.10] 92: [:3 1]“ Hum y=l3 21X A system is represented by the state and ““9!le
output equations that follow. Without  solving the state equation, ﬁnd [Section:
4.10] 37. 38. 39. 40. 209 Problems a. the characteristic equation;
b. the poles of the system 0
32:0
1 #CNN y=ll 2. 0iX Given the following statespace repre
sentation of a system, ﬁnd Y(s):
[Section: 4.10] . 1 2 1 .
x._ [_3 _1]x+['1]sm3t J. y=11 21x; x(0)= [f] stale Space Given the following system represented gpie Space
in state Space, solve for Y(s) using the
Laplace transform method for solution
of the state equation: [Section: 4.10] J. =0 1 0 0
it: —2 W4 1 x+ 0 e"
0 0 —6 1
,0
y=[1 0 0]x; x(0)= O
O Solve the following state equation and
output equation for y(t), where u(t) is the unit step. Use the Laplace
transform method. [Section: 4.10] i=[j_?]x+[1]u(x) [0 1]x; x(0)z[é] 5913 Space y: gate Spece Solve for y(r) for the following system
represented in state space, where u(t)
is the unit step. Use the Laplace l 210 41. 42. 43. 45. Chapter 4 Time Response transform approach to solve the state equation. [Section:
4.10] —3 1 0 0
it: 0 —6 1 x+ 1 u(t)
o 0 —5 1 45'
0
y=[0 1 ilx; x(O)= 0
0 Use MATLAB to plot the step response
of Problem 40. [Section: 4.10] Repeat Problem 40 using MATLAB'S
Symbolic Math Toolbox and Eq. (4.96).
In addition, run your program '1
withan initial condition, x(0) = 1
[Section: 4.10] O gimbalic 114....M Using classical (not Laplace) methods
only, solve for the statetransition matrix,
the state vector, and the output of the
system represented here: [Section: 4.11] wieyPLUS Quittmi Salnil”, 59“ Space . 0 1
x=[_1 {5]x; y=[1 2lx; men Using classical (not Laplace) methods
only, solve for the statetransition matrix,
the statevector, and the output of the
system represented here, where u(t) is the unit
step: [Section: 4.11]. it = [j] 3]“ mun) Ha 21x: x<0>= [8] Solve for y(t) for the following system
represented in state space, where u(t)
is the unit step. Use the classical
approach to solve the state equation.
[Section: 4.11] gm“ Space —2 1 O 1
k: 0 0 1 x+ 0 u(t)
. 0 —6 _1 0 50. 47. 48. 49. y=[1 O 0]X; Repeat Problem 45 using MATLAB‘s
Symbolic Math Toolbox and Eq.
(4.109). In addition, run your
program with an initial condition,
1
x(0) = i
0 Usingmethodsdescribedin AppendixG.1
located at www.wi1ey.com/college/nise
simulate the following system and plot
the step response. Verify the expected
values of percent overshoot, peaktime, and settling
time. “when: Mags . [Section: 4.11]. state 511a:e ‘I
— s2 —i— 0.85 + 1
Using methods described in Appendix G. 1" 93“ Space
located at www.wiley.comlcollege/nise simulate the following system and plot
the output, y(t), for a step input: T(5) O 1 O 0
it: —10 —7 1 X+ O u(t)
O O —2 1
—1
y(t) = [i 1 0]x; x(0) = O
F 0 A human responds to a visual one With a physical
response, as shown in Figure P4.10. The transfer
function that relates the output physical response,
P(s), to the input visual command, V(s), is P(s) A (s + 0.5) V(s) (s + 2)(s + 5) (Stefani, 1973). Do the following: G(s) = a. Evaluate the output response for a unit step input
using the Laplace transform. state Space b. Represent the transfer function in
state space. c. Use MATLAB to simulate the system
and obtain a plot of the step
response. J Industrial robots are used for myriad applications;
Figure P4.11 shows a robot used to move 55pourid 211 Problems Step 1: Light source on Step 2: Recognize light source Step 3: Respond to light source Figure P4.10 Steps in determining the transfer function relating output physical response to the input vishal command Figure P4.11 Vacuum robot lifts two bags of.salt bags of salt pellets; a vacuum head lifts the bags before positioning. The robot can move as many as 12
bags per minute (Schneider; 1992). Assume a model
