# State propagation from uncertain control input

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*}

• Is there no need to add the term $2\nabla_x f(\mu_x(k),\mu_u(k))\Sigma_{xu}(k)\nabla_u f(\mu_x(k),\mu_u(k))^\top$? – Astghik Hakobyan Jul 8 at 4:14
• Usually $\Sigma_{xu}$ is zero. You'll have to include it if it's not. – holmeski Jul 8 at 13:27