State space representation

        A blessing from above

Cool we know systems now, but it's a bit of an ugly system representation to be honest. Luckily we live in a universe where the state space representation exists and we can recast any system as a vector first order ODE and an output equation. This representation is of the form $$ \dot x = f(x, u)$$ $$ y = h(x, u),$$ where $x$ is the state of the system, $u$ the input, and $y$ the output. The state dynamics are described with $f: \mathbb{R}^n\times\mathbb{R}^p\rightarrow\mathbb{R}^n$ and the output measurements with $h: \mathbb{R}^n\times\mathbb{R}^p\rightarrow\mathbb{R}^q$.

What black magic do we perform to get these first order ODEs? Well, suppose you have a second order ODE in $v$, $\ddot v = \dot v + v$, then this is equivalent to the first order ODE $$ \begin{bmatrix}\dot v\\\ddot v\end{bmatrix} = \begin{bmatrix}\dot v\\ \dot v + v\end{bmatrix} = \begin{bmatrix}0&1\\1&1\end{bmatrix}\begin{bmatrix}\dot v\\ v\end{bmatrix}.$$

Last but not least of the amazing aspects of the state space representation: there are many nice numerical integrators to simulate them. Think forward Euler or Runge-Kutta.