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## Examples treated using Matlab software

First example of Kalman filtering

The objective is to estimate an unknown constant drowned in noise.

This constant is measured using a noise sensor.

The noise is centered, Gaussian and of variance equal to 1.

The initial conditions are equal to 0 for the estimate and equal to 1 for the variance of the estimation error.

```clear
t=0:500;
R0=1;
constant=rand(1);
n1=randn(size(t));
y=constant+n1;

subplot(2,2,1)
%plot(t,y(1,:));
plot(t,y,’k’);% in B&W

grid
title(‘sensor’)
xlabel(‘time’)
axis([0 500 -max(y(1,:)) max(y(1,:))])

R=R0*std(n1)^2;% variance of noise measurement

P(1)=1;%initial conditions on variance of error estimation
x(1)=0;

for i=2:length(t)
K=P(i-1)*inv(P(i-1)+R);
x(i)=x(i-1)+K*(y(:,i)-x(i-1));
P(i)=P(i-1)-K*P(i-1);
end
err=constant-x;
subplot(2,2,2)
plot(t,err,’k’);
grid
title(‘error’);
xlabel(‘time’)
axis([0 500 -max(err) max(err)])

subplot(2,2,3)
plot(t,x,’k’,t,constant,’k’);% in W&B
title(‘x estimated’)
xlabel(‘time’)
axis([0 500 0 max(x)])
grid

subplot(2,2,4)
plot(t,P,’k’);% in W&B
grid,axis([0 100 0 max(P)])
title(‘variance error estimation’)
xlabel(‘time’)```

Figure 7.3. Line graph of measurement, error, best filtration and variance of error

Second example of Kalman filtering

The objective of this example is to extract a dampened sine curve of the noise.

The state vector is a two component column vector:

The system noise is centered, ...

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