The following second order system represents a linear model for the steering of a vehicle with nonslipping wheels:

$\mathrm{sys1}≔\mathrm{StateSpace}\left(\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[\mathrm{\alpha }\right]\,\left[1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\right]\right]\right)\,\mathrm{Matrix}\left(1\,1\right)\right)\:$

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

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

Checking whether the system is observable with alpha = 1/2.

$\mathrm{Observable}\left(\mathrm{sys1}\,'\mathrm{parameters}'=\left\{\mathrm{\alpha }=\frac{1}{2}\right\}\right)$

$\mathrm{true}$

Defining the characteristic polynomial with damping ratio $Z$ and natural frequency $\mathrm{\omega }$.

$\mathrm{cp}≔s^2+2Z\mathrm{\omega }s+{\mathrm{\omega }}^2$

$\mathrm{cp}≔2Z\mathrm{\omega }s+{\mathrm{\omega }}^2+s^2$

The corresponding poles are:

$\mathrm{poles}≔\left[\mathrm{solve}\left(\mathrm{cp}\,s\right)\right]$

$\mathrm{poles}≔\left[\left(-Z+\sqrt{Z^2-1}\right)\mathrm{\omega }\,\left(-Z-\sqrt{Z^2-1}\right)\mathrm{\omega }\right]$

Designing the state feedback controller gains ($\mathrm{Kc}$ and $\mathrm{Kr}$) by  pole placement:

$\mathrm{Kc1},\mathrm{Kr1}≔\mathrm{StateFeedback}:-\mathrm{PolePlacement}\left(\mathrm{sys1}\,\left[a\,b\right]\,'\mathrm{return\_Kr}'\right)\:$

$\mathrm{Kc1}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{Kc1}\,\left[a=\mathrm{poles}\left[1\right]\,b=\mathrm{poles}\left[2\right]\right]\right)\right)$

$\mathrm{Kc1}≔\left[\begin{array}{cc}{\mathrm{\omega }}^2& -{\mathrm{\omega }}^2\mathrm{\alpha }+2\mathrm{\omega }Z\end{array}\right]$

$\mathrm{Kr1}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{Kr1}\,\left[a=\mathrm{poles}\left[1\right]\,b=\mathrm{poles}\left[2\right]\right]\right)\right)$

$\mathrm{Kr1}≔\left[\begin{array}{c}{\mathrm{\omega }}^2\end{array}\right]$

Designing the state observer with gain $L$ by pole placement as well.

$\mathrm{L1}≔\mathrm{StateObserver}:-\mathrm{PolePlacement}\left(\mathrm{sys1}\,\left[a\,b\right]\right)\:$

$\mathrm{L1}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{L1}\,\left[a=\mathrm{poles}\left[1\right]\,b=\mathrm{poles}\left[2\right]\right]\right)\right)$

$\mathrm{L1}≔\left[\begin{array}{c}2\mathrm{\omega }Z\\ {\mathrm{\omega }}^2\end{array}\right]$

The controller-observer subsystem is obtained next. Note that the state, input, and output variables of the returned system object are renamed accordingly.

$\mathrm{cosys1}≔\mathrm{ControllerObserver}\left(\mathrm{sys1}\,\mathrm{Kc1}\,\mathrm{L1}\,':-\mathrm{Kr}'=\mathrm{Kr1}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{cosys1}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 2\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{y1\_ref}\left(t\right)\,\mathrm{y1\_meas}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{u1}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1\_est}\left(t\right)\,\mathrm{x2\_est}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}-{\mathrm{\omega }}^2\mathrm{\alpha }-2\mathrm{\omega }Z& 1-\mathrm{\alpha }\left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2\mathrm{\omega }Z\right)\\ -2{\mathrm{\omega }}^2& {\mathrm{\omega }}^2\mathrm{\alpha }-2\mathrm{\omega }Z\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cc}{\mathrm{\omega }}^2\mathrm{\alpha }& 2\mathrm{\omega }Z\\ {\mathrm{\omega }}^2& {\mathrm{\omega }}^2\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}-{\mathrm{\omega }}^2& {\mathrm{\omega }}^2\mathrm{\alpha }-2\mathrm{\omega }Z\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cc}{\mathrm{\omega }}^2& 0\end{array}\right]\end{array}\right.$

The closed-loop equations of the state feedback control system with observer are obtained using the closedloop = true option. Using stateerror = true, augment_output = true, and outputtype = de, we obtain the closed-loop equations of sys1 with state feedback controller with Kc and Kr gains and observer with gain L. The returned system object is a set of differential equations, where the differential variables are x1, x2, x1_err and x2_err, and the outputs are y1 and u1 (controller output).

