Sensors & Transducers 04 by IA Publshng, S. L. http://www.sensorsportal.com Estmaton of System Models by Swarm Intellgent Method,* Xaopng XU, Ququ ZHU, Feng WANG, Fuca QIAN, Fang DAI School of Scences, X an Unversty of Technology, X an, 70054, Chna School of Mathematcs Statstcs, X an Jaotong Unversty, X an, 70049, Chna School of Automaton Informaton Engneerng, X an Unversty of Technology, X an, 70048, Chna Tel.: 86 9 8888, fa: 86 9 8888 * E-mal: up@aut.edu.cn Receved: September 04 /Accepted: 8 November 04 /Publshed: December 04 Abstract: System dentfcaton s the theory methods of establshng mathematcal models of the systems. As one of the ey ssues of system control scence, system dentfcaton has been wdely appled to the desgn analyss of the control system. Accordngly, system dentfcaton becomes one of the current actve subects. Currently, parameter estmaton of the nonlnear system models s a very mportant problem n the area of system dentfcaton. In ths paper, the parameter estmaton problems of the nonlnear system models are converted nto a nonlnear functon optmzaton problem over parameter space ntally. Then, the estmates of the nonlnear system model parameters are gotten by usng a modfed fsh swarm algorthm. Fnally, n smulaton, the presented dentfcaton method s used to several dfferent nonlnear systems, the smulaton results ndcated that the presented method s feasble reasonable. Copyrght 04 IA Publshng, S. L. Keywords: Nonlnear system model, System dentfcaton, Parameter estmaton, Optmzaton problem, Swarm ntellgent algorthm.. Introducton System dentfcaton s the theory methods of establshng mathematcal models of the systems. The feld of the dentfcaton process parameter estmaton has developed rapdly durng the past decade. At present, the dentfcaton theory of the lnear system has been very mature [-4]. To our best nowledge, n vew of nonlnear systems wdely est n people s producton lfe, one have plunged nto researchng them n the socal scences natural scences [5-0]. In research on the method of the nonlnear system dentfcaton, because many engneerng obects are very comple, the dentfcaton methods of the nonlnear systems show remarable advantage n the analyss of the engneerng obect. However, because of the nherent complety dversty of the nonlnear systems, the current research on the nonlnear far not reached the degree of maturty; t has not been fully revealed that the essence problem of nonlnear s contaned behnd the comple phenomenon. Thereby, the comple nonlnear obect recogntons are stll not well solved va the tradtonal dentfcaton methods; the dentfcaton of the nonlnear system s the man topcs n the current nternatonal dentfcaton felds. Consequently, ths paper puts forward a parameter dentfcaton approach for the nonlnear system based on the swarm ntellgent algorthm. http://www.sensorsportal.com/html/digest/p_578.htm 9
Creatng a new swarm ntellgent method creates a new branch n ths feld, whch often catches scholarly nterest. The fsh swarm algorthm s a new populaton-based/swarm ntellgent evolutonary computaton technque proposed by L et al. [] that was nspred by the natural schoolng behavor of the fsh. Fsh swarm algorthm presents a strong ablty to avod local mnmums n order to acheve global optmzaton. It has been proofed n the functon optmzaton [], Parameter estmaton [], combnatoral optmzaton [], least squares support vector machne [4] geotechncal engneerng problems [5], among others. Of course, the fsh swarm algorthm can be used to the feld of the nonlnear system dentfcaton. In ths paper, am at the dversty of the nonlnear system models, a parameter estmaton method, whch s the same wth a varety of the nonlnear system model, s proposed by usng a modfed fsh swarm algorthm. Fnally, three nds of the nonlnear models are taen as the smulaton eamples to show the effectveness of the proposed parameter estmaton approach. The rest of ths paper s organzed as follows. The net Secton descrbes the problem formulaton for the nonlnear system models. In Secton, we ntroduce a modfed fsh swarm algorthm wth the partcle swarm optmzaton algorthm. A parameter estmaton method for the