Page 346 - Modern Control Systems

320 Chapter 5 The Performance of Feedback Control Systems

The time constant for the exponential decay is T = l/(£&>„) in seconds. The
number of cycles of the damped sinusoid during one time constant is

(cycles/time) X T =
27r£&)„ 27rg(o n 2TT£

Assuming that the response decays in n visible time constants, we have

cycles visible = -—-. (5.19)

For the second-order system, the response remains within 2% of the steady-state
value after four time constants {AT). Hence, n = 4, and

2 112
4(3 4(1 - C ) ., 0.55
cycles visible = —— = —— =* —7- (5.20)
ITTC, 2TTQ C
for 0.2 < £ < 0.6.
As an example, examine the response shown in Figure 5.5(a) for £ = 0.4. Use
y{t) = 0 as the first minimum point and count 1.4 cycles visible (until the response
settles with 2% of the final value).Then we estimate

0.55 0.55 nf%n
cycles 1.4

We can use this approximation for systems with dominant complex poles so that
(ol
T(s)
S + 2t,03 nS + (*) n .2'

Then we are able to estimate the damping ratio £ from the actual system response of
a physical system.
An alternative method of estimating £ is to determine the percent overshoot for
the step response and use Figure 5.8 to estimate £. For example, we determine an
overshoot of 25% for £ = 0.4 from the response of Figure 5.5(a). Using Figure 5.8,
we estimate that £ = 0.4, as expected.

5.5 THE s-PLANE ROOT LOCATION AND THE TRANSIENT RESPONSE

The transient response of a closed-loop feedback control system can be described in
terms of the location of the poles of the transfer function. The closed-loop transfer
function is written in general as

^ R(s) A(5) '

where A (s) = 0 is the characteristic equation of the system. For the single-loop sys-
tem of Figure 5.4, the characteristic equation reduces to 1 + G(s) = 0. It is the