# Offline State Estimation

Imagine I have a nonlinear system: $$\frac{\text{d} x}{\text{d} t} = f(x,t)\\ y = g(x).$$ I can design a nonlinear observer to estimate the state $$x$$ based on the measurements $$y$$. However, this estimation does not need to be done online. Meaning that I have the whole measurement signal $$y(t=0- T_{\text{end}})$$ and would like to estimate the state in the corresponding time interval [$$0, T_{\text{end}}$$]. Is there an offline method for doing so?

I am asking this question, because having the whole measurement signal (previous and future steps for any given state $$x(t)$$) might help improving the estimation accuracy. Something like Matlab filtfilt, which removes the potential filtering phase shifts by doing filtering from both time ends.

• You tag this as "non-linear", but the basic Kalman filter and Luenberger observer are linear systems. Is your system, indeed, nonlinear? Jan 19 at 19:25
• Yes, my system is nonlinear. You are right, I will edit the question.
– Ali
Jan 19 at 19:31