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As engineers, we have control over the \+ spring constant " }{TEXT 256 1 "k" }{TEXT -1 25 " and the damper const ant " }{TEXT 257 1 "c" }{TEXT -1 18 ". Given the mass " }{TEXT 259 2 "m " }{TEXT -1 79 "of the car and the expected amplitude of a typical \+ bump, our task is to choose " }{TEXT 260 1 "k" }{TEXT -1 5 " and " } {TEXT 258 1 "c" }{TEXT -1 90 " so that the suspension system's respons e is as close as possible to the desired response." }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 143 " Our approach to this engineering problem \+ is to measure the discrepancy between the desired and actual motion of the system as a function of " }{TEXT 262 1 "k" }{TEXT -1 5 " and " } {TEXT 261 4 "c. " }{TEXT -1 137 "After deriving the actual response b y solving the system's differential equation, we use an optimization a lgorithm to find the values of " }{TEXT 264 1 "k" }{TEXT -1 5 " and " }{TEXT 263 1 "c" }{TEXT -1 67 " that minimize this discrepancy. In th is application, we will use " }{TEXT 299 55 "Newton's Method for uncon strained nonlinear programming" }{TEXT -1 31 " as the optimization alg orithm." }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 3 " " }{TEXT 265 46 "B elow is a diagram of the suspension system. " }{TEXT 266 4 "x(t)" } {TEXT 267 64 " denotes the amplitude of the automobile as a function o f time. 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::::::::::::::::J:<:vY::::J::yA::::Z::xI::::::::B:Zy=:::::::J::yA::::Z ::xI::::::::;:vY:::::<:ry:::::>:jy;::::;:vY::::::::J::yA:::::::Z::xI:: :::;:vY::::::::<:ry::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::Z:vYxI:::::::::::::::::::::::::::B:;:xI::::B:Zy=::::J::yA:::::: ::>:jy;:::::::B:Zy=::::J::yA:::::::Z::xI:::::;:vY:::::<:ry::::Z::xI::: :::::B:Zy=:::::::J::yA::::Z::xI::::::::;:vY::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::Z::yA ::::>:jy;::::B:Zy=::::::::<:ry::::::::>:jy;::::B:Zy=:::::::J::yA::::Z: :xI:::::;:vY::::J::yA::::::::>:jy;:::::::B:Zy=::::J::yA:::::::Z::xI::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::B:yay=:::::::::: :::::::::::::::::::jy;::::::::;:vY::::::::<:ry:::: :>:jy;:::::::B:Zy=::::J::yA::::Z::xI::::B:Zy=::::::::<:ry::::::::>:jy; ::::B:Zy=:::::::J::yA::::::::::::::::::::::::::::::::::::::::::::::::: :::::::Z:vYxI:::::::::::::::::::::::::::::;B:jy;::::;:vY:::::<:ry::::: :::Z::xI::::::::;:vY:::::<:ry::::::::>:jy;::::B:Zy=::::J::yA::::>:jy;: :::::::;:vY::::::::<:ry:::::>:jy;:::::::B:Zy=::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::; ::::::::::::::::::::::::::::::::::::::::::::::::::::::F:Y:<::::::AB::<::::::wAB:::::::::: :::yay=J:B:::::::::::::::::::jysy:>:<::::::::::2:" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "An