Kr = Matrix or 0

Specify a feedforward gain $K_r$ (reference input). If Kr = 0, there are no reference inputs $r$ and the returned closed-loop system is autonomous. The default value is 0.

augment_output = true or false

True means append the sys inputs to the output vector of the closed-loop system. The default is false.

outputtype = tf, coeff, zpk, ss, or de

Determines the subtype of the returned system object.  The default return type is based on the type of the system object specified in the sys parameter.

parameters = {list, set}(name = complexcons)

Specifies numeric values for the parameters of sys. These values override any parameters previously specified for sys. The numeric value on the right-hand side of each equation is substituted for the name on the left-hand side in the sys equations. The default is the value of sys given by DynamicSystems:-SystemOptions(parameters).

The StateFeedbackClosedLoop command calculates the closed-loop system equations of the state feedback controller with $K_c$ gain and plant sys.

When the option Kr is specified, the state feedback controller is governed by the control law $u_c=-K_c·x+K_r·r$, where $x$ is the state vector of sys, $r$ is the reference vector, $K_c$ is the feedback gain and $K_r$ is the direct or feedforward gain.

The sys is a SISO (single input, single output) or MIMO (multiple input, multiple output) linear system object created using the DynamicSystems package. The system object can be of types: transfer function (TF), zero-pole-gain (ZPK), coefficients (Coeff), state-space (SS), and diff-equation (DE). It is assumed that the state feedback controller gains are obtained using the sys representation obtained by DynamicSystems[StateSpace].

The StateFeedbackClosedLoop command returns a system object whose type is the same as the type of sys, unless the option outputtype is specified.

The closed-loop system inputs are

the reference inputs $r=y_{\mathrm{\_ref}}$ (when a non-zero Kr Matrix is specified)

The reference vector $r$ contains a reference signal for each sys output $y$.

The closed-loop system outputs are

the outputs $y$ of sys

the controller outputs $u_c$ (if augment_output = true)

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

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

$\mathrm{eqs}≔\left[L\mathrm{diff}\left(i\left(t\right)\,t\right)+Ri\left(t\right)=V\left(t\right)-Kw\left(t\right)\,J\mathrm{diff}\left(w\left(t\right)\,t\right)+bw\left(t\right)=Ki\left(t\right)-T\left(t\right)\right]\:$

$\mathrm{sysde}≔\mathrm{DiffEquation}\left(\mathrm{eqs}\,':-\mathrm{inputvariable}'=\left[V\left(t\right)\,T\left(t\right)\right]\,':-\mathrm{outputvariable}'=\left[w\left(t\right)\,i\left(t\right)\right]\right)\:$

$\mathrm{sys}≔\mathrm{StateSpace}\left(\mathrm{sysde}\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 2\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[V\left(t\right)\,T\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[w\left(t\right)\,i\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}-\frac{R}{L}& -\frac{K}{L}\\ \frac{K}{J}& -\frac{b}{J}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{cc}\frac{1}{L}& 0\\ 0& -\frac{1}{J}\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}\right.$

$\mathrm{sys}≔\mathrm{Subsystem}\left(\mathrm{sys}\,\left\{1\right\}\,\left\{1\right\}\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[V\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[w\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}-\frac{R}{L}& -\frac{K}{L}\\ \frac{K}{J}& -\frac{b}{J}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}\frac{1}{L}\\ 0\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}0& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}0\end{array}\right]\end{array}\right.$

$\mathrm{Kc},\mathrm{Kr}≔\mathrm{StateFeedback}:-\mathrm{PolePlacement}\left(\mathrm{sys}\,\left[p\left[1\right]\,p\left[2\right]\right]\,'\mathrm{return\_Kr}'\right)$

$\mathrm{Kc},\mathrm{Kr}≔\left[\begin{array}{cc}-\frac{JL{p}_1+JL{p}_2+JR+Lb}{J}& \frac{J^{2}L{p}_1{p}_2+JLb{p}_1+JLb{p}_2-J{K}^2+L{b}^2}{JK}\end{array}\right],\left[\begin{array}{c}\frac{LJ{p}_1{p}_2}{K}\end{array}\right]$

$\mathrm{clsys}≔\mathrm{StateFeedbackClosedLoop}\left(\mathrm{sys}\,\mathrm{Kc}\,':-\mathrm{Kr}'=\mathrm{Kr}\right)\:$

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

$\left[\begin{array}{l}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s);\; 2\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{w\_ref}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[w\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\right]\\ \mathrm{a}\=\left[\begin{array}{cc}-\frac{R}{L}+\frac{JL{p}_1+JL{p}_2+JR+Lb}{LJ}& -\frac{K}{L}-\frac{J^{2}L{p}_1{p}_2+JLb{p}_1+JLb{p}_2-J{K}^2+L{b}^2}{LJK}\\ \frac{K}{J}& -\frac{b}{J}\end{array}\right]\\ \mathrm{b}\=\left[\begin{array}{c}\frac{J{p}_1{p}_2}{K}\\ 0\end{array}\right]\\ \mathrm{c}\=\left[\begin{array}{cc}0& 1\end{array}\right]\\ \mathrm{d}\=\left[\begin{array}{c}0\end{array}\right]\end{array}\right.$

