PID Parameters Optimization by Using Genetic Algorithm. Andri Mirzal, Shinichiro Yoshii, Masashi Furukawa

Transcription

1 PID Paramr Opimizaion by Uing Gnic Algorihm Andri Mirzal, Shinichiro Yohii, Maahi Frkawa Grada School o Inormaion Scinc and chnology Hokkaido Univriy Sapporo, Japan andri, yohii, mack@complx.ng.hokdai.ac.jp ABSRAC im dlay ar componn ha mak im-lag in ym rpon. hy ari in phyical, chmical, biological and conomic ym, a wll a in h proc o marmn and compaion. In hi work, w implmn Gnic Algorihm GA in drmining PID conrollr paramr o compna h dlay in Fir Ordr Lag pl im Dlay FOLPD and compar h rl wih Iraiv Mhod and Ziglr-Nichol rl rl. yword: im dlay, gnic algorihm, iraiv mhod, ziglr-nichol rl 1. INRODUCION In h prvio work [1], ahor hav implmnd and compard wo ning mhod, Iraiv Mhod and Ziglr-Nichol rl, o compna h c o dlay in abiliy o ym and howd ha Iraiv Mhod ha prior prormanc in analyzd ca, FOLPD Fir Ordr Lag pl im Dlay. B hr ar om ca whr w can h wo ning mhod, i.. h dynamic plan which i paramr ar conanly changing. In hi or o ym, w hav o do rning in ral im, which can b applid by h ning mhod bca w hav o ak h ym olin ir in ordr o i paramr. In hi work, w xnd or prvio work [1] by implmning Gnic Algorihm GA in drmining PID Conrollr paramr o compna h dlay in ordr on FOLPD and compar h rl wih Iraiv Mhod and Ziglr-Nichol rl rl.

2 Figr 1. h PID conrollr gnral rcr whr plan ha dlay componn i 1 R X - E c d X U G p p Y 2. HE OBJECIVE FUNCIONS FINESS VALUES h mo crcial p in applying GA i o choo h objciv ncion ha ar d o vala in o ach chromoom. Som work [3] [4] prormanc indic a h objciv ncion. In [3] ahor Man o h Sqard Error MSE, Ingral o im mliplid by Abol Error IAE, Ingral o Abol Magnid o h Error IAE, and Ingral o h Sqard Error ISE, whil in [4] ahor ISE, IAE, and IAE. Hr w all or prormanc indic ad abov and Ingral o im mliplid by h Sqard Error ISE o minimiz h rror ignal E and compar hm o ind h mo iabl on. h prormanc indic ar dind a ollow [2]: 1 2 MSE d, IAE d, IAE d 2 ISE d, and 2 ISE d 1 Whr i h rror ignal in im domain. h PID conrollr i d o minimiz h rror ignal, or w can din mor rigoroly, in h rm o rror criria: o minimiz h val o prormanc indic mniond abov. And bca h mallr h val o prormanc indic o h corrponding chromoom h ir h chromoom will b, and vic vra, w din h in o h chromoom a: 1 in val 2 prormanc indx 3. DELAY COMPONEN Dlay in conrol ym can b dind a im-inrval bwn an vn ha ar in on poin wih i op in anohr poin wihin ym [5]. Dlay i alo rcognizd a ranpor lag,

3 dadim, and im lag. Bca dlay alway rdc abiliy o minimm pha ym ym which don hav pol and zro in h righ hal o -plan, i i imporan o analyi abiliy o ym wih im dlay. Figr 2. Dlay c on ym Dlay - W can dlay c in a ym which ca im hi a ym op rom igr abov. h rlaionhip bwn and - can b wrin a: ] [ d l 3 whr i ni p. L -, d d 4 am or <, F d d d d 5 So, w g : ] [ ] [ F l l 6 In ordr o do ning proc ing GA, w approxima h dlay wih Dirc Frqncy Rpon DFR ri. Acally in Malab, hr i im dlay bil in ncion, Pad ri approximaion, b w choo o DFR ri bca ir, i ha bn hown in [1] ha hi ri ha h mall avrag rror among h ohr vn ri and h cond i whil h dlay block ncion in Conrol Sym oolbox d o imla Iraiv Mhod and Ziglr-Nichol

