Can anyone tell me how to implement the following time-varying state space model

Answer 1

[nick]

Hi Bella,

I understand that you are facing trouble in implementation of state space model with additive white gaussian noise.

The “kalman” and “ss” function can be used for such an implementation. 

% Define the system matrices

A = [0.8, 0.2; -0.3, 0.5]; %state transition matrix

B_ = [zeros(size(A,1),1) eye(size(A,1))]; % there is no effect of input, so B=0 and G is identity; B_=[B G]

C = [1 0]; %measurement matrix

D_ = [zeros(size(C,1),1) zeros(size(C,1),size(A,1))]; %H is 0 of appropriate dimension according to question and again due to similar reasons in B_, D=0 ; D_= [D H]

Q = 0.1*eye(size(A,1)); % assumed process covar

R = 0.2*eye(size(C,1)); % assumed measurement covar

N = zeros(size(A,1), size(C,1)); % Nw by Ny matrix ; considered as 0 as there is no correlation between process noise and measurement noise

sys = ss(A,B_,C,D_); % since your state-space model is input independent B=0 and D=0

[kalmf,l,p] = kalman(sys,Q,R,N) %kalmf is state space model that produces state estimates and true plant output estimates

Kindly refer to the below documentations of “kalman” and “ss” function :

https://www.mathworks.com/help/control/ref/ss.kalman.html#:~:text=Plant%20model%20with%20process%20noise%2C%20specified%20as%20a%20state%2Dspace%20(ss)%20model.%20The%20plant%20has%20known%20inputs%20u%20and%20white%20process%20noise%20inputs%20w.%20The%20plant%20output%20yt%20does%20not%20include%20the%20measurement%20noise.

https://www.mathworks.com/help/control/ref/ss.html?searchHighlight=ss&s_tid=srchtitle_support_results_1_ss#:~:text=example-,sys%20%3D%20ss(A%2CB%2CC%2CD,y%3DCx%2BDu,-For%20instance%2C%20consider

Hope it helps.

Regards,

Neelanshu