Page 58 - Anatomy of a Robot

02_200256_CH02/Bergren 4/17/03 11:23 AM Page 43
CONTROL SYSTEMS 43
Varying Frequency Only: Damping = .2
Frequency = 2.5, 1.5, 1, .5
1.8
1.6
1.4
1.2
1
x x x x
0.8
0.6
0.4
0.2
0
0 2 4 6 8 10
Time tTime t
Time t
Time t
FIGURE 2-17 The second-order system responds differently as the
frequency is varied.

Response time Take a look at Figure 2-17 entitled Varying Frequency Only. It
was made holding the damping parameter d constant and varying the frequency v
(we’ll get into how to do that soon). The point is, the curves rise toward the final
value of 1 at varying speeds. A few ways are available for measuring the response
time, including
Time from 0 to first crossing of 1
Time from 0 to first peak overshoot (the maximum value)

The system has a different response time for different values of damping. If we
look at the time from 0 to the first crossing, the four curves vary in rise time from
3/4 seconds to almost 4 seconds. These four curves vary in frequency from 2.5 to
0.5 radians per second. A circle has 2p radians. Frequency is related to radians in
the following way:

F 2 p v

where F is in Hertz (cycles per second), v is in radians per second, and p is
3.14159 ....