Consider a state-space system corresponding to a DC Motor:

$\mathrm{sys\_a}≔\mathrm{Matrix}\left(\left[\left[-\frac{d}{J}\,\frac{K}{J}\,0\right]\,\left[-\frac{K}{L}\,-\frac{R}{L}\,0\right]\,\left[1\,0\,0\right]\right]\right)\:$

$\mathrm{sys\_b}≔\mathrm{Matrix}\left(\left[\left[0\,\frac{1}{J}\right]\,\left[\frac{1}{L}\,0\right]\,\left[0\,0\right]\right]\right)\:$

$\mathrm{sys\_c}≔\mathrm{Matrix}\left(\left[\left[0\,1\,0\right]\,\left[1\,0\,0\right]\,\left[0\,0\,1\right]\right]\right)\:$

$\mathrm{sys\_d}≔\mathrm{Matrix}\left(\left[\left[0\,0\right]\,\left[0\,0\right]\,\left[0\,0\right]\right]\right)\:$

$\mathrm{sys1}≔\mathrm{StateSpace}\left(\mathrm{sys\_a}\,\mathrm{sys\_b}\,\mathrm{sys\_c}\,\mathrm{sys\_d}\,'\mathrm{inputvariable}'=\left[V\left(t\right)\,T\left(t\right)\right]\,'\mathrm{outputvariable}'=\left[\mathrm{i\_out}\left(t\right)\,\mathrm{omega\_out}\left(t\right)\,\mathrm{theta\_out}\left(t\right)\right]\,'\mathrm{statevariable}'=\left[\mathrm{\omega }\left(t\right)\,i\left(t\right)\,\mathrm{\theta }\left(t\right)\right]\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys1}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{3\; output(s);\; 2\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[V\left(t\right)\,T\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{i\_out}\left(t\right)\,\mathrm{omega\_out}\left(t\right)\,\mathrm{theta\_out}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{\omega }\left(t\right)\,i\left(t\right)\,\mathrm{\theta }\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}-\frac{d}{J}& \frac{K}{J}& 0\\ -\frac{K}{L}& -\frac{R}{L}& 0\\ 1& 0& 0\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cc}0& \frac{1}{J}\\ \frac{1}{L}& 0\\ 0& 0\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]\end{array}\right.$

The numeric values of the model parameters are as follows:

$\mathrm{params1}≔\left\{J=0.01\,K=0.01\,L=0.5\,R=1\,d=0.1\right\}\:$

Extract a subsystem with the desired input and output.

$\mathrm{subsys1}≔\mathrm{Subsystem}\left(\mathrm{sys1}\,\left\{1\right\}\,\left\{2\right\}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{subsys1}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[V\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{omega\_out}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{\omega }\left(t\right)\,i\left(t\right)\,\mathrm{\theta }\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}-\frac{d}{J}& \frac{K}{J}& 0\\ -\frac{K}{L}& -\frac{R}{L}& 0\\ 1& 0& 0\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}0\\ \frac{1}{L}\\ 0\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}0\end{array}\right]\end{array}\right.$

Check if the subsystem is controllable:

$\mathrm{Controllable}\left(\mathrm{subsys1}\,'\mathrm{parameters}'=\mathrm{params1}\right)$

$\mathrm{true}$

Compute the poles for a desired time constant of 1 second:

$\mathrm{\tau }≔1\:$

$\mathrm{p1}≔\mathrm{ComputePoles}\left(\mathrm{subsys1}\,\mathrm{\tau }\,'\mathrm{parameters}'=\mathrm{params1}\right)$

$\mathrm{p1}≔\left[\mathrm{-1.}\,\mathrm{-1.}\,\mathrm{-1.}\right]$

Use the previous results to compute the state feedback gain:

$\mathrm{Kc}≔\mathrm{StateFeedback}:-\mathrm{Ackermann}\left(\mathrm{subsys1}\,\mathrm{p1}\,'\mathrm{parameters}'=\mathrm{params1}\right)$

$\mathrm{Kc}≔\left[\begin{array}{ccc}36.49000000& \mathrm{-4.500000000}& 0.5000000000\end{array}\right]$

Consider the following state-space system

$\mathrm{Am}≔\mathrm{Matrix}\left(\left[\left[\mathrm{\delta }\,1\,-\mathrm{\phi }\,\mathrm{\sigma }\right]\,\left[3\,0\,0\,3\right]\,\left[1\,0\,-1\,1\right]\,\left[0\,0\,0\,0\right]\right]\right)\:$

