Using Kalman Filter to estimate the time-varying parameters

Hi Matlab friends,

I wondered if it is even possible to use Kalman filter to estimate the parameter which varies in time. I tried to follow the equations, but still did not achieve a good fit. I produced a sample curves like:

%--------------------------------

figure;

ti=(1:0.25:36)/12; ome=2*pi;

y=cos(ome*ti)+sin(ome*ti); % true signal

yn=cos(ome*ti)+sin(ome*ti)+randn(1,length(ti)); % signal + noise

plot(ti,yn,'g'); hold on; plot(ti,y,'b');

%-----------------------------------

and would love to do the fit to get back the sinusoidal (blue line by) using a Kalman filter. What I did is:

%-----------------------------------

x_t=[1;1]; % initial value

A=[cos(ome) 0;0 sin(ome)]; % design matrix of states

Ex=eye(2); % state error

E_t=Ex; % predicted error

Ez=(1e3)^2; % observation error

z_t=yn; % observation

C=[1 1]; % design matrix of observation

for t = 1:length(ti)

    x_t = A * x_t;

    E_t = A * E_t * A' + Ex;

    K_t=E_t * C' /(C * E_t * C' + Ez); % Kalman gain

    x_t = x_t + K_t *(z_t(t) - C*x_t); % Update state

    E_t = (eye(2) - K_t * C)*E_t; % Update covariance

    % Store parameter

    xHat_t1(t)=x_t(1);

    xHat_t2(t)=x_t(2);

end

y2=xHat_t1.*cos(ome*ti)+xHat_t2.*sin(ome*ti); % construct the fit

plot(ti,y2,'r')

%-----------------------------------

The red fit does not really give me back the original (blue). Any Kalman filter expert please help. Any suggestion would be really helpful here.

[John Petersen]

What are you trying to estimate? Are you trying to estimate the coefficients of the two signals in y? I think your A matrix is wrong. It is constant and probably should be changing with t. Also your covariance values are very large. Probably should choose something smaller.