Qi Kang*, Lei Wang and Qidi Wu

Transcription

1 In. J. Bo-Inspred Compaon, Vol., Nos. /, Swarm-based approxmae dynamc opmzaon process for dscree parcle swarm opmzaon sysem Q Kang*, Le Wang and Qd W Deparmen of Conrol Scence and Engneerng, ongj Unversy, 4800 Caoan Gongl, Shangha 0804, Chna E-mal: q.kang07@gmal.com E-mal: wangle_j@6.com E-mal: wqd@moe.ed.cn *Correspondng ahor Absrac: hs paper presens a convergence analyss of parcle swarm opmzaon sysem by reang as a dscree-me lnear me-varan sysem frsly. And hen, based on he resls of sysem convergence condons, dynamc opmal conrol of a deermnsc PSO sysem for parameers opmzaon s sded by sng dynamc programmng; and an approxmae dynamc programmng algorhm swarm-based approxmae dynamc programmng (swarm-adp) s proposed n hs paper. Fnally, nmercal smlaons proved he valdaed of hs presened dynamc opmzaon mehod. Keywords: parcle swarm opmzaon; PSO; approxmae dynamc programmng; dynamc opmzaon. Reference o hs paper shold be made as follows: Kang, Q., Wang, L. and W, Q.D. (009) Swarm-based approxmae dynamc opmzaon process for dscree parcle swarm opmzaon sysem, In. J. Bo-Inspred Compaon, Vol., Nos. /, pp Bographcal noes: Q Kang s crrenly a PhD sden of he Conrol Deparmen n ongj Unversy, Shangha, Chna. Hs crren research neress nclde nellgen compaon and approxmae dynamc programmng. Le Wang receved hs PhD n Conrol Deparmen from ongj Unversy n Shangha n 998. He s crrenly a Professor of he Conrol Deparmen n ongj Unversy, Shangha, Chna. Hs crren research neress nclde nellgen conrol, nellgen compaon, CIMS and sysem engneerng. Qd W s a Professor of Conrol Engneerng and Managemen Scence a ongj Unversy, Shangha, Chna. She receved her PhD n Conrol Scence and Engneerng n 986 from EH, Zrch and Swzerland. Her crren research focses on nellgen conrol, nellgen compaon, CIMS and managemen scence. Inrodcon he parcle swarm opmzaon (PSO) sysem s a poplaon-based hersc global opmzaon echnology frs nrodced by Kennedy and Eberhar (995). I belongs o he caegory of swarm nellgence mehods, and s basc dea s based on he smlaon of smplfed anmal socal behavors sch as fsh schoolng, brd flockng, ec. Drng he las decade, PSO ganed ncreasng poplary de o s effecveness n performng dffcl opmzaon asks (Eberhar and Sh, 00), as well as smplcy of mplemenaon and he ably o qckly converge o a reasonably good solon. here has been a consderable amon of work done n developng PSO, hrogh emprcal smlaons (Wang e al., 004; Yang e al., 007; Asanga e al., 004), n he negraon of s self-adapaon, parameer selecon, swarm opology and negrang wh oher nellgen mehods. In addon, has been effecvely appled o power sysem opmzaon (Esmn e al., 005; Abdo, 00), prodc schedlng (Xa e al., 004), raffc plannng (Zhang and L, 007), comper nework opmzaon (Yan e al., 004), neral nework ranng (Cha-feng, 004), and mlple objecves opmzaon (Carlos e al., 004), ec. Recenly, he analyss of he sably and convergence of he parcle swarm sysem s geng more aenon by researchers n compaonal nellgence. Kennedy carred o he frs analyss of he smplfed parcles behavor (Kennedy, 998), who showed he dfferen parcle rajecores for a range of desgn choces for he gan Copyrgh 009 Inderscence Enerprses Ld.

