Wireless Sensor and Actuator Networks for Lighting Energy ...

The time update equations and measurement update equations can be derived as (5.3)
and (5.6) respectively.
k
x k +1
= Ax k
k
(5.3)
k
P k +1
= AP k k A T + Q (5.4)
K k +1
= P k k +1
C T CP k
k +1
C T + R
k
x +1 k
k +1
= x k +1
( ) 1 (5.5)
k
+ K k +1 ( y k +1
Cx k +1 ) (5.6)
k
P +1 k
k +1
= ( I K k +1
C)P k +1
(5.7)
where
k
x k +1
is the predicted state of the system for time k+1 using measurements up to k;
x k k is the estimated state of the system at time k given measurements up to k;
k
P k +1
k
( )( x k +1
ˆx k +1 ) T
= E
k
x k +1
ˆx
k +1
using measurements up to time k;
is the error covariance matrix for time k+1
P k k
= E
k
x k
ˆx
k
k
( )( x k
ˆx k ) T
is the error covariance matrix of time k given
measurements up to time k;
K k+1 is the Kalman gain for time k+1.
The prediction and corresponding prediction error generated by the Kalman
filtering approach on a set of daylight data is shown in Figure 5-3, where the blue solid
line in (a) is the true daylight data sampled every ten seconds and the green dashed line
is the prediction. The mean square error (MSE) is 1800.2, which is large because the
Kalman filtering method was started with random guesses of initial values. The MSE is
22.7 if the initial guesses are not used.
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