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Unformatted text preview: 5 Chapter; uynauutnwpmw Thickness T Figure 3.51
Continuous rolling mill l : N
Gear ratio
Vertically
adjustable
roller
Motion of
WW’ material
. out of rollers Thickness x
Fixed
roller determine the following:
(a) The DC gain; (b) The final value to a step input. tinuous rolling mill depicted in Fig. 3.51. Suppose that the motion of 3.18 Consider the con
the adjustable roller has a damping coefﬁcient 19, and that the force exerted by the rolled table roller is proportional to the material’s change in thickness: material on the adjus
F s :: C(T ~— x). Suppose further that the DC motor has a torque constant K, and a back emf constant Kg. and that the rack—andpinion has effective radius of R. (a) What are the inputs to this system? The output? (b) Without neglecting the effects of gravity on the adjustable roller, draw a block dia— gram of the system that explicitly shows the followrng quantities: VS (5), 10 (s), F (s)
(the force the motor exerts on the adjustable roller), and X (s). (c) Simplify your block diagram as much as possible while still ide
each input separately. nti‘fying output and Probicmsfbr Section 3.2: System Modeling Diagrams 3.19 Consider the block diagram shown in Fig. 3.52. Note that ai and bl are constantsi Compute the transfer function for this system. This special structure is called the “conth canonical form” and will be discussed further in Chapter 7. %
E:
i
i n igmub allWm Block diagram for
Problem 3.19 Figure 3.53 Block diagrams for
Problem 3.20 O Y(s)
RO
(a)
i a G7
+ +
+
RC » G1 G3 > G4 ’ G6 + Y
G2 1 (.5 .j
(b)
r G7
> G6 +
+
R G’ 2 ’G2 ’Ga 2 G4 Gs><Jr :HY 3.21 Find the transfer functions for the block diagrams in Fig. 3.54. using the ideas of block diagram simpliﬁcation The s ' ‘ '
. . peeial structure in F1 . 3.54 b ' “
canonlcal form” and will be discussed in Chapter 7. g ( ) 15 called the ObserVer 5 Chapter 5 Uyndmu. [\CDPUIrox. (a) (b) (c) R(s) A(s) > l)(s) ___.__.._..J 3(5) _ Y(S) H (s) L,_,__.._. Figure 3.54
Block diagrams for Problem 3.21 3.22 Use block—diagram algebra t Fig. 3.55. (d) C(s) i‘ 0 determine the transfer function between R(s) an d Y(s) in figure 309;»
Block diagram for
Problem 3.22 Figure 3.56
Circuit for Problem 3.23 Figure 3.57 Unity feedback system
for Problem 3.24 Prablemsfor Section 3.3: Effect ofPole Locations 3.23 For the electric circuit shown in Fig. 3.56, ﬁnd the following:
(a) The time—domain equation relating i(t) and v10);
(b) The time—domain equation relating i (t) and v20); (c) Assuming all initial conditions are zero, the transfer function ‘Zf—ESS; and the damping ratio g“ and undamped natural frequency ton of the system; (d) The values of R that will result in v2 (2‘) having an overshoot of no more than 25%,
assuming v1 (t) is a unit step, L = 10 mH, and C = 4 ,uF. L R
+ +
v10) 2(1) C v20) 3.24 For the unity feedback system shown in Fig. 3.57, specify the gain K of the proportional controller so that the output 37(1) has an overshoot of no more than 10% in response to
a unit step. O Y(s) 3.25 For the unity feedback system shown in Fig. 3.58, specify the gain and pole location
of the compensator so that the overall closed—loop response to a unit—step input has an overshoot of no more than 25%, and a 1% settling time of no more than 0.1 sec. Verify
your design using MATLAB. Chapter 3 Dynamic Response ire 3.58 ty feedback system
Problem 3.25 Figure 3.59 Unity feedback system
for Problem 3.28 Figure 3.60
Desired closed—loop
pole locations for
Problem 3.28 Compensator Plant
J" K g g 100
Rmo T 5+4; s+25 Problems for Section 3 3.26 Suppose you desire the
Draw the region in the s .4: Time— peak time of a . . ,
speciﬁcation tp < tp. 3.27 A certain servomech
and no ﬁnite zeros. overshoot (M p), and settling time (a) Sketch the region in the s anism s ‘
The time—domain O Yts) plane that corresponds to v ystem has dy nami speci Domain Specification ' ‘ to b
iven second—order system
g alues of the poles that meet the es dominated b _
ﬁcations on the rise time (ts) are given by tr < 0.6 sec, Mp g 17%, ts g 9.2 sec. will meet all three speciﬁcations. (b) Indicate on your smallest rise—time and als
sign a unity feedback ‘ .
