Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 37
Vilà-Valls et al.
Figure 6.
F-PLL loop architecture block diagram.
with α = α1 + α2 + α3 and β = β1 + β2.
Using the knowledge on the equivalence between loops, the
most obvious and direct application of KFs to emulate the F-PLL
makes use of the same architecture by replacing both PLL and
FLL with two cooperative KFs. The first KF would be in charge
(1)
(2)
of x k = [θ k f k f k ] and the second one of x k = [ f k f k ] , with an
interaction between both using the output of the second-filter as a
frequency input to the first KF.
But from an optimal filtering point of view, this in not necessary because a single filter can optimally solve the estimation
problem, that is, the sequential estimation of xk = [θ k f k f k ] using
the available observations up to time k. To obtain a fair comparison, the KF should use both phase and frequency discriminators;
therefore, the observation equation should be modified to account
for both measurements. The resulting KF predictor form is
xˆ k +1| k
r (1)
= Fk xˆ k | k −1 + Fk K k k(2) ,
rk
(35)
where rk(1) and rk(2) are the outputs of the KF phase and frequency
discriminators, respectively, and the Kalman gain
α1, k
K k = α 2, k
α 3, k
0
β1, k .
β 2, k
JULY 2017, Part II of II
(36)
The corresponding predicted phase is given by
θˆk +1|k = θˆk |k −1 + Ts fˆk |k −1 +
ˆ
Ts2 fk | k −1
2
+ α k rk(1) + β k rk(2) ,
(37)
with αk = α1,k + Tsα2,k + Ts2a3/2 and β = Tsβ1 +Ts2β2/2. Analyzing (34)
and (37), it is straightforward to see that the F-PLL equivalent is
obtained with a time-invariant Kalman gain equal to
α1
K = α 2 / Ts
2α 3 / Ts2
0
β1 / Ts ,
2 β 2 / Ts2
(38)
which leads to the following final phase prediction:
θˆk +1|k = θˆk |k −1 + fˆk | k −1 +
ˆ
fk | k −1
2
+ α rk(1) + β rk(2) ,
(39)
again with α = α1 + α2 + α3 and β = β1 + β2. This result confirms the
expression for the time-invariant equivalent Kalman gain previously introduced, and it is equivalent to the compact F-PLL expression in (34). Notice that the gains in the F-PLL are adjusted
heuristically, while the KF sequentially computes the optimal gain
and provides the optimal solution to the problem. Therefore, the
proposed KF is equivalent to the F-PLL formulation, except for the
values of the time-variant coefficients, which, in fact, are optimally
IEEE A&E SYSTEMS MAGAZINE
37
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