Page 356 - Modern Control Systems

330 Chapter 5 The Performance of Feedback Control Systems

The loop transfer function of the equivalent unity feedback system is Z(s). It follows
that the error constants for nonunity feedback systems are given as:

2
= limZ(s), = lim sZ(s), and = lim s Z(s).
K p K v K a
s-*0 .v—»0 s—*[)
Note that when H(s) = 1, then Z(s) = G c(s)G(s) and we maintain the unity feedback
error constants. For example, when H(s) = 1, then K p = lim Z(s) = lim G c(s)G(s),
as expected. s ~* s_ *

5.7 PERFORMANCE INDICES

Increasing emphasis on the mathematical formulation and measurement of control
system performance can be found in the recent literature on automatic control.
Modern control theory assumes that the systems engineer can specify quantitatively
the required system performance. Then a performance index can be calculated or
measured and used to evaluate the system's performance. A quantitative measure of
the performance of a system is necessary for the operation of modern adaptive con-
trol systems, for automatic parameter optimization of a control system, and for the
design of optimum systems.
Whether the aim is to improve the design of a system or to design a control sys-
tem, a performance index must be chosen and measured.

A performance index is a quantitative measure of the performance
of a system and is chosen so that emphasis is given
to the important system specifications.

A system is considered an optimum control system when the system parameters
are adjusted so that the index reaches an extremum, commonly a minimum value.
To be useful, a performance index must be a number that is always positive or zero.
Then the best system is defined as the system that minimizes this index.
A suitable performance index is the integral of the square of the error, ISE,
which is defined as
2
ISE = / e (t) dt. (5.37)
Jo
The upper limit T is a finite time chosen somewhat arbitrarily so that the integral
approaches a steady-state value. It is usually convenient to choose T as the settling
time T s. The step response for a specific feedback control system is shown in Figure
5.25(b), and the error in Figure 5.25(c). The error squared is shown in Figure 5.25(d),
and the integral of the error squared in Figure 5.25(e). This criterion will discriminate
between excessively overdamped and excessively underdamped systems. The mini-
mum value of the integral occurs for a compromise value of the damping. The perfor-
mance index of Equation (5.37) is easily adapted for practical measurements because a
squaring circuit is readily obtained. Furthermore, the squared error is mathematically
convenient for analytical and computational purposes.