2 6 Q. Kang e al. hrogh smlaons. Ozcan and Mohan (999) showed ha a parcle n a smple one-dmensonal PSO sysem follows a rajecory defned by a snsodal wave, randomly decdng on boh s amplde and freqency, nder he premse of me-nvaran. Under he same premse, Clerc and Kennedy (00) formally sded he analyss of he sably properes of he algorhm, whch presened hree models of consrcon facor mehod and analysed a parcle s rajecory n dscree me, hen progressed o he vew of n connos me. Emara and Faah (004) presened a knd of connos PSO sysem model, and proved s sably sng Lyapnov mehod. Vsakan e al. (006) provded a sably analyss of he sochasc parcle dynamcs, by represenng he parcle dynamcs as a non-lnear feedback conrol sysem, who he assmpon ha all parameers are non-random, as formlaed by Lre. Jn e al. (007) presens a sffcen condon for he PSO sysem mean-sqare o be sable, based on he assmpon of he prevos arcles and he heory of sochasc processes. In hs paper, we wll provde convergence analyss of PSO by reang as a dscree-me lnear me-varan sysem grond on above resls frsly; and hen, based on he resls of sysem convergence condon, dynamc opmal conrol for PSO sysem s sded; and an approxmae dynamc opmzaon mehod swarm-based approxmae dynamc programmng (swarm-adp) s proposed n hs paper. Fnally, nmercal smlaons proved he valdaed of hs presened dynamc opmzaon mehod. he paper s organsed as follows: n Secon, he sandard PSO algorhm s gven. In Secon 3, he convergence analyss of he PSO sysem s dscssed o oban he admssble conrol space. In Secon 4, he swarm-based approxmae dynamc opmzaon sed o parameer selecon for a dscree PSO sysem s sded. In Secon 5, llsrave nmercal smlaons and resls are gven, followed by he conclson of he paper. he PSO sysem he PSO sysem s a poplaon-based hersc global opmzaon echnology frs nrodced by Kennedy and Eberhar n 995. he algorhm manans a poplaon of parcles, where each parcle represens a poenal solon o an opmzaon problem. In he parcle swarm algorhm, he rajecory of each ndvdal n he search space s adjsed by dynamcally alerng he velocy of each parcle, accordng o s own flyng experence and he flyng experence of he oher parcles n he search space. Assme an m-dmensonal ( m N ) search space m S R, and le n denoe he swarm sze ( n N ). Each parcle ( = n) can be represened as an objec wh several characerscs. hen, he flgh velocy of parcle ( = n) for he nex fness evalaon n dd ( = m) dmensonal sbspace s calclaed sng eqaon (): v ( + ) = ω v ( ) + c r ( ) p ( ) x ( ) + d, d,, d d, d, ^ cr, d() pgd, () x, d() Where, v () s he parcle velocy a he h eraon, x () s he parcle poson a he h eraon. p s he personal bes poson of parcle acheved p o he crren eraon. p g s he global bes poson obaned so far by all ndvdals, represens he collaborave effec of he ndvdals. ω > 0 s called nera wegh. he acceleraon coeffcens c and c deermne he relave nflence of he socal and cognon componens, and are ofen boh se o he same vale o gve each componen (he cognon and socal learnng raes) eqal wegh. r U (0,) and r U (0,) are elemens from nform random seqences n he range (0,). he new poson for ndvdals s he addon of he poson a me and he dsance ha ndvdals wll fly wh he new velocy. he synchronos pdae of poson s hs: () x ( + ) = x () + v ( + ) () Each parcle wll compe her fness vale ( ()) f x, and hen, he personal bes poson of each ndvdal s pdaed sng he followng eqaon: p(), f f( x( + )) f( p()) p ( + ) = (3) x( + ), f f( x( + )) < f( p( )) he global bes poson fond by any parcle drng all prevos seps, p s defned as: g p ( + ) = arg mn f ( p ( + )), n (4) g p he psedo code for PSO algorhm s descrbed n Fgre. Fgre he psedo code of PSO algorhm Begn Inalse poplaon: boh poson x and velocy v; Se parameers, ω, c and, c ec; Ieraon=; Whle (ermnaon creron s no me) Do Evalae swarm sng correspondng fness fncon; Updae he personal bes poson p (personal hnkng) and he global bes poson p (swarm collaboraon); g Updae he velocy v sng eqaon (); Updae he poson x sng eqaon (); Ieraon = Ieraon + ; End Do (nl ermnaon creron s me) End

3 Swarm based approxmae dynamc opmzaon process 63 3 he convergence analyss of PSO sysem In hs secon, we wll presen convergence analyss of PSO by reang as a dscree-me lnear me-varan sysem. o smplfy he noaon, he dervaon wll be performed n only one dmenson n he whole paper, sng a sngle parcle. hs allows s o drop boh and j sbscrps, frher mplyng ha x denoes he vale of x a me, raher han he vale x assocaed wh parcle nmber. he sochasc componen wll also be removed emporarly wh he sbsons φ = cr (), φ = cr () and φ = φ+ φ. Frher, he poson of parcle wll be consdered n dscree me only, reslng n he followng eqaons: φ p + φ pg x+ = x + ω v + φ ( x ) φ φ p + φ pg v+ = ω v + ( φ )( x ) φ Consderng he varey of parameer ω, φ, a dscree-me lnear me-varan PSO sysem s derved. Here, we rea he PSO dynamc sysem as a feedback conrol sysem, n whch s ω sll consan. he eqaons governng he dynamcs can be expressed as: x ω x = v+ ω v y x = ( 0) v φ = φ ( y p), where, p = p + φ pg φ + φ heorem : (Vsakan e al., 006): Le he parcle dynamcs be represened by eqaons (6) (8) and sasfy wh an eqlbrm pon a he orgn. hen, he orgn s asympocally sable f ( ω + ω ) ω <, ω 0, Φ : = sp( φ ) < + ω Proof: Consder he Lyapnov fncon V( X) = X QX (0) Where, Q s a symmerc posve defne marx. he decrease n he sysem energy as represened by he Lyapnov fncon beween wo dscree-me nsans s gven by ( ) ( ) ( ) ( ) φ φ Δ V = V X V X = X QX X QX = X A QA Q X y B QAX + y B QB (5) (6) (7) (8) (9) () Snce φy( φy Φy) 0, f s added o he rgh-hand sde of he eqaon, we ge ( ) φ + φ y BB φ y ( φ y Φy ) ( ) ( φ ) ( φ y ) ( B QB) ΔV X A QA Q X y B QAX = X A QA Q X y B QA ΦC X () he rgh-hand sde s negave by compleng a sqare erm f he followng marx eqaons are sasfed: A QA Q = U U B QA =ΦC W U B QB = W W (3) Comparng hese wh he relaonshp esablshed n he Dscree-me Posve Real Lemma ndcaes ha f and only f he lnear sysem wh he ransfer fncon H ( z ) sasfes all he condons saed n he Posve Real Lemma. H( z) = ΦC( zi A) B+ D, D = (4) I s sraghforward hen o show ha H ( z) sasfes he condons n he Posve Real Lemma f j { Ge θ } ω <, ω 0 and +ΦR ( ) > 0 (5) { } j Ge ( θ + ω R ) > ( ω + ω ) Here, for each θ [0, π ). R{} ndcaes he real par of s argmen. I hen leads o ( ω + ω ) K < + ω Conseqenly, ς ς ( ) φ ς φ ( Uς φ y W) ( Uς φ y W) Uς φ y W 0 ΔV U U yw U y W W = = (6) (7) Snce he dfference Δ V n he Lyapnov fncon s nonncreasng, he parcle dynamc are garaneed o be sable, accordng o he Lyapnov sably heorem. 4 Dynamc opmzaon of PSO sysem In Secon 3, he convergence of me-varan PSO sysem s dscssed; and correspondng parameer admssble regons ensre sysem sable s worked o. In hs secon, nder he correspondng parameer resrcon, we wll perform he dynamc parameer opmzaon based on he feedback PSO sysem sng dynamc programmng heory, on prpose o fnd he opmal rajecory of PSO parameers ncldng feedback conrol for parameer seng. Dynamc programmng was developed by R.E. Bellman n he laer 950s (Bellman, 957). I can be sed o solve conrol problems for non-lnear, me-varyng sysems and

