Modern Control Engineering

Section 8–7 / Zero-Placement Approach to Improve Response Characteristics 601

and the remaining closed-loop pole is real and is located at

s=–c

Note that in this problem there are three requirements. The first requirement is that the

response to the step disturbance input damp out quickly. The second requirement is that the max-

imum overshoot in the response to the unit-step reference input be between 19%and 2%and the

settling time be less than 1 sec. The third requirement is that the steady-state errors in the re-

sponses to both the ramp and acceleration reference inputs be zero.

A set (or sets) of reasonable values of a, b,andcmust be searched using a computational

approach. To satisfy the first requirement, we choose the search region for a, b,andcto be

2 a6, 2 b6, 6 c 12

This region is shown in Figure 8–34. If the dominant closed-loop poles s=–a_jbare located

anywhere in the shaded region, the response to a step disturbance input will damp out quickly. (The

first requirement will be met.)

Notice that the denominator of Y(s)/D(s)can be written as

=s^3 +(2a+c)s^2 +Aa^2 +b^2 +2acBs+Aa^2 +b^2 Bc

=(s+a+jb)(s+a-jb)(s+c)

=s^3 +(1+10K)s^2 +10K(a+b)s+10Kab

s^2 (s+1)+10K(s+a)(s+b)

0

j 6

j 4

j 2

• j 6


• j 4


• j 2


• 12 – 10 – 8 – 6 – 4 – 2 2 s

Region for

a and b

Region for c

jv

Figure 8–34
Search regions for
a, b,andc.