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

sys1 := StateSpace(Matrix([[0, 1], [0, 0]]), Matrix([[alpha], [1]]), Matrix([[1, 0]]), Matrix(1, 1)):

PrintSystem(sys1);

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

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

Observable(sys1, 'parameters' = {alpha = 1/2});

$\mathrm{true}$

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

cp := s^2 + 2*Z*omega*s + omega^2;

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

The corresponding poles are:

poles := [solve(cp, s)];

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

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

Kc1, Kr1 := StateFeedback:-PolePlacement(sys1, [a, b], 'return_Kr'):

Kc1 := simplify(eval(Kc1, [a = poles[1], b = poles[2]]));

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

Kr1 := simplify(eval(Kr1, [a = poles[1], b = poles[2]]));

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

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

L1 := StateObserver:-PolePlacement(sys1, [a, b]):

L1 := simplify(eval(L1, [a = poles[1], b = poles[2]]));

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

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

cosys1 := ControllerObserver(sys1, Kc1, L1, ':-Kr' = Kr1):

PrintSystem(cosys1);

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

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).

clsys1 := ControllerObserver(sys1, Kc1, L1, ':-Kr' = Kr1, ':-closedloop' = true, ':-stateerror' = true, ':-augment_output' = true, ':-outputtype' = de):

PrintSystem(clsys1);

$\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}\[\frac{\ⅆ}{\ⅆt}\mathrm{x1}\left(t\right)\=\{\begin{array}{l}-{\mathrm{\omega }}^2\mathrm{\alpha }\mathrm{x1}\left(t\right)+\left(1-\mathrm{\alpha }\left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2Z\mathrm{\omega }\right)\right)\mathrm{x2}\left(t\right)\\ +{\mathrm{\omega }}^2\mathrm{\alpha }\mathrm{x1\_err}\left(t\right)+\mathrm{\alpha }\left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2Z\mathrm{\omega }\right)\mathrm{x2\_err}\left(t\right)\\ +{\mathrm{\omega }}^2\mathrm{\alpha }\mathrm{y1\_ref}\left(t\right),\end{array}\\ \frac{\ⅆ}{\ⅆt}\mathrm{x2}\left(t\right)\=\{\begin{array}{l}-{\mathrm{\omega }}^2\mathrm{x1}\left(t\right)+\left({\mathrm{\omega }}^2\mathrm{\alpha }-2Z\mathrm{\omega }\right)\mathrm{x2}\left(t\right)+{\mathrm{\omega }}^2\mathrm{x1\_err}\left(t\right)\\ \left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2Z\mathrm{\omega }\right)\mathrm{x2\_err}\left(t\right)+{\mathrm{\omega }}^2\mathrm{y1\_ref}\left(t\right),\end{array}\\ \frac{\ⅆ}{\ⅆt}\mathrm{x1\_err}\left(t\right)\=-2Z\mathrm{\omega }\mathrm{x1\_err}\left(t\right)+\mathrm{x2\_err}\left(t\right),\\ \frac{\ⅆ}{\ⅆt}\mathrm{x2\_err}\left(t\right)\=-{\mathrm{\omega }}^2\mathrm{x1\_err}\left(t\right),\\ \mathrm{y1}\left(t\right)\=\mathrm{x1}\left(t\right),\\ \mathrm{u1}\left(t\right)\=\{\begin{array}{l}-{\mathrm{\omega }}^2\mathrm{x1}\left(t\right)+\left({\mathrm{\omega }}^2\mathrm{\alpha }-2Z\mathrm{\omega }\right)\mathrm{x2}\left(t\right)+{\mathrm{\omega }}^2\mathrm{x1\_err}\left(t\right)\\ \left(-{\mathrm{\omega }}^2\mathrm{\alpha }+2Z\mathrm{\omega }\right)\mathrm{x2\_err}\left(t\right)+{\mathrm{\omega }}^2\mathrm{y1\_ref}\left(t\right)\]\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.

a2 := Matrix([[-R/L, -K/L, 0], [K/J, -B/J, 0], [0, 1, 0]]):

b2 := Matrix([[1/L, 0, 0],[0, -1/J , 0],[0, 0, -1]]):

c2 := Matrix([[1, 0, 0],[0, 1, 0],[0, 0, 1]]):

d2 := Matrix(3, 3):

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:

params := {J = 0.01, K = 0.01, L = 0.5, R = 1, B = 0.1}:

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 $z\left(t\right)$.

sys2 := StateSpace(a2, b2, c2, d2, 'inputvariable' = [V(t), T(t), wref(t)], 'outputvariable' = [i(t), w(t), z(t)]):

PrintSystem(sys2);

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

The state feedback controller and observer gains are given next:

Kc2 := Matrix([[1.89107, 7.35829, 70.71068]]):

Kr2 := Matrix(1, 1):

L2 :=  Matrix([[0.08028, -0.00148], [990.05009, .98010], [.98010, 10.09948]]):