4 ning rl Pad ri, h dlay componn modld by.m ncion d o conrc ranr ncion o a ym i ill in h complx rqncy domain rprnaion qaion 1, o ha w hav o ranla i ino polynomial ri rprnaion. Frhrmor, or imlaion prpo, w cond ordr DFR ri o circmvn nncary complxiy, bca a h ordr o h ri ar ging highr, no only h calclaion bcom diicl b alo i inrodc nw pol and zro which mak h ym mch mor liv. 2 2 DFR ri : GENEIC ALGORIHM PID conrollr paramr will b opimizd by applying GA. Hr w Malab Gnic Algorihm oolbox [6] o imla i. h ir and h mo crcial p i o ncoding h problm ino iabl GA chromoom and hn conrc h poplaion. Som work rcommnd 2 o 1 chromoom in on poplaion. h mor h chromoom nmbr, h br h chanc o g h opimal rl. Howvr, bca w hav o conidr h xcion im, w 8 or 1 chromoom in ach gnraion. Encoding i don in ral nmbr rahr han binary ncoding bca h lar dicard h paramr val i i xcd i prciion capabiliy. Each chromoom compri o hr paramr, d, p, i, wih val bond varid dpnd on h dlay and objciv ncion d. Ar many xprimn, w ind ha h val bond hold b according o h Iraiv Mhod and Ziglr-Nichol rl val rang o nr h convrgnc hr ar many ca which h convrgnc can b rachd i w h paramr val bond arbirarily, vn hogh h opimal rl incldd in ho bond rang. h poplaion in ach gnraion i rprnd by 8 x 4 or 1 x 4 marix, dpnd on h chromoom nmbr in poplaion, which ach row i on chromoom ha compri d, p, i val and h la colmn addd o accommoda in val F o corrponding chromoom... d1 d 2 dn p1 p2.. pn i1 i1.. in F1 F 2.. F n chromoom1 chromoom 2.. chromoomn 8

5 W maximm gnraion rminaion maxgnrm.m o rmina h program rahr han conidring h b chromoom in val changing ra bca w wan o conrol h xcion im. Howvr, h b chromoom in val changing ra i alo bing conidrd by rnning h program nil h b in val op incraing, hn w ha poin a h maximm gnraion. Ar vral xprimn i hown ha hr i no viibl improvmn ar 3 h gnraion, o w 3 a h maximm gnraion. Malab GA oolbox [6] provid hr lcion chniq, ornamn Slcion, Rol Whl Slcion and Normaliz Gomric Slcion. ornamn Slcion rqir mor xcion im whil Rol Whl Slcion allow h wakr chromoom o b lcd many im, o w choo Normalizd Gomric Slcion o choo h parn. Ar parn bing lcd, h croovr opraion will b don. W arihmic croovr arihxovr.m ncion bca i i pciically bing d or loaing poin nmbr and provid mor han on croovr poin. And w or croovr poin bca or chromoom compri o hr alll, on poin croovr can no accommoda hr alll in on opraion. Maion i don by ing maion probabiliy arond.1 prcn. In gnral maion opraion hold no b don oo on bca h arching proc will chang ino random arch a h maion probabiliy ging highr. 5. APPLYING HE GENEIC ALGORIHM hr ar vral variabl d a h andard o mar ym prormanc. In gnral, ni p inp i d o h ym, and h op ignal i characrizd by om andard prormanc mar: ling im, prcn ovrhoo, rror ignal, ri im, pak im, and abiliy margin. All h mar ar dind in im domain rpon. Figr 3 blow dcrib andard prormanc mar o a ypical ym drivn by ni p inp. Prcn ovrhoo i dind a h poin whr h ym rpon rach h pak, in hi ca 53%. hr ar vral criria or ling im, or xampl 1% cririon, 2% cririon, and 5% cririon. Hr w 5% cririon ling im. And or h ri im, acally in gnral, i mard a h im ndd by ym o rach rom o 1% o inal val or rom 1% o 9% o inal val. B, or marmn impliciy, w - 95% cririon. Pak im i h poin whr h maximm val rachd ovrhoo a 3.2 cond. And rror ignal i h dirnc bwn h inp ignal magnid and ym rpon inal magnid. In hi work, w G 1/1, dlay i in h rang o.1 o 1 cond. And bca h ym ar