$\mathrm{Bm}≔\mathrm{Matrix}\left(\left[\left[5\,z\,-1\right]\,\left[1\,3\,-x\right]\,\left[0\,2\,y\right]\,\left[0\,0\,z\right]\right]\right)\:$

$\mathrm{Cm}≔\mathrm{Matrix}\left(\left[\left[1\,0\,3\,5\right]\,\left[-3\,0\,\mathrm{\sigma }\,7\right]\right]\right)\:$

$\mathrm{Dm}≔\mathrm{Matrix}\left(\left[\left[1\,0\,0\right]\,\left[0\,1\,1\right]\right]\right)\:$

The numeric values of the system parameters are

$\mathrm{params2}≔\left\{\mathrm{\delta }=2\,\mathrm{\phi }=7\,\mathrm{\sigma }=3\,z=4\right\}\:$

Create the state-space model, using the parameters option so that they do not have to be later passed to subsequent function calls as was done in the previous example.

$\mathrm{sys2}≔\mathrm{StateSpace}\left(\mathrm{Am}\,\mathrm{Bm}\,\mathrm{Cm}\,\mathrm{Dm}\,'\mathrm{parameters}'=\mathrm{params2}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{sys2}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 3\; input(s);\; 4\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\,\mathrm{u2}\left(t\right)\,\mathrm{u3}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\,\mathrm{x4}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cccc}\mathrm{\delta }& 1& -\mathrm{\phi }& \mathrm{\sigma }\\ 3& 0& 0& 3\\ 1& 0& \mathrm{-1}& 1\\ 0& 0& 0& 0\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{ccc}5& z& \mathrm{-1}\\ 1& 3& -x\\ 0& 2& y\\ 0& 0& z\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cccc}1& 0& 3& 5\\ \mathrm{-3}& 0& \mathrm{\sigma }& 7\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\end{array}\right]\end{array}\right.$

Extract a subsystem with the desired input and output.

$\mathrm{subsys2}≔\mathrm{Subsystem}\left(\mathrm{sys2}\,\left\{1\right\}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{subsys2}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 1\; input(s);\; 4\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\,\mathrm{x4}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cccc}\mathrm{\delta }& 1& -\mathrm{\phi }& \mathrm{\sigma }\\ 3& 0& 0& 3\\ 1& 0& \mathrm{-1}& 1\\ 0& 0& 0& 0\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}5\\ 1\\ 0\\ 0\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cccc}1& 0& 3& 5\\ \mathrm{-3}& 0& \mathrm{\sigma }& 7\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}\right.$

Check if the system is controllable:

$\mathrm{Controllable}\left(\mathrm{subsys2}\right)$

$\mathrm{false}$

$\mathrm{Observable}\left(\mathrm{subsys2}\right)$

$\mathrm{true}$

The subsystem is observable but uncontrollable. Try removing structural uncontrollable states

$\mathrm{rsys}≔\mathrm{ReduceSystem}\left(\mathrm{subsys2}\right)\:$$\mathrm{PrintSystem}\left(\mathrm{rsys}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 1\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}\mathrm{\delta }& 1& -\mathrm{\phi }\\ 3& 0& 0\\ 1& 0& \mathrm{-1}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}5\\ 1\\ 0\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}1& 0& 3\\ \mathrm{-3}& 0& \mathrm{\sigma }\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}\right.$

$\mathrm{Controllable}\left(\mathrm{rsys}\right)$

$\mathrm{true}$

$\mathrm{Observable}\left(\mathrm{rsys}\right)$

$\mathrm{true}$

Compute the poles for a desired time constant of 1 second:

$\mathrm{\tau }≔1\:$

$\mathrm{p2}≔\mathrm{ComputePoles}\left(\mathrm{rsys}\,\mathrm{\tau }\right)$

$\mathrm{p2}≔\left[\mathrm{-1.049223528}+0.2188673555\mathrm{I}\,\mathrm{-1.049223528}-0.2188673555\mathrm{I}\,\mathrm{-0.043808373}\right]$

$\mathrm{Kc}≔\mathrm{StateFeedback}:-\mathrm{Ackermann}\left(\mathrm{rsys}\,\mathrm{p2}\right)$

$\mathrm{Kc}≔\left[\begin{array}{ccc}0.5898446404& 0.1930322270& \mathrm{-1.400163308}\end{array}\right]$