I want to estimate F (2*2) and H (N*2) in a state space model. The system is considered free of noise and the states (for t=1,2,...) are given
x(t+1)=Fx(t)
y(t)=Hx(t)
y (N*1) are the observations. I am not sure whether it's still called a SSM. Could anyone please suggest me the best way to estimate F and H? Any help is appreciated.
Many Thanks,
Ahammed
Ahammed,
This is a very well studied problem. A variety of techniques have been studied and reported.
I would suggest a classic book for you to start with (unless you have already used it):
Lennart Ljung, System Identification: Theory for the User, Prentice Hall.
-Sanjay
Good comments and suggestions from the Sanjay college on the identification of enough known systems.
Thank you very much, Morteza, Prof. Sanjay and Mohamed-Mourad for your valuable comments and suggestions
First write the system equations in matricial form (I will use Matlab notation):
[x(t+1);y(t)]=[F;H*]x(t)
Then define
Z=[x(2) ... x(N); y(1) ... y(N-1)]
X=[x(1) ... x(N-1)]
The parameters [F;H] are related with Z and X by the equation
Z=[F;H]*X
and [F;H] can be estimated by solving this equation in the least squares sense:
[F;H]=Z*X'(X'*X)^(-1)
Thank you very much, Prof. P. Lopes dos Santos. Wonderful answer!
The last line is [F;H]=Z*X'(X*X')^(-1), I guess.
Sorry, aain, I meant right, not write, it was the stupid iPad that changed the word.
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