6 compnad by PID conrollr, h rror ignal ar alway zro. In addiion o h iv ym andard prormanc mar dcribd abov, in Iraiv Mhod and Ziglr-Nichol rl, w calcla h ym prormanc indic dcribd by qaion 1 alo. hi i don bca w wan o compar i wih h rl o GA, which i bing opimizd in h rm o prormanc indic. Idally, w can xpc corrponding GA prormanc indic hold b alway br han wo ning rl. o calcla prormanc indic, w approxima h ingral in qaion 1 wih addiion igma and.1 cond ampling im and h igma ppr limi wih 15 cond or all analyzd ca, no mar how qick i rach convrgnc val. 6. SIMULAION RESULS AND ANALYSIS Figr 3. Sandard prormanc Mar 6 5 Ziglr Nichol Iraiv Mhod MSE IAE ISE IAE ISE Prcn Ovrhoo Sling im 5% cririon Dlay in cond Ziglr Nichol 1 Iraiv Mhod MSE IAE.5 ISE IAE ISE Dlay in cond a. h comparion o prcn ovrhoo PO. b. h comparion o ling im S. Ri im in cond logarihmic cal Ziglr Nichol Iraiv Mhod MSE IAE ISE IAE ISE Pak im in cond logarihmic cal Ziglr Nichol Iraiv Mhod MSE IAE ISE IAE ISE Dlay in cond Dlay in cond c. h comparion o ri im R. d. h comparion o pak im P.

7 Sabiliy margin logarihmic cal Ziglr Nichol Iraiv Mhod MSE IAE ISE IAE ISE Dlay in cond. h comparion o abiliy margin SM. abl 1. Avrag val o andard prormanc mar. Paramr Ziglr- Iraiv Opimizd Opimizd Opimizd Opimizd Opimizd Nichol Mhod by MSE by IAE by ISE by IAE by ISE %OV 38% 15% 1% 6% 11% 1% 8% S5% R P SM Sandard prormanc mar. Prcn Ovrhoo Whil igr 3a mmariz h val chang o prcn ovrhoo wih rpc o h im dlay, abl 1 giv i avrag val. Ziglr-Nichol rl giv h bigg val or all im dlay, conqnly i avrag val i h bigg alo, 38%. Hr h dirnc bwn wo ning mhod and GA mhod can b n: whil ning mhod hav almo h am parn, i val dcraing a h im dlay growing biggr, xcp or mall val o dlay whr Ziglr-Nichol rl giv incraing val, h GA mhod giv almo conan val, arond 1% a long h im dlay i no oo mall, xcp or ca opimizd by IAE, whr h prcn ovrhoo val lca and dcraing a im dlay incraing. abl 1 how ha GA prodc mch br prcn ovrhoo han wo ohr ning mhod, pcially i opimizd by IAE cririon. So i can b concldd ha GA can b d o opimizd prcn ovrhoo. Sling im h val o ling im 5% cririon all ovr h im dlay i mmarizd by igr 3b, whr i can b n ha almo all mhod, xcp or GA opimizd by IAE and IAE, all ndr almo h am raigh lin wih poiiv lop. I man ha a h dlay incraing, h ling im will