Example Desired Response" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 375 "For illustration purposes, suppose \+ our target response is a decaying exponential function. That is, when the automobile hits a bump of unit amplitude, we want it to spring up and down with an exponentially shrinking amplitude. (In practice, the desired response need not be an algebraic function. It could simply \+ be a discrete set of ordered pairs of amplitudes and times.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Target:=t->exp(-. 2*t)*cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'TargetGR6#%\"tG6\"6 $%)operatorG%&arrowGF(*&-%$expG6#,$9$$!\"#!\"\"\"\"\"-%$cosG6#F1F5F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(Target(t), t = 0 ..20, title=`Target Response`, thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7hp7$$\"\"!F)$\"\"\"F )7$$\"3emmm;arz@!#=$\"3y&>P2Q(*oM*F/7$$\"39LLLL3VfVF/$\"3XCtg:g&yI)F/7 $$\"3]****\\i&*)fD'F/$\"3cqGOYNx_rF/7$$\"3%pmm;H[D:)F/$\"3qv,P@[?DeF/7 $$\"3#QL$e9w)*=#*F/$\"3&3L6O=\\b-&F/7$$\"3'****\\PpU&G5!#<$\"3[RRG'[`6 ?%F/7$$\"3am;/Em=N6FI$\"3%*z+LgseiLF/7$$\"3MLLLe0$=C\"FI$\"3j?&*)zcU,_ #F/7$$\"3SLL$eR\"=\\8FI$\"318*yvmB#y;F/7$$\"3[LLLLA`c9FI$\"3kI)=)GVB?& )!#>7$$\"3cLL$32$)Qc\"FI$\"3-D-'G&pSc]!#?7$$\"3iLLL3RBr;FI$!3!)ym:3R,y rFjn7$$\"3C++vV^\"\\)=FI$!3OM_nQ$p$>@F/7$$\"3imm;zjf)4#FI$!3^obK'=S+J$ F/7$$\"3-++]Piq'H#FI$!3a@Al&yCL>%F/7$$\"3ULL$e4;[\\#FI$!3OaI!3C?`%[F/7 $$\"37++]P%*4`TL&F/7$$\"3`LL$3-8D!GFI$!3Cym>gU9%Q&F/7$ $\"3)omm;%ez`GFI$!3\\UQQ@ua=aF/7$$\"3!)****\\i'y]!HFI$!3c\"oMr'fePaF/7 $$\"3#pmm\"Ha6eHFI$!3!y=zn@\"RTaF/7$$\"3gLL$e>_6,$FI$!3fTUEn(p$HaF/7$$ \"3G++]i*)=kIFI$!3\")R$R(H'))=S&F/7$$\"3'pmm\"HdA$GFjn7$$\"3]+++]1!>+&FI$\"3)))RMxrQ)\\5F/7$$\"3T++++FZ=_FI$ \"3y(oQ(\\G22(=hB!)=GF/7$$\"3yL$e9YAs%f FI$\"3)on43cGP(GF/7$$\"3Qnm\"Hd&)>/'FI$\"3`Yy@]gH+HF/7$$\"3uLekGrO*3'F I$\"3czrR3+>.HF/7$$\"35+]P%o[n8'FI$\"3%G2XzO;$**GF/7$$\"3MnT5S-8%='FI$ \"3)G*44U6\")))GF/7$$\"3qLL$ez6:B'FI$\"3ci2DDJ#=(GF/7$$\"3Q++D1o(oX'FI $\"3+R5\"*=Gc2FF/7$$\"3/nmm;=C#o'FI$\"3W)fmaj.8U#F/7$$\"3Mnmmm#pS1(FI$ \"3UTv!4[c#Hq)Fjn7$$\"3<+++DZ5Q&)FI$!3 7h^U2IyX6F/7$$\"3!3++v3zMu)FI$!3?\\*>#=*f:N\"F/7$$\"3sn;a)QA1&))FI$!3i L))[A]-I9F/7$$\"3kMLe*olx&*)FI$!3I;[_tF])[\"F/7$$\"3c,]i!**3\\1*FI$!36 (zX&\\*[r_\"F/7$$\"3Yomm\"H_?<*FI$!3k#R*)yKWja\"F/7$$\"3An;a)GV/F*FI$! 3W(GTl)HMZ:F/7$$\"3unmT&GM)o$*FI$!3xg'3^\")RI`\"F/7$$\"3Go;H#GDsY*FI$! 33S&35VmT]\"F/7$$\"3-nm;zihl&*FI$!3E44ZS*p:Y\"F/7$$\"33++vVA(yx*FI$!32 -:ER'3#)Fjn 7$$\"31+](=GB2F\"F]`l$\"3#)otn=zB(z(Fjn7$$\"3+++]2'HKH\"F]`l$\"3k?Otis ;IqFjn7$$\"3`mmmwanL8F]`l$\"3y2N=4;-$)\\Fjn7$$\"35+++v+'oP\"F]`l$\"3#> b**GtHYH#Fjn7$$\"3CLLeR<*fT\"F]`l$!3_G=C^LzR8F`o7$$\"3C+++&)Hxe9F]`l$! 