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02_200256_CH02/Bergren 4/17/03 11:24 AM Page 63

FIGURE 2-30 Gauss and Legendre CONTROL SYSTEMS 63

The cost function increases in magnitude as the square of the errors. The step size,
used to recover from errors, then increases linearly proportional to the error.
Specifically then, since

X1t 12 X1t2 S1t2 1d C1X1t22>d X1t22

we have the two elements iterated as follows:

X11t 12 X11t2 S11t2 1X11t2 X1d1t22

X21t 12 X21t2 S21t2 1X21t2 X2d1t22

If we were to set step sizes S1(t) S2(t) 0.1, then

X11t 12 0.9 X11t2 0.1 X1d1t22

X21t 12 0.9 X21t2 0.1 X2d1t22

Thus, X1 and X2 slowly seek the values of X1d and X2d. Also, X(t) slowly seeks the
value of Xd(t).
Before we look at cost functions other than LMS, let’s finish answering some of the
other questions we posed earlier.