Genec Algorhm n Parameer Esmaon of Nonlnear Dynamc Sysems E. Paeraks manos@egnaa.ee.auh.gr V. Perds perds@vergna.eng.auh.gr Ah. ehagas kehagas@egnaa.ee.auh.gr hp://skron.conrol.ee.auh.gr/kehagas/ndex.hm Auomaon and Robocs Lab hp://skron.conrol.ee.auh.gr Deparmen of Elecrcal and Compuer Engneerng Arsole Unversy -Thessalonk 54006 GREECE
Genec Algorhm n Parameer Esmaon of Nonlnear Dynamc Sysems E. Paeraks V. Perds A. ehagas Deparmen of Elecrcal and Compuer Engneerng Arsole Unversy Thessalonk - Greece.Inroducon We nroduce a parameer esmaon mehod for nonlnear dynamc sysems. A smple genec algorhm (GA) s used wh a recursve probably selecon mechansm. The mehod s appled o nonlnear sysems wh known srucure and unknown parameers. The nonlnear dynamc sysem has o be denfable. The selecon probables have o sasfy an enropy creron so as o enable he genec algorhm o avod poor soluons. Ths s a new feaure ha enhances he performance of he GA on he parameer esmaon problem. Numercal smulaons are gven concernng he parameer esmaon of a planar roboc manpulaor wh four (4) parameers.
. Parameer Esmaon Problem The esmaon problem consss n deermnng he member of he se ha bes descrbes he daa accordng o a gven creron. The purpose of he esmaon has o be oulned. If he fnal goal s a conrol sraegy for a parcular sysem he accuracy of esmaon should be udged on he bass of he me response. However, f he fnal goal s o analyze he properes of a sysem he accuracy of esmaon should be udged on he bass of devaons n he model parameers. Such problems can be found n engneerng problems and also n bology, economy, medcne ec. The daa are produced by he acual sysem I when operaes under a gven expermenal q condon. The oupu of he sysem a me wll be denoed by y() (where y() R ), and a sequence of oupus y( 0 ),, y() wll be denoed by y. The vecors u() (where m u () R ), u denoe he npu a me and he sequence of appled npus, respecvely. n Smlarly ϕ() (where ϕ() R ), ϕ denoe he sysem saes a me and he sequence of sysem saes, respecvely. The vecor ϑ (where ϑ R d ) denoes he parameer vecor. Consder now he followng famly of dscree me dynamc sysems I( fgy,,,,, u, ) ϕ ϑ 0 : () = f( ( ), u() ; 0 ), y () = gϕ(), u (); ϑ0 ϕ ϕ ϑ ( ) () In he sysem of ype () he funcons f() and g() are known. The npu u and he r nal condons are also known. The oupu y s measurable and only he parameer vecor ϑ 0 s unknown. All he possble ϑ ha can be appled o I consruc a model se whch wll be denoed by M. The model se s ndexed by he fne dmensonal parameer vecor ϑ, so ha a parcular model n he model se wll be denoed by M ( ϑ ). The vecor ϑ ranges over a se denoed by D M. (where D M. R d ), whch generally wll be assumed o be compac. I s also assumed ha ϑ 0 D M.. Takng no accoun nosy measuremens, as n real world problems, he I changes o: ( 0 ) ϕ() = f ϕ( ),..., ϕ( r), u() ; ϑ, ~ y () = y () + w () () Now!y s he measurable oupu and w() s whe, zero mean oupu nose. Samplng he nosy sysem oupu ~ y () for T me seps, a daa record C can be consruced, where dmc=q T and q s he number of oupus. Noe ha he dynamc sysem requres ha a suffcen number of oupu are measured oherwse he sysem may be undenfable. An exhausve search of M has o be precluded n realsc suaons where M has a vas number of members. On he oher hand GA s a global search scheme akng no accoun only a small par of M. The need for a global opmum search scheme arses from he fac ha he fness funcon may have local mnma, eher nheren or due o nosy measuremens (Lung, 87), (Södersröm and Soca 89). Also, GA has no he drawbacks of he sandard esmaon echnques n case of dsconnuous obecve funcons. Hence, he use of hs evoluonary scheme s very aracve for parameer esmaon of nonlnear dynamcal sysems. In he GA, an evolvable populaon P of poenal soluons has o be defned akng values over he D M. Hence as P (where P D M ) s denoed a se of dsnc values of P = ϑ, ϑ,..., ϑ, where s a fxed ϑ D M. The se P s fne dmensonal, namely { }
