Page 499 - Mechanical Engineers' Handbook (Volume 2)

490 Closed-Loop Control System Analysis

2
J e (t) dt (98)
0
A system designed by this criterion tends to show a rapid decrease in a large initial
error. Hence the response is fast and oscillatory. Thus the system has poor relative stability.
2
The ISE is of practical significance because e (t) dt resembles power consumption for some
systems.

7.6 Integral of Time-Multiplied Absolute-Error (ITAE) Criterion

According to the ITAE criterion, the optimum system is the one that minimizes the perform-
ance index:

J t e(t) dt (99)
0
This criterion weighs large initial errors lightly, and errors occurring late in the transient
response are penalized heavily. A system designed by use of the ITAE has a characteristic
that the overshoot in the transient response is small and oscillations are well damped.

7.7 Comparison of Various Error Criteria 2
Figure 34 shows several error performance curves. The system considered is

C(s) 1
(100)
R(s) s 2 s 1
2
The curves of Fig. 34 indicate that 0.7 corresponds to a near-optimal value with
respect to each of the performance indices used. At 0.7 the system given by Eq. (100)
results in rapid response to a step input with approximately 5% overshoot.
Table 5 summarizes the coefficients that will minimize the ITAE performance criterion
for a step input to the closed-loop transfer function 5
C(s) a 0
(101)
n
R(s) s a n 1 s n 1 as a 0
1
2
Table 6 summarizes the coefficients that will minimize the ITAE performance criterion
for a ramp input applied to the closed-loop transfer function:
C(s) bs b 0
1
(102)
n
R(s) s b s n 1 bs b
n 1 1 0
Figure 35 shows the response resulting from optimum coefficients for a step input ap-
plied to the normalized closed-loop transfer function given in Eq. (101), for ISE, IAE
(integral absolute error), and ITAE.
8 SIMULATION FOR CONTROL SYSTEM ANALYSIS
Often in control system design the governing differential equations are reduced to a form
that facilitates the shaping of a controller. The model order reductions involve things such
as linearization, neglecting fast dynamics, parasitics, time delays, and so on. Once a con-
troller is synthesized for a system, after making simplifying assumptions, it becomes nec-