4 64 Q. Kang e al. s sraghforward o program, whch s based on Bellman s prncple of opmaly. 4. Fne horzon dynamc opmzaon for deermnsc nbonded PSO sysem Consder he PSO dynamc sysem descrbed by eqaons (6) (8), whch has he assocaed performance ndex: N J; N( X) = mn Obj( X N) + Obj( X) + (8) = Where, Obj() s defned as he objecve fncon, N denoes he maxmm erave mes. he framework of dynamc opmser for PSO sysem dynamc opmzaon s descrbed as Fgre. Fgre o begn, le PSO sysem opmzaon sng dynamc programmng (see onlne verson for colors) N N = N and wre J ( X ) = Obj( X ) (9) I s he evalaon of objecve fncon o be opmsed a me N. Decremen o and N wre J Obj X Obj X N = ( N) + ( N ) + N N (0) Fnd N () by mnmsng (0). o do hs, se he sae eqaon o wre JN = mn { Obj( XN ) + N N } + Obj( XN) N = mn { Obj( X N ) + N N } N + Obj( AX + B ) N N } () If here are no consrans, he mnmm of J N s fond by J Obj( AX + B ) 0 = = + N N N N N N () Obj( AX N + BN ) N = arg N + = 0 (3) N N If we conned o decremen and apply he opmaly prncple, he resl for each = N,,, 0 s Obj( AX B) arg + = + = 0 4. Swarm-ADP for deermnsc PSO sysem (4) Dynamc programmng s a very sefl ool n solvng opmzaon and opmal conrol problems. However, s ofen compaonally nenable o rn re dynamc programmng de o he backward nmercal process reqred for s solon,.e., as a resl of he well-known crse of dmensonaly (Powell, 007). One has o fnd a seres of conrol acons ha ms be aken n seqence. hs seqence wll gve he opmal performance cos, b he oal cos of hose acons s nknown nl he end of ha seqence. Consderng he crse of dmensonaly of dynamc programmng, many approxmae mehods are proposed. A very general algorhm s based on he approxmaon of he cos-o-go fncon by means of some fxed-srcre paramerc archecre, whch s raned on he bass of sample pons comng from he dscresaon of he sae space. here are many examples of sch approach, where dfferen srcres are employed, polynomal approxmae, splnes, mlvarae adapve regresson splnes and neral neworks. here are several synonyms sed ncldng adapve crc desgns (ACD), approxmae dynamc programmng, neral dynamc programmng, renforcemen learnng and so on. For he comper mplemen of boh dscree-me PSO sysem and connos-me PSO sysem, we canno ge he opmal general solon lke eqaon (4), b a seral * dscree daa of φ k, whch s sed o f o he dynamc of * φ k. In addon, s hard o oban he resls by solvng he HJB eqaons drecly, becase dfferen problems have dsnc objecve fncons, and s sally non-lnear; frhermore, he conrol resrcon redces he compaon complexy, b canno avod crse of dmensonaly radcally. herefore, we propose a swarm-adp for opmzaon of PSO sysem. In he swarm-adp, he hersc sochasc poplaon searchng mehod s adoped o ry conrol laws n s admssble errory o fnd a near opmal conrol for PSO dynamc sysem. If conrol varable s defned n a ml-dmensonal space, swarm-adp wll p p more advanages becase of s characerscs of sochasc search and swarm nellgence. Insead of solvng HJB dfferenal eqaon, swarm-adp fnds he mnmal by drec calclaons sng some learnng cells, sch as neral neworks, ec. We wll sdy he sysem sep by sep o fnd he relaonshp of opmal conrols and opmal