nonlnear system model s gven n Secton 4 based on the modfed fsh swarm algorthm. Secton 5 presents the llustratve numercal smulatons to llustrate the feasblty of the presented dentfcaton method. Fnally, Secton 6 summarzes the contrbuton of ths paper conclusons.. Problem Descrpton In ths paper, general nonlnear system model can be epressed as the followng form. y ( = f ( u(, t, θ ), () where y( s the system output, u( s the system nput, θ=(θ, θ,, θ ) T s the unnown parameter vector, the epresson form of the functon f s nown, u( s gven. Moreover, the actual measurement values (y 0 (, t=,,, n) of the system output y( are nown. It s necessary that the parameter s estmated by usng y 0 (. To the purpose of estmaton, the nonlnear system models, whch are represented by Eq. (), must satsfy the followng assumptons. ) y( must be measurable; ) Each parameter must be related wth the output y( of the system. That s to say, the parameters can be estmated based on the observed data. ) As long as the parameters are determned, the value of the system output y( can be obtaned by the system smulaton. 4) The system dose not dverges n fnte tme. That s to say, the value of y( dose not tend to nfnty.. Modfed Fsh Swarm Algorthm wth Partcle Swarm Optmzaton.. Partcle Swarm Optmzaton Partcle swarm optmzaton [6, 7] s a swarm ntellgence algorthm typcally employed n numercal optmzaton problems that has ganed n popularty n recent years due to ts effcency effectveness at addressng scence engneerng problems. Smlar to the genetc algorthm, partcle swarm optmzaton s based on a romly ntalzed populaton searches for optmal generaton-to-generaton updatng. Whereas obects n the genetc algorthm are called ndvduals, they are called partcles n the partcle swarm optmzaton. Each partcle moves (fles) at a certan velocty. The velocty vector gves momentum to a partcle, wth the specfc amount updated by the behavor of the two varables,.e., the memory (cogntve behavor) current percepton (socal behavor) for each partcle. Gven suffcent tme (teratons), partcles can be epected to floc to a place optmally suted to ther needs. The aforementoned behavors, fundamental to partcle swarm optmzaton, are formularzed as v ( ) = v ( ) ϕ ( )( pbest ( ) ( )), () ϕ ( )( gbest( ) ( )) φ () = cr(), φ () = cr(), () ( ) = ( ) v( ), (4) where denotes the partcle nde; the teraton nde; the partcle poston. Correspondng partcle velocty s represented by v. pbest represents ts own prevous best poston; gbest represents the prevous best poston of the entre swarm. In Eq. (), the second term on the rght h sde s a cogntve term the thrd s a socal term. c c are the cogntve socal acceleraton constants, respectvely. r r are two rom varables generated unformly wthn [0, ], respectvely... Fsh Swarm Algorthm Fsh swarm algorthm, frst proposed n 00, s a new populaton-based optmzaton technque nspred by the natural feedng behavor of fsh. A fsh s represented by ts D-dmensonal poston 94
X = (,,,,, D ), food satsfacton for the fsh s represented as. Ths paper targets mnmzaton. The relatonshp between two fsh s denoted by ther Eucldean dstance d = X X. Another parameters nclude: vsual (representng the vsual dstances of fsh), step (mamum step length), δ (a crowd factor). n s used to represent the sze of the fsh populaton. All fsh try to dentfy locatons able to satsfy ther food needs usng three dstnct behavors. These nclude ) Searchng behavor. Searchng s a basc bologcal behavor adopted by fsh loong for food. It s based on a rom search, wth a tendency toward food concentraton. It s epressed mathematcally as = R( S) X <, X (5) = R( S), (6) where represents the th element of fsh poston X. We romly select for fsh X a new poston X wthn ts vsual. If the correspondng s satsfed, Eq. (5) s then employed at the net poston X. If s not satsfed after try number trals, a rom poston wthn the step range wll be drectly adopted as Eq. (6). In the above equatons, R(S) R(S) represent rom varables wthn [0, step] [-step, step], respectvely. ) Swarmng behavor. Fsh assemble n several swarms to mnmze danger. Obectves common to all swarms nclude satsfyng food ntae needs, entertanng