How can I find the initial state vector from state space model?

Question

Assume that we have input $u(k)\in \Re $ and output $y(k) \in \Re$ and we estimate the black box model by using subspace identification method.

$$x (k+1) = Ax (k) + Bu(k) $$ $$y (k) = Cx (k) $$

If we know $A \in \Re^{2x2} ,B \in \Re^{2x1},C \in \Re^{1x2}$ and the input and output.

How can we find $x (k) \in \Re^{2}$ at $y (k) $ when $k > 0$ ?

Answer 1

I assume you do not know the initial condition and/or have uncertainty and noise. Otherwise you can use the state equations to propagate the state vector. If the system is observable

https://en.wikipedia.org/wiki/Observability,

then you can use an observer:

https://en.wikipedia.org/wiki/State_observer.

If you have noise in the output and uncertainty in the model, you can use the Kalman filter:

https://en.wikipedia.org/wiki/Kalman_filter

Better resource on KF:

http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf

If your system is not observable, then the state cannot be estimated from the information you have.