5 Swarm based approxmae dynamc opmzaon process 65 performance coss sng parcle swarm o deermne he opmal conrol seqence. Defnon : Admssble sae space Ω AS : he admssble rajecores ms belong smlaneosly o hese reachable n n domans R and conrollable domans R n ΩR ΩC n-dmensonal space. ha s, a any gven me, he orgn ms be reachable from he presen sae, and he presen sae ms be reachable from he se of gven nal saes. hs, he admssble sae se s defned as n Ω : =Ω Ω R. AS R C Defnon : Admssble polcy space Ω : correspondng o he admssble sae space Ω AS, he polcy appled o he sysem make p of a l-dmensonal admssble polcy space l Ω R. Inally, defne parcle s codng forma and nalse he l poplaon n admssble polcy space Ω R. A nal sage 0, nalse V ( x ) and parcle s fness vale s calclaed by cos fncon Vx ( 0) = mn { rx ( 0, 0) + Vx ( ( 0, 0)) }. 0 Ω Once he opmal polcy * s fond, s saved by he 0 learnng cell 0 ( x 0 ),one b of he opmal polcy nework OPN 0 ( x ), composng opmal polcy seqence nework OPSN ( x. ) Meanwhle, V( x 0) s sored o he opmal cos nework OCN J ( x ), a hs sage, s recorded as J 0 ( x ). herefore, when solvng he Bellman eqaon V( x ) = mn r( x, ) + V( ( x, )), he vale of { } Ω Vx ( (, )) for correspondng sae x = ( x, ) can be + derved from J ( x ). However, only fne nmercal vales for dscree saes are recorded by neworks x ( ) and J ( x ), n each sage of solvng he Bellman eqaon. Usally, he ransfer saes () canno mach wh he saes n J ( x ) accraely. herefore, a knd of lnear nerpolaon approxmae mehod s adoped n ranng neworks. Usng he same mehod, we compe V( x ) by parcle swarm o fnd he near-opmal conrol law n accordngly, we can ge * and V( x ) Ω,, recorded by cell ( x ) and OCN J ( x ) sng lnear nerpolaon, respecvely. he res may be dedced by analogy, we ge J ( x), J ( x), J ( x),, nl he seqence converge. Meanwhle, he opmal polcy seqence x ( ) = { ( x), ( x), ( x), } 0 0 mappng { J ( x), J ( x), J ( x), } J( x) s obaned. Afer geng he opmal polcy seqence ( x ), we shold carry o he daa fng accordng o feedback conrol mode of he PSO dynamc sysem, o approxmae opmal parameers accordngly. he resls wll help o nsrc he parameer seng for achevng beer algorhm performance. Algorhm : Swarm-ADP Sep Inalse he sae vecor X 0 R n one-dmensonal admssble sae space AS randomly, ( ) X : = X (), X (),, X ( n) ncldes n sample saes, and se he nal cos fncon V ( X 0 ) = 0 n ; accordngly, he OCN J : J 0 ( X ) = 0 (nework nalsaon ) for connos sae X Ω AS. here are wo nalsaon sraeges: V( X 0 ) = 0 n ; esmae V( X 0) by applyng fne sable conrols. Sep Perform sae ranson X X + for each sample sae X () s, s =,,, n, n he general sage accordng o he followng PSO dynamc fncon (sae fncon): ω X+ () s = X() s + () s 0 ω [ φ ] (5) () s = φ y () s = F X () s = 0 X () s (6) Here, conrol polcy () s Ω brngs a sae ranson. Sep 3 Hersc sochasc opmzaon of conrol polcy () s Ω for he PSO dynamc feedback sysem descrbed by eqaons (5) (6) sng PSO: 3. Parcle codng and nalsaon. For sample sae X () s, defne parcle swarm PS() s, whch ncldes N parcles. he poson x () s of parcle ( =,,, N ) represens he polcy appled o sae () s a hs sage, whch s nalsed randomly n X Ω. For a l-dmensonal conrol sysem, parcle shold be represened as x(, s j): =, (, s j), h j =,,, l denoes he j sb-conrol n admssble space Ω. he velocy of parcle v () s s defned as

