Controller Tuning : NMinimize of Mathematica versus ga command of Matlab
Authors: Housam Binous and Ahmed Bellagi

MATLAB code given below uses genetic algorithms for PI controller tuning

MATHEMATICA code below uses NMINIMIZE to fetch best PI controller parameters

In[]:=

Gc[KP_?NumericQ,TAUI_?NumericQ]:=KP+TAUI/s;GT=1/(s+1)/(s+2)/(s+3)

Out[]=

1

(1+s)(2+s)(3+s)

In[]:=

G[KP_?NumericQ,TAUI_?NumericQ]:=GTGc[KP,TAUI]/(1+GTGc[KP,TAUI])

In[]:=

y[KP_?NumericQ,TAUI_?NumericQ]:=InverseLaplaceTransform[1/sG[KP,TAUI],s,t]//Chop;

ISE CRITERIA

In[]:=

f1[KP_?NumericQ,TAUI_?NumericQ]:=Re[Quiet@NIntegrate[(1-y[KP,TAUI])^2,{t,0,100}]//Chop];(*IAE*)

In[]:=

NMinimize[f1[KP,TAUI],{KP,TAUI},MaxIterations->10]

Out[]=

{0.696163,{KP20.2936,TAUI5.69175}}

OUTPUT FOR MATLAB code

IAE CRITERIA

In[]:=

f2[KP_?NumericQ,TAUI_?NumericQ]:=Re[Quiet@NIntegrate[Abs[1-y[KP,TAUI]],{t,0,100}]//Chop];(*IAE*)

In[]:=

NMinimize[f2[KP,TAUI],{KP,TAUI},MaxIterations->10]

Out[]=

{1.28738,{KP11.8884,TAUI5.97855}}

OUTPUT FOR MATLAB code

ITAE CRITERIA

f3[KP_?NumericQ,TAUI_?NumericQ]:=Re[Quiet@NIntegrate[tAbs[1-y[KP,TAUI]],{t,0,100}]//Chop];(*IAE*)

In[]:=

NMinimize[f3[KP,TAUI],{KP,TAUI},MaxIterations->10]

Out[]=

{1.52359,{KP7.21523,TAUI4.9157}}

OUTPUT FOR MATLAB code