method = MZN or AMIGO

Specifies the tuning method. MZN uses a modified version of the open loop Ziegler-Nichols tuning method based on the FOTD model. The AMIGO method calculates the controller gains using the AMIGO tuning rule and does not apply to the P controller. The default is MZN.

controller = P, PI or PID for the MZN method; PI or PID for the AMIGO method.

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 ZNTimeModified command calculates the controller gains based on the Ziegler-Nichols time domain (open-loop) tuning methods as specified in the options, using the given  first order time delay model parameters.

The ZNTimeModified command returns a list of controller gains $\mathrm{Kp}$, $\mathrm{Ki}$ and $\mathrm{Kd}$ according to the selected controller type P, PI or PID, respectively, 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: $C\left(s\right)=\mathrm{Kp}+\frac{\mathrm{Ki}}{s}$

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

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

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

$\mathrm{sys}≔\mathrm{NewSystem}\left(\frac{2.5}{s^2+3s+5}\right)\:$

$\mathrm{Params}≔\mathrm{ParameterIdentify}:-\mathrm{TimeDomain}\left(\mathrm{sys}\right)$

$\mathrm{Params}≔\left[0.5000000000\,0.2000000000\,0.3645751311\right]$

$\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{MZN}\,'\mathrm{controller}'=P\right)$

$\left[1.097167540\right]$

$\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{MZN}\,'\mathrm{controller}'=\mathrm{\Pi }\right)$

$\left[0.9874507860\,0.9028324590\right]$

$\mathrm{pid}≔\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{MZN}\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{factored}'=\mathrm{true}\,'\mathrm{returntype}'=\mathrm{record}\right)$

$\mathrm{pid}≔{\mathrm{Record}}_{\mathrm{packed}}\left(K=1.316601048\,\mathrm{Ti}=0.7291502622\,\mathrm{Td}=0.1822875656\right)$

$\mathrm{pidsys}≔\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{MZN}\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{returntype}'=\mathrm{system}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{0.2399999999s^2+1.316601048s+1.805664917}{s}\end{array}\right.$

$\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{AMIGO}\,'\mathrm{controller}'=P\right)$

Error, (in ControlDesign:-ZNTimeModified) invalid controller type; the AMIGO method does not apply to the 'P' controller

$\mathrm{PIpars}≔\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{AMIGO}\,'\mathrm{controller}'=\mathrm{\Pi }\,'\mathrm{returntype}'=\mathrm{record}\right)$

$\mathrm{PIpars}≔{\mathrm{Record}}_{\mathrm{packed}}\left(\mathrm{Kp}=0.4330243721\,\mathrm{Ki}=1.879995098\right)$

$\mathrm{PIsys}≔\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{AMIGO}\,'\mathrm{controller}'=\mathrm{\Pi }\,'\mathrm{returntype}'=\mathrm{system}\right)\:$

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

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{0.4330243721s+1.879995098}{s}\end{array}\right.$

$\mathrm{ZNTimeModified}\left(\mathrm{Params}\left[1\right]\,\mathrm{Params}\left[2\right]\,\mathrm{Params}\left[3\right]\,'\mathrm{method}'=\mathrm{AMIGO}\,'\mathrm{controller}'=\mathrm{PID}\right)$

$\left[0.8937253934\,3.082606536\,0.1053196425\right]$

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