6 66 Q. Kang e al. he reglaon of conrol v(): s =Δ, (), s hrogh whch o carry o polcy search n space l. 3. A he general sage, mplemen swarm opmzaon of bonded conrol () s Ω for each sample sae X (), s s =,, n. he flgh velocy of parcle and s synchronos pdae of poson are calclaed as: Δ, (, sk+ ) = ω Δ, (, sk) + cr (, sk)[ p, (, sk), (, sk)] ^ + cr (, sk) pg, (, sk),(, sk) (7), (,k s + ) =, (,k) s +Δ, (,k s + ), Where, a maxmm velocy Δ max () s for each modls of he velocy vecor s defned n order o conrol excessve roamng of parcles osde he ser defned space. Whenever Δ, () s exceeds he defned lm, s se as Δ max () s. For parcle, s fness fncon s defned as follows: (, (, sk) ) = mn { rx ( ( s),, ( sk, )) + VX ( ( ( s),, ( sk, ))) } F (,) s k Ω (8) Here, rx ( ( s), ( sk, )) : = ObjX ( ( s)) + ( sk, ) ( sk, );,,, n whch, Obj() represens he objecve fncon; and ( ( ( ),, (, ))) ( ( ( ),, (, ))) V X s s k = J X s s k, whch s obaned from he OCN J ( X ) descrbed laer. Compe each parcle s fness vale (, (, )) F s k, and hen, he personal bes poson of each ndvdal s pdaed sng he followng eqaon: p, (, s k), f F(, (, s k+ )) F( p, (, s k)) p, (, s k+ ) = (9), (, s k+ ), f F(, (, s k+ )) < F( p, (, s k)) he global bes poson fond by any ndvdal drng all prevos seps, p, g( s, k ) s defned as: g,, p, ( ) p (, s k) = argmn F p (, s k), n (30) Perform swarm opmzaon eraons, le k = k+. Afer schedled eraons k = C, he global opmal max * conrol s fond, ( s) = p (, max ) g s C for all sample saes X (), s s =,, n. I s sored by he opmal polcy cell () s, whch s nclded n he OPN : =. () () ( n) For = 0,,,, he opmal cos V+ ( X ( s )) s pdaed accordng o he Bellman dynamc programmng fncon: { ( ) ( )} V ( X ( s)) = mn r X ( s), ( s) + V ( X ( s), ( s)) +,,, Ω ( ) * * * = Obj( X()) s + () s () s + V ( (), ()) X s s Here, V ( ( ( ), * ( ))) ( ( ( ), * X s s J X ( ))) s s =, whch s obaned from he opmal cos nework J ( X ). A he general sage, he opmal cos V( X( s )) s obaned and sored as J( X( s )), by applyng he opmal polcy seqence * * * * (): s = 0(), s (), s, () s n rn, whch s { } sored as he OPSN. Accordngly, * * * * : = () () ( n) * * * 0() () () * * * 0() () () = * * * 0( n) ( n) ( n) Sep 4 Approxmae he cos for all sae X, denoed by he OCN J ( X ) (nework learnng), hrogh lnear nerpolaon of V ( X ( s)), s =,, n. (3) (3) Le = +, repea Sep 3, nl V+ ( X ) V ( X ) < ε, ε s an arbrary posve real nmber; accordngly, J ( X ) J( X ), or he maxmm sage nmber s me = max. Op he opmal polcy seqence * * * * : () () ( ) = n. For nfne horzon opmal conrol, s possble o approxmae he J ( X ) hrogh few saes afer adeqae eraons, becase V( X ) can raverse almos all saes when. Sep 5 Daa fng and analyss of he polcy seqence * * * * : = () () ( n), and he resls and crves wll edce he correspondng opmal parameers, whch wll be help o nsrc parameer seng of PSO algorhm. he PSO sysem adopng he opmsed parameers wll be opmsed crclarly n hs same way.