$\mathrm{clsys1}≔\mathrm{ControllerObserver}\left(\mathrm{sys1}\,\mathrm{Kc1}\,\mathrm{L1}\,':-\mathrm{Kr}'=\mathrm{Kr1}\,':-\mathrm{closedloop}'=\mathrm{true}\,':-\mathrm{stateerror}'=\mathrm{true}\,':-\mathrm{augment\_output}'=\mathrm{true}\,':-\mathrm{outputtype}'=\mathrm{de}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{clsys1}\right)$

$\left[\begin{array}{l}\mathbf{Diff.\; Equation}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{y1\_ref}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{u1}\left(t\right)\right]\\ \mathrm{de}\=\{\begin{array}{l}\[\frac{\ⅆ}{\ⅆt}\mathrm{x1}\left(t\right)\=\{\begin{array}{l}-{\mathrm{\omega }}^2\mathrm{\alpha }\mathrm{x1}\left(t\right)+\left(1-\mathrm{\alpha }\left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2\mathrm{\omega }Z\right)\right)\mathrm{x2}\left(t\right)\\ +{\mathrm{\omega }}^2\mathrm{\alpha }\mathrm{x1\_err}\left(t\right)+\mathrm{\alpha }\left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2\mathrm{\omega }Z\right)\mathrm{x2\_err}\left(t\right)\\ +{\mathrm{\omega }}^2\mathrm{\alpha }\mathrm{y1\_ref}\left(t\right),\end{array}\\ \frac{\ⅆ}{\ⅆt}\mathrm{x2}\left(t\right)\=\{\begin{array}{l}-{\mathrm{\omega }}^2\mathrm{x1}\left(t\right)+\left({\mathrm{\omega }}^2\mathrm{\alpha }-2\mathrm{\omega }Z\right)\mathrm{x2}\left(t\right)+{\mathrm{\omega }}^2\mathrm{x1\_err}\left(t\right)\\ \left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2\mathrm{\omega }Z\right)\mathrm{x2\_err}\left(t\right)+{\mathrm{\omega }}^2\mathrm{y1\_ref}\left(t\right),\end{array}\\ \frac{\ⅆ}{\ⅆt}\mathrm{x1\_err}\left(t\right)\=-2\mathrm{\omega }Z\mathrm{x1\_err}\left(t\right)+\mathrm{x2\_err}\left(t\right),\\ \frac{\ⅆ}{\ⅆt}\mathrm{x2\_err}\left(t\right)\=-{\mathrm{\omega }}^2\mathrm{x1\_err}\left(t\right),\\ \mathrm{y1}\left(t\right)\=\mathrm{x1}\left(t\right),\\ \mathrm{u1}\left(t\right)\=\{\begin{array}{l}-{\mathrm{\omega }}^2\mathrm{x1}\left(t\right)+\left({\mathrm{\omega }}^2\mathrm{\alpha }-2\mathrm{\omega }Z\right)\mathrm{x2}\left(t\right)+{\mathrm{\omega }}^2\mathrm{x1\_err}\left(t\right)\\ \left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2\mathrm{\omega }Z\right)\mathrm{x2\_err}\left(t\right)+{\mathrm{\omega }}^2\mathrm{y1\_ref}\left(t\right)\]\end{array}\end{array}\end{array}\right.$

The following DC motor example demonstrates the use of the parameters, controlled_inputs, measured_inputs, controlled_outputs and  measured_outputs options.

$\mathrm{a2}≔\mathrm{Matrix}\left(\left[\left[-\frac{R}{L}\,-\frac{K}{L}\,0\right]\,\left[\frac{K}{J}\,-\frac{B}{J}\,0\right]\,\left[0\,1\,0\right]\right]\right)\:$

$\mathrm{b2}≔\mathrm{Matrix}\left(\left[\left[\frac{1}{L}\,0\,0\right]\,\left[0\,-\frac{1}{J}\,0\right]\,\left[0\,0\,-1\right]\right]\right)\:$

$\mathrm{c2}≔\mathrm{Matrix}\left(\left[\left[1\,0\,0\right]\,\left[0\,1\,0\right]\,\left[0\,0\,1\right]\right]\right)\:$

$\mathrm{d2}≔\mathrm{Matrix}\left(3\,3\right)\:$

The numeric values for the DC motor stator inductance $L$, stator resistance $R$, electromotive force (emf) constant $K$, rotor moment of inertia $J$, and damping ratio $B$ are given next:

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

The input variables are the source voltage $V\left(t\right)$, the torque load $T\left(t\right)$ and the rotor angular speed reference $\mathrm{wref}\left(t\right)$, and the output variables are the stator current $i\left(t\right)$, the rotor angular speed $w\left(t\right)$ and the integral of the rotor speed error $z\left(t\right)$.

