[ Cross-posted from: https://math.stackexchange.com/questions/164169/estimating-the-input-to-a-system-from-a-system-state ]
I have a system for which I have obtained a non-linear time-varying state-space representation. For this system I am able to measure one of the states. I would like to estimate the input from this.
In my parameter estimation notes I have found a scribbled aside "[the extended Kalman Filter] can be used to estimate the system input if the output is known". So far I've not found any references or sections in my notes that describe the use of the EKF in this way. Can anyone please suggest a reference (or provide) an explanation of how to perform this task?
The system is: $$x(t)= \begin{bmatrix} x_1(t)\\ x_2(t) \end{bmatrix}; \quad \dot{x}(t)= \begin{bmatrix} x_2(t) \\ \alpha(t) x^2_2(t) + \beta(t) + \gamma(t) u_1(t) \end{bmatrix}; \quad y=x_2(t) $$ Where: $\alpha(t)$ and $\gamma(t)$ are constant parameters; $\beta(t)$ is a time-varying parameter; $y=x_2(t)$ is a known (measured) time-varying state; and $u(t)$ is the unknown system input. I have made a simplifying assumption that $\beta(t)$ is constant initially. After proving against this simplified system I want to move to a system where: $$x(t)= \begin{bmatrix} x_1(t)\\ x_2(t) \end{bmatrix}; \quad u(t)= \begin{bmatrix} u_1(t)\\ u_2(t) \end{bmatrix}; \quad y=x_2(t)\\ \quad \dot{x}(t)= \begin{bmatrix} x_2(t) \\ \alpha(t) x^2_2(t) + \beta(t) + \gamma(t) u_1(t) \end{bmatrix}; \quad \beta(t) = f(u_2(t)) $$ Where: $u_1(t)$ is the unknown system input ($u(t)$ in part 1); $u_2(t)$ and $f(u_2(t))$ are a known input and a known function. If someone can point me at a reference for the first part (constant $\beta(t)$) or give a brief explanation of how to approach this then I hope to be able to work to a solution for the second part.