11 learn in Chapter 4, the configurati
troller.) You are to design the contr
wn in Fig. 3.60. .
haded regions in Fig. 3.59? (A Simple 3.28 Suppose you are to de
Fig. 3.59. (As you wr ro ortional—integral con .
p P in the shaded regions she can and g correspond to the s poles lie with (a) What values of estimate from t
z a 2: 2. Find values for K an (in) Let Ka .
system he w1 6(1) sketch the speci he ﬁgure is sufﬁci 0 meet the set —plane where the pole ﬁe locations (de .
tling time speciﬁcation exa order plant depicted in
on shown is referred to as a
oller so that the closed—loop ent.) thin the shaded regions. controller for a ﬁrst— d K] so that the poles of e less than t}. y a pair of complex poles (t r), percent s could be placed so that the system noted by x) that will have the
ctly. the closedloop .MW (c) Prove that no matter what the values of Ka and a are, the controller provides enough
ﬂexibility to place the poles anywhere in the complex (lefthalf) plane. 3.29 The open—loop transfer function of a unity feedback system is Gm : Sls + 2)’ The desired system response to a step input is speciﬁed as peak time tp = 1 sec and
overshoot MP 2 5%. (3) Determine whether both speciﬁcations can be met simultaneously by selecting the
right value of K. (b) Sketch the associated region in the splane where both speciﬁcations are met, and
indicate what root locations are possible for some likely values of K. (c) Relax the specifications in part (a) by the same factor and pick a suitable value for
K, and use MATLAB to verify that the new speciﬁcations are satisﬁed. 3.30 The equations of motion for the DC motor shown in Fig. 2.32 were given in Eqs. (2.52—
2.53) as " KK . K
.1QO + (19+ 1 (3)9,” = —tVa. Assume that m = 0.01kg.m2, b = 0.001 N'mtsee,
Ke = 0.02 Vsec,
Kt = 0.02 N.m/A’
Ra = 10 52. (a) Find the transfer function between the applied voltage va and the motor speed 9m. (b) What is the steadystate speed of the motor after a voltage Va 2: 10 V has been
applied? (c) Find the transfer function between the applied voltage Va and the shaft angle 9m. (d) Suppose feedback is added to the system in part (c) so that it becomes a position
servo device such that the applied voltage is given by Va : K(9r “' 9m), where K is the feedback gain. Find the transfer function between 9, and 9,”. (e) What is the maximum value of K that can be used if an overshoot Mp < 20% is
desired? (D What values of K will provide a rise time of less than 4 sec? (Ignore the Mp
constraint.) (g) Use MATLAB to plot the step response of the position servo system for values of
the gain K = 0.5, l. and 2. Find the overshoot and rise time for each of the three step
responses by examining your plots. Are the plots consistent with your calculations
in parts (e) and (f)? 3.31 You wish to control the elevation of the satelliteetracking antenna shown in Figs. 3.61
and 3.62. The antenna and drive parts have a moment of inertia J and a damping B; (c) What is the maximum value of K that can be used if you wish to have an overshoot 1 Chapter 3 Dynamic Response
Mp < 10%? .. 1 . . . .
we: 3 6 _ (d) What values of K W111 provrde a rise time of less than 80 sec? (Ignore the Mp
:elhte—trackmg constraint.)
tenna A (e) Use MATLAB to plot the step response of the antenna system for K = 200, 400,
Irce: Courtesy Space 1000, and 2000. Find the overshoot and rise time of the four step responses by
tems/ LO’ at examining your plots. Do the plots conﬁrm your calculations in parts (c) and (d)? 3.32 Show that the secondorder system
_ y + 2:010 + wiy = 0, MO) 2 ya. ya» = 0,
has the response
e—ot
YO) = y0— sin(wdt + cos‘1 g).
t/ 1 — {2
Prove that, for the underdamped case (g < l), the response oscillations decay at a
predictable rate (see Fig. 3.63) called the logarithmic decrement
2
5 lny—0 lne0rd 0rd 7“:
g M [1  C2
‘ Am An
Ft are 3&2 =1 _.._ 2 ln——,
9 . YI yi
Schematic of antenna
for Problem 331 where
271 27r
17d : — : is the damped natural period of vibration. The damping coefﬁcient in terms of the
logarithmic decrement is then ut mostly from the Figure 3.63 ‘ Deﬁnition of
logarithmic decrement bearing and aerodynamic friction, b
f motion are tent from back emf of the DC drive motor. The equations 0 these arise to some ex 10‘ +30 2 TC, ’ at
where TC is the torque from the drive motor. Assume th J = 600.000 kgm2 B = 20,000 Nmsec. and the antenna angle 6.
rence command 9r NY d the transfer function between the applied torque Tc a) Fin ‘ f
Eb) Suppose the applied torque is computed so that 9 tracks a re e
according to the feedback law
Tc : Kwr " 9)» e e . W d .
h 1‘ K is h t edba k g 1D In 11 t dnbfer functlon bet Ben r (in
W 1 e e (2 a F (l t 5 rr W 6 , 0 ...
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 Spring '07
 Balakrishnan
 Digital Signal Processing, Signal Processing

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