$\mathrm{sys2}≔\mathrm{StateSpace}\left(\mathrm{a2}\,\mathrm{b2}\,\mathrm{c2}\,\mathrm{d2}\,'\mathrm{inputvariable}'=\left[V\left(t\right)\,T\left(t\right)\,\mathrm{wref}\left(t\right)\right]\,'\mathrm{outputvariable}'=\left[i\left(t\right)\,w\left(t\right)\,z\left(t\right)\right]\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{sys2}\right)$

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{3\; output(s);\; 3\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[V\left(t\right)\,T\left(t\right)\,\mathrm{wref}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[i\left(t\right)\,w\left(t\right)\,z\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}-\frac{R}{L}& -\frac{K}{L}& 0\\ \frac{K}{J}& -\frac{B}{J}& 0\\ 0& 1& 0\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{ccc}\frac{1}{L}& 0& 0\\ 0& -\frac{1}{J}& 0\\ 0& 0& \mathrm{-1}\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\end{array}\right.$

The state feedback controller and observer gains are given next:

$\mathrm{Kc2}≔\mathrm{Matrix}\left(\left[\left[1.89107\,7.35829\,70.71068\right]\right]\right)\:$

$\mathrm{Kr2}≔\mathrm{Matrix}\left(1\,1\right)\:$

$\mathrm{L2}≔\mathrm{Matrix}\left(\left[\left[0.08028\,-0.00148\right]\,\left[990.05009\,0.98010\right]\,\left[0.98010\,10.09948\right]\right]\right)\:$

The controlled input is the source voltage $V\left(t\right)$. The measured input is $\mathrm{wref}\left(t\right)$. The controlled output is $z\left(t\right)$. The measured outputs are $w\left(t\right)$ and $z\left(t\right)$.

The controller-observer subsystem equations are obtained next:

$\mathrm{cosys2}≔\mathrm{ControllerObserver}\left(\mathrm{sys2}\,\mathrm{Kc2}\,\mathrm{L2}\,'\mathrm{Kr}'=\mathrm{Kr2}\,'\mathrm{parameters}'=\mathrm{params}\,'\mathrm{controlled\_inputs}'=\left\{1\right\}\,'\mathrm{measured\_inputs}'=\left\{3\right\}\,'\mathrm{controlled\_outputs}'=\left\{3\right\}\,'\mathrm{measured\_outputs}'=\left\{2\,3\right\}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{cosys2}\right)$

It is also possible to obtain an LQG controller by computing LQR and Kalman gains. First, compute the weighting matrices:

$W≔\mathrm{ComputeQR}\left(\mathrm{sys2}\,0.4\,'\mathrm{parameters}'=\mathrm{params}\right)$

$W≔\mathrm{Record}\left(Q=\left[\begin{array}{ccc}1.& 0.& 0.\\ 0.& 1.& 0.\\ 0.& 0.& 1.\end{array}\right]\,R=\left[\begin{array}{ccc}0.321321227565641& \mathrm{-0.332291362282656}& \mathrm{-0.00321457404413542}\\ \mathrm{-0.332291362282655}& 210.045523435307& 1.20713539723679\\ \mathrm{-0.00321457404413543}& 1.20713539723679& 0.160000000000000\end{array}\right]\right)$

$\mathrm{Kc3},\mathrm{Kr3}≔\mathrm{LQR}\left(\mathrm{sys2}\,W:-Q\,W:-R\,'\mathrm{parameters}'=\mathrm{params}\,'\mathrm{return\_Kr}'\right)$

$\mathrm{Kc3},\mathrm{Kr3}≔\left[\begin{array}{ccc}1.03313325178664& \mathrm{-0.00143259581815793}& 0.0122156987864645\\ 0.0000388509514481089& \mathrm{-0.0225671357809562}& \mathrm{-0.000745437890855909}\\ \mathrm{-0.0176293731317396}& \mathrm{-0.0279650446417151}& \mathrm{-2.49392402957048}\end{array}\right],\left[\begin{array}{ccc}2.03313325178664& 0.00856740418184207& 0.0122156987864645\\ 0.0100388509514481& \mathrm{-0.122567135780956}& \mathrm{-0.000745437890855909}\\ \mathrm{-0.0176293731317396}& 0.972034955358285& \mathrm{-2.49392402957049}\end{array}\right]$

$\mathrm{G1}≔\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(3\right)\:$

$\mathrm{H1}≔\mathrm{LinearAlgebra}:-\mathrm{ZeroMatrix}\left(3\right)\:$

$\mathrm{Q1}≔\mathrm{Matrix}\left(\left[\left[5\,0\,0\right]\,\left[0\,1\,0\right]\,\left[0\,0\,1\right]\right]\right)\:$

$\mathrm{R1}≔0.001\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(3\right)\:$

$\mathrm{Kgain}≔\mathrm{Kalman}\left(\mathrm{sys2}\,\mathrm{G1}\,\mathrm{H1}\,\mathrm{Q1}\,\mathrm{R1}\,'\mathrm{parameters}'=\mathrm{params}\right)\left[1\right]$

$\mathrm{Kgain}≔\left[\begin{array}{ccc}68.7357198244512& 0.656996670591518& 0.00405274599764882\\ 0.656996670591518& 23.1776195213554& 0.357647329351719\\ 0.00405274599764882& 0.357647329351719& 31.6320623200853\end{array}\right]$

$\mathrm{LQG}≔\mathrm{ControllerObserver}\left(\mathrm{sys2}\,\mathrm{Kc3}\,\mathrm{Kgain}\,'\mathrm{Kr}'=\mathrm{Kr3}\,'\mathrm{parameters}'=\mathrm{params}\right)\:$

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