State-space system matrix Amat and input matrix Bmat (5 states and 3 inputs)

$\mathrm{Amat}≔\mathrm{Matrix}\left(\left[\left[1\,-2\,3\,-5\,12\right]\,\left[0\,4\,-1\,2\,-3\right]\,\left[2\,5\,7\,-4\,3\right]\,\left[-2\,4\,3\,8\,7\right]\,\left[19\,14\,1\,-4\,7\right]\right]\right)\:$

$\mathrm{Bmat}≔\mathrm{Matrix}\left(\left[\left[3\,6\,3\right]\,\left[9\,7\,5\right]\,\left[5\,2\,1\right]\,\left[4\,-1\,7\right]\,\left[6\,-8\,2\right]\right]\right)\:$

Desired poles

$\mathrm{p1}≔\left[-1+\mathrm{I}\,-0.1\,-3.581\,-1-\mathrm{I}\,-5.7720\right]\:$

Obtain the state feedback gain Kc:

$\mathrm{Kc}≔\mathrm{StateFeedback}:-\mathrm{PolePlacement}\left(\mathrm{Amat}\,\mathrm{Bmat}\,\mathrm{p1}\right)$

$\mathrm{Kc}≔\left[\begin{array}{ccccc}4.90010228021979& 1.35723979032960& \mathrm{-0.298845142567305}& \mathrm{-3.75886722413410}& 2.98040260420070\\ \mathrm{-0.769663041613290}& \mathrm{-0.805638413823460}& 0.0343911471969793& \mathrm{-1.04028874033607}& \mathrm{-1.15733500000763}\\ \mathrm{-4.12917582839681}& \mathrm{-0.310095814574346}& 0.515200545031575& 4.00754085025251& \mathrm{-2.27773489827046}\end{array}\right]$

Verify the closed-loop system matrix has the desired poles

$\mathrm{LinearAlgebra}:-\mathrm{Eigenvalues}\left(\mathrm{Amat}-\mathrm{Bmat}·\mathrm{Kc}\right)$

$\left[\begin{array}{c}\mathrm{-5.77200000000001}+0.\mathrm{I}\\ \mathrm{-3.58100000000069}+0.\mathrm{I}\\ \mathrm{-0.100000000000702}+0.\mathrm{I}\\ \mathrm{-0.999999999999428}+\mathrm{I}\\ \mathrm{-0.999999999999428}-\mathrm{I}\end{array}\right]$

System object sys with symbolic entries (3 states, 3 inputs, 3 outputs)

$\mathrm{sys}≔\mathrm{StateSpace}\left(\mathrm{Matrix}\left(\left[\left[1\,2\,3\right]\,\left[x\,4\,-1\right]\,\left[2\,5\,7\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[3\,6\right]\,\left[9\,7\right]\,\left[5\,2\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\,0\right]\,\left[0\,1\,0\right]\,\left[0\,0\,1\right]\right]\right)\,\mathrm{Matrix}\left(3\,2\right)\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{3\; output(s);\; 2\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\,\mathrm{u2}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\,\mathrm{y3}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}1& 2& 3\\ x& 4& \mathrm{-1}\\ 2& 5& 7\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cc}3& 6\\ 9& 7\\ 5& 2\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]\end{array}\right.$

Desired symbolic poles

$\mathrm{p2}≔\left[-4\,-b+c\mathrm{I}\,-b-c\mathrm{I}\right]\:$

Obtain the state feedback gain Kc and the feed-forward gain Kr:

$\mathrm{Kc},\mathrm{Kr}≔\mathrm{StateFeedback}:-\mathrm{PolePlacement}\left(\mathrm{sys}\,\mathrm{p2}\,'\mathrm{return\_Kr}'=\mathrm{true}\,'\mathrm{parameters}'=\left[x=0\right]\right)\:$

$\mathrm{evalf}\left(\mathrm{Kc}\right)$

$\left[\begin{array}{ccc}-0.01105243808b^2-0.01105243808c^2-0.07399413568b-0.3147790352& 0.01492746525b^2+0.01492746525c^2-0.05451317909b-0.8774754121& -0.01297231488b^2-0.01297231488c^2-0.2777819673b-1.055971500\\ 0.03282239188b^2+0.03282239188c^2+0.2197401605b+0.9347983468& -0.04433004833b^2-0.04433004833c^2+0.1618876227b+2.605836072& 0.03852384419b^2+0.03852384419c^2+0.8249282666b+3.135915362\end{array}\right]$