3m+kVSOVaBFjn7$$\"3gmm\"H!o-*\\\"F]`l$!37=@\\YQ)yv$Fjn7$$\"3:++DTO5T:F ]`l$!3g>i&Rx=^Q%Fjn7$$\"3emmmT9C#e\"F]`l$!3?XO:,`'f>%Fjn7$$\"3\"****\\ i!*3`i\"F]`l$!3)=ekcv(Q8LFjn7$$\"3_LLL$*zym;F]`l$!3+j\")*Q)=sX?Fjn7$$ \"3fLL$3N1#4F]`l$\"3' 3P7UN4>1#Fjn7$$\"3A++v.Uac>F]`l$\"3px+TArT2:Fjn7$$\"#?F)$\"3!z=W6o$Guu F`o-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!6\"-%&TITL EG6#%0Target~ResponseG-%*THICKNESSG6#\"\"$-%%VIEWG6$;F(F^hl%(DEFAULTG " 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Deriving the Actual Response" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 159 "To derive the actual response of the system to a un it bump, we must solve the differential equation governing the system' s behavior with the initial condition " }{TEXT 270 8 "x(0) = 1" } {TEXT 271 102 ". Below is a more general equation describing the syst em under a periodic disturbance with frequency " }{XPPEDIT 18 0 "omega ;" "6#%&omegaG" }{TEXT 272 15 " and amplitude " }{XPPEDIT 18 0 "F[0]; " "6#&%\"FG6#\"\"!" }{TEXT 273 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "System_Equation := diff( x(t),`$`(t,2))+2*zeta*omega[n]*diff(x(t),t)+omega[n]^2*x(t) = F[0]/m*s in(omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0System_EquationG/,( -%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"**F1F2%%zetaGF2&%&omegaG6# %\"nGF2-F(6$F*F-F2F2*&)F5F1F2F*F2F2*&*&&%\"FG6#\"\"!F2-%$sinG6#*&F6F2F -F2F2F2%\"mG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where:" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "w[n] = sqrt(k/m);" "6#/&%\"wG6#%\"nG-%% sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "zeta \+ = c/c[c];" "6#/%%zetaG*&%\"cG\"\"\"&F&6#F&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "c[c] = 2*m*omega[n]; " "6#/&%\"cG6#F%*(\"\"#\"\"\"%\"mGF)&%&omegaG6#%\"nGF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 274 108 "To solve this differential equation, we've developed a package sp ecifically for analyzing vibrating systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "libname:=\"c:/myl ib/vibsys\",libname:\nwith(plots):\nwith(VibratingSystem);" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/%*AmplitudeG%=Amplitude_Respon se_AnimationG%8Amplitude_Response_PlotG%1Critical_DampingG%2Natural_Fr equencyG%8Operating_Damping_RatioG%&PhaseG%9Phase_Response_AnimationG% 4Phase_Response_PlotG%3Resonant_AmplitudeG%/Resonant_PhaseG%-Solve_Sys temG%3Time_Response_PlotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 275 61 "Our initial conditions are unit amplitude and zero veloc