User`s guide

7 Recursive Techniques for Model Identification
For models that do no t hav e the linear regression form, it is not possible to
compute exactly the predicted output and the gradie nt
ψ t
()
for the current
parameter estimate
ˆ
θ t −
()
1
. To learn how you can compute approx imatio n for
ψ t
()
and
ˆ
θ t −
()
1
for general model structures, see the section on recurs ive
prediction-error methods in System Identification: Theory for the User by
Lennart Ljung (Prentice Hall PTR, Upper Saddle River, NJ, 1999).
Kalman Filter Algorithm
• “Mathematics of the Kalman Filter Algorithm” on page 7-8
• “Using the Kalman FilterAlgorithm”onpage7-9
Mathematics of the Kalman Filter Algorithm
The following set of equations sum m a ri zes the Kalman filter adaptation
algorithm:
ˆˆ
ˆ
θθt t Kt yt yt
()
=−
()
+
() ()
−
()
()
1
ˆ
ˆ
yt t t
T
()
=
()
−
()
ψθ1
Kt Qt t
()
=
() ()
ψ
Qt
Pt
RtPt t
T
()
=
−
()
+
()
−
()()
1
1
2
ψψ
Pt Pt R
Pt t t Pt
RtPt t
T
T
()
=−
()
+−
−
( ) () ()
−
()
+
()
−
()()
1
11
1
1
2
ψψ
ψψ
This formu la t ion a ssumes the linear-reg ression fo rm of the model:
yt t t et
T
()
=
() ()
+
()
ψθ
0
7-8