State estimation with Kalman Filter

F. Haugen: Kompendium for Kyb. 2 ved Høgskolen i Oslo 104
The observability matrix is (n =2)
⎡ £
·
¸
c1 0 ¤ ⎤
· ¸
C
M obs =
CA 2−1 = ⎢ −−−−−−−−−−−
= CA ⎣ £
c1 0 ¤ · ¸ ⎥
1 a ⎦ = c1 0
c 1 ac 1
0 1
(8.10)
The determinant of M obs is
det (M obs )=c 1 · ac 1 − c 1 · 0=ac 1
2
(8.11)
The system is observable only if ac 1 2 6=0.
• Assume that a 6= 0which means that the first state variable, x 1 ,
contains some non-zero information about the second state variable,
x 2 . Then the system is observable if c 1 6=0,i.e.ifx 1 is measured.
• Assume that a =0which means that x 1 contains no information
about x 2 . In this case the system is non-observable despite that x 1 is
measured.
[End of Example 18]
8.3 The Kalman Filter algorithm
The Kalman Filter is a state estimator which produces an optimal
estimate in the sense that the mean value of the sum (actually of any
linear combination) of the estimation errors gets a minimal value. In other
words, The Kalman Filter gives the following sum of squared errors:
a minimal value. Here,
E[e x T (k)e x (k)] = E £ e x1 2 (k)+···+ e xn 2 (k) ¤ (8.12)
e x (k) =x est (x) − x(k) (8.13)
is the estimation error vector. (The Kaman Filter estimate is sometimes
denoted the “least mean-square estimate”.) This assumes actually that the
model is linear, so it is not fully correct for nonlinear models. It is assumed
the that the system for which the states are to be estimated is excited by
random (“white”) disturbances ( or process noise) and that the