ity: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Xinit:=1; Xdotinit:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&XinitG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)XdotinitG \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 65 "We now s olve the system and define the response as a function of " }{TEXT 277 1 "k" }{TEXT 278 2 ", " }{TEXT 279 2 "c " }{TEXT 293 4 "and " }{TEXT 294 2 "t," }{TEXT 280 6 " with " }{TEXT 281 5 "m=100" }{TEXT 282 5 " a nd " }{TEXT 283 7 "F0 = 0." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "System_Response:=una pply(Solve_System(Xinit,Xdotinit),(k,c,m,omega,F0,t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0System_ResponseGR6(%\"kG%\"cG%\"mG%&omegaG%#F0G %\"tG6\"6$%)operatorG%&arrowGF-,(*&*(9(\"\"\"9&F5,**,9'F5F6F5-%$cosG6# *&F9F59)F5F59%F5-%%sqrtG6#,&*$)F?\"\"#F5F5*(\"\"%F59$F5F6F5!\"\"F5\"\" )**FFF5-%$sinGFF5F5F6FJFjnF5F5* &-FA6#FCF5,**&F_oF5FUF5F5*&FEF5FTF5F5**FFF5FIF5F6F5FTF5FJ*$F\\pF5F5F5F JF5*&#F5FFF5*&*&,:F[pFJ*,FFF5F?F5FIF5F6F5FTF5FJF^oF5FfoF5F]pF5**FFF5F4 F5FUF5FboF5F5FjoF5FioFJF`oFJFhoFJ*,FFF5F4F5F6F5F9F5FIF5FJ*,FFF5F@F5FIF 5F6F5FTF5F5F5-F_p6#,$*&*&,&F?F5FepF5F5F>F5F5F6FJ#FJFFF5F5*&-FA6#FCF5Fi pF5FJF5FJF-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Actual:= (k, c, t)->System_Response(k, c, 100, 1, 0, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ActualGR6%%\"kG%\"cG%\"tG6\"6$%)operatorG%&arrowGF*- %0System_ResponseG6(9$9%\"$+\"\"\"\"\"\"!9&F*F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 284 52 "Let's now overlay plots of the act ual response with " }{TEXT 285 4 "k=10" }{TEXT 286 5 " and " }{TEXT 287 4 "c=10" }{TEXT 288 57 ", and the target response. Notice that th ese choices of " }{TEXT 289 2 "k " }{TEXT 290 4 "and " }{TEXT 291 1 "c " }{TEXT 292 117 " do not provide a good match to the target. One app roach to improving the match is to try different combinations of " } {TEXT 295 2 "k " }{TEXT 296 4 "and " }{TEXT 297 2 "c " }{TEXT 298 105 "until we're satisfied. However, we use a more systematic approach th at utilizes optimization techniques." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "plot([ Target(t), Actual (10, 10, t) ], t=0..20, thickness=3, \n title=`Target response vs. Actual response for k = 10 and c = 10`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7hp7$$\"\"!F)$\"\"\"F )7$$\"3emmm;arz@!#=$\"3y&>P2Q(*oM*F/7$$\"39LLLL3VfVF/$\"3XCtg:g&yI)F/7 $$\"3]****\\i&*)fD'F/$\"3cqGOYNx_rF/7$$\"3%pmm;H[D:)F/$\"3qv,P@[?DeF/7 $$\"3#QL$e9w)*=#*F/$\"3&3L6O=\\b-&F/7$$\"3'****\\PpU&G5!#<$\"3[RRG'[`6 ?%F/7$$\"3am;/Em=N6FI$\"3%*z+LgseiLF/7$$\"3MLLLe0$=C\"FI$\"3j?&*)zcU,_ #F/7$$\"3SLL$eR\"=\\8FI$\"318*yvmB#y;F/7$$\"3[LLLLA`c9FI$\"3kI)=)GVB?