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I have a generic function: $g(s,u)$. Now I want to have a local approximation near the point $(s^{\star}, u^{\star})$ in the quadratic form $$s^{T} Q s + u^{T} R u$$ to apply an optimal control algorithm named LQR. I have to apply a second order Taylor expansion which will result in $$g(s^{\star}, u^{\star}) + g_s(s^{\star}, u^{\star})(x-x^{\star}) + g_u(s^{\star}, u^{\star})(u-u^{\star}) + \frac{1}{2} \left[g_{ss}(s^{\star}, u^{\star})(s-s^{\star})^2 + g_{su}(s^{\star}, u^{\star})2(s-s^{\star})(u-u^{\star}) + g_{uu}(s^{\star}, u^{\star})(u-u^{\star})^2 \right]$$

Now my question is which partial derivatives form the matrices $Q$ and $R$.

I know how to derive this matrices for an example with one variable, then $$\frac{1}{2} x^T Q x + x^T q + q_0$$ can be written as $$\begin{bmatrix}s \\ 1\end{bmatrix}^T \begin{bmatrix}Q & q \\ q^T & q_0\end{bmatrix} \begin{bmatrix}s \\ 1\end{bmatrix}$$ which is a quadratic form. But how to do this with two variables?

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    Rephrased the question here: http://math.stackexchange.com/questions/198088/quadratic-approximation-of-a-cost-function-with-a-taylor-expansion2012-09-17

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