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:

cosys2 := ControllerObserver(sys2, Kc2, L2, 'Kr' = Kr2, 'parameters' = params, 'controlled_inputs' = {1}, 'measured_inputs' = {3}, 'controlled_outputs' = {3}, 'measured_outputs' = {2, 3}):

PrintSystem(cosys2);

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 3\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{wref}\left(t\right)\,\mathrm{w\_meas}\left(t\right)\,\mathrm{z\_meas}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[V\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1\_est}\left(t\right)\,\mathrm{x2\_est}\left(t\right)\,\mathrm{x3\_est}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}\mathrm{-5.78214000000000}& \mathrm{-14.8168600000000}& \mathrm{-141.419880000000}\\ 1.& \mathrm{-1000.05009000000}& \mathrm{-0.980100000000000}\\ 0.& 0.0199000000000000& \mathrm{-10.0994800000000}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{ccc}0.& 0.08028& \mathrm{-0.00148}\\ 0.& 990.05009& 0.98010\\ \mathrm{-1.}& 0.98010& 10.09948\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}\mathrm{-1.89107000000000}& \mathrm{-7.35829000000000}& \mathrm{-70.7106800000000}\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]\end{array}\right.$

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

W := ComputeQR(sys2, 0.4, 'parameters'=params);

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

Kc3, Kr3 := LQR(sys2, W:-Q, W:-R, 'parameters'=params,'return_Kr');

$\mathrm{Kc3},\mathrm{Kr3}≔\left[\begin{array}{ccc}1.03313325178664& \mathrm{-0.00143259581815696}& 0.0122156987864642\\ 0.0000388509514481088& \mathrm{-0.0225671357809562}& \mathrm{-0.000745437890855909}\\ \mathrm{-0.0176293731317396}& \mathrm{-0.0279650446417151}& \mathrm{-2.49392402957048}\end{array}\right],\left[\begin{array}{ccc}2.03313325178664& 0.00856740418184304& 0.0122156987864642\\ 0.0100388509514481& \mathrm{-0.122567135780956}& \mathrm{-0.000745437890855909}\\ \mathrm{-0.0176293731317396}& 0.972034955358285& \mathrm{-2.49392402957049}\end{array}\right]$

G1 := LinearAlgebra:-IdentityMatrix(3):

H1 := LinearAlgebra:-ZeroMatrix(3):

Q1 := Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]]):

R1 := 0.1e-2*LinearAlgebra:-IdentityMatrix(3):

Kgain := Kalman(sys2, G1, H1, Q1, R1, 'parameters'=params)[1];

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

LQG := ControllerObserver(sys2, Kc3, Kgain, 'Kr'=Kr3, 'parameters'=params):

PrintSystem(LQG);

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{3\; output(s);\; 6\; input(s);\; 3\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{i\_ref}\left(t\right)\,\mathrm{w\_ref}\left(t\right)\,\mathrm{z\_ref}\left(t\right)\,\mathrm{i\_meas}\left(t\right)\,\mathrm{w\_meas}\left(t\right)\,\mathrm{z\_meas}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[V\left(t\right)\,T\left(t\right)\,\mathrm{wref}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1\_est}\left(t\right)\,\mathrm{x2\_est}\left(t\right)\,\mathrm{x3\_est}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{ccc}\mathrm{-72.8019863280245}& \mathrm{-0.674131478955204}& \mathrm{-0.0284841435705773}\\ 0.346888424553293& \mathrm{-35.4343330994510}& \mathrm{-0.432191118437309}\\ \mathrm{-0.0216821191293884}& 0.614387626006566& \mathrm{-34.1259863496558}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cccccc}4.06626650357328& 0.0171348083636861& 0.0244313975729285& 68.7357198244512& 0.656996670591518& 0.00405274599764882\\ \mathrm{-1.00388509514481}& 12.2567135780956& 0.0745437890855909& 0.656996670591518& 23.1776195213554& 0.357647329351719\\ 0.0176293731317396& \mathrm{-0.972034955358285}& 2.49392402957049& 0.00405274599764882& 0.357647329351719& 31.6320623200853\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{ccc}\mathrm{-1.03313325178664}& 0.00143259581815696& \mathrm{-0.0122156987864642}\\ \mathrm{-0.0000388509514481088}& 0.0225671357809562& 0.000745437890855909\\ 0.0176293731317396& 0.0279650446417151& 2.49392402957048\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cccccc}2.03313325178663895& 0.00856740418184303958& 0.0122156987864642380& 0& 0& 0\\ 0.0100388509514481080& \mathrm{-0.122567135780956196}& \mathrm{-0.000745437890855909276}& 0& 0& 0\\ \mathrm{-0.0176293731317396021}& 0.972034955358285102& \mathrm{-2.49392402957048542}& 0& 0& 0\end{array}\right]\end{array}\right.$