8 incra linarly. In abl 1 w can ha h ling im avrag val i no o dirn among all mhod, in h rang 1.37 cond o 1.54 cond, xcp or GA opimizd by IAE.745 cond and IAE 1.1 cond. Howvr, bca h ohr GA mhod rl ar in agr wih wo ning mhod, w can rally dirnia h rl. So i can b aid ha h ling im i no opimizd by GA mhod. Ri im h hird variabl i ri im, which plod in igr 3c ing logarihmic cal in y axi. W can rong parn which all h rl, xcp or Iraiv Mhod, hav almo h am val all ovr h im dlay. And i hold no b rpriing i w g almo h am avrag val or all rl, arond.44 cond o.59 cond, xcp or Iraiv Mhod,.912 cond. Anohr inring hing i, in gnral, almo in all rang o im dlay h crv kp hir ranking nchangd, wih ordr rom h bigg val: Iraiv Mhod, IAE, IAE, ISE, ISE, MSE, and Ziglr-Nichol rl. From h avrag val abl 1, h b rl i givn by Ziglr-Nichol rl,.444 cond and h wor on i Iraiv Mhod.912 cond, and all GA mhod prodc almo h am avrag val. B bca GA rl ar no o dirn compard o Ziglr-Nichol rl, i can b concldd ha GA can opimiz h ri im. Pak im Almo h am parn, a in ri im plo, i hown on h pak im plo on igr 3d, xcp Iraiv Mhod, in larg dlay rang, nd o divrg, whr all mhod how almo h am val or all ovr im dlay. B w m pay anion on GA opimizd by IAE bca hr i a rang which i val i biggr han h ohr. Excp or Iraiv Mhod which i 3.43 cond, all ohr mhod prodc pak im arond.57 cond o.83 cond. h b val ar givn by GA opimizd by MSE and ISE,.576 cond. Lik ri im, in gnral, almo in all rang o im dlay h crv kp hir ranking nchangd, wih ordr rom h bigg val: Iraiv Mhod, IAE, IAE, Ziglr-Nichol, ISE, MSE, and ISE. And bca h GA mhod prodc pak im plo which only br han Iraiv Mhod, no Ziglr-Nichol, h pak im can b opimizd by GA mhod. Sabiliy Margin h la andard prormanc mar i abiliy margin igr 3. Sabiliy margin i h maximm gain ha can b bor ym rpon go ino inoidal cycl. In h imlaion, hi i don by imply incraing val o c nil inoidal cycl happn, and h abiliy margin o corrponding ym i c a inoidal cycl.

9 hi i h ir rl ha how conincy all ovr im dlay rang, which all crv all ndr almo h am lin. So hi i h rong parn, and bca h rongr h parn, h l h abiliy o GA mhod o opimiz corrponding prormanc mar, w can GA mhod o opimiz h abiliy margin. Convrly, h GA mhod ar likly o prodc l abl ym. Lik ri im and pak im, in gnral, almo in all rang o im dlay h crv kp hir ranking nchangd, wih ordr rom h bigg val: Iraiv Mhod, IAE, Ziglr-Nichol, IAE, ISE, ISE, and MSE. B nlik ling im, ri im, and pak im, abiliy margin val rdc a h im dlay incra Prormanc Indic W dirnia h rm andard prormanc mar and prormanc indic hr, whr andard prormanc mar hav alrady bn dicd abov, prormanc indic ar: MSE, IAE, ISE, IAE, and ISE. abl 2 blow mmariz prormanc indic rom imlaion rl. A xpcd, h avrag val o GA prormanc indic ar alway mallr han i corrponding Ziglr-Nichol and Iraiv Mhod. Morovr Ziglr-Nichol rl prodc mallr avrag prormanc indic val han Iraiv Mhod do or all im dlay val rang. abl 2. Prormanc indic o ning mhod and GA mhod. Dlay Ziglr-Nichol Iraiv Mhod MSE IAE ISE IAE ISE MSE IAE ISE IAE ISE Av Opimizd by MSE Opimizd by IAE Opimizd by ISE Opimizd by IAE Opimizd by ISE Alhogh i can b n ha MSE ha h mall avrag val or all hr mhod, and IAE ha h larg avrag val or Ziglr-Nichol and GA mhod, and only cond in Iraiv Mhod, i don imply ha on m MSE and m avoid ing IAE a a objciv ncion in GA bca h prormanc indic, a hown by qaion 1, hav dirn diniion and canno b compard. Morovr, a hown in abl 1, GA mhod opimizd by MSE don prodc h b rl or all analyzd andard prormanc mar, only or pak im. o g mor inigh abo h comparion among h mhod, w plo val chang o ach prormanc indic wih rpc o im dlay blow.

