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Unformatted text preview: Problems 167 em. 3.42 Suppose that unity feedback is to be applied around the listed openloop systems. Use
Routh’s stability criterion to determine whether the resulting closedloop systems will
of an aircraft :; be stable. 40+”
and the pitch (33 KG“) = s(s§+2.t2+33+4)
.5 . _ 2 5+4)
(b) [(60) _ s2 (H1) _ Ma
(Q) KGB) — 82(33+232—.\‘—1) 3.43 Use Routh’s stability criterion to determine how many roots with positive real parts the
following equations have: (a) s4 + 853 + 32x2 4 so: + 100 = 0. ' (b) s5 + 10:4 + 3053 + 8032 + 344.: + 480 = 0.
(c) :4 + 2:3 + 7.92 m 2.3 + 8 = 0. (t1)r3+r2 +205+7S =0. (e) s4 + 652 + 25 = 0. degrees. The
:ording to the than 10% and
ning the step
lty assOciated r_4._..__.,____,__ _,_.,. Mvhcw :ms. 3.44 Find the range of K for which all the roots of the following polynomial are in the LHP:
:5 + 534 +1053 +1059 + is +K : 0.
Use MATLAB to verify your answer by plotting the roots of the polynomial in the
splane for various values of K.
3.45 The transfer function of a typical tapedrive system is given by
Ks+4
Go) : c_u_w
slts + 0.5)(s + 1) (S" + 0.45 + 4)]
where time is measured in milliseconds. Using Routh’s stability criterion, determine
the amount of t the range of K for which this system is stable when the characteristic equation is
3), given some 'i 1 —l» ((5) = O.
' 3.46 Consider the closedloop magnetic levitation system shown in Fig. 3.67. Determine the
conditions on the system parameters (a, K, 2, p. K0) to guarantee closedloop system
i stability.
Figure 3.67 + e K(s+z) it XD
. . . R ”r 1’
Magnetic lewtation (5+p) (Em?)
3W that system _ 1
3.47 Consider the system shown in Fig. 3.68.
(3) Compute the closadloop characteristic equation.
(b) For what values of (72A) is the system stable? Hint: An approximate answer may
be found using
l 9‘“ s 1 4 Ta
Figure 3.68 Control system for
Problem 3.47 tte errors but high
)1 provides robust
ystem less stable. These three kinds and describes the tables 4.2 and 4.3.
to replace a given
: transfer 'fu nction
ids to uIIkTS) then id C2d. tracking input and
: rate of 40°C/ sec.
error constant (KP 3
ts? system? : rather than in the roller. )r at if the approx
height e(kT_;) and Figure 4.23 Threeampliﬁer topologies For
Problem 4.2 Problems 207
PROBLEMS Probiemsfor Section 4.] .' The Basic Equations of Control 4.1 4.2 4.3 if S is the sensitivity of the unity feedback system to changes in the plant transfer function
and '1‘ is the transfer function from reference to output, show that S + T = 1. We deﬁne the sensitivity of a transfer function G to one of its parameters k as the ratio
of percent change in G to percent change in k.
