Is there a way to transform state-space matrices $A$ and $C$ into canonical forms for stochastic systems?

Question

I have a stochastic MIMO state space system as follows: \begin{align*} x_{k+1} &= Ax_k + w_k\\ y_k &= Cx_k + v_k \end{align*}

I want to find the transformation matrix $T$ for converting $A$ to observable canonical form $A_o$ using relation \begin{equation*} A_o = T^{-1} A T \end{equation*} and with that, $C$ gets transformed to $C_o = CT$. I tried using MATLAB canon function, but it needs the knowledge of $B$ matrix. If $B$ is set to zero matrix then canon throws out an error. Is there any way to get the transformation matrix $T$ without the need for matrix $B$?

Any reference would be greatly appreciated. Thanks!

Answer 1

Observer forms do not depend on the matrix B. Pick any.

Whether the system is stochastic or deterministic isn't important from this point of view.

Note however that there isn't one form that can be called THE canonical form for multivariable systems. The method you use may pick one, but there's a degree of freedom. It has to do with observability indices.