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<div class=Section1>
<p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p>
<div>
<div style='border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm'>
<p class=MsoNormal><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>De :</span></b><span
style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> Perrichon Pierre
[mailto:perrichon.pierre@wanadoo.fr] <br>
<b>Envoyé :</b> mardi 6 avril 2010 08:33<br>
<b>À :</b> 'users@lists.scilab.org'<br>
<b>Objet :</b> <o:p></o:p></span></p>
</div>
</div>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal>Dear,<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal><span lang=EN-US>I am studying different methods to translate
un simple transfer function from %s to %z, and try different methods to track
the better one, with a frequency approach.<o:p></o:p></span></p>
<p class=MsoListParagraph style='text-indent:-18.0pt;mso-list:l0 level1 lfo2'><![if !supportLists]><span
lang=EN-US><span style='mso-list:Ignore'>-<span style='font:7.0pt "Times New Roman"'>
</span></span></span><![endif]><span lang=EN-US>Continuous (this is the base in
%s)<o:p></o:p></span></p>
<p class=MsoListParagraph style='text-indent:-18.0pt;mso-list:l0 level1 lfo2'><![if !supportLists]><span
lang=EN-US><span style='mso-list:Ignore'>-<span style='font:7.0pt "Times New Roman"'>
</span></span></span><![endif]><span lang=EN-US>Euler method<o:p></o:p></span></p>
<p class=MsoListParagraph style='text-indent:-18.0pt;mso-list:l0 level1 lfo2'><![if !supportLists]><span
lang=EN-US><span style='mso-list:Ignore'>-<span style='font:7.0pt "Times New Roman"'>
</span></span></span><![endif]><span lang=EN-US>Bilinear method<o:p></o:p></span></p>
<p class=MsoListParagraph style='text-indent:-18.0pt;mso-list:l0 level1 lfo2'><![if !supportLists]><span
lang=EN-US><span style='mso-list:Ignore'>-<span style='font:7.0pt "Times New Roman"'>
</span></span></span><![endif]><span lang=EN-US>Impulse response<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Then, I plot the different graphs using the
BODE instruction.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>The plot is apparently not correct, because
It shows that the Impulse method is the better one, except that the continuous
curve has been shifted (see and compare graph 1 and 2)<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I had to redraw all the graphs using the
repfreq, dbphi, and plot2d instructions to get correct results, which show that
the Bilinear method is the best one (graph 4)<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>So I think there is a bug in the Bode plot,
or a method error (SIMO // SISO ?)<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Could you confirm ?<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Here is my program (also in the attached
file)<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>// Programme P2_2<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>// 06-04-2010<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//----------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>s=%s;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>z=%z;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>Te=0.0003;
// Sample time<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>Fmin=1;
//Hertz<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>Fmax=1000;
// Hertz<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>K=0.43;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>T=0.0075;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>a=0.2;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT0=K*(1+T*s)/(1+a*T*s);
// Continuous system<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT0 = syslin('c', FT0);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>FT10_0 = tf2ss(FT0);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>pz=(1-z^-1)/Te;
// Euler Transform<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>FT1 =
K*(1+T*pz)/(1+a*T*pz);
<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT1 = syslin(Te, FT1 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT10_1 = tf2ss(FT1);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>pz=2/Te*(1-z^-1)/(1+z^-1);
// Bilinear transform<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>FT2 = K*(1+T*pz)/(1+a*T*pz);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT2 = syslin(Te, FT2 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT10_2 = tf2ss(FT2);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>b=1/(a*T);
// Invariance impulsionnelle<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>c=1/a;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>EXP=exp(-b*Te);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>N=c*(z-1) + 1 - EXP;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>D=z-EXP;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT3=K*(N/D);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT3 = syslin(Te, FT3 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>FT10_3 = tf2ss(FT3);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>scf(1);clf();
// Create and clear screen 1<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>bode([FT10_0],Fmin,Fmax);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>legends(['Continu';], [1;],
with_box=%t, opt=4 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>scf(2);clf();
// Create and clear screen 2<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>bode([FT10_0;FT10_3],Fmin,Fmax);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>legends(['Continu';'Invariance'],
[1;2], with_box=%t, opt=4 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>scf(3);clf();
// Create and clear screen 3<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>bode([FT10_0;FT10_1;FT10_2;FT10_3],Fmin,Fmax);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>legends(['Continu';'Euler';'Bilineaire';'Invariance';],
[1;2;3;4], with_box=%t, opt=4 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//--------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>frq=Fmin:0.01:Fmax;<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[frq0,rep0]
=repfreq(FT10_0,frq);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[db0,phi0]=dbphi(rep0);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[frq1,rep1]
=repfreq(FT10_1,frq);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'>[db1,phi1]=dbphi(rep1); <o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span style='font-size:10.0pt;
font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[frq2,rep2]
=repfreq(FT10_2,frq);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[db2,phi2]=dbphi(rep2); <o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[frq3,rep3]
=repfreq(FT10_3,frq);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>[db3,phi3]=dbphi(rep3); <o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//Affichage<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>//------------<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>scf(4);clf();
// Create and clear screen 4<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>xgrid();<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>subplot(2,1,1);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,db0,style=1);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,db1,style=2);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,db2,style=3);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,db3,style=4);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>legends(['Continu';'Euler';'Bilineaire';'Invariance';],
[1;2;3;4], with_box=%t, opt=4 );<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>xtitle('', 'Fréquence
Hz','Magnitude (Db)');<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>subplot(2,1,2);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,phi0,style=1);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,phi1,style=2);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,phi2,style=3);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>plot2d('ln',frq,phi3,style=4);<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>xgrid();<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'>xtitle('', 'Fréquence
(Hz)','Phase (degré)');<o:p></o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal style='text-autospace:none'><span lang=EN-US
style='font-size:10.0pt;font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Best regards<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal>P.Perrichon<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal><b><i><span style='font-size:10.0pt;font-family:"Arial","sans-serif";
color:blue'><o:p> </o:p></span></i></b></p>
<p class=MsoNormal style='margin-bottom:12.0pt'><b><i><span style='font-size:
8.0pt;font-family:"Arial","sans-serif";color:blue'>Nous protégeons notre
environnement : n'imprimez ce mail que si nécessaire.<br>
</span></i></b><span style='font-size:7.5pt;font-family:"Arial","sans-serif";
color:silver'>:._______________<br>
<br>
</span><span style='font-size:8.0pt'><o:p></o:p></span></p>
<p class=MsoNormal><o:p> </o:p></p>
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