swarm members attractng new swarm members. Mathematcally, c = R( S), X c X c < < δ n s. (7) A fsh located at X has neghbors wthn ts vsual. X c dentfes the center poston of those neghbors s used to descrbe the attrbutes of the entre neghborng swarm. If the swarm center has a greater concentraton of food than s avalable at the fsh s current poston X (.e., c < ), f the swarm (X c ) s not overly crowded (n s /n < δ), the fsh wll move from X to net X, toward X c. Here, n s represents number of ndvduals wthn the X c s vsual. Swarmng behavor s eecuted for a fsh based on ts assocated X c ; otherwse, searchng behavor guarantees a net poston for the fsh. ) Followng behavor. When a fsh locates food, neghborng ndvduals follow. Mathematcally, mn = R( S), X mn X mn < f < δ. (8) Wthn a fsh s vsual, certan fsh wll be perceved as fndng a greater amount of food than others, ths fsh wll naturally try to follow the best one (X mn ) n order to ncrease satsfacton (.e., gan relatvely more food [ mn < ] less crowdng [n f /n < δ]). n f represents number of fsh wthn the vsual of X mn. Searchng behavor commences f followng behavor s unable to determne a fsh s net poston. Besdes, the fsh swarm algorthm should provde a bulletn that records the optmal state current performance of fsh durng teratons... Modfed Fsh Swarm Algorthm The fsh swarm algorthm vsual provdes local search attrbutes. A small vsual restrcts a fsh to nteracton wth a relatvely small number of companons. The fsh swarm algorthm step lmts mamum step length, wth a small step lmtng fsh to searchng a small area ncreasng the rs of wastng tme. The step values are set based on Eucldean dstance calculatons are senstve to fsh swarm algorthm performance (refer to Eqs. (5), (7) (8)). As settngs are dffcult, ths paper employs the partcle swarm optmzaton formulaton to mnmze the mpact of the step factor. As a result, artfcal fsh are able to swm le a partcle n the partcle swarm optmzaton, subect to the vsual factor, but not the step. All the fsh swarm algorthm equatons have been modfed. ) Searchng behavor = ϕ ( <, ) (9) = R( ), (0) V = c r ϕ, () where φ s the unform rom number wthn [0, ] wth a mean value of one. c s, a unform rom number wthn [0, ]. X s stll a new poston wthn X s vsual (same as fsh swarm algorthm). Therefore, Eq. (9) uses the partcle swarm 95
optmzaton formulae to releases step settngs n Eq. (5). Snce Eq. (9) s free to step, ths paper further modfed Eq. (6) as Eq. (0), provdng such wth a vsual range. R(V) s a rom varable wthn [-vsual, vsual]. When step s smaller than vsual n fsh swarm algorthm, the modfed fsh swarm algorthm allows fsh to swm for greater lengths than permtted by the fsh swarm algorthm. Modfed searchng behavor s totally free to step. ) Swarmng behavor = ϕ 4 ( c ), c < s < δ, () where the defnton of φ 4 s the same as φ defnton of X c s same as that n the fsh swarm algorthm. Modfed swarmng behavor s therefore free to step. ) Followng behavor = 5 ( mn ϕ, mn < f < δ, ) () where the defnton of φ 5 s the same as φ the defnton of X mn s same as that used n the fsh swarm algorthm. Modfed followng behavor s therefore free to step. Ths modfed fsh swarm algorthm s free of the step parameter of the fsh swarm algorthm. All postons of X, X c, X mn, the net poston X are dependent upon vsual only. The maor advantage of ths s to release a step parameter. 4. Parameter Estmaton Approach Usng Modfed Fsh Swarm Algorthm The specfc steps of the estmaton procedure are as follows. Step. Parameter θ s taen as a fsh. Step. Determne the ftness functon: On the bass of nowng the values of the parameters, the system outputs y( can be obtaned based on Eq. () from the smulaton eperment. The purpose of the dentfcaton s to mae the dentfcaton of the system output as close as possble the nown system output true value, whch should mae these parameters correspondng to the fshes wth a smaller ftness value. So we tae the followng crteron functon as the ftness functon []. = y( y0 ( ] t [, (4) Step. Intalze the fsh swarm: Let populaton sze s n, ntal poston of fshes are romly set n the range allowed, the ndvdual etreme coordnates of each