7 Swarm based approxmae dynamc opmzaon process 67 Fgre 3 he framework of swarm-adp for -D PSO sysem (see onlne verson for colors) 5 Nmercal smlaons y x Le p = 0, y = x p = x, X =, v = v deermnsc PSO sysem can be represened: ω = 0.7 X+ = X + = ( X, ) 0 ω = 0.7 he (33) Accordng o he resls of relaonshp, assre of sysem convergence, ( ω + ω ) ω <, ω 0, Φ : = sp( φ ) <, we can + ω ge Φ ( ω = 0.7) < Accordngly, we can oban he admssble bond by = φx : Ω : = sp( ) = x (34)

8 68 Q. Kang e al. Fgre 4 he admssble bond Ω (see onlne verson for colors) * ( f) ( 0.x x x 3.54x 0.x + 6.4x 0.06) = * ( f) 0 = (0.4 x + 0.6x x 3.34x 0.7x x+ 0.0) (36) (37) Fgre 5 Smlaon resls for Sphere fncon (a) cos vale crve (b) opmal polcy rajecory (c) polcy fng (see onlne verson for colors) For he objecve fncon, defned n he assocaed performance ndex, wo benchmarks are adoped n he nmercal smlaon, whch s presened n able. able Benchmarks for smlaons Fncon Mahemacal represenaon Range of search (a) Sphere n f( x) = x = [,] Generalsed Grewank n n x f() x = cos( ) 4000 x x + = = [,] Bellman s dynamc programmng eqaon: J+ ( X( s)) = (35) mn Obj( X( s)) + ( s) + J ( ( ( ), ),, ( )), X s s Ω he parameers of or swarm dynamc programmng algorhm s se as follows: he sze of he parcle swarm s N = 50; he range of nera wegh facor ω s se as [ ωmn, ω max ] = [0.3,.0], and s lnear dynamc redced drng he whole opmzaon process; acceleraon consans c = c =.0; Vmax = Xmax ;.he maxmm eraon s C max = 00. he algorhm ermnae condon s sage = max, and he maxmm sage nmber for dynamc programmng s max = 00. Fgres 5 6 presens he smlaon resls for Sphere and Generalsed Grewank fncon, n whch, he resls nclde he cos vale, opmal polcy rajecory and polcy fng resls, ec. he fng fncons for Sphere ( f ) and Grewank ( f ) are presened respecvely as follows: (b) (c)

9 Swarm based approxmae dynamc opmzaon process 69 Fgre 6 Smlaon resls for Generalsed Grewank (a) cos vale crve (b) opmal polcy rajecory (c) polcy fng (see onlne verson for colors) he PSO opmal conrol problem dscssed n hs paper s parameer opmzaon, n whch he reqremen of resl precson s no so src; herefore, hs knd of precson sacrfce s deserved, whch redce he complexy grealy. In he work of hs paper, random varables n PSO sysem are gnored; n addon, s grond on he premse of sngle ndvdal flyng n one-dmensonal space. herefore, s or fre work and promsng projec o sdy approxmae dynamc opmzaon for a sochasc PSO sysem, as well as ml-dmensonal PSO sysem. References 6 Conclsons (a) (b) (c) hs paper provdes he convergence analyss of PSO by reang as a dscree me-varan feedback sysem. Based on he resls of sysem convergence condons, dynamc opmal conrol for a deermnsc PSO sysem s sded for parameers opmzaon; and a swarm-adp s proposed. Nmercal smlaons proved he valdaed of hs presened mehod. In swarm-adp, approxmae s sed o redce he compaon complexy; he fnal resls may have devaon from he precse vales. However, he objecve of