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[ Cross-posted from: http://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.

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Is $\beta(t)$ known? My initial stab at approaching this would be to see if there is a good way to reformulate the problem such that the state $x_2(t)$ that you know can be modeled as the input to the system, and the unknown $u_1(t)$ is the state that you're trying to estimate. You would start by trying to solve for $u_1(t)$ in terms of the measurement $x_2(t)$. –  Jason R Aug 13 '12 at 17:18
    
It is (or at least I can assume it is, just as I'm assuming that it's constant to get myself started)! The system is a vehicle, so $x_1$ is distance and $x_2$ is velocity (or speed). I'm trying to get to a point where I can estimate the input power to the vehicle (via the force acting on it, $u_1(t)$, given some known characteristics and the speed. From memory (notes not with me) $\alpha$ is $\frac{\rho C_d A}{2m}$, $\gamma$ is $1/m$ and $m \beta$ is the combination of rolling resistance and gravitational force if the vehicle is on a slope (I'm assuming zero slope for the sake of my sanity!). –  IainCunningham Aug 13 '12 at 21:03
    
Didn't quite have room to say, 1) thanks for responding, 2) that $u_2$ is the slope the vehicle is travelling up/down. –  IainCunningham Aug 13 '12 at 21:09

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