User`s guide

7 Recursive Techniques for Model Identification
To spec ify the Kalman filter algorithm, set adm to 'kf' and adg to the value
of the drift matrix R
1
(described in “Mathematics of the Kalman Filter
Algorithm” on page 7-8).
Forg etting Factor Algorithm
• “Mathematics of the Forgetting Factor Algorithm” on page 7 -10
• “Using the Forgetting Factor Algorithm” on page 7-11
Mathematics of the For getting Factor Algorithm
The following set of equations summarizes the forgetting factor adaptation
algorithm:
ˆˆ
ˆ
θθt t Kt yt yt
()
=−
()
+
() ()
−
()
()
1
ˆ
ˆ
yt t t
T
()
=
()
−
()
ψθ1
Kt Qt t
()
=
() ()
ψ
Qt Pt
Pt
tPt t
T
()
=
()
=
−
()
+
()
−
()()
1
1λψ ψ
Pt Pt
Pt t t Pt
tPt t
T
T
()
=−
()
−
−
( ) () ()
−
()
+
()
−
()()
⎛
⎝
⎜
⎜
⎞
⎠
1
1
11
1
λ
ψψ
λψ ψ
⎟⎟
⎟
To obtain Q(t), the following function is minimized at time t:
λ
tk
k
t
ek
−
=
∑
()
2
1
This approach discounts old measurements exponentially such that an
observation that is
τ
samples old carries a weight that is equal to
λ
τ
times
the weight of the most recent observation.
τ
λ
=
−
1
1
represents the memory
7-10