Page 345 - Modern Control Systems

Section 5.4 Effects of a Third Pole and a Zero on the Second-Order System Response 319

X t v 4

"-J2
- 6 -3
K>
-6.25 -2.5
-J2
FIGURE 5.16
The poles and X
zeros on the -J*
s-plane for a
third-order system.

zero are shown on the s-plane in Figure 5.16. As a first approximation, we neglect
the real pole and obtain

m 10(5 + 2.5)
s 2 + 6s + 25

We now have £ = 0.6 and co n = 5 for dominant poles with one accompanying zero
for which a/(£(a n) = 0.833. Using Figure 5.13(a), we find that the percent overshoot
is 55%. We expect the settling time to within 2% of the final value to be
4
Z = = 1.33 s.
0.6(5)
Using a computer simulation for the actual third-order system, we find that the per-
cent overshoot is equal to 38% and the settling time is 1.6 seconds. Thus, the effect
of the third pole of T{s) is to dampen the overshoot and increase the settling time
(hence the real pole cannot be neglected). •

The damping ratio plays a fundamental role in closed-loop system performance.
As seen in the design formulas for settling time, percent overshoot, peak time, and
rise time, the damping ratio is a key factor in determining the overall performance.
In fact, for second-order systems, the damping ratio is the only factor determining
the value of the percent overshoot to a step input. As it turns out, the damping ratio
can be estimated from the response of a system to a step input [12]. The step re-
sponse of a second-order system for a unit step input is given in Equation (5.9),
which is

y{t) = I - y~^< 1 sm{u> nBt + 0),

where B £ , and 6 = cos £,. Hence, the frequency of the damped sinu-
soidal term for £ < 1 is
2 1/2
o> = <o n(\ - £ ) = a)J3,
and the number of cycles in 1 second is w/(2ir).