& )!#>7$$\"3cLL$32$)Qc\"FI$\"3-D-'G&pSc]!#?7$$\"3iLLL3RBr;FI$!3!)ym:3R,y rFjn7$$\"3C++vV^\"\\)=FI$!3OM_nQ$p$>@F/7$$\"3imm;zjf)4#FI$!3^obK'=S+J$ F/7$$\"3-++]Piq'H#FI$!3a@Al&yCL>%F/7$$\"3ULL$e4;[\\#FI$!3OaI!3C?`%[F/7 $$\"37++]P%*4`TL&F/7$$\"3`LL$3-8D!GFI$!3Cym>gU9%Q&F/7$ $\"3)omm;%ez`GFI$!3\\UQQ@ua=aF/7$$\"3!)****\\i'y]!HFI$!3c\"oMr'fePaF/7 $$\"3#pmm\"Ha6eHFI$!3!y=zn@\"RTaF/7$$\"3gLL$e>_6,$FI$!3fTUEn(p$HaF/7$$ \"3G++]i*)=kIFI$!3\")R$R(H'))=S&F/7$$\"3'pmm\"HdA$GFjn7$$\"3]+++]1!>+&FI$\"3)))RMxrQ)\\5F/7$$\"3T++++FZ=_FI$ \"3y(oQ(\\G22(=hB!)=GF/7$$\"3yL$e9YAs%f FI$\"3)on43cGP(GF/7$$\"3Qnm\"Hd&)>/'FI$\"3`Yy@]gH+HF/7$$\"3uLekGrO*3'F I$\"3czrR3+>.HF/7$$\"35+]P%o[n8'FI$\"3%G2XzO;$**GF/7$$\"3MnT5S-8%='FI$ \"3)G*44U6\")))GF/7$$\"3qLL$ez6:B'FI$\"3ci2DDJ#=(GF/7$$\"3Q++D1o(oX'FI $\"3+R5\"*=Gc2FF/7$$\"3/nmm;=C#o'FI$\"3W)fmaj.8U#F/7$$\"3Mnmmm#pS1(FI$ \"3UTv!4[c#Hq)Fjn7$$\"3<+++DZ5Q&)FI$!3 7h^U2IyX6F/7$$\"3!3++v3zMu)FI$!3?\\*>#=*f:N\"F/7$$\"3sn;a)QA1&))FI$!3i L))[A]-I9F/7$$\"3kMLe*olx&*)FI$!3I;[_tF])[\"F/7$$\"3c,]i!**3\\1*FI$!36 (zX&\\*[r_\"F/7$$\"3Yomm\"H_?<*FI$!3k#R*)yKWja\"F/7$$\"3An;a)GV/F*FI$! 3W(GTl)HMZ:F/7$$\"3unmT&GM)o$*FI$!3xg'3^\")RI`\"F/7$$\"3Go;H#GDsY*FI$! 33S&35VmT]\"F/7$$\"3-nm;zihl&*FI$!3E44ZS*p:Y\"F/7$$\"33++vVA(yx*FI$!32 -:ER'3#)Fjn 7$$\"31+](=GB2F\"F]`l$\"3#)otn=zB(z(Fjn7$$\"3+++]2'HKH\"F]`l$\"3k?Otis ;IqFjn7$$\"3`mmmwanL8F]`l$\"3y2N=4;-$)\\Fjn7$$\"35+++v+'oP\"F]`l$\"3#> b**GtHYH#Fjn7$$\"3CLLeR<*fT\"F]`l$!3_G=C^LzR8F`o7$$\"3C+++&)Hxe9F]`l$! 3m+kVSOVaBFjn7$$\"3gmm\"H!o-*\\\"F]`l$!37=@\\YQ)yv$Fjn7$$\"3:++DTO5T:F ]`l$!3g>i&Rx=^Q%Fjn7$$\"3emmmT9C#e\"F]`l$!3?XO:,`'f>%Fjn7$$\"3\"****\\ i!*3`i\"F]`l$!3)=ekcv(Q8LFjn7$$\"3_LLL$*zym;F]`l$!3+j\")*Q)=sX?Fjn7$$ \"3fLL$3N1#4F]`l$\"3' 3P7UN4>1#Fjn7$$\"3A++v.Uac>F]`l$\"3px+TArT2:Fjn7$$\"#?F)$\"3!z=W6o$Guu F`o-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7W7$F($\"+'*********!#57$$\"+< arz@F_il$\"+!>Dk(**F_il7$$\"+M3VfVF_il$\"+=0\\1**F_il7$$\"+j&*)fD'F_il $\"+YB&*3)*F_il7$$\"+#H[D:)F_il$\"+%R2$y'*F_il7$$\"+e0$=C\"!\"*$\"+[LE p#*F_il7$$\"+3RBr;Fgjl$\"+-#z#3()F_il7$$\"+zjf)4#Fgjl$\"+!*>`+&Fgjl$\"+I?j;8F_il7$$\"+]Z /NaFgjl$\"+-Ia1D!#67$$\"+]$fC&eFgjl$!+%>iSQ(F\\^m7$$\"+'z6:B'Fgjl$!+MH ,\"f\"F_il7$$\"+<=C#o'Fgjl$!+!yIO`#F_il7$$\"+n#pS1(Fgjl$!+u'z3E$F_il7$ $\"+j`A3vFgjl$!+g!yM,%F_il7$$\"+n(y8!zFgjl$!+mI8)e%F_il7$$\"+j.tK$)Fgj l$!+7f28^F_il7$$\"+)3zMu)Fgjl$!+C*yf]&F_il7$$\"+#H_?<*Fgjl$!+'ex@!eF_i l7$$\"+!G;cc*Fgjl$!+)pj9(fF_il7$$\"+WA(yx*Fgjl$!+Y\\JAgF_il7$$\"+4#G,* **Fgjl$!+K@EXgF_il7$$\"+]-1@5!\")$!+GA2SgF_il7$$\"+!o2J/\"F\\bm$!+9%of +'F_il7$$\"+%Q#\\\"3\"F\\bm$!+?@.!)eF_il7$$\"+;*[H7\"F\\bm$!+]h;acF_il 7$$\"+qvxl6F\\bm$!+WAtI`F_il7$$\"+`qn27F\\bm$!+51)[$\\F_il7$$\"+cp@[7F \\bm$!