Modeling of a Class of Nonlinear Dynamic System

Transcription

1 Sensors & ransducers, Vol. 69, Issue 4, Aprl 4, pp Sensors & ransducers 4 by IFSA Publshng, S. L. Modelng of a Class of Nonlnear Dynamc System Guangjun LIU, Xaopng XU, Feng WANG School of Scences, X an Unversty of echnology, X an, 754, Chna School of Mathematcs and Statstcs, X an Jaotong Unversty, X an, 749, Chna el.: , fax: E-mal: gjluxaut@6.com Receved: 3 December 3 Accepted: 7 March 4 Publshed: 3 Aprl 4 Abstract: A modelng approach for a nd of the nonlnear dynamc system s studed n ths paper. At frst, suppose that the orgnal nonlnear dynamc system s expressed by the nonlnear Hammersten model, the transfer functon of the model can then be changed nto a lnear form by usng the functon expanson, accordngly, an ntermedate model s generated. Secondly, the parameters of the ntermedate model are attaned based on a fsh swarm optmzaton algorthm. hrdly, through the parameters relatonshps of the ntermedate model and the Hammersten model, we derve the parameters of the Hammersten model; consequently, the modelng of the orgnal nonlnear dynamc system s acheved. Fnally, the feasblty of the presented modelng method for the nonlnear dynamc system s llustrated by the numercal smulaton experments. Copyrght 4 IFSA Publshng, S. L. Keywords: Nonlnear system, Dynamc system, Hammersten model, Modelng, Evolutonary algorthm.. Introducton Mathematcal models are the bass of all control problems; the movement law of thngs descrbed by equatons s the mathematcal model. Consequently, system modelng s a very actve branch of the cybernetcs at present. he realstc systems are almost nonlnear systems, thus t s mportant to study the modelng of the nonlnear systems []. Recently, the modelng methods of the nonlnear systems have been pad attenton by many scholars [-5]. Moreover, the modelng of the nonlnear dynamc system was one of the man problems for the modelng [4, 5]. Due to lac of descrbng unform mathematcal model for the dssmlar nonlnear systems, the modelng approach of the nonlnear systems s often amed at the specfc system [3]. In practcal applcaton, many nonlnear dynamc systems are descrbed by a nonlnear statc subunt followed by a lnear dynamc subunt [3]. he man purpose of ths paper s to nvestgate the modelng method for a nd of nonlnear dynamc system. Frstly, the model of the orgnal nonlnear dynamc system s supposed to be expressed by the Hammersten model, whch s represented by a nonlnear statc subunt followed by a lnear dynamc subunt. hrough the functon expanson, the nonlnear transfer functon of the Hammersten model can be changed nto a lnear form, thus generatng an ntermedate model. hen, the fsh swarm algorthm s used to obtan the parameters of the ntermedate model. Next, through the parameter relatonshps of the ntermedate model and the Hammersten model, the modelng of the nonlnear dynamc system s realzed. Fnally, smulaton results show the ratonalty and feasblty of the presented method. he rest of the paper s organzed as follows. he next secton ntroduces the fsh swarm algorthm. In secton 3, we descrbe the process of the modelng for Artcle number P_4 53