for the open—loop swivel controller and plant of w s K 62(5): 0()=H‘ 2 V,‘(S) (s+10)(s +45 +10)
where (00(3) is the Laplace transform of the robot’s
output swivel velocity and ltd(5) is the voltage applied
to the controller. a. Evaluate percent overshoot, settling time, peak
time, and rise time of the response of the open—
loop swivel velocity to a stepvoltage input. Justify
all secondorder assumptions. 5W“ ”3‘9 b. Represent the openloop system in
state space. si. 52. c. Use MATLAB or any other computer “WW? program to simulate the system
and compare your results to (a). Anesthesia induces muscle relaxation (paralysis) and
unconsciousness in the patient. Muscle relaxation can
be monitored using electromyogram signals from
nerves in the hand; unconsciousness can be monitored
—using the cardiovascular system’s mean arterial pres
sure. The anesthetic drug is a mixture of isoﬂurane and
atracun'um. An approximate model relating muscle relaxation to the percent isoﬂurane in thinﬁxmrejs P(s) 7.63 x 102 ( Ul lj 52+1.155+0.28 where P(s) is muscle relaxation_measured_asﬁa frac
tion of total paralysis (nonnaliaed to unity) and U (s)
is the percent mixture of isoﬂurane(Linkens, 1992).
[Section: 4.6] a. Find the damping ratio and the natural frequency
of the paralysis transient response. b. Find the maximum possible percent paralysis if a
2% mixture of_ isoflurane is used. c. Plot the step response of paralysis if a 1% mixture
of 'isoﬂurane is used. (I. What percent isoﬂurane would have to be used for
100% paralysis? 0 treat acute asthma, the drug theophylline is infused
intravenously. The rate of change of the drug cort
cehtration in the blood is equal to the difference
between the infused concentration and the eliminated
concentration. The infused concentration is i(t)/Vd,
where 1'0) is the rate of ﬂow of the drug by weight and
Vd is the apparent volume and depends on the patient.
The eliminated concentration is given by kwcm, , 212 Chapter 4 Time Response where C(t) is the current concentration of the drug
in the blood and kro is the elimination rate constant.
The theophylline concentration in the blood is
critical—if it is too low, the drug is ineffective: if
too high, the drug is toxic (Jannett, 1992). You will
help the doctor with your calculations. a. Derive an equation relating the desired blood
concentration, CD, to the required infusion rate
by weight of the drug, IR. b. Derive an equation that will tell how long the drug
must be administered to reach the desired blood
concentration. Use both rise time and settling time. c. Find the infusion rate of theophylline if VD =
600 ml, km = 0.07 111, and the required blood
level of the drug is 12 meg/ml (“mcg" means
micrograms). See Jannert ( I 992) for a description
of parameter values. d. Find the rise and settling times for the constants in (c). 53. Upper motor neuron disorder patients can beneﬁt and
regain useful function through the use of functional
neuroprostheses. The design requires a good under
standing of muscle dynamics. in an experiment to determine muscle‘responses, the identiﬁed transfer
function was (Zhou, 1995) 2.53"0‘008‘(1 + 0.1725)(1 + 0.0085) (1 + 0.07s)2(l + 0.05:)2
Find the unit step response of this transfer function. M(s) = 54. When electrodes are attached to the mastoid bones
(right behind the ears) and current pulses are applied, found that the transfer function from the current to
the subject’s angle (in degrees) with respect to the
vertical is given by (Nashua; 1974) 9(5) _ 5.8(0.3s + Uta—0‘15
W ‘ (imam a. Determine whether a dominant pole approxima
tion can be applied to this transfer function. h. Find the body sway caused by a 250 p.A pulse of
150 rnsec duration. 55. A MOEMS (optical MEMS) is a MEMS (Micro
Electromechanical Systems) with an optical ﬁber
channel that takes light generated from a laser diode.