+/Oe'[%F_il7$$\"+3'HKH\"F\\bm$!+gfeDRF_il7$$\"+xanL8F\\bm$!+')Q# fP$F_il7$$\"+v+'oP\"F\\bm$!+)[$p`FF_il7$$\"+S<*fT\"F\\bm$!+Ci))o@F_il7 $$\"+&)Hxe9F\\bm$!+o'H$=:F_il7$$\"+.o-*\\\"F\\bm$!+?Wqg!*F\\^m7$$\"+TO 5T:F\\bm$!+=!Rpw#F\\^m7$$\"+U9C#e\"F\\bm$\"+ofn#=$F\\^m7$$\"+1*3`i\"F \\bm$\"+S,e'4*F\\^m7$$\"+$*zym;F\\bm$\"+$*==S9F_il7$$\"+^j?4F_il7$$\"+jMF^@s#F_ il7$$\"+9@BM=F\\bm$\"+#zZu0$F_il7$$\"+`v&Q(=F\\bm$\"+/fZ&H$F_il7$$\"+O l5;>F\\bm$\"+wBD$[$F_il7$$\"+/Uac>F\\bm$\"+Y0w)f$F_il7$F^hl$\"+QDi`OF_ il-Fchl6&FehlF(FfhlF(-%+AXESLABELSG6$Q\"t6\"Q!6\"-%&TITLEG6#%ZTarget~r esponse~vs.~Actual~response~for~k~=~10~and~c~=~10G-%*THICKNESSG6#\"\"$ -%%VIEWG6$;F(F^hl%(DEFAULTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 60 "Measuring the Error between the Desired and Actual Res ponses" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 300 577 "Squared differences are a common measure of error between two functions. Ide ally, we would measure error as the integral of the squared difference between the desired and actual response. However, evaluating this in tegral in this application is extremely difficult, due to the high com plexity of the actual response function. As an approximation, we samp le the functions at key times and sum up the squared differences betwe en the functions at these points. In this case as an illustration, we sample at the peaks, zeros and valleys of the target function over tw o periods." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "period:=2*Pi; num_periods:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'periodG,$%#PiG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,num_periodsG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " time_sample:=seq( i * period / 4, i=1..4 * num_periods );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,time_sampleG6*,$%#PiG#\"\"\"\"\"#F',$F'#\"\" $F*,$F'F*,$F'#\"\"&F*,$F'F-,$F'#\"\"(F*,$F'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "Error:=add( (Actual(k, c, time_sample[i])\n \+ - Target(time_sample[i]))^2, \n i = 1..4 * num _periods);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ErrorG,2*$),&*&*&,2% \"cG\"&++\"*&F-\"\"\"-%%sqrtG6#,&*$)F,\"\"#F/F/*&\"$+%F/%\"kGF/!\"\"F/ F/*(\"$+#F/F,F/F9F/F:*(FF/FDF/*&F-F/F0F/F:FAF:*(F " 0 "" {MPLTEXT 1 0 38 "Error:=subs( \{k=x[1],c=x[2]\} , Error):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "infolevel['UnconstrainedNew ton']:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%4Unconst rainedNewtonG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 307 31 "The algorithm makes use of the " }{TEXT 310 13 "LinearAlgebra " }{TEXT 311 5 " and " }{TEXT 312 6 "linalg" }{TEXT 313 31 " packages, so we load them now." