Page 310 - Modern Control Systems

284 Chapter 4 Feedback Control System Characteristics

.Gripper Compute the steady-state error to a unit step input as
a function of the parameter p.
E4.6 A unity feedback system has the loop transfer function
1 OK
L(s) = G c(s)G(s) = ^ - T 7 .
Camera s(s + b)
Determine the relationship between the steady-state
error to a ramp input and the gain K and system para-
meter b. For what values of K and b can we guarantee
that the magnitude of the steady-state error to a ramp
input is less than 0.1?
E4.7 Most people have experienced an out-of-focus slide
projector. A projector with an automatic focus adjusts
for variations in slide position and temperature distur-
bances [11]. Draw the block diagram of an autofocus sys-
tem. and describe how the system works. An unfocused
u> slide projection is a visual example of steady-state error.
E4.8 Four-wheel drive automobiles are popular in regions
where winter road conditions are often slippery due to
snow and ice. A four-wheel drive vehicle with antilock
• Y(s)
to
Desired Gripper brakes uses a sensor One keep each wheel rotating to
traction.
maintain
Figure
system
is
shown
in
gripper position
positior E4.8. Find the closed-loop response of this system as
it attempts to maintain a constant speed of the wheel.
(b) Determine the response when R{s) = A/s.
FIGURE E4.3 Robot fruit picker.
5{s + 3)
s(s + 15) Wheel speed
step change in the desired input? (b) Calculate the
required K in order to yield a steady-state error of
0.1 mm for a ramp input of 10 cm/s.
= 0; K = 100
Answers: e si
FIGURE E4.8 Four-wheel drive auto.
Desired
position E4.9 Submersibles with clear plastic hulls have the poten-
tial to revolutionize underwater leisure. One small
submersible vehicle has a depth-control system as illus-
trated in Figure E4.9.
(a) Determine the closed-loop transfer function
7(s) = Y{s)/R(s).
(b) Determine the sensitivity Sjjj and S' K.
(c) Determine the steady-state error due to a distur-
bance T,i(s) - 1/s.
(d) Calculate the response y(t) for a step input
R(s) = 1/s when/C = K 2 = landl < Jfj < 10.
Motor Select Kt for the fastest response.
E4.10 Consider the feedback control system shown in
FIGURE E4.4 Disk drive control. Figure E4.10. (a) Determine the steady-state error for a
step input in terms of the gain, K. (b) Determine the
E4.5 A feedback system has the closed-loop transfer overshoot for the step response for 40 =£ K < 400. (c)
function given by Plot the overshoot and the steady-state error versus K.
s 1 + ps + 10 E4.ll Consider the closed-loop system in Figure E4.ll,
7(.y) = - ; ir 1 .
S 3 + 2ps- + 4.v + (3-p) where
Compute the sensitivity of the closed-loop transfer K 14
-
function to changes in the parameter p, where p > 0, G(s) = s + 10 and H(s) = + 5s + 6