Page 827 - The Mechatronics Handbook

066_Frame_C26 Page 12 Wednesday, January 9, 2002 1:59 PM

10
8
6
4
2
Imag Axis 0
−2
−4
−6
−8
−10
−12 −10 −8 −6 −4 −2 0 2 4
Real Axis

FIGURE 26.10 Root locus for Example 2(a).

4
3
2
1
Imag Axis 0

−1
−2
−3
−4
−10 −8 −6 −4 −2 0
Real Axis
FIGURE 26.11 Root locus for Example 2(b).

Since the poles −0.2 ± j1.99 are canceled by a pair of zeros at the same point in the closed-loop system
−1
transfer function T = G(1 + G) , the dominant poles are at −0.56 and −0.94 ± j1.61 (they have relatively
large negative real parts and the damping ratio is about 0.5).
Now, suppose that this controller is fixed and the complex poles of the plant are slightly modified by
taking ζ = 0.09 and ω 1 = 2.2. The root locus corresponding to this system is as shown in Fig. 26.11. Since
lightly damped complex poles are not perfectly canceled, there are two more branches near the imaginary
axis. Moreover, for the same value of K = 600, the closed-loop system poles are

±
±
{ – 10.78 j2.57, 1.21 j1.86, 0.05 ± j1.93, 0.51}
–
–
In this case, the feedback system is unstable.
Example 3
One of the most important examples of mechatronic systems is the DC motor. An approximate transfer
function of a DC motor [8, pp. 141–143] is in the form

K m
P m s() = --------------------------, t m > 0
(
ss + 1/t m )
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