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with(LinearAlgebra): with(linalg): " }}{PARA 7 "" 1 "" {TEXT -1 104 "Warning, the previous binding of the name GramSchmidt has been removed and it now has an assigned value\n " }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "libname:=\"C:/mylib/nlp\", libname;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6%Q-C:/mylib/nlp6\"Q0c:/mylib/vibsys F'Q@C:\\Program~Files\\Maple~6.01/libF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with( NonlinearProgramming );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%)OptimizeG%5PrimalDualLogBarrierG%4UnconstrainedNewt onG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 302 62 "The parame ters to Newton's Method are the objective function (" }{TEXT 316 10 "E rror(k,c)" }{TEXT 317 255 "), the number of decision variables (two), \+ the starting point for the search (<10, 10>), the convexity of the obj ective function (nonconvex, in this case), and the data type of the va riables (complex floats). We use complex floats instead of real becaus e " }{TEXT 314 10 "Error(k,c)" }{TEXT 315 107 " can take on complex va lues. In this case, the algorithm takes the real part of all functio n evaluations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 9 "(Because " }{TEXT 318 10 "Error(k,c)" }{TEXT 319 147 " is very complicated, the algorithm takes about 10 minutes to run on a PC . The bottle neck is computing the Hessian matrix of second derivativ es.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "UnconstrainedNewton( Error, 2, <10,10>, 'nonconvex', \+ 'complex[8]' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"xG6#\"\"\"$ \"3m0%y8e=58\"!#:/&F&6#\"\"#$\"3\\7)Gx@u7+%!#;" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 18 "Testing the Result" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 304 41 "Newton's Method found a local minimum at \+ " }{TEXT 305 23 "k = 113.101, c = 40.013" }{TEXT 306 102 ". We now ov erlay the desired and actual plots with these values and see that the \+ match is quite good." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([Target(t), Actual(113, 40, t) ], t= 0..20, thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7hp7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#=$\" 3y&>P2Q(*oM*F/7$$\"39LLLL3VfVF/$\"3XCtg:g&yI)F/7$$\"3]****\\i&*)fD'F/$ \"3cqGOYNx_rF/7$$\"3%pmm;H[D:)F/$\"3qv,P@[?DeF/7$$\"3#QL$e9w)*=#*F/$\" 3&3L6O=\\b-&F/7$$\"3'****\\PpU&G5!#<$\"3[RRG'[`6?%F/7$$\"3am;/Em=N6FI$ \"3%*z+LgseiLF/7$$\"3MLLLe0$=C\"FI$\"3j?&*)zcU,_#F/7$$\"3SLL$eR\"=\\8F I$\"318*yvmB#y;F/7$$\"3[LLLLA`c9FI$\"3kI)=)GVB?&)!