number. The se P ncludes he poenal soluons a he generaon. The oupu y () corresponds o he parameer vecor ϑ P where s he ndex of he model and s he generaon ndex. The model valdaon s acheved hrough he dscrepancy beween he nosy measured oupu ~ y () and he model oupu y () a me sep and s gven by E (), where denoes Eucldean norm: E () = y () ~ y() (3) The sum of E () for every me sep, where 0 T, defnes he mean square error (MSE) funcon w ( T) for he parcular model wh parameer vecor ϑ a he generaon: T (4) w ( T) E ( ) = = 0 The funcon w ( T) s posve so can sand as a fness funcon. The esmae of ϑ 0 a he -h generaon sϑ * : * ϑ = arg mn w ( T) (5) s s=,,... Therefore he parameer esmaon problem can be saed as a sac opmzaon problem of fndng he value ϑ * ha mnmzes he fness funcon ( ) wt.
3. Producon of Selecon Probables Based on he fness funcon of (4), a probably s assgned o each member of P. Ths probably expresses he smlary of he model oupu o sysem oupu and s relave o he performance of each oher member of P. Several producon schemes of selecon probables have been proposed n leraure. One of hem s he followng: w ( T) p ( T) = (6) w ( T) m= m The ral soluons wh parameer vecors ϑ P where have o be evaluaed over he whole measuremen daa record C n order o calculae hese selecon probables. Anoher selecon probably scheme s he followng: p ( T) = e w ( T) (7) ws ( T) s e = Subsung w ( T) from (4) we have he Bolzmann selecon probably mechansm where σ s he emperaure parameer, whch s usually consan. T E ( ) σ = e (8) p ( T) = T Es ( ) σ = s= e Here, we propose a new mehod of producon of he selecon probables. The mehod s based on he selecon scheme (8), alhough has wo dfferences. Frs, he probables are calculaed recursvely and second, here s no need o ake no accoun all he measuremen record C n order o evaluae he ral soluons. The probably updae rule for he me sep s defned as: p () = ( ) p e E ( ) σ ( ) E ( ) σ ( ) s s = ps( ) e Where ( ) s 0 p = and s =... s he pror probably dsrbuon. The man pon ha makes hs mehod dfferen from he ohers s ha he evaluaon of each model selecon probably s performed for a oal of T T me seps. The number of T me seps are defned by he enropy creron, whch s explaned n Secon. The role of he enropy creron s wofold. Frs, prevens premaure convergence of he algorhm reanng hus he requred populaon dversy. Second, makes he algorhm able o use fewer oupu measuremens han would oherwse be necessary ( T T) mes = 0, =,..., = T fnally one can oban: p p ( T ) = s= () 0 p s e () 0 T = 0 e E ( ) σ ( ) T = 0 E ( ) σ ( ) (9) (0). Repeang for
4. Enropy Creron Genec algorhm requres each generaon o conan a suffcen varey of models. If one of he selecon probables s allowed o end o one and he ohers near zero, he nex generaon of models wll only conan he bes model of he prevous generaon. Thus, he genec algorhm may ge suck wh a poor model, whch provdes only a local mnmum of esmaon error. On he oher hand, s clear ha f he selecon probables do no concenrae suffcenly on he mos promsng models, hen he search of he parameer space wll be essenally random search. To avod eher of he above suaons we nroduce a novel creron based on he enropy of he probables p (), p (),.., p (). The enropy H () of p (), p (),.., p () a me sep s defned by H () = p () ( p () s= s log s ) () The maxmum value of H () s log( ), and s acheved when p () = p () =..= p () =. The mnmum value of H () s zero, and s acheved for p () =, p () s =0, s,.e. when all probably s concenraed on one model. Now consder he followng dynamc hreshold H () such as H () ( ) () = log T where T n me seps. Ths s smply an ncreasng funcon wh respec o. I s clear ha he nequaly H () < H() (3) wll be sasfed for some T. The above nequaly express he enropy creron; when he enropy of p (), p (),.., p () falls bellow H(), hen probably updae sops and he nex generaon of models s produced. Termnaon wll ake place a some value of enropy beween 0 and log( ), ensurng ha he selecon probables are neher oo concenraed, nor oo dffuse. Ths means ha probably updae s performed for a varable number of me seps deermned by he enropy creron.