It also has a photodetector that measures light intensity
variations and outputs voltage variations proportional
. to small mechanical device deﬂections. Additionally, a a person will sway forward and backward. It has been , OpenLoop Responses _ Openloop Response _ _
(simulated, d = 0.8)
I _ _ ,_ _____ Openloop Response _ (experimental) xlnm]: WV] 0 500 at“, 1000 1500 Figure P4.12 voltage input is capable of deflecting the'device. The
apparatus can be used as an optical switch or as'
a variable optical attenuator, and it does not exceed
2000 pm in any dimension. Figure P4.12 shows input
output signal pairs used to identify the parameters of
the system. Assume a secondorder transfer function
and ﬁnd the system’s transfer function (Borovic, 2005). 56. response of the deﬂection of a ﬂuidﬁlled catheter
to changes in pressure can be modeled ‘using a second:
order model. Knowledge of the parameters of the
model' rs important because 1n cardiovascular applica tions the undamped natural frequency should be close to ﬁve times the heart rate. However, due to sterility ,
and other considerations, measurement of the para
meters is difﬁcult A method to obtain transfer func ,
tions using measurements of thfjmplitudes of two i i
consecutive peaks of the response_ and their timing [ has been developed (Glantz, 1979). Assume that Fig
ure P4.13 is obtained from catheter 'r’neasurements 1
Using the information shown and assuming a second— i ' order model excited by a unit step input, ﬁnd the
corresponding transfer function. i 57. Several factors affect the workings of the kidneysFori 
example, Figure P4.l4 shows how a step change in.
arterial flow pressure affects renal blood flow in rats. In the “hot tail" part of the experiment, peripheral
thermal receptor stimulation is achieved by inserting
the rat’s tail in heated water. Variations between
different test subjects are indicated by the vertical
lines. It has been argued that the “control” and “hot tail" responses are identical except for their steady
state values (DiBona, 2005). a.» Using Figure P4:14, obtain the normalized
(Cﬁnal = I) transfer functions for both responses. 1 213 . .! Problems Step Response I System: T i i him: (we): 0.0505:
iAmletude: Us I I. ......... I . I :l I . , ......... .. I _
uSymmﬂ H j i
_ _ ___  _ {Timuru): 0.0616. ‘ ' H I . .Ampliludc:0i923 , ' I ' I ........... l' I l
 l
l
r
' l Amplitude 0 0.05 0. l 0. l5
Time (see) Figure P4.13 ”30 9.7 x 10%2 414400.: +.106.6 x 106) (s2 + 3800s + 23.86 x 106%2 + 240.: + 2324.3 x 103) g; , ‘ org) =
Hﬁtnuauuuaaauttn 0.025
Use a dominantpole argument to ﬁnd an equivalent transfer function with the same numerator but only
three poles. Use MATLAB to ﬁnd the actual size and i
l
i
l
l
l
approximate system unit step responses, plotting 0.020 them on the same graph. Explain the differences
between both responses given that both pairs of poles
are so far apart. Step reaponse
o
o
_.
til 59. At some point in their lives most people will suffer from
at least one onset of low back pain. This disorder can
trigger excruciating pain and temporary disability, but
its causes are hard to diagnose. It is well known that low
back pain alters motor trunk patterns; thus it is of
interest to study the causes for these alterations and
their extent. Due to the different possible causes of this  O  HOT TAIL
+ CONTROL —l 0 l 2 3 4 5 6
Time. seconds Figure P414 NtATLAg b. Use MATLAB to prove or disprove
the assertion about the ”control"
and “hot tail“ responses. 58. The transfer function of a nanopositioning device capable of translating biological samples within a few
pm uses a piezoelectric actuator and a linear variable
differential transformer (LDVT) as a displacement
sensor. The transfer function from input to displace
ment has been found to be (Salapaka, 2002) type of pain, a “control” group of people is hard to
obtain for laboratory studies. However, pain can be
stimulated in healthy people and muscle movement
ranges can be compared. Controlled back pain can be
induced by injecting saline solution directly into related
muscles or ligaments. The transfer function from infu—
sion rate to pain re5ponse was obtained experimentally
by injecting a 5% saline solution at six different infusion
rates over a period of 12 minutes. Subjects verbally
rated their pain every 15 sec on a scale from O to 10; 214 Constant
infusion ﬁle 60. 61. Chapter 4 Time ResponSe with 0 indicating no pain and 10 unbearable pain.