8‘3 _ dG/G A dlnG _ it: .1156
k ‘ dk/k — dlnt: ‘ Gttk' The purpose of this prohlem is to examine the effect of feedback on sensitivity. In particular, we would like to compare the topologies shown in Fig. 4.23 for connecting three ampliﬁer stages with a gain of —K into a single ampliﬁer with a gain of — 10. (a) For each topology in Fig. 4.23, compute ﬁ; so that ifK : 10, Y : —lOR. (b) For each topology, compute .3? when G = Y/R. [Use the respeCLiVe ,6, valnes found
in part (a).'_[ Which case is the least sensitive? (c) Compute the sensitivities of the systems in Fig. 4.23(h,c) to ,5; and 63. Using your
results, comment on the relative need for precision in sensors and actuators. :32 (b) Compare the two structures shown in Fig. 4.24 wilh respect to sensitivity to changes in
the overall gain due to changes in the ampliﬁer gain. Use the relation idlnF_KctF S _ # _ _
a’ In K F dK
as the measure. Select H1 and H2 so that the nominal system outputs satisfy Fl 2 F2,
and assume KH1 > O. 208 Chapter 4 A First Analysis of Feedback Figure 4.24
Block diagrams for Problem 4.3 4.4 A unity feedback control system has the openloop transfer function A (a) Compute the sensitivity of the closcdeloop transfer function to changes in the
parameter A. lb) Compute the sensitiVity of the closed100p transfer function to changes in the
parameter a. (c) Ifthe unity gain in the feedhack changes to a value of t5 94 l. compute the sensitivity
of the closedloop transfer function with respect to ,8. 4.5 Compute the equation for the system error for the ﬁltered feedback system shown in
Fig. 4.4. 4.6 if S is the sensitivity of the filtered feedback system to changes in the plant transfer function and T is the transfer function from reference [0 output, compute the sum of
S+T.ShowthatS+T= i iszH. (:1) Compute the sensitivity of the ﬁltered feedback system shown in Fig. 4.4 with
respect to changes in the plant transfer function, G. (b) Compute the sensitivity of the ﬁltered feedback system shown in Fig. 4.4 with
respect to changes in the controller transfer function, Dd (c) Compute the sensitivity of the ﬁltered feedback system shown in Fig. 4.4 with
respect to changes in the ﬁlter transfer function, F, (d) Compute the sensitivity of the ﬁltered feedback system shown in Fig. 4.4 with
respect to changes in the sensor transfer function. H. Problemsfor Section 4.2: Control of SteadyStare Error 4.7 Consider the DC—motor c0ntrol system with rate (tachometer) feedback shown in
Fig. 425(3).
(a) Find values for K’ and it; so that the system of Fig. 4.25(b) has the same transfer
function as the system of Fig. 4.25(a).
(1)) Determine the system type with reSpect to tracking 6, and compute the system K1;
in terms of parameters K’ and id .
(c) Does the addition of tachometer feedback with positive it, increase or decrease K”? 4.8 Consider the system shown in Fig. 4.26, where (3+4)2
0(3) = Km Figure 4.25
Control system fl Figure 4.26 Control system f
Problem 4.8 Figure 4.2? Control system
Problem 4.9 :lt‘Ol cm are shown in
cgleetcd arid the to a step distur—
rder to make the .ccurately sketch
gain K computed :s corresponding
shootMp 5 5%. speciﬁcations. the new control
bance torque be 3 tracking? ice rejection? i Figure 4.41 DC Motor speedcontrol
block diagram for
Problems 4.29 and 4.30 4.29 Problems 217 (3) Determine the system type and error constant with respect to tracking polynomial
reference inputs of the system for P [D = kp], PD [D = kp + [(03], and PID [D = kp + kI/t + [cps] controllers. Let kg = 19, it; 2 0.5, and kD = 4/19.
(b) Determine the system type and error constant of the system with respect to distur bance inputs for each of the three regulators in part (a) with respect to rejecting
polynomial disturbances WU) at the input to the plant. (c) Is this system better at tracking references or rejecting disturbances? Explain your
response brieﬂy. (d) Verify your results for parts (a) and (b) using MATLAB by plotting unit step and
ramp responses for both tracking and disturbance rejection. The DCsmotor speed control shown in Fig. 4.41 is described by the differential equation
it + 6sz = 600%;  lSOOw, where y is the motor speed, va is the armature voltage, and w is the load torque. Assume
the armature voltage is computed using the PI control law t
vﬂ = — (kpe + [:1]; edit), where e = r — y. (a) Compute the transfer function from W to 1’ as a function of ﬁg; and Jr}. (b) Compute values for kp and it; so that the characteristic equation of the closedsloop
system will have roots at —60 :l: 60}. 4.30 4.31 For the sysrern in Problem 4.29. compute the following steadystate errors;
(a) to a unit—step reference input; (h) to a unitramp reference input; (c) to a unit—step disturbance input; (d) for a unitramp disturbance input. (e) Verify your answers to (a) and (d) using MATLAB. Note that a ramp response can
be generated as a step response of a system modiﬁed by an added integrator at the
reference input. COnsider the satelliteattitude centrol problem shown in Fig. 4.42 where the normalized
parameters are I : 10 spacecraft. inertia, NlTlSBC2/I'Etd , : reference satellite attitude, rad. ...
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 Fall '09

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