10 Figr 4. Prormanc Indic GA Opimizd by IAE IAE o Iraiv Mhod IAE o Ziglr-Nichol MSE val.4 ISE val GA Opimizd by MSE MSE o Iraiv Mhod MSE o Ziglr-Nichol Dlay in cond a. h comparion o MSE val Dlay in cond b. h comparion o IAE val GA Opimizd by IAE IAE o Iraiv Mhod IAE o Ziglr-Nichol ISE val 6 IAE val GA Opimizd by ISE ISE o Iraiv Mhod ISE o Ziglr-Nichol Dlay in cond c. h comparion o ISE val Dlay in cond d. h comparion o IAE val ISE val GA Opimizd by ISE ISE o Iraiv Mhod ISE o Ziglr-Nichol Dlay in cond. h comparion o ISE val

11 All iv igr abov conidnly how ha GA mhod giv h mall val o all prormanc indic analyzd or all rang o im dlay. So no only in avrag val, b alo or all mard val do GA mhod prodc h mall corrponding prormanc indic. Howvr, h dirnc bwn GA mhod and wo ning mhod rl, xcp or IAE objciv ncion whr h dirnc incra a h im dlay incra, ar no impriv nogh o com ino conclion ha GA mhod i mch br han wo ohr mhod in minimizing rror criria. Bid, w m conidr h convrgnc problm ari in applying GA, which in hi work h xprimn don alway com ino dird olion. Evn hogh w h val bond bad on h prvio rl rom wo ning mhod, i only improv h probabiliy ha h imlaion com ino convrgnc rl rom 45 ca, hr wo im aild o rach convrgnc rl. 7. CONCLUSIONS 1. Gnic Algorihm applid in PID conrollr improv FOLPD ranin rpon compard o wo ning mhod. hi i hown by avrag prcn ovrhoo rdcion, mor han 7% and 3% wih rpc o h Ziglr-Nichol rl and Iraiv Mhod, whil kp h ri im and pak im almo nchangd and improv h ling im. Howvr, hr i payo in h abiliy margin which rdc lighly compard o wo ning mhod. 2. h avrag val o GA prormanc indic, a xpcd, ar alway mallr han i corrponding Ziglr-Nichol and Iraiv Mhod. Morovr Ziglr-Nichol rl prodc mallr avrag prormanc indic val han Iraiv Mhod do or all im dlay val rang. Howvr, h dirnc bwn GA mhod and wo ning mhod rl, xcp or IAE objciv ncion whr h dirnc incra a h im dlay incra, ar no impriv nogh o com ino conclion ha GA mhod i mch br han wo ohr mhod in minimizing rror criria. 3. hr ar convrgnc problm ha ari in applying GA, which in hi work h xprimn don alway com ino dird olion. Evn hogh w h val bond bad on h prvio rl rom wo ning mhod, i only improv h probabiliy ha h imlaion com ino convrgnc rl. Morovr, h val bond ing bad on ning mhod rl dicard h poibiliy o ind opimm rl rom ohr val rang.

12 Rrnc 1. Andri Mirzal, Shinichiro Yohii, Maahi Frkawa, Approximaion and Compnaion o Dlay in Analog Conrol Sym, 精密工学会北海度支部学術講演会, Sapporo, Japan, Richard C. Dor, Robr H. Bihop, Modrn Conrol Sym 1 h Ediion, Paron Prnic Hall, Ian Griin, On-lin PID Conrollr ning ing Gnic Algorihm, Dblin Ciy Univriy, O Mahony & CJ. Downing Cork Ini o chnology, Irland, ladiz Fala Wroclaw Univriy o chnology, Poland, Gnic Algorihm or PID Paramr Opimizaion, Minimizing Error Criria. 5. O Dwyr, A., h Eimaion and Compnaion o Proc wih im Dlay, Ph.D. hi, School o Elcronic Enginring, Dblin Ciy Univriy, C. R. Hock, J. Join. and M.ay. A Gnic Algorihm or Fncion Opimizaion: A Malab Implmnaion. ACM ranacion on Mahmaical Sowar, Proil Andri Mirzal i a dn a Grada School o Inormaion Scinc and chnology, Hokkaido Univriy. H obaind hi Bachlor dgr Honor in Elcrical Enginring majoring in Conrol Enginring rom Bandng Ini o chnology, Indonia, 23. Hi rarch inr ar in opimizaion mhod, conrol ym, and nwork hori. H i now carrying o hi rarch program ndr prviion o Proor Maahi Frkawa, Aonomo Sym Enginring Laboraory, Grada School o Inormaion Scinc and chnology, Hokkaido Univriy. Shinichiro Yohii Maahi Frkawa