$\mathrm{clsys1}≔\mathrm{StateFeedbackClosedLoop}\left(\mathrm{sys}\,\mathrm{Kc}\,':-\mathrm{Kr}'=\mathrm{Kr}\,':-\mathrm{outputtype}'=\mathrm{tf}\,':-\mathrm{augment\_output}'=\mathrm{true}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{w\_ref}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[w\left(s\right)\,V\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{p_1{p}_2}{s^2+\left(-p_1-p_2\right)s+p_1{p}_2}\\ {\mathrm{tf}}_{2,1}\=\frac{LJ{p}_1{p}_2{s}^2+p_1{p}_2\left(JR+Lb\right)s+p_1{p}_2\left(K^2+Rb\right)}{K{s}^2+K\left(-p_1-p_2\right)s+K{p}_1{p}_2}\end{array}\right.$

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

$Z≔0.98\:$$\mathrm{\omega }≔\frac{5}{Z}\:$

$\mathrm{poles}≔\left[\mathrm{solve}\left(s^2+2Z\mathrm{\omega }s+{\mathrm{\omega }}^2\,s\right)\right]\:$

$p\left[1\right]≔\mathrm{poles}\left[1\right]\;$$p\left[2\right]≔\mathrm{poles}\left[2\right]$

$p_1≔\mathrm{-4.999999998}+1.015293305\mathrm{I}$

$p_2≔\mathrm{-4.999999998}-1.015293305\mathrm{I}$

$\mathrm{Kc}≔\mathrm{map}\left(\mathrm{simplify}\,\mathrm{eval}\left(\mathrm{Kc}\,\mathrm{params}\right)\,'\mathrm{zero}'\right)$

$\mathrm{Kc}≔\left[\begin{array}{cc}\mathrm{-1.000000002}& 13.00541026\end{array}\right]$

$\mathrm{Kr}≔\mathrm{map}\left(\mathrm{simplify}\,\mathrm{eval}\left(\mathrm{Kr}\,\mathrm{params}\right)\,'\mathrm{zero}'\right)$

$\mathrm{Kr}≔\left[\begin{array}{c}13.01541024\end{array}\right]$

$\mathrm{clsys}≔\mathrm{StateFeedbackClosedLoop}\left(\mathrm{sys}\,\mathrm{Kc}\,':-\mathrm{Kr}'=\mathrm{Kr}\,':-\mathrm{parameters}'=\mathrm{params}\right)\:$

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

$\mathrm{ResponsePlot}\left(\mathrm{clsys}\,8\mathrm{Step}\left(\right)\,'\mathrm{duration}'=10\,'\mathrm{numpoints}'=200\,'\mathrm{labels}'=\left[''t''\,''DC\; Motor\; Rotational\; Speed''\right]\,'\mathrm{labeldirections}'=\left[\mathrm{horizontal}\,\mathrm{vertical}\right]\,'\mathrm{gridlines}'=\mathrm{true}\,'\mathrm{view}'=\left[0..10\,0..10\right]\right)$

$\mathrm{prop}≔\mathrm{StepProperties}\left(\mathrm{clsys}\right)$

$\mathrm{prop}≔1.,\left[0.103866687817983\,0.100000000000000\right],\left[0.231175892242102\,0.333333333333333\right],\left[0.441908334624695\,0.666666666666667\right],\left[0.742737847117827\,0.900000000000000\right],\left[\mathrm{undefined}\,\mathrm{undefined}\right],\left[1.09872655595021\,0.980000000000000\right]$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{plot}\left(\left[\mathrm{prop}\left[2..7\right]\right]\,'\mathrm{style}'=\mathrm{point}\,'\mathrm{symbol}'=\mathrm{cross}\,'\mathrm{color}'=\mathrm{blue}\,'\mathrm{symbolsize}'=30\right)\,\mathrm{plot}\left(\left[\left[0\,\mathrm{prop}\left[1\right]\right]\,\left[10\,\mathrm{prop}\left[1\right]\right]\right]\,'\mathrm{color}'=\mathrm{gray}\right)\,\mathrm{ResponsePlot}\left(\mathrm{clsys}\,\mathrm{Step}\left(\right)\,'\mathrm{duration}'=10\,'\mathrm{numpoints}'=200\right)\,'\mathrm{gridlines}'=\mathrm{true}\,'\mathrm{title}'=''Step\; response''\,'\mathrm{labels}'=\left[''t''\,''DC\; Motor\; Rotational\; Speed''\right]\,'\mathrm{labeldirections}'=\left[\mathrm{horizontal}\,\mathrm{vertical}\right]\,'\mathrm{view}'=\left[0..10\,0..1.2\right]\right)$