Final_1.pdf

Kalman Filtering 59

Kalman gain must be such that the error variance of the final estimated
state is minimized. Writing out the variance for the final state estimate we
have

(4.8)

The task is now to find the value of kthat minimizes the variance. In-
stead of doing the derivation mathematically, we will arrive at the value of
kby analogy to some high school circuit theory. Consider the parallel circuit
as shown in Figure 4.2.
The fraction of current flowing into each arm of the circuit kand 1 – k
is shown in Equation 4.10. According to Ohm’s law, the current flow
chooses the path of least resistance, and the flow on each arm is inversely
proportional to the resistance of that arm. According to Kirchoff’s law, the
sum of the current flowing through each arm of the circuit must be equal to
the total current flow I. Also, the flow of current is such that minimum en-
ergy is expended in the process. The energy of the circuit is given as

(4.9)

Equating , it is now easy to see the similarity between
the two situations. The fraction of current and the Kalman gain is given as

k= (4.10)

+

σ

σσ

ε

ηε

2

22

RRεε ηη==σσ^22 and

EI=−()kR kR+







2 1 2 2

εη

var()xkxkykkˆii++ 11 ||=−()()varˆii+ + var()i=−(),,i+ i

2

1

112 2 σσ^222

εη

FIGURE 4.2 A Parallel Circuit.

Re

Rh

I(1-k)

I.k

I