We recommend obtaining the transformation matrix via one of the following commands:
[sysc,T] = canon(sys,'companion');
or
[sysc,T] = canon(sys,'canonical');
Please note that "sysc=ssform(sys,'Form','canonical')" uses "[sysc,T] = canon(sys,'companion')" to compute 'sysc'. As noted in our documentation, the companion canonical form is the same as the observable canonical form (see https://www.mathworks.com/help/control/ref/dynamicsystem.canon.html#mw_78697154-9c08-4155-81e2-425d89456ca7). In both cases, the transformation matrix is the controllability matrix constructed from the state matrix (A) and the first column of the input matrix (B). The transformation matrix is computed in the following file, which you can open and inspect for more information about the algorithm:
C:\Program Files\MATLAB\R2021a\toolbox\shared\controllib\engine\+ltipack\@ssdata\canon.m
Here is an example code snippet demonstrating how to use the transformation matrix for your convenience:
%transform to canonical form
sysc=ssform(sys,'Form','canonical');
[sysc,T] = canon(sys,'companion');
%compute the transform matrix
T = ctrb(sys.A,sys.B(:,1));
% Check transformation of state matrix
T\sys.A*T
sysc.A
% Check transformation of output matrix
sys.C*T
sysc.C
% Check transformation of input matrix
T\sys.B
sysc.B
