form = ControlStaircase, ObserveStaircase, ControlCanon, ObserveCanon, ModalCanon, or Balanced

Specifies the form of the transformed Matrix. The default is ControlStaircase.

output = T, Tinv, A, B, C, D, r, or list of same

Specifies the values returned.  The default is T. If output is assigned a name, then the value corresponding to that name is returned. If output is a assigned a list of names, then the values corresponding to each of the names in the list is returned, in that order.

The values for T and Tinv are the transformation matrix and its inverse, respectively.

The values for A, B, C, and D are the transformed state-space matrices.

The value for r is the dimension of the system Matrix. For the ControlStaircase and ObserveStaircase transforms, r is the dimension of the controllable and observable subsystems, respectively. For the other transforms, r is the dimension of A.

discrete = truefalse

True means the system is discrete, false means it is continuous. The default is the value of the variable discrete in the DynamicSystems[SystemOptions] command.

returnlist = truefalse

True means return a list; false means return an expression sequence. The default is false.

The SSTransformation command performs a selected similarity transformation on state-space matrices.

The similarity transformations are ControlStaircase, ObserveStaircase, ControlCanon, ObserveCanon, ModalCanon, and Balanced. The transformation is selected with the named option form.

For a state-space system defined by the state and output equations $\frac{\ⅆ}{\ⅆt}x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$ and $y\left(t\right)=Cx\left(t\right)+\mathrm{D}u\left(t\right)$, the computed similarity transformation matrix T transforms the input state-space system as follows:

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$\mathrm{Amat}≔⟨⟨-5\left|1\right|0⟩\,⟨0\left|-2\right|1⟩\,⟨0\left|0\right|-1⟩⟩\:$

$\mathrm{Bmat}≔⟨⟨0\,0\,1⟩⟩\:$

$\mathrm{Cmat}≔⟨⟨1\left|0\right|0⟩⟩\:$

$\mathrm{Dmat}≔⟨⟨0⟩⟩\:$

$\mathrm{SSTransformation}\left(\mathrm{Amat}\,\mathrm{Bmat}\,\mathrm{Cmat}\,\mathrm{Dmat}\,\mathrm{form}=\mathrm{ControlCanon}\,\mathrm{output}=\left['A'\,'B'\,'C'\,'\mathrm{D}'\right]\right)$

$\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ \mathrm{-10}& \mathrm{-17}& \mathrm{-8}\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],\left[\begin{array}{ccc}1& 0& 0\end{array}\right],\left[\begin{array}{c}0\end{array}\right]$

$\mathrm{SSTransformation}\left(\mathrm{Amat}\,\mathrm{Bmat}\,\mathrm{Cmat}\,\mathrm{Dmat}\,\mathrm{form}=\mathrm{ObserveCanon}\,\mathrm{output}=\left['A'\,'B'\,'C'\,'\mathrm{D}'\right]\right)$

$\left[\begin{array}{ccc}0& 0& \mathrm{-10}\\ 1& 0& \mathrm{-17}\\ 0& 1& \mathrm{-8}\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 1\end{array}\right],\left[\begin{array}{c}0\end{array}\right]$

$\mathrm{SSTransformation}\left(\mathrm{Amat}\,\mathrm{Bmat}\,\mathrm{Cmat}\,\mathrm{Dmat}\,\mathrm{form}=\mathrm{ModalCanon}\,\mathrm{output}=\left['A'\,'B'\,'C'\,'\mathrm{D}'\,'T'\right]\right)$

$\left[\begin{array}{ccc}\mathrm{-1.}& 0& 0\\ 0& \mathrm{-2.}& 0\\ 0& 0& \mathrm{-5.}\end{array}\right],\left[\begin{array}{c}\frac{1}{4}\\ \frac{1}{3}\\ \frac{1}{12}\end{array}\right],\left[\begin{array}{ccc}1& \mathrm{-1}& 1\end{array}\right],\left[\begin{array}{c}0\end{array}\right],\left[\begin{array}{ccc}1& \mathrm{-1}& 1\\ 4& \mathrm{-3}& 0\\ 4& 0& 0\end{array}\right]$