Page 365 - Modern Control Systems

Section 5.8 The Simplification of Linear Systems 339

Table 5.7 The Optimum Coefficients of T(s) Based
on the ITAE Criterion for a Ramp Input
2
s 2 + 3.2(o ns + (o T
2
5 3 + 1.75co ;i s + 325afe + a? n
5 4 + 2Al(o„s 3 + 4.93cols 2 + 5.14w;V + w 4
4
2
5 5 + 2.19(0,/ + 6.5Q(o / + 6.30<ojy + 5.24co s + cof,
The locations of the closed-loop roots dictated by the ITAE system are shown
in Figure 5.32. The damping ratio of the complex roots is £ = 0.44. However, the
complex roots do not dominate. The actual response to a step input using a comput-
er simulation showed the overshoot to be only 2% and the settling time (to within
2% of the final value) to be equal to 0.75 second.
For a ramp input, the coefficients have been determined that minimize the
ITAE criterion for the general closed-loop transfer function [6]

b\S + bn
5
T(s) = r • ( -50
v
s" + />„_i* + ••• + b vs + b Q
This transfer function has a steady-state error equal to zero for a ramp input. The
optimum coefficients for this transfer function are given in Table 5.7. The transfer
function, Equation (5.50), implies that the process G(s) has two or more pure inte-
grations, as required to provide zero steady-state error. •

5.8 THE SIMPLIFICATION OF LINEAR SYSTEMS

It is quite useful to study complex systems with high-order transfer functions by
using lower-order approximate models. For example, a fourth-order system could be
approximated by a second-order system leading to a use of the performance indices
in Figure 5.8. Several methods are available for reducing the order of a systems
transfer function.
One relatively simple way to delete a certain insignificant pole of a transfer
function is to note a pole that has a negative real part that is much more negative
than the other poles. Thus, that pole is expected to affect the transient response
insignificantly.
For example, if we have a system with transfer function

G(s) = K
s(s + 2)(s + 30)'

we can safely neglect the impact of the pole at s = -30. However, we must retain
the steady-state response of the system, so we reduce the system to

r( G(s) ^ = {K/30)
s(s + 2)'