Page 361 - Modern Control Systems

Section 5.7 Performance Indices 335

FIGURE 5.29
The performance
indices of the
telescope control 7 8 910
system versus K 3.

and 5.7, we found that the optimum setting varied for different performance
indices.
The coefficients that will minimize the ITAE performance criterion for a step
input have been determined for the general closed-loop transfer function [6]

Y(s) = bj>
T(s) = 1 (5.47)
R(s) / + V / " + ••• +bis + h

This transfer function has a steady-state error equal to zero for a step input. Note
that the transfer function has n poles and no zeros. The optimum coefficients for the
ITAE criterion are given in Table 5.6. The responses using optimum coefficients for
a step input are given in Figure 5.30 for ISE, IAE, and ITAE. The responses are pro-
vided for normalized time co„t. Other standard forms based on different perfor-
mance indices are available and can be useful in aiding the designer to determine
the range of coefficients for a specific problem. A final example will illustrate the
utility of the standard forms for ITAE.

EXAMPLE 5.8 Two-camera control
A very accurate and rapidly responding control system is required for a system that
allows live actors to appear as if they are performing inside of complex miniature
sets. The two-camera system is shown in Figure 5.31(a), where one camera is trained
on the actor and the other on the miniature set. The challenge is to obtain rapid and
accurate coordination of the two cameras by using sensor information from the

Table 5.6 The Optimum Coefficients of 7"(s) Based on the
ITAE Criterion for a Step Input
S + (O n
2
s 2 + \Ao) ns + w ,
s 3 + 1.15(0,/ + 2A5(ols + a>l
2 2
x 4 + 2.1<o„s 3 + 3.4(o ns + 2.1 (ols + <4
A
4
s 5 + 2.8<o ns + 5.Q(o;/ + 5.5(of/ + 3.4a> „s + to*
/ + 3.25(0,/ + 6.ffi(x>ls A + 8.60(o;V + 7.45^,5 2 + 3.95a>j|s + <o,, 6