Page 358 - Modern Control Systems

332 Chapter 5 The Performance of Feedback Control Systems

The performance index ITAE provides the best selectivity of the performance
indices; that is, the minimum value of the integral is readily discernible as the system
parameters are varied. The general form of the performance integral is

1 = /(e(0, r(t), y(/), 0 dt % (5.41)

where / is a function of the error, input, output, and time. We can obtain numerous
indices based on various combinations of the system variables and time. Note that
the minimization of IAE or ISE is often of practical significance. For example, the
minimization of a performance index can be directly related to the minimization of
fuel consumption for aircraft and space vehicles.
Performance indices are useful for the analysis and design of control systems.
Two examples will illustrate the utility of this approach.

EXAMPLE 5.6 Performance criteria
A single-loop feedback control system is shown in Figure 5.26, where the natural
frequency is the normalized value, o) n = 1. The closed-loop transfer function is then

1
T(s) = 2 (5.42)
s + 2£s + 1

Three performance indices—ISE, ITAE, and ITSE—calculated for various values
of the damping ratio £ and for a step input are shown in Figure 5.27. These curves
show the selectivity of the ITAE index in comparison with the ISE index. The value
of the damping ratio £ selected on the basis of ITAE is 0.7. For a second-order sys-
tem, this results in a swift response to a step with a 4.6% overshoot. •

Ms) O •"

(a)

FIGURE 5.26 1 1 • Y(s)
Single-loop k s + 2£ s
feedback control
system, (a) Signal-
flow graph.
(b) Block diagram
model. (b)