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Consider a nonlinear system $x(k+1)=f(x(k),u(k))$, where $x(k)\in\mathbb{R}^{n}$ is the state, $u(k)\in\mathbb{R}^m$ is the control input. Here $u(k)$ is normally distributed RV with mean $\mu_u(k)$ and variance $\Sigma_u(k)$. I want to find the distribution of $x(k)$ starting from some deterministic $x(0)$ for some horizon $K$. Then can the ordinary EKF update equation be used for the prediction in the below way?

\begin{align*} \mu_x(k+1)&=f(\mu_x(k),\mu_u(k)),\\ \Sigma_x(k+1)&=\nabla_x f(\mu_x(k),\mu_u(k))\Sigma_x(k)\nabla_x f(\mu_x(k),\mu_u(k))^\top+\nabla_u f(\mu_x(k),\mu_u(k))\Sigma_u(k)\nabla_u f(\mu_x(k),\mu_u(k))^\top \end{align*}

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Yes. This approach is commonly used when IMUs are present because they measure the rate of state change with some uncertainty.

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  • $\begingroup$ Is there no need to add the term $2\nabla_x f(\mu_x(k),\mu_u(k))\Sigma_{xu}(k)\nabla_u f(\mu_x(k),\mu_u(k))^\top$? $\endgroup$
    – Astghik Hakobyan
    Jul 8 '20 at 4:14
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    $\begingroup$ Usually $\Sigma_{xu}$ is zero. You'll have to include it if it's not. $\endgroup$
    – holmeski
    Jul 8 '20 at 13:27
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    $\begingroup$ It being non zero would mean the random control input is correlated with your state. $\endgroup$
    – holmeski
    Jul 8 '20 at 13:28

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