Proposition (Linearization of Nonlinear Control System)

The linearization of a control system for $t>t_{0}$ around $$\tilde{p}=(\tilde{x}(t),\tilde{u}(t))$$ of$$\begin{align*} \dot{x}(t)&=f(x(t),u(t),t)\\ \dot{y}(t)&=g(x(t),u(t),t) \end{align*}$$is given by$$\begin{cases} \dot{x}(t)=\frac{ \partial f }{ \partial x } |_{\tilde{p}(t)}x(t)+\frac{ \partial f }{ \partial u } |_{\tilde{p}(t)}u(t) \\ \dot{y}(t)=\frac{ \partial g }{ \partial x } |_{\tilde{p}(t)}x(t)+\frac{ \partial g }{ \partial u } |_{\tilde{p}(t)}u(t) \end{cases}$$