On possiblity to go from state space formulation to an ODE or system of such

Question

I seen the canonical way one goes from a nth order ODE to a state space formulation. Is the always a corresponding way back given i.e given any state space model can we find an ODE or system thereof, maybe on some condition on choice of state variables?

Answer 1

If you start of with a single input linear time invariant state space model defined as

$$ \dot{x} = A\,x + B\,u \\ y = C\,x + D\,u $$

then you can write it as one $n$th order ODE if the pair $(A,B)$ in controllable. With $n$ the dimension of $x$. In order to achieve this you have to perform a similarity transformation which brings the state space model into the controllable canonical form. If the state space model has a single input then this transformation $z = T\,x$ can be found with

$$ \vec{v} = \begin{bmatrix}0 & \cdots & 0 & 1\end{bmatrix} \begin{bmatrix}B & A\,B & \cdots & A^{n-1} B\end{bmatrix}^{-1}, $$

$$ T = \begin{bmatrix}\vec{v} \\ \vec{v}\,A \\ \vdots \\ \vec{v}\,A^{n-1}\end{bmatrix}. $$

So that

$$ \dot{z} = T\,A\,T^{-1}z + T\,B\,u \\ y = C\,T^{-1}z + D\,u $$

with

$$ T\,A\,T^{-1} = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & 0 & 1 \\ a_1 & \cdots & \cdots & \cdots & a_n \end{bmatrix}, $$

$$ T\,B = \begin{bmatrix}0 \\ \vdots \\ 0 \\ 1\end{bmatrix}. $$

The new states $z$ can be seen as

$$ z = \begin{bmatrix}z_1 \\ z_2 \\ \vdots \\ z_n\end{bmatrix} = \begin{bmatrix}z_1 \\ \frac{d}{dt}z_1 \\ \vdots \\ \frac{d^{n-1}}{dt^{n-1}}z_1\end{bmatrix}, $$

so substituting this into the system of linear first order differential equations then its last equation can be seen as a $n$th order differential equation.

PS: I am not sure if in general there also exists such similarity transformation if there are multiple inputs.