How to write a second-order dynamical system in first order form?

Question

Given a second-order dynamical system: $$ \ddot{x} = f(t,x,u)$$

How do I convert it the equivalent first-order system? $$ \dot{\vec{z}} = \vec{g}(t,\vec{x},u)$$

Answer 1

Define two new state variables: $$ x_1 = x $$ $$ x_2 = \dot{x} $$

Now you can rewrite the original system: $$ \ddot{x} = f(t,x,u) \quad \to \quad \dot{x}_2 = f(t,x_1,u)$$

Combining these equations yields a system of first-order system: $$ \dot{x}_1 = x_2 $$ $$ \dot{x}_2 = f(t,x_1,u) $$

Writing in vector form: $$ \vec{z} = \begin{bmatrix}x_1\\x_2\end{bmatrix} $$

$$ \dot{\vec{z}} = \vec{g}(t,\vec{z},u) = \begin{bmatrix}x_2\\f(t,x_1,u)\end{bmatrix} $$