Page 420 - Mechanical Engineers' Handbook (Volume 2)

7 Further Criteria for Gain Selection 411

J e(t) dt (IAE Index) (25)
0

J t e(t) dt (ITAE Index) (26)
0

J [e(t)] dt (ISE Index) (27)
2
0

J t[e(t)] dt (ITSE Index) (28)
2
0
where e(t) is the system error. This error usually is the difference between the desired and
the actual values of the output. However, if e(t) does not approach zero as t → , the
preceding indices will not have finite values. In this case, e(t) can be defined as e(t) c( )
c(t), where c(t) is the output variable. If the index is to be computed numerically or
experimentally, the infinite upper limit can be replaced by a time t large enough that e(t)is
ƒ
negligible for t t .
ƒ
The integral absolute-error (IAE) criterion (25) expresses mathematically that the de-
signer is not concerned with the sign of the error, only its magnitude. In some applications,
the IAE criterion describes the fuel consumption of the system. The index says nothing about
the relative importance of an error occurring late in the response versus an error occurring
early. Because of this, the index is not as selective as the integral-of-time-multiplied absolute-
error (ITAE) criterion (26). Since the multiplier t is small in the early stages of the response,
this index weights early errors less heavily than later errors. This makes sense physically.
No system can respond instantaneously, and the index is lenient accordingly, while penalizing
any design that allows a nonzero error to remain for a long time. Neither criterion allows
highly underdamped or highly overdamped systems to be optimum. The ITAE criterion
usually results in a system whose step response has a slight overshoot and well-damped
oscillations.
The integral squared-error (ISE) and integral-of-time-multiplied squared-error (ITSE)
criteria are analogous to the IAE and ITAE criteria, except that the square of the error is
employed for three reasons: (1) in some applications, the squared error represents the sys-
tem’s power consumption; (2) squaring the error weights large errors much more heavily
than small errors; (3) the squared error is much easier to handle analytically. The derivative
of a squared term is easier to compute than that of an absolute value and does not have a
discontinuity at e 0. These differences are important when the system is of high order
with multiple error terms.
The closed-form solution for the response is not required to evaluate a performance
index. For a given set of parameter values, the response and the resulting index value can
be computed numerically. The optimum solution can be obtained using systematic computer
search procedures; this makes this approach suitable for use with nonlinear systems.

7.2 Optimal-Control Methods
Optimal-control theory includes a number of algorithms for systematic design of a control
law to minimize a performance index, such as the following generalization of the ISE index,
called the quadratic index:

T
J (xQx uRu) dt (29)
T
0