5. Descrpon of he Smple Genec Algorhm Havng he populaon P and he selecon probables p (), p (),.., p () we are ready o make he fnal sep n order o pass o he nex generaon (+). Frs of all, he ral soluons n P have o be encoded by srngs of bs. Noe ha n he nal generaon he 0 0 models ϑ P have arbrary parameer values bu always n D M. Each parameer vecor ϑ P s ranslaed as a chromosome and each parameer no as a gene. Specfcally, a mappng s akng place from D M o a dscree se of b srngs. Assume ha a gene s represened by n bs he chromosome occupes n d bs. Thus, here are n d parameer combnaons resulng n n d models. Wha s mporan here s he change of he model se M o a dscree model se. The dscrezaon has o be done akng no accoun he rade off beween slowness of he algorhm and desrable accuracy of soluon. All he chromosomes are poenal parens wh a selecon probably for mang. A seepes ascen hll-clmbng operaor s appled o he chromosome wh he maxmum selecon probably and a elsm operaor s appled. A roulee wheel s used o selec he par of parens. Genec operaors such as crossover and muaon are appled o seleced parens. The whole genec process s sopped when ndvduals have been produced. Then a re-mappng s made from he dscree se of b srngs o D M and a new se of soluons P + s ready o be evaluaed. Schemacally he flow of he parameer esmaon algorhm s he followng. Creae a populaon wh members from a model se wh arbrary parameer vecors. Assgn an nal probably o each model START he parameer esmaon loop. START he producon of selecon probables loop. Updae recursvely he selecon probables. IF he enropy creron s no sasfed hen CONTINUE n he loop ELSE EXIT from loop. START he producon of he nex generaon The seepes ascen hll-clmbng operaor s appled o he parameer vecor wh he maxmum selecon probably and he elsm operaor s appled. Selecon of parens and applcaon o hem genec operaors such as crossover and muaon REPEAT he producon loop unl ndvduals have been produced. IF he algorhm converges o a soluon or he maxmum number of generaons s exceeded STOP ELSE CONTINUE n he parameer esmaon loop.