2 Sensors & ransducers, Vol. 69, Issue 4, Aprl 4, pp a nd of nonlnear dynamc system. Secton 4 presents the numercal smulatons to llustrate the feasblty of the presented method. Secton 5 summarzes the contrbuton of ths paper and conclusons.. Fsh Swarm Algorthm he fsh swarm algorthm (FSA) s a new populaton-based/swarm ntellgent evolutonary computaton technque proposed by L et al. [6] that was nspred by the natural schoolng behavor of fsh. FSA presents a strong ablty to avod local mnmums n order to acheve global optmzaton. It has been proofed n functon optmzaton [6], combnatoral optmzaton [7], least squares support vector machne [8] and geotechncal engneerng [9] problems, among others. FSA mtates three typcal behavors, defned to nclude searchng for food, swarmng n response to a threat, and followng to ncrease the chance of achevng a successful result. hree major parameters nvolved n FSA nclude vsual dstance (vsual), maxmum step length (step), and a crowd factor. FSA effectveness seems prmarly nfluenced by the former two (vsual and step). A fsh s denoted by ts D-dmensonal poston X (x, x,, x,, x D ), and food satsfacton for the fsh s represented as FS. hs paper targets FS mnmzaton. he relatonshp between two fsh s denoted by ther Eucldean dstance d j X X j. Another parameters nclude: vsual (representng the vsual dstances of fsh), step (maxmum step length), and 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. hese nclude: a) Searchng behavor Searchng s a basc bologcal behavor adopted by fsh loong for food. It s based on a random search, wth a tendency toward food concentraton. It s expressed mathematcally as: satsfyng food ntae needs, entertanng swarm members and attractng new swarm members. Mathematcally, x + x + R(S) ( x c - x ) / X c -X, FS c < FS and (n s / n) < δ. A fsh located at X has neghbors wthn ts vsual. X c dentfes the center poston of those neghbors and 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., FS c < FS ), and f the swarm (X c ) s not overly crowded (n s /n < δ), the fsh wll move from X to next X +, toward X c. Here, n s represents number of ndvduals wthn the X c s vsual. Swarmng behavor s executed for a fsh based on ts assocated X c ; otherwse, searchng behavor guarantees a next poston for the fsh. c) Followng behavor When a fsh locates food, neghborng ndvduals follow. Mathematcally, x + x + R(S) ( x mn - x ) / X mn -X, FS mn < FS and (n f / n) < δ. Wthn a fsh s vsual, certan fsh wll be perceved as fndng a greater amount of food than others, and ths fsh wll naturally try to follow the best one (X mn ) n order to ncrease satsfacton (.e., gan relatvely more food [FS mn < FS ] and 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 next poston. Besdes, FSA should provde a bulletn that records the optmal state and current performance of fsh durng teratons. o execute the aforementoned behavors, FSA mechansm must follow the process shown n Fg.. x x x FS < FS, j + x + R( S), X j X j () x + x + R( S), () where x represents the th element of fsh poston X. We randomly select for fsh X a new poston X j wthn ts vsual. If the correspondng FS j s satsfed, Eq. () s then employed at the next poston X +. If FS j s not satsfed after try number trals, a random poston wthn the step range wll be drectly adopted as Eq. (). In the above equatons, R(S) and R(S) represent random varables wthn [, step] and [-step, step], respectvely. b) Swarmng behavor Fsh assemble n several swarms to mnmze danger. Objectves common to all swarms nclude Fg.. Flow chart of FSA. 3. Modelng of Nonlnear Dynamc System Suppose that the orgnal nonlnear dynamc system s expressed by the nonlnear Hammersten model as shown n Fg.. Let u(,y( and v( be a measurement nput, the system output and a nose respectvely, x( be an ntermedate nput sgnal. 54

