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).

controlled_input = posint

Specifies the controlled input ($u_c$) index for sys, for the MIMO case. The default is 1.  

controlled_output = posint

Specifies the controlled output ($y_c$) index for sysfor the MIMO case. The  default is 1.

augment_output = true or false

True means append the controlled_input (controller output) to the output vector. The default is false.

The PIDClosedLoop command calculates the closed-loop system equations of a system sys with a PID controller pid in the direct path.

The system 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).

The controller pid is a SISO (single input, single output) system object created using the DynamicSystems package. The system object is generally a transfer function (TF).

If sys is a MIMO system, only one input and one output can be selected to form the closed-loop system with the PID controller pid.

If $u_c$ is the $i^{\mathrm{th}}$ input of sys, $y_{\mathrm{\_ref}}$ is the $i^{\mathrm{th}}$  input of the closed-loop.

If $y_c$ is the $j^{\mathrm{th}}$ output of sys, it remains the $j^{\mathrm{th}}$  output of the closed-loop.  

The rest of the inputs $u_{\mathrm{nc}}$ (non-controlled inputs) and outputs $y_{\mathrm{nc}}$ (non-controlled outputs) of sys are also included in the inputs and outputs of the closed-loop system.

The pid output ($u_c$) is appended to the output vector of the closed-loop if the option augment_output = true is specified.

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

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

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

$\mathrm{eq1}≔J\mathrm{diff}\left(\mathrm{\omega }\left(t\right)\,t\right)+b\mathrm{\omega }\left(t\right)-Ki\left(t\right)=0\:$

$\mathrm{eq2}≔L\mathrm{diff}\left(i\left(t\right)\,t\right)+Ri\left(t\right)=V\left(t\right)-K\mathrm{\omega }\left(t\right)\:$

$\mathrm{eq3}≔\mathrm{diff}\left(\mathrm{\theta }\left(t\right)\,t\right)=\mathrm{\omega }\left(t\right)\:$

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

$\mathrm{systf}≔\mathrm{TransferFunction}\left(\left[\mathrm{eq1}\,\mathrm{eq2}\,\mathrm{eq3}\right]\,\left[V\left(t\right)\right]\,\left[i\left(t\right)\,\mathrm{\omega }\left(t\right)\,\mathrm{\theta }\left(t\right)\right]\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{3\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[V\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[i\left(s\right)\,\mathrm{\omega }\left(s\right)\,\mathrm{\theta }\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{Js+b}{JL{s}^2+\left(JR+Lb\right)s+K^2+Rb}\\ {\mathrm{tf}}_{2,1}\=\frac{K}{JL{s}^2+\left(JR+Lb\right)s+K^2+Rb}\\ {\mathrm{tf}}_{3,1}\=\frac{K}{JL{s}^3+\left(JR+Lb\right)s^2+\left(K^2+Rb\right)s}\end{array}\right.$

$\mathrm{subsystf}≔\mathrm{Subsystem}\left(\mathrm{systf}\,'\mathrm{all}'\,\left\{3\right\}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[V\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{\theta }\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{K}{JL{s}^3+\left(JR+Lb\right)s^2+\left(K^2+Rb\right)s}\end{array}\right.$

$\mathrm{\tau }≔0.7$

$\mathrm{\tau }≔0.7$

$\mathrm{PIDgains}≔\mathrm{PIDAuto}\left(\mathrm{subsystf}\,\mathrm{\tau }\,'\mathrm{parameters}'=\mathrm{params}\right)$

$\mathrm{PIDgains}≔{\mathrm{Record}}_{\mathrm{packed}}\left(\mathrm{Kp}=10.2115133154945\,\mathrm{Ki}=0.000879895477927287\,\mathrm{Kd}=1.01771326328058\,\mathrm{Tf}=0\right)$

$\mathrm{Kc}≔\mathrm{PIDgains}\left['\mathrm{Kp}'\right]+\frac{\mathrm{PIDgains}\left['\mathrm{Ki}'\right]}{s}+\frac{\mathrm{PIDgains}\left['\mathrm{Kd}'\right]s}{1+\frac{1}{100}\mathrm{PIDgains}\left['\mathrm{Kd}'\right]s}$

$\mathrm{Kc}≔10.2115133154945+\frac{0.000879895477927287}{s}+\frac{1.01771326328058s}{1+0.0101771326328058s}$

$\mathrm{sysKc}≔\mathrm{TransferFunction}\left(\mathrm{Kc}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1.121637189s^2+10.21152227s+0.0008798954779}{0.01017713263s^2+1.s}\end{array}\right.$

$\mathrm{feedback}≔\mathrm{PIDClosedLoop}\left(\mathrm{systf}\,\mathrm{sysKc}\,'\mathrm{parameters}'=\mathrm{params}\,'\mathrm{controlled\_output}'=3\,'\mathrm{augment\_output}'=\mathrm{true}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{4\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{theta\_ref}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[i\left(s\right)\,\mathrm{\omega }\left(s\right)\,\mathrm{\theta }\left(s\right)\,V\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{5.707525218\times {10}^{\mathrm{-7}}{s}^7+0.00007383461575s^6+0.001938610656s^5+0.01992714700s^4+0.08375976566s^3+0.1022235877s^2+8.807753734\times {10}^{\mathrm{-6}}s}{2.589350714\times {10}^{\mathrm{-9}}{s}^8+5.710010486\times {10}^{\mathrm{-7}}{s}^7+0.00003768910326s^6+0.0006954648583s^5+0.004914660338s^4+0.01347328969s^3+0.01737375949s^2+0.01022227069s+8.807753734\times {10}^{\mathrm{-7}}}\\ {\mathrm{tf}}_{2,1}\=\frac{5.707525218\times {10}^{\mathrm{-7}}{s}^6+0.00006812709054s^5+0.001257339750s^4+0.007353749497s^3+0.01022227069s^2+8.807753734\times {10}^{\mathrm{-7}}s}{2.589350714\times {10}^{\mathrm{-9}}{s}^8+5.710010486\times {10}^{\mathrm{-7}}{s}^7+0.00003768910326s^6+0.0006954648583s^5+0.004914660338s^4+0.01347328969s^3+0.01737375949s^2+0.01022227069s+8.807753734\times {10}^{\mathrm{-7}}}\\ {\mathrm{tf}}_{3,1}\=\frac{0.01121637189s^2+0.1021152227s+8.798954779\times {10}^{\mathrm{-6}}}{0.00005088566315s^5+0.005610627958s^4+0.06101873098s^3+0.1113163719s^2+0.1021152227s+8.798954779\times {10}^{\mathrm{-6}}}\\ {\mathrm{tf}}_{4,1}\=\frac{0.005608185945s^5+0.1183558427s^4+0.7249716183s^3+1.022226173s^2+0.00008807753734s}{0.00005088566315s^5+0.005610627958s^4+0.06101873098s^3+0.1113163719s^2+0.1021152227s+8.798954779\times {10}^{\mathrm{-6}}}\end{array}\right.$

$\mathrm{ResponsePlot}\left(\mathrm{feedback}\,1\,'\mathrm{duration}'=10\,'\mathrm{gridlines}'=\mathrm{true}\,'\mathrm{parameters}'=\mathrm{params}\,'\mathrm{output}'=\mathrm{\theta }\right)$