6. Resuls Our algorhm s now appled o he problem of esmang he parameers of he MIT Seral Lnk Drec Drve Arm. Ths s confgured as a wo-lnk horzonal manpulaor by lockng he azmuh angle a 80 0. The npu o he manpulaor s he angle sequence ω = ( ω, ω). A PD conroller conrols he manpulaor so on angles ω, ω rack he reference npu ω. Therefore he problem s parameer esmaon under closed loop condons. Ignorng gravy forces, he equaons descrbng he manpulaor and conroller are he followng p 0 ω () () ω u 0 () () 0 p () () + ω" ω" 0 = ω ω u ω" () () ω" I + I + mll cos( ω) + ( ml + ml ) + ml I + mll cos( ω) + ml 4 4 () () I + mll ( ) + ml I + ml ω"" + ω"" cos ω 4 4 ( ω ) ( ) () () F m l l () sn mll sn( ) F ω" mll ω ω " () + sn ω 0 ω" ω" The neral parameers are I = 8. 095N m s rad, I = 0. 53N m s rad. The coeffcen of frcon s F= 0. 0005N m s rad. The gan parameers of he PD conroller are p = 500, u = 300 for he frs on, and p = 400, u = 30 for he second on. The mass and he lengh parameers, namely m = 0. gr, m =. gr, l = 0. 46m, l = 0. 445m, are consdered unknown for he purposes of hs example. The sysem equaons are dscrezed n me (wh a dscrezaon sep d = 00. sec ) o oban equaons of he form φ() = f( φ( ), u() ; ϑ0 ), y () = g( ϕ() ; ϑ0 ): ω () ω ( ) m () ( ) ω ω ω () m ϕ() = y () = u () = ϑ ω ( ) l ( ω ( )) l ( ω ( ) ω ( )) + ω () =,,, l cos cos ω ( ) l ( ω ( )) + l ( ω ( ) ω ( ) sn sn ) l where ϕ(), y (), u() and ϑ are he sae, oupu, npu and parameer vecor, respecvely; he las wo erms n y() are he coordnaes of he manpulaor end pon. Ths complees he descrpon of he sysem. The parameer esmaon algorhm s appled usng genec algorhm parameers as follows: number of bs per parameer n=3, crossover probably q=0.8, populaon sze =50, number of crossover and muaon pons p=4, number of generaons I max = 5000. The vecor ϑ ranges over he se D M. The lower bound of D M a he each drecon chosen o be 30% of he rue value of he correspondng parameer, and he upper bound chosen o be 00% of he rue value of he correspondng parameer. Several expermens are run wh he above seup, varyng he level of he nose presen n he measuremens (nose free, ±0.5 0, ±0.50 0, ± 0, ± 0, ±3 0 and ±5 0 ); nose s whe, zeromean and unformly dsrbued. For every choce of he model ype and nose level one
hundred expermens are run; he accuracy of he parameer esmaes for every expermen s expressed by he followng quany: δm δm δl δl + + + m m l l S = 4 where δm s he error n he esmae of m and smlarly for he remanng parameers. In oher words, S s he average relave error n he parameer esmaes. The resuls usng recursve selecon probables are presened n Fg. Each curve s he cumulave hsogram of S for one hundred expermens a a gven nose level. We also appled a genec algorhm usng he mean square error (MSE) o deermne he selecon probables (see equ.6). The cumulave resuls of MSE selecon probables are presened n Fg. As can be seen, hese resuls are sgnfcanly worse han he resuls of recursve selecon probably mechansm. Fnally, mus be menoned ha he average duraon of one run of he recursve parameer esmaon mehod s 7 mnues on HP Apollo 735 worksaon. Runs nvolvng models wh recursve selecon probably requre around 500 generaons, each generaon requrng around 50 npu/oupu measuremens, whle runs nvolvng MSE selecon probably requre around 00 generaons, each generaon requrng exacly 300 npu/oupu measuremens.
7. Conclusons We have appled a GA on parameer esmaon problem and esed s performance on he MIT Seral Lnk Drec Drve Arm. The es problem s hard because of he srong nonlneares of he sysem, he nsensvy o mass values, he presence of nose n he daa and he exsence of he PD conroller. The performance of he algorhm s que good, even for hgh nose levels. A low nose levels, he number of models ha are close o he rue sysem wh accuracy beer han % s que hgh. The success of our mehod s arbued o he followng feaures. Recursve Probably Selecon Mechansm. The use of an exponenal error funcon, and he resulng use of a compeve and mulplcave scheme, accenuae he compeon beween rval models and faclaes he selecon of he bes model n a search subse. Enropy Creron. I ensures an approprae balance beween exreme and nsuffcen dversy, such ha he algorhm s no rapped a local mnma. The recursve naure of he probably selecon mechansm and he use of he enropy creron reduce he compuaon requred for he algorhm o oban parameer esmaes whn he specfed accuracy. An addonal aracve feaure of our algorhm s s generaly; no parcular assumpons have been made abou he form of he sysem equaons (), nor abou he probably dsrbuon funcon of he nose on he measuremens. In parcular, f() and g() need no be connuous.