Abdo, M.A. (00) Opmal power flow sng parcle swarm opmzaon, Elecrcal Power and Energy Sysem, Vol. 4, pp Asanga, R., Saman, K.H. and Harry, C.W. (004) Self-organzng herarchcal parcle swarm opmzer wh me-varyng acceleraon coeffcens, IEEE ransacon on Evolonary Compaon, Vol. 8, No. 3, pp Bellman, R.E. (957) Dynamc Programmng, Prnceon Unversy Press, Prnceon, 957. Carlos, A.C., Gregoro,.P. and Maxmno, S.L. (004) Handlng mlple objecves wh parcle swarm opmzaon, IEEE ransacons on Evolonary Compaon, Vol. 8, No. 3, pp Cha-feng, J. (004) A hybrd of genec algorhm and parcle swarm opmzaon for recrren nework desgn, IEEE ransacon on Sysems, Man and Cybernecs, Par B: Cybernecs, Vol. 34, No., pp Clerc, M. and Kennedy, J. (00) he parcle swarm-exploson, sably and convergence n a mldmensonal complex space, IEEE ransacons on Evolonary Compaon, Vol. 6, No., pp Eberhar, R.C. and Sh, Y. (00) Parcle swarm opmzaon: developmens, applcaons and resorces, Proc. Congress on Evolonary Compaon, IEEE, Pscaaway, pp Emara, H.M. and Faah, A.H.A. (004) Connos swarm opmzaon echnqe wh sably analyss, Proceedngs of Amercan Conrol Conference, Caro, Egyp, pp Esmn, A., Lamber-orres, G. and Zambron de Soza, A.C. (005) A hybrd parcle swarm opmzaon appled o loss power mnmzaon, IEEE ransacons on Power Sysems, Vol. 0, No., pp Jn, X.L., Ma, L.H. and W,.J. (007) Convergence analyss of he parcle swarm opmzaon based on sochasc processes, ACA Aomaca Snca, Vol. 33, No., pp Kennedy, J. (998) he behavor of parcle, Proc. 7h Ann. Conf. Evolon Programmng, March, San Dego, CA, pp Kennedy, J. and Eberhar, R. (995) Parcle swarm opmzaon, Proc. Inernaonal Conference on Neral Ne-works, IEEE, Perh, Asrala, pp Ozcan, E. and Mohan, C.K. (999) Parcle swarm opmzaon: srfng he waves, Proceedngs of Congress on Evolonary Compa, IEEE, Washngon D.C., USA, pp Powell, W.B. (007) Approxmae dynamc programmng: solvng he crses of dmensonaly, Wley-Inerscence, 007.

10 70 Q. Kang e al. Vsakan, K., Krsnaplla, S. and Flemng, P.J. (006) Sably analyss of he parcle dynamcs n parcle swarm opmzer, IEEE ransacons on Evolonary Compaon, Vol. 0, No. 3, pp Wang, L., Kang, Q. and W, Q.D. (004) Groped-and-delayed broadcasng mechansm for opmm nformaon n parcle swarm opmzaon, Proceedngs of he IEEE Inernaonal Conference on Cybernecs and Inellgen Sysems, Dec. 3, Sngapore, pp Xa, W.J., W, Z.M., Zhang, W., e al. (004) Applyng parcle swarm opmzaon o job-shop schedlng problem, Chnese Jornal of Mechancal Engneerng (Englsh Edon), Vol. 7, No. 3, pp Yang, X., Yan, J., Yan, J. and Mao, H. (007) A modfed parcle swarm opmzer wh dynamc adapaon, Appled Mahemacs and Compaon, Vol. 89, No., pp Yan, P., J, C.L. and Zhang, Y.Y. (004) Opmal mlcas rong n wreless ad hoc sensor neworks, IEEE Inernaonal Conference on Neworkng, Sensng and Conrol, pp Zhang, Z.X. and L, C.W. (007) Applcaon of he mproved parcle swarm opmzer o vehcle rong and schedlng problems, IEEE In. Conf. on Grey Sysems and Inellgen Servces, pp.50 5.