fsh s set as the current poston, calculate the correspondng ndvdual etreme (. e., the ndvdual s ftness value), whle the global etreme value (. e., the global ftness value) s the best among the ndvduals etreme, record the fsh s seral number of the best value, set the global etremum as the current best fsh poston, many others. Step 4. Calculate the ftness value of each fsh. Step 5. For each fsh, ts ftness value s compared wth the ndvdual s etreme, f t s ecellent, then the current ndvdual etreme s updated. Step 6. For each fsh, ts ftness value s compared wth the global etreme, f t s ecellent, then the current global etreme s updated. Step 7. Each fsh s swarmng behavor s carred out by usng Eq. (). Step 8. Each fsh s followng behavor s carred out based on Eq. (). Step 9. Each fsh s searchng behavor s carred out accordng to Eqs. (9), (0) (). Step 0. If preset stop crtera (usually t s the mamum number of teratons or mnmum error) dose not reach, then the procedure returns to Step 4. Otherwse, the procedure end, the optmal parameter value θ of the nonlnear system model s obtaned. 5. Smulaton Study In order to demonstrate the effectveness of the presented dentfcaton method for the nonlnear system models, the followng llustratve eamples are gven. Here, the MA represents the modfed fsh swarm algorthm, the A denotes the fsh swarm algorthm, the PSO descrbes the partcle swarm optmzaton algorthm the GA represents the genetc algorthm. Eample. Consder the followng transfer functon model. y( u( g at = -τt e. (5) We can see that ths s a pure delay plus nerta segment model, where the dentfed parameters are the proporton coeffcent g, the nerta coeffcent a the lag factor τ. Eample. Consder the followng state space model. ( ) ( ) ( t ) l t t 0 =, (6) ( t ) l ( ) u( t y( = l ( l ( ), (7) 4 t 96
(0), (8) = (0). (9) = Eample. Consder the followng Hammersten model. A( z ) y( = B( z ) ( C( z ) w(, (0) of the nonlnear system models parameters are obtaned by usng the MA are almost appromate to the actual values of the parameters. That s to say, the precson of the presented dentfcaton method by usng the MA s remarably hgh compared wth other methods. Table. The estmates of Eample. where A ( z ) = az az, () B ( z ) = b z b z, () C ( z ) = cz, () Parameters l l l l4 True values 0.5 0.. 0.6 MA 0.500 0.989.0 0.599 A 0.4895 0.877.907 0.609 PSO 0.497 0.065.884 0.5880 GA 0.4869 0.07.060 0.5897 Table. The estmates of Eample. ( = u( ru (, (4) 4 r u ( r u ( w( s a Gaussan whte nose sequence wth zero mean, stard varance σ=0.0. Although the above three cases s not the same form, but they can be attrbuted to the form of Eq. () through a certan transformaton. In the smulaton, the parameters of the dentfcaton algorthm are set as follows. The number of artfcal fsh n=0, the mamum step length step=50, the vsual dstances of fsh vsual=5, try number s 0, crowd factor δ=0.68, the trust parameters c =c =, the mamum number of teratons t ma =50, the parameter X ma =, the mamal rate (boundary value) V ma =. In the Eample, the ntalzed search ranges of the parameters are [, 7]. In the Eample, the ntalzed search ranges of the parameters are [0.,.]. In the Eample, the ntalzed search ranges of the parameters are [-,.4]. The estmates of the parameters of the nonlnear system models by usng the MA are shown n Tables,, respectvely. In order to show the valdty of the presented dentfcaton method, we further adopted the A (or PSO, or GA) to dentfy the parameters of the above mentoned nonlnear system models, respectvely. And the smulaton results are also gven n Table, Table Table, respectvely. Table. The estmates of Eample. Parameters g a τ True values 6 4 MA 6.0004.0008.9979 A 5.979.00.8986 PSO 5.9607.8975 4.0069 GA 5.889.00 4.00 From the smulaton eperment results of the Eamples,, t can be seen that the estmates Parameters a a b b True values -0.9.0. 0.6 MA -0.8986.000.0 0.5990 A -0.8879 0.9769.899 0.5966 PSO -0.905.009.0 0.589 GA -0.90 0.8987.879 0.605 Parameters c r r r True values. 