In physics, it is common in a system of this sort: $x' = b + A x$

Let $k$ be the size of original $x$ therefore

$y' = b k + A_{k \ast} x$

$x$ as a column vector whose last entry is $y$

$\begin{bmatrix}\vdots \\y\end{bmatrix}' =\begin{bmatrix}\vdots\\b_k\end{bmatrix}+\begin{bmatrix}\vdots\\A_{k \ast}\end{bmatrix} x$

And I want to extend $y'$s derivative in the vector to some larger vector like $\begin{bmatrix}\vdots\\y\\y'\\ \vdots \\y^{(n)}\end{bmatrix}$ $y_1 \equiv y' = b_k + A_{k \ast} x$

Therefore, $y_1' = A_{k \ast} x'= A_{k \ast} (b + A x) = A_{k \ast} b + A_{k \ast} A x$

Then now system is: $\begin{bmatrix}\vdots\\y\\y_1\end{bmatrix}' = \begin{bmatrix}\vdots \\0\\A_{k \ast} b\end{bmatrix} + \begin{bmatrix}\vdots &0\\0&1\\A_{k \ast} A&\end{bmatrix} \begin{bmatrix}x\\y_1\end{bmatrix}$

$y_2 \equiv y_1' = y'' = A_{k \ast} b + A_{k \ast} A x$

Therefore, $y_2' = A_{k \ast} A x' = A_{k \ast} A (b + A x) = A_{k \ast} A b + A_{k \ast} A^2 x$

Then now system is: $\begin{bmatrix}\vdots \\y\\y_1\\y_2\end{bmatrix}'= \begin{bmatrix}\vdots \\0\\0\\A_{k \ast} A b\end{bmatrix} + \begin{bmatrix}\vdots &0&0\\0&1&0\\0&0&1\\A_{k \ast} A^2&&\end{bmatrix} \begin{bmatrix}x\\y_1\\y_2\end{bmatrix}$

And the general formula is $y_s \equiv y_{s-1}'= y^{(s)} = A_{k \ast} A^{s - 2} b + A_{k \ast} A^{s - 1} x$

Applicable when $s \geq$ 2 as $y'$ is already covered in the original system. The extended system has wider dynamics than the original system. Unless the initial conditions of the new system are set by the same formula, the new system will have different dynamic than the original system.