Several trials were averaged and the data was ﬁtted
to the following transfer function: 9.72 x 103(s + 0.0001) G(s) = 22—
(s + 0.009) (s + 0.018s + 0.0001) Infusion Pump Human Response
Constant
M (5) 6(3) back pain
Figure P4.15 For experimentation it is desired to build an automatic
dispensing system to make the pain level constant as
shown in Figure P4. 15. It follows that ideally the
injection system transfer function has to be Mm=ch .to obtain an overall transfer function M (s)G(s) m 1. However, for implementation purposes M(s) must
have at least one more pole than zeros (Zedka,
1999). Find a suitable transfer function, M (s), by
inverting 6(5) and adding poles that are far from the
imaginary axis. An artiﬁcial heart works in closed loop by varying its
pumping rate according to changes in signals from the
recipient’s nervous system. For feedback compensation
design it is important to know the heart’s open—loop
transfer function. To identify this transfer function, an
artiﬁcial heart is implanted in a calf while the main parts
of the original heart are left in place. Then the u
atnal pumping rate in the original heart is mea— w
sured while step input changes are effected on the it =
artificial heart. It has been found that, in general, .
the obtained response closely resembles that of a 9 _3 _3 _3 _3
secondorder system. In one such experiment it 4392 x '0 0'56 X 10 ‘1'0 X '0 ”'3'79 x 10 "
was found that the step response has a ‘0'347 x 103 ‘11'7 x 103 *0347 X 10—3 0 "'
%OS = 30% and a time of ﬁrst peak T = 0251 "203 x 10'3 ‘956 X 103 0 ‘1 0 0 l 0 8 127 sec (Nakamura, 2002). Find the correspond
ing transfer function. An observed transfer function fr0m voltage potential
to force in skeletal muscles is given by (Ionescu, 2005) _ 450
_ (s + 5)(s + 20) a. Obtain the system’s impulse response. T05”) b. Integrate the impulse response to ﬁnd ‘the step
response. 62. 52(5) = _ (52 + 3.99 x 103 s + 3.97 x 103)(s2 + 4.215 +1823) 63. l
as l
l 1:. Verify the result in Part b by obtaining the step
response using Laplace transform techniques. In typical conventional aircraft, longitudinal ﬂight model
linearization results in transfer functions with two pairs
of complex conjugate poles. Consequently, the natural
response for these airplanes has two modes in their
natural response. The “short period” mode is relatively
wellidamped and has a hi gh—frequency oscillation. 'Ihe
“plugoid mode” is lightly damped and its oscillation
frequency is relatively low. For example, in a speciﬁc
aircraft the transfer function from win g elevator deﬁes
tion to nose angle (angle of attack) is (McRuer: 1973) 26.12(s + 0.0098)(s + 1.371) 3. Find which of the poles correspond to the shun
period mode and which to the phugoid mode. pole approximation), retaining the two poles andij
the zero closest to the; waxis. ‘ c. Use MATLAB to compare the step
.responses of the original transfer
function and the appr0ximation. “ATLAs Using wind tunnel tests, insect ﬂight dynamics canbe
studied in a very similar fashion to that of manmade
aircraft. Linearized longitudinal ﬂight equations fort;
bumblebee have been found in the unforced case to be where u = forward velocity, w = vertical velocity.
q = angular pitch rate at center of mass, and 6 = pitch angle between the ﬂight direction and the horizontal?
(Sun, 2005). NtATLAs 3. Use MATLAB to obtain the system’s
eigenvalues. b. Write the general form of the statetransition matrix.