3 Sensors & ransducers, Vol. 69, Issue 4, Aprl 4, pp v( u ( x ( w ( y( f ( u( ) G(z) Fg.. Structure of Hammersten model. he nonlnear statc gan can be approxmately expressed by the followng p-order polynomal. x( f ( u( ) r u ( p R U (. (3) he followng form of the transfer functon can express the lnear dynamc system. B( z ) b + b z + + b z G( z), (4) A( z ) + a z + + a m m n nz where m, n are the polynomal s order, and m n generally. he lnear transfer functon of Eq. (4) can be descrbed by the followng dfference equaton. A ( z ) y( B( z ) x( + e(, (5) where e(a(z - )v( can be nterpreted as a random fttng error. It follows from Eq. (3) and (5) that A( z ) y( m p j j z u ( e(, (6) j + where j r b j,,,, p, j,,, m. From Eq. (6), we have the followng formula. y( n m a y( t ) u + j ( t j) + e(, p j (7) And Eq. (7) can be expressed by followng vectors form. y ( θϕ ( + e(, (8) where θ (-a, -a,, -a n, a, a,, a m, a, a,, a m,, a p, a p,, a pm ), φ( (y(t-), y(t-),, y(t-n), u(, u(t-),, u(t-m), u (, u (t-),, u (tm),, u p (, u p (t-),, u p (t-m)). Obvously, y( s a functon of the power of u(, and s called an ntermedate model throughout ths paper. Here θ s a parameter vector of ntermedate model. It s assumed that the assessment value of the parameter vector θ n the ntermedate model s hus the devaton of the assessment can be judged by the followng crteron functon []. J h + h [ y( ) yˆ( )], () where h s the wndow wdth of dentfcaton, ŷ() are the nput values of the obtaned assessment model. One can solve the mnmum of Eq. () and obtan the correspondng parameter vector θ of the ntermedate model usng the above mentoned FSA, because solvng the mnmum of Eq. () s an optmzaton problem. he parameters of Hammersten model are obtaned by these ntermedate parameters as follows. In order to smplfy problem, wthout loss of generalty, suppose that the fnal gan of the lnear dynamc subsystem of Hammersten model s, and then we have b + b + + bm G( ) + a + a + + a n. () hereby, Eq. () can be expressed by the followng matrx form. C B C A, () where C (,,, ), A (, a,, a n ), B (b, b,, b m ). hus t follows from j r b j that: where R (r, r,, r p ), and Η p H RB, (3) p m m pm, (4) Rght-multplyng C n each sde of Eq. (3) gves HC RB C. (5) Consequently, t follows from Eq. (5), B CC B and Eq. () that HC R. (6) C A ransposng each sde of Eq. (3) yelds H BR. (7) ˆ θ ( aˆ ˆ ˆ ˆ ˆ ˆ ˆ, a,, a n,,,, m,,. ˆ,, ˆ,, ˆ, ˆ,, ˆ ) m p p pm (9) Rght-multplyng C n each sde of Eq. (7) gves H C BRC. (8) 55

4 Sensors & ransducers, Vol. 69, Issue 4, Aprl 4, pp hereby, by Eq. (8), R CC R and Eq. (6), we have C AH C B. (9) C HC o sum up, the man steps of the proposed dentfcaton algorthm can be brefly summarzed as follows. Step. Solve Eq. () to get the parameter vector θ of the ntermedate model usng the FSA, Step. Assessments of A and H can be realzed accordng to Eqs. () and (3). Step 3. Assessments of R and B are fulflled from Eqs. (6) and (9). Sequentally, the parameter estmates of the Hammersten model are obtaned. hat s to say, the modelng for the nonlnear dynamc system s completed. 5. Smulatons o demonstrate the valdty of the presented method for a nonlnear dynamc system, now, we consder Hammersten model wth -order nonlnear subsystem and 3-order lnear dynamc subsystem. he mathematcal model s H, A ( ). Subsequently, the parameter vector R can be obtaned on the bass of Eq. (6). R ( ). Fnally, accordng to Eq. (8) the parameter vector B gves B ( ). So far, we have mplemented the estmaton of Hammersten model based on the proposed method. he outputs of the estmated model usng the proposed method are plotted aganst tme n Fg. 3. Note that Fg. 3 also ncludes the response of the smulated nonlnear dynamc system. And the correspondng estmaton error s shown n Fg Respones of smulated system Estmated result of system x ( u( +.7u (. ().4.6z +.9z G ( z). 3 + z +.3z.5z () Output.5 Because an mpulse sgnal s often appled to calbratng experments of dynamc system, ths example also adopts t as an nput sgnal, u(, and the response of the smulated nonlnear dynamc system s plotted aganst tme, as shown n Fg. 3 and Fg. 5. he set of tranng samples s consttuted accordng to the nput and output response sgnal. he ntermedate parameter vector, θ, could be solved by the FSA. In ths example, the ntal values of the smulated model are y(, t,,3. he parameter values of the modelng algorthm are set as follows. Let wndow wdth of estmaton: h5, some parameters n FSA are set: the number of artfcal fsh s 3, the maxmum teratve steps are 5, step s 4.5, vsual s 5, try number s, δ s.68, and system model parameters ntal values are all pced out randomly from [-.7,.]. he parameter vector θ of the ntermedate model can be obtaned usng the FSA as follows. θ (.9978,.367,.4955,.37,.4995,..799,.7,.3478,.5563) hen, the followng parameter matrces H, A are formed accordng to Eqs. (3) and (). Error me Fg. 3. Comparson the response of smulated system wth those for estmated models. 6 x me Fg. 4. Error curve. Obvously, n the absence of nose, the estmates mply that the result of modelng s very good. he curve of smulated nonlnear dynamc response and that of the estmated model output nearly concde, as shown n Fg. 3 and Fg. 4. o further llustrate the ant-dsturbance ablty of the proposed method, we 56