#>7$$\"3cLL$32$)Qc\" FI$\"3-D-'G&pSc]!#?7$$\"3iLLL3RBr;FI$!3!)ym:3R,yrFjn7$$\"3C++vV^\"\\)= FI$!3OM_nQ$p$>@F/7$$\"3imm;zjf)4#FI$!3^obK'=S+J$F/7$$\"3-++]Piq'H#FI$! 3a@Al&yCL>%F/7$$\"3ULL$e4;[\\#FI$!3OaI!3C?`%[F/7$$\"37++]P%*4`TL&F/7$$\"3`LL$3-8D!GFI$!3Cym>gU9%Q&F/7$$\"3)omm;%ez`GFI$!3\\ UQQ@ua=aF/7$$\"3!)****\\i'y]!HFI$!3c\"oMr'fePaF/7$$\"3#pmm\"Ha6eHFI$!3 !y=zn@\"RTaF/7$$\"3gLL$e>_6,$FI$!3fTUEn(p$HaF/7$$\"3G++]i*)=kIFI$!3\") R$R(H'))=S&F/7$$\"3'pmm\"HdA$GFjn7$$\"3 ]+++]1!>+&FI$\"3)))RMxrQ)\\5F/7$$\"3T++++FZ=_FI$\"3y(oQ(\\G22(=hB!)=GF/7$$\"3yL$e9YAs%fFI$\"3)on43cGP(GF/7$$ \"3Qnm\"Hd&)>/'FI$\"3`Yy@]gH+HF/7$$\"3uLekGrO*3'FI$\"3czrR3+>.HF/7$$\" 35+]P%o[n8'FI$\"3%G2XzO;$**GF/7$$\"3MnT5S-8%='FI$\"3)G*44U6\")))GF/7$$ \"3qLL$ez6:B'FI$\"3ci2DDJ#=(GF/7$$\"3Q++D1o(oX'FI$\"3+R5\"*=Gc2FF/7$$ \"3/nmm;=C#o'FI$\"3W)fmaj.8U#F/7$$\"3Mnmmm#pS1(FI$\"3UTv!4[c#Hq)Fjn7$$\"3<+++DZ5Q&)FI$!37h^U2IyX6F/7$$\"3!3+ +v3zMu)FI$!3?\\*>#=*f:N\"F/7$$\"3sn;a)QA1&))FI$!3iL))[A]-I9F/7$$\"3kML e*olx&*)FI$!3I;[_tF])[\"F/7$$\"3c,]i!**3\\1*FI$!36(zX&\\*[r_\"F/7$$\"3 Yomm\"H_?<*FI$!3k#R*)yKWja\"F/7$$\"3An;a)GV/F*FI$!3W(GTl)HMZ:F/7$$\"3u nmT&GM)o$*FI$!3xg'3^\")RI`\"F/7$$\"3Go;H#GDsY*FI$!33S&35VmT]\"F/7$$\"3 -nm;zihl&*FI$!3E44ZS*p:Y\"F/7$$\"33++vVA(yx*FI$!32-:ER'3#)Fjn7$$\"31+](=GB2F\"F]` l$\"3#)otn=zB(z(Fjn7$$\"3+++]2'HKH\"F]`l$\"3k?Otis;IqFjn7$$\"3`mmmwanL 8F]`l$\"3y2N=4;-$)\\Fjn7$$\"35+++v+'oP\"F]`l$\"3#>b**GtHYH#Fjn7$$\"3CL LeR<*fT\"F]`l$!3_G=C^LzR8F`o7$$\"3C+++&)Hxe9F]`l$!3m+kVSOVaBFjn7$$\"3g mm\"H!o-*\\\"F]`l$!37=@\\YQ)yv$Fjn7$$\"3:++DTO5T:F]`l$!3g>i&Rx=^Q%Fjn7 $$\"3emmmT9C#e\"F]`l$!3?XO:,`'f>%Fjn7$$\"3\"****\\i!*3`i\"F]`l$!3)=ekc v(Q8LFjn7$$\"3_LLL$*zym;F]`l$!3+j\")*Q)=sX?Fjn7$$\"3fLL$3N1#4F]`l$\"3'3P7UN4>1#Fjn7$$\"3A++ v.Uac>F]`l$\"3px+TArT2:Fjn7$$\"#?F)$\"3!z=W6o$GuuF`o-%'COLOURG6&%$RGBG $\"#5!\"\"F(F(-F$6$7hp7$F($\"+++++5!\"*7$$\"+S&)G\\a!#6$\"+e$[L)**!#57 $$\"+3x&)*3\"Ffil$\"+%GGR$**Ffil7$$\"+ilyM;Ffil$\"+)G5E&)*Ffil7$$\"+NS(*Ffil7$$\"+DJdpKFfil$\"+_'\\sU*Ffil7$$\"+M3VfVFfil$\"+ 9w#Q+*Ffil7$$\"+j&*)fD'Ffil$\"+EH&\\.)Ffil7$$\"+#H[D:)Ffil$\"+g4UBoFfi l7$$\"+;w)*=#*Ffil$\"+Gg%)fgFfil7$$\"+%pU&G5F_il$\"+9,+__Ffil7$$\"+Em= N6F_il$\"+cZ)=T%Ffil7$$\"+e0$=C\"F_il$\"+eeS^NFfil7$$\"+'R\"=\\8F_il$ \"+')HYwEFfil7$$\"+LA`c9F_il$\"+Lh@/=Ffil7$$\"+qI)Qc\"F_il$\"+C>ec%*Fc il7$$\"+3RBr;F_il$\"+)Gd96\"Fcil7$$\"+EX2yF_il$!+[*)\\Y@Ffil7$$\"+zjf)4#F_il$!+ '3#G&z#Ffil7$$\"+Qiq'H#F_il$!+A#G&HQFfil7$$\"+'4;[\\#F_il$!+!**)HAYFfi l7$$\"+!QZ**p#F_il$!+-^_s^Ffil7$$\"+j'y]!HF_il$!++:@WaFfil7$$\"+Ia6eHF _il$!+]%Q+Z&Ffil7$$\"+(>_6,$F_il$!+=S7yaFfil7$$\"+j*)=kIF_il$!++_\")oa Ffil7$$\"+IdAPFcil7$$\"+]1!>+&F_il$\"+'48&z6Ffil7$$\"++FZ=_F_il$\"+`lmj=Ffil7$$ \"+]Z/NaF_il$\"+1e%oR#Ffil7$$\"+]?v