State Space Model in Controllable Canonical Form

Question

Mathematica by default puts state space model realizations in controllable companion form, as seen here:

tfsys = TransferFunctionModel[(b1 s^2 + b2 s + b3)/(s^3 + a1 s^2 + 
      a2 s + a3 ), s];
StateSpaceModel[tfsys]

Which outputs a block matrix like: $$ \begin{bmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -a_3 & -a_2 & -a_1 & 1 \\ b_3 & b_2 & b_1 & 0\end{bmatrix} $$ However, I want it in controllable canonical form, which should look like: $$ \begin{bmatrix}-a_1 & -a_2 & -a_3 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ b_1 & b_2 & b_3 & 0\end{bmatrix} $$ StateSpaceModel offers the StateSpaceRealization option but it only has ControllableCompanion and ObservableCompanion, neither of which is what I want. Is there a simple way of getting the right state space form?

Answer 1

StateSpaceModel[TransferFunctionModel[(b1 s^2 + b2 s + b3)/(s^3 + a1 s^2 + a2 s + a3), s]]

And we can get the form you want by selecting the states in the desired order.

SystemsModelExtract[%, All, All, Reverse@Range@3]

There doesn't seem to be a consensus on which of the above is the correct 'controllable canonical form'. Here are some sources (link, link) that call what is returned by Mathematica as the controllable canonical form.