controller = P, PI, PD, or PID

Specifies the controller type. The default value is PID.

factored  = true or false

True means return the factored version of the controller gains ($K=\mathrm{Kp}$, $\mathrm{Ti}=\frac{\mathrm{Kp}}{\mathrm{Ki}}$, $\mathrm{Td}=\frac{\mathrm{Kd}}{\mathrm{Kp}}$). False means return the controller gains ($\mathrm{Kp}$, $\mathrm{Ki}$, $\mathrm{Kd}$). The default is false.

returntype =  list or record or system

Specifies the returned type. If list (or record) is specified, a list (or record) containing the controller parameters is returned. If system is specified, a System (DynamicSystems system object) containing the transfer function of the specified controller type (P, PI, PID) is returned. The default value is list.

The CohenCoon command calculates the controller gains based on the Cohen-Coon tuning method using the given first-order time-delay model parameters.

The CohenCoon command returns a list of controller gains $\mathrm{Kp}$, $\mathrm{Ki}$ and $\mathrm{Kd}$ according to the selected controller type (P, PI, PD, PID), where $\mathrm{Kp}$ is the proportional gain, $\mathrm{Ki}$ is the integral gain, and $\mathrm{Kd}$ is the derivative gain.

The controller transfer function is then obtained as:

P: $C\left(s\right)=\mathrm{Kp}$ 

PI: $C\left(s\right)=\mathrm{Kp}+\frac{\mathrm{Ki}}{s}$

PD: $C\left(s\right)=\mathrm{Kd}s+\mathrm{Kp}$

PID: $C\left(s\right)=\mathrm{Kp}+\frac{\mathrm{Ki}}{s}+\mathrm{Kd}s$

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

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

$\mathrm{sys1}≔\mathrm{NewSystem}\left(\frac{s+1}{s^2+5s+12}\right)\:$

$\mathrm{Params1}≔\mathrm{evalf}\left(\mathrm{ParameterIdentify}:-\mathrm{TimeDomain}\left(\mathrm{sys1}\right)\right)$

$\mathrm{Params1}≔\left[0.08333333333\,0.9965217283\,0.9331168468\right]$

$\mathrm{Pgain}≔\mathrm{CohenCoon}\left(\mathrm{Params1}\left[1\right]\,\mathrm{Params1}\left[2\right]\,\mathrm{Params1}\left[3\right]\,'\mathrm{controller}'=P\right)$

$\mathrm{Pgain}≔\left[17.01539475\right]$

$\mathrm{PIgains}≔\mathrm{CohenCoon}\left(\mathrm{Params1}\left[1\right]\,\mathrm{Params1}\left[2\right]\,\mathrm{Params1}\left[3\right]\,'\mathrm{controller}'=\mathrm{\Pi }\right)$

$\mathrm{PIgains}≔\left[12.52745528\,11.47249211\right]$

$\mathrm{PDgains}≔\mathrm{CohenCoon}\left(\mathrm{Params1}\left[1\right]\,\mathrm{Params1}\left[2\right]\,\mathrm{Params1}\left[3\right]\,'\mathrm{controller}'=\mathrm{PD}\right)$

$\mathrm{PDgains}≔\left[17.82548949\,2.753995614\right]$

$\mathrm{PIDgains}≔\mathrm{CohenCoon}\left(\mathrm{Params1}\left[1\right]\,\mathrm{Params1}\left[2\right]\,\mathrm{Params1}\left[3\right]\,'\mathrm{controller}'=\mathrm{PID}\right)$

$\mathrm{PIDgains}≔\left[20.21678293\,11.46866675\,5.925668011\right]$

$\mathrm{PIDrec}≔\mathrm{CohenCoon}\left(\mathrm{Params1}\left[1\right]\,\mathrm{Params1}\left[2\right]\,\mathrm{Params1}\left[3\right]\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{returntype}'=\mathrm{record}\right)$

$\mathrm{PIDrec}≔{\mathrm{Record}}_{\mathrm{packed}}\left(\mathrm{Kp}=20.21678293\,\mathrm{Ki}=11.46866675\,\mathrm{Kd}=5.925668011\right)$

$\mathrm{PIDsys}≔\mathrm{CohenCoon}\left(\mathrm{Params1}\left[1\right]\,\mathrm{Params1}\left[2\right]\,\mathrm{Params1}\left[3\right]\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{returntype}'=\mathrm{system}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{5.925668011s^2+20.21678293s+11.46866675}{s}\end{array}\right.$

$\mathrm{sys2}≔\mathrm{NewSystem}\left(\frac{s+a}{s^2+5s+12}\right)\:$

$\mathrm{Params2}≔\mathrm{ParameterIdentify}:-\mathrm{TimeDomain}\left(\mathrm{sys2}\right)$

$\mathrm{Params2}≔\left[\frac{a}{12}\,\frac{\sqrt{\left|\frac{a^2}{144}-1\right|}}{a}\,\frac{\left|\frac{-12+5a-12\sqrt{\left|\frac{a^2}{144}-1\right|}}{a}\right|{\left|a\right|}^2+\left|\sqrt{2}\sqrt{\left|13a^2-72\left|\frac{a^2}{144}-1\right|-60a\right|}a-12\sqrt{\left|\frac{a^2}{144}-1\right|}\left|a\right|\right|}{24{\left|a\right|}^2}\right]$

$\mathrm{PIDparams}≔\mathrm{CohenCoon}\left(\mathrm{op}\left(\mathrm{Params2}\right)\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{factored}'=\mathrm{true}\,'\mathrm{returntype}'=\mathrm{record}\right)\:$

$\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{PIDparams}:-K\,a=1\right)\right)$

$20.21678292$

$\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{PIDparams}:-\mathrm{Ti}\,a=1\right)\right)$

$1.762784060$

$\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{PIDparams}:-\mathrm{Td}\,a=1\right)\right)$

$0.2931063776$

$\mathrm{PIDgains}≔\mathrm{CohenCoon}\left(\mathrm{op}\left(\mathrm{Params2}\right)\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{returntype}'=\mathrm{record}\right)\:$

$\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{PIDgains}:-\mathrm{Kp}\,a=1\right)\right)$

$20.21678292$

$\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{PIDgains}:-\mathrm{Ki}\,a=1\right)\right)$

$11.46866674$

$\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{PIDgains}:-\mathrm{Kd}\,a=1\right)\right)$

$5.925668009$

[1] K. J. Astrom and T. Hagglund, Advanced PID Control, ISA, 2006.