How many constants would have to be found? r3) . wweiww—rwW ww—  i 54‘. A dcdc converter is a device that takes iii An IPMC (Ionic PolymerMetal Com 5““ Space as an input an unregulated dc voltage
and provides a regulated dc voltage as its
output. The output voltage may be lower (buck con
verter), higher (boost converter), or the same as the
input voltage. Switching dcdc converters have a
semiconductor active switch (BJT or PET) that is
closed periodically with a duty cycle d in a pulse
width modulated (PWM) manner. For a boost con
verter, averaging techniques can be used to arrive at the
following state equations (Van Dijk, 1995): LE: —(l —d)uC+E5
aTHC . uc C—= 1 —d ——
dt ( M R where L and C are respectively the‘values of internal
inductance and capacitance; 1]; is the current through
the internal inductor; R' 15 the resistive load connected
to the converter; E is the dc input voltage; and the
capacitor voltage rig, is the converter’ 5 output. a. Write the converter’s equations in the form x =an + Bu
y = Cl‘. assuming d is a constant. b. Using the A, B, and C matrices calf Clan a, obtain
S E; (s) the converter’ 5 transfer function state Sinite posite) is a Naﬁon sheet plated with gold
on both sides. An IPMC bends when an
electric ﬁeld is applied across its thickness. IPMCs
have been used as robotic actuators in several appli—
cations and as active catheters in biomedical applica
tions. With the aim of improving actuator settling
times, a statespace model has been developed
for a 20 mm x 10 mm x02 mm polymer sample
(Mallavarapu, 2001): iii =i281'34 iizﬁllil + iii“
y = [12.54 2.26] [‘2] where u is the applied input voltage and y is the
deﬂection at one of the material’s tips when the
sample is tested in a cantilever arrangement. Problems 3. Find the state transition matrix for the system. b. From Eq. (4.109) in the text it follows that if a
system has zero initial conditions the system
output for any input can be directly calculated
from the state— —space representation and the state
transition matrix using y(1‘) = me e f0<1>tr — r)Bu (T) at Use this equation to ﬁnd the zero initial condition unit'step response of the IPMC material sample.
“MlAs c. Use MATLAB to verify that your
step response calculation in Part b
is correct. Design Problems 66. 67. 68. Find an equation that relates 2% settling Mamas
time to the value of fv for the translational . "
mechanical system shown in Figure i. J “3
P416. Neglect the mass of all compo cum, :3quer
nents. [Section_: 4.6] f0) 2 N/m
Figure P4.16 Consider the translational mechanical system shown
in Figure P4.17..A lpound force, f (I), is applied at
t = 0. If fv = 1, ﬁnd K and M such that the response
is characterized by a 2second settling time and a
lsecond peak time. Also, what is the resulting per
cent overshoot? [Section: 4.6] f0) Figure P4.17 Given the translational mechanical system of Figure
P417, where K = 1 and f (t) is a*unit step, ﬁnd the
values of M and fv to yield a response with 30% over
shoot and a settling time of 10 seconds. [Section: 4.6] l 21 6 Chapter 4 Time Response 69. Find J and K in the rotational system shewn in Fig—
ure P418 to yield a 30% overshoot and a settling time
of 4 seconds for a step input in torque. [Section: 4.6] 1 _ E El 5’ m. .«
K Figure P4.18 TU) 70. Given the system shown in Figure P4.19, ﬁnd the
damping, D, to yield a 30% overshoot in output angular
displacement for a step input in torque. [Section: 4.6] TO) 91(3) :
N2=5 N3=10
D N1=25 N4 = 5 % Nmfrad
Morn— 75. Figure P4.19 71 . For the system shown in Figure P420, ﬁnd WBVPEUS
N1 /N2 so that the settling time for a step $21M _
torque input is 16 seconds. [Section: 4.6] gr  ’ 5 no lintrnl Soluti‘l'1 : N. l Nm/rad 1 Nms/rad Figure P420 72. Find M and K, shownin the system of Figure P421,
to yield x(t) with 10% overshoot and 10 seconds tun
J: l kg—m2
N2=20 For the motor: J“ = 1 legm2
D“: l Nm—s/rad
RR = 1 9 Kb = 1 V—s/rad
K, = 1 Nm/A Figure P421 73. 74. 76. 77. settling time for a step input in motor torque, Tm(t).