5 Sensors & ransducers, Vol. 69, Issue 4, Aprl 4, pp now compare the result of the estmated n the presence of nose. A Gaussan nose wth zero mean and devaton, σ.4, s added to the nonlnear dynamc response sgnal y(. he Hammersten model s estmated by the same method and parameter values as mentoned above, and the followng results are obtaned. θ (.84,.96,.499,.959,.487,,.7754,.37,.357,.5598) H, A ( ), R ( ), B ( ). he parameter values of the Hammersten model can stll be well estmated usng the proposed algorthm. he outputs of the estmated model are plotted aganst tme n Fg. 5. Note that the Fg. 5 also ncludes the response of the smulated nonlnear dynamc system. And the correspondng estmaton error s shown n Fg. 6. Fg. 5 and Fg. 6 show that the curve of the estmated model s n agreement wth that of the smulated nonlnear dynamc response. hereby, the proposed modelng method has a strong ant-dsturbance ablty. Output Error Respones of smulated system Estmated result of system me Fg. 5. Comparson the response of smulated wth nose wth those for the estmated models me Fg. 6. Error curve. And the correspondng estmaton error s shown n Fg. 6. Fg. 5 and Fg. 6 show that the curve of the estmated model s n agreement wth that of the smulated nonlnear dynamc response. hereby, the proposed modelng method has a strong antdsturbance ablty. 6. Conclusons hs paper presents a method of modelng for a nd of nonlnear dynamc system, whch s expressed by the Hammersten model. he approach s as follows. he Hammersten model s castled to be a lnear form based the functon expanson. hen, the parameters of the ntermedate model are solved usng the FSA. he relatonshp of the parameters of the ntermedate model and those of the Hammersten model s formulated to realze modelng for nonlnear dynamc systems. he results of the numercal smulatons have llustrated that the proposed method s feasble. Acnowledgements hs wor s supported by the Scentfc Research Program Funded by Shaanx Provncal Educaton Department (Program No. 3JK698), the Doctoral Scentfc Research Start-up Funds of eachers of X an Unversty of echnology of Chna (Grant No. 8-6), the Project of Shaanx Provncal Natural Scence Foundaton of Chna, and echnology Project of X an Cty of Chna (Grant No. CXY345(3)). References []. F. Dng, System dentfcaton. Part B: basc models for system descrpton, Journal of Nanjng Unversty of Informaton Scence and echnology: Natural Scence Edton, Vol. 3, Issue,, pp []. J. Sjöberg, Q. Zhang, L. Ljung, et al., Nonlnear blac-box modelng n system dentfcaton a unfed overvew, Automatca, Vol. 3, Issue, 995, pp [3]. M. Haastrup, M. Hansen, M. K. Ebbesen, Ø. Mourtsen, Modelng and parameter dentfcaton of deflectons n planetary stage of wnd turbne gearbox, Modelng, Identfcaton and Control, Vol. 33, Issue,, pp. -. [4]. J. Sjöberg, J. Schouens, Intalzng Wenerhammers-ten models based on parttonng of the best lnear approxmaton, Automatca, Vol. 48, Issue,, pp [5]. J. Bentsman, M. Mller, Y. Rubnovch, Dynamcal systems wth actve sngulartes: nput/state/output modelng and control, Automatca, Vol. 44, Issue 7, 8, pp [6]. J. Wang, C. Cheng, A fast multplayer neural networ tranng algorthm based on the layer-by-layer optmzng procedures, IEEE ransacton on Neural Networs, Vol. 7, Issue 4, 996, pp

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