Figure 3.47: Basic AFC scheme for RRO compensation at single frequency ω i .
The continuous-time adaptive control to adjust the estimates of a i (t)andb i (t)
consists the following update laws as shown in Figure 3.47,
dt a i (t)=g i y(t)cos(ω i t),
dt b i (t)=g i y(t)sin(ω i t),
where y(t) is the position error signal (PES), g i > 0 is the adaptation gain and
ω i is the desired compensation frequency.
Based on Laplace transform analysis, Bodson et al  showed that the
adaptive control scheme of equations 3.94 and 3.95 is equivalent to the scheme
based on internal model principle
s 2 +ω i
C r (s)=g i
in the sense that given the same disturbance d i (t), the responses y(t)are
identical for zero initial conditions. Hence the stability of the AFC system, for
both single and multiple frequency runout compensation cases, can be verified
by checking the stability of the closed-loop system consisting of O and C r .
Furthermore, the adaptive control is stable for all g if O is positive real; when