[Section: 4.6] If 12,0) is a step voltage in the network shown in
Figure P422, ﬁnd the value of the resistor such that a
20% overshoot in voltage will be seen across the
capacitor if C, = 10—6 F and L = 1 H. [Section: 4.6] R L WU) C Figure P422 If v,(t) is a step voltage in the network
shown in Figure P422, ﬁnd the values
of R and C to yield a 20% overshoot
and a 1 ms settling time for vc(t) if L 2 1H. [Section: 4.6] «iieyPLUs . 5
“mused” Given the circuit of Figure P422, where C = 10 11F,
ﬁnd R and L to yield 15% overshoot with a settling
time of 2 ms for the capacitor voltage. The input, v(t},
is a unit step. [Section: 4.6] For the circuit shown in Figure P423, ﬁnd the values
of R2 and C to yield 15% overshoot with a settling
time of 1 ms for the voltage across the capacitor, with
v,(r) as a step input. [Section: 4.6] 1H R2 “H
+ CAM!) V5“) 9 Figure P423 Hydraulic pumps are used as inpiits to
hydraulic circuits to supply pressure,
just as voltage sources supply potential
to electric circuits. Applications for
hydraulic circuits can be found in the robotics and
aireraft industries, where hydraulic actuators are used
to move component parts. Figure P424 shows the
internal parts of the pump. A barrel containing equally
spaced pistons rotates about the iaxis. A swashplate,
set at an angle, causes the slippers at the ends of
the pistons to move the pistons in and out. When
the pistons are moving across the intake port, they are
extending, and whenthey are moving across the dis~
charge port, they are retracting and pushing ﬂuid from
the port. The large and small actuators at the top and _ A}! Problems 217 Figure P424 Pump diagram bottom, respectively, control the angle of the swash
plate, or. The swashplate angle affects the piston stroke
length. Thus, by controlling the swashplate angle, the
pump discharge ﬂow rate can be regulated. Assume the
state equation for the hydraulic pump is x = [(3.45 — 14000Kc) —0.255 x 109]x 0.499 x 10“ —3.68
+ —3.45 +14000K, a "G
—0.499 x 10ll 0’
where x = [if] and Pd is the pump discharge pressure (Manring,
1996).,Find the value of controller ﬂow gain, KC, so
that the damping ratio of the system’s poles is 0.9. {Progressive Analysis and Design Problems 18. Highspeed rail pantograph. Problem 66(c) in 19. Control of HIV/AIDS. In Chapter 3, Chapter 2 asked you to ﬁnd 6(3) = (11(3) 4
Yea,(s))/ Fup(s) (O’Connor; 1997). a. Use the dominant poles from this transfer function
and estimate percent overshoot, damping ratio, nat
ural frequency, settling time, peak time, and rise time. b. Determine if the secondorder approximation is
“Ml—As valid. c. Obtain the step response of 6(5) and
compare the results to Part a. Problem 30, we developed a linearized
statespace model of HIV infection. The model assumed that two different drugs Were used to
combat the spread of the HIV virus. Since this book
focuses on singleinput, singleoutput systems, only one
of the two drugs will be considered. We will assume that
only RTIs are used as an input. Thus, in the equations of
Chapter 3,sProblem 30, u2 = 0 (Craig, 2004). a. Show that when using only RTIs in the linearized
system of Problem 30 and substituting the
typical parameter values given in the table of
Problem 30(c), the resulting statespace represen
tation for the system is given by \.. fr —0.04167 0 —0.0058
,T* = 0.0217 —0.24 0.0058
i; 0 100 —2.4 r 5,2
x T* + —5.2 ul
v 0
T
y=[0 0 1] T*
V b. Obtain the transfer function from RTI efﬁciency Y (S) 111(5). (1. Assuming RTIs are 100% effective, what will be
the steadystate change of virus count in a given
infected patient? Express your answer in virus
copies per mL of plasma. Approximately how
much time will the medicine take to reach its
maximum possible effectiveness? to virus count; namely ﬁnd ...
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