0.8 0.5 0. MA.979 0.80 0.5009 0.985 A.076 0.8094 0.4880 0.79 PSO.8 0.779 0.509 0.00 GA.0 0.7899 0.4887 0.054 To our best nowledge, the above mentoned nonlnear system models have a certan representaton n a varety of the nonlnear system model. Consequently, for most nonlnear system models, the proposed parameter dentfcaton method based on the modfed fsh swarm algorthm s effectve feasble from the above smulaton results. 6. Conclusons A parameter dentfcaton method for the nonlnear system model s proposed based on the modfed fsh swarm algorthm n ths paper. Moreover, compared wth other methods, the satsfactory dentfcaton results are gotten. In smulaton, the results show that the modfed fsh swarm algorthm has the advantages of mult-pont optmzaton, smple, easy so on. In partcular, due to t dose not depend on the model form n the search optmzaton process, t s wdely used n varous model parameter estmaton. Accordngly, t s shown that the presented method s vald reasonable. Acnowledgements Ths wor s supported by Natonal Natural Scence Foundaton of Chna under Grant 97
No. 677 9004, Specalzed Research Fund for the Doctoral Program of Hgher Educaton under Grant No. 0680008, the Scentfc Research Program Funded by Shaan Provncal Educaton Department under Grant No. 4JK58, the Proect Supported by Shaan Provncal Natural Scence Foundaton research of Chna under Grant No. 04JM85, the Doctoral Scentfc Research Start-up Funds of Teachers of X an Unversty of Technology of Chna under Grant No. 08-006 the Technology Proect of X an Cty of Chna under Grant No. CXY45(). References []. L. Lung, System dentfcaton: theory for the user, nd ed., Prentce Hall, Englewood Clffs, 999. []. J. Söberg, Q. Zhang, L. Lung, et al., Nonlnear blac-bo modelng n system dentfcaton a unfed overvew, Automatca, Vol., Issue, 995, pp. 69-74. []. E. W. Ba, Identfcaton of Systems wth hard nput nonlneartes, n Perspectves n Control, Ed. R. Moheman, Sprnger Verlag, New Yor, 00. [4]. N. R. Xu, W. Z. Song, A. B. Xa, System dentfcaton, Southeast Unversty Press, Nanng, 99. [5]. J. Roll, A. Nazn, L. Lung, Nonlnear system dentfcaton va drect weght optmzaton, Automatca, Vol. 4, Issue, 005, pp. 475-490. [6]. F. Gr, E. W. Ba, Bloc orented nonlnear system dentfcaton, Sprnger-Verlag, Berln, 00. [7]. A. Wlls, T. B. Schön, L. Lung, B. Nnness, Identfcaton of Hammersten-Wener models, Automatca, Vol. 49, Issue, 0, pp. 70-8. [8]. Y. Han, R. A. De Callafon, Hammersten system dentfcaton usng nuclear norm mnmzaton, Automatca, Vol. 48, Issue 9, 0, pp. 89-9. [9]. W. D. Chang, Nonlnear system dentfcaton control usng a real-coded genetc algorthm, Appled Mathematcal Modellng, Vol., Issue, 007, pp. 54-550. [0]. C. S. James, Identfcaton for systems wth bnary subsystems, IEEE Transactons on Automatc Control, Vol. 59, Issue, 04, pp, -7. []. X. L, Z. Shao, J. Qan, An optmzng method base on autonomous anmates: fsh swarm algorthm, Systems Engneerng Theory Practce, Vol., Issue, 00, pp. -8. []. X. L, Y. Xue, F. Lu, G. H. Tan, Parameter estmaton method based on artfcal fsh school algorthm, Journal of Shong Unversty, Vol. 4, Issue, 004, pp. 84-87. []. X. L, F. Lu, G. Tan, J. Qan, Applcatons of artfcal fsh school algorthm n combnatoral optmzaton problems, Journal of ShanDong Unversty, Vol. 4, Issue 5, 004, pp. 64-67. [4]. X. Chen, D. Sun, J. Wang, J. Lang, Tme seres forecastng based on novel support vector machne usng artfcal fsh swarm algorthm, n Proceedngs of the 4 th Internatonal Conference on Natural Computaton, Beng, Chna, 0- August 008, pp. 06-. [5]. Y. M. Cheng, L. Lang, S. C. Ch, W. B. We, Determnaton of the crtcal slp surface usng artfcal fsh swarms algorthm, Journal of Geotechncal Geoenvronmental Engneerng, Vol. 4, Issue, 008, pp. 44-5. [6]. J. Kennedy, R. C. Eberhart, Partcle swarm optmzaton, n Proceedngs of the IEEE Internatonal Conference on Neural Networs, Pscataway, New Jersey, 5-8 May 995, pp. 94-948. [7]. R. C. Eberhart, J. F. Kennedy, A new optmzer usng partcle swarm theory, n Proceedngs of the Internatonal Symposum on Mcromachne Human Scence, Toyo, Japan, 0- July 995, pp. 9-4. 04 Copyrght, Internatonal Frequency Sensor Assocaton (IA) Publshng, S. L. All rghts reserved. (http://www.sensorsportal.com) 98
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