$\mathrm{evalf}\left(\mathrm{Kr}\right)$

$\left[\begin{array}{ccc}-0.008550987593b^2-0.008550987593c^2-0.001731055306b+0.04885958325& 0.01803167490b^2+0.01803167490c^2+0.03516269174b-0.9924780905& -0.003750099818b^2-0.003750099818c^2-0.01136627340b+0.3208166598\\ 0.02539384194b^2+0.02539384194c^2+0.005140709696b-0.07434464063& -0.05354861031b^2-0.05354861031c^2-0.1044225391b+1.510152606& 0.01113666007b^2+0.01113666007c^2+0.03375438766b-0.4881539649\end{array}\right]$

Verify the closed-loop system matrix has the desired poles

$\mathrm{sys}:-a≔\mathrm{eval}\left(\mathrm{sys}:-a\,x=0\right)\:$

$\mathrm{eigen}≔\mathrm{LinearAlgebra}:-\mathrm{Eigenvalues}\left(\mathrm{sys}:-a-\mathrm{sys}:-b·\mathrm{Kc}\right)$

$\mathrm{eigen}≔\left[\begin{array}{c}\mathrm{-4}\\ -b+\mathrm{I}c\\ -b-\mathrm{I}c\end{array}\right]$

$\mathrm{sys1}≔\mathrm{StateSpace}\left(\mathrm{Matrix}\left(\left[\left[4\,0\,-3\,5\right]\,\left[3\,0\,0\,3\right]\,\left[1\,0\,-1\,1\right]\,\left[0\,0\,0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[5\,1\,-1\right]\,\left[1\,3\,-1\right]\,\left[0\,2\,6\right]\,\left[0\,0\,z\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\,3\,5\right]\,\left[-3\,0\,5\,7\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\,0\right]\,\left[0\,1\,1\right]\right]\right)\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 3\; input(s);\; 4\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\,\mathrm{u2}\left(t\right)\,\mathrm{u3}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\,\mathrm{x4}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cccc}4& 0& \mathrm{-3}& 5\\ 3& 0& 0& 3\\ 1& 0& \mathrm{-1}& 1\\ 0& 0& 0& 0\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{ccc}5& 1& \mathrm{-1}\\ 1& 3& \mathrm{-1}\\ 0& 2& 6\\ 0& 0& z\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cccc}1& 0& 3& 5\\ \mathrm{-3}& 0& 5& 7\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\end{array}\right]\end{array}\right.$

Get a subsystem considering the first two inputs.

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

$\mathrm{PrintSystem}\left(\mathrm{subsys1}\right)$

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

Desired poles

$\mathrm{p3}≔\left[-1+\mathrm{I}\cdot 3\,-1-\mathrm{I}\cdot 3\,-7+\mathrm{I}\cdot 9\,-7-\mathrm{I}\cdot 9\right]$

$\mathrm{p3}≔\left[\mathrm{-1}+3\mathrm{I}\,\mathrm{-1}-3\mathrm{I}\,\mathrm{-7}+9\mathrm{I}\,\mathrm{-7}-9\mathrm{I}\right]$

Get the state-feedback gain for subsys1

$\mathrm{Kc}≔\mathrm{StateFeedback}:-\mathrm{PolePlacement}\left(\mathrm{subsys1}\,\mathrm{p3}\right)$

Error, (in ControlDesign:-StateFeedback:-PolePlacement) the given state-space realization is not controllable

Remove the uncontrollable states if possible. ReduceSystem will remove the structural uncontrollable states.

$\mathrm{rsubsys}≔\mathrm{ReduceSystem}\left(\mathrm{subsys1}\,'\mathrm{reducedtype}'=\mathrm{controllable}\right)\:$

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

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

Verify the resulting subsystem is controllable

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

$\mathrm{true}$

Desired poles for the controllable subsystem

$\mathrm{p4}≔\left[-1\,-7+\mathrm{I}\cdot 9\,-7-\mathrm{I}\cdot 9\right]\:$

$\mathrm{Kc}≔\mathrm{StateFeedback}:-\mathrm{PolePlacement}\left(\mathrm{rsubsys}\,\mathrm{p4}\right)$

$\mathrm{Kc}≔\left[\begin{array}{ccc}65.5315096909962& \mathrm{-75.9744508280517}& \mathrm{-190.798821593246}\\ 28.5444171884827& \mathrm{-32.6665742925171}& \mathrm{-82.1138959689311}\end{array}\right]$