Proceedngs o the 009 IEEE Internatonal Conerence on Systems, Man, and Cybernetcs San Antono, TX, USA - October 009 Delay onlnear System Predctve Control On MPSO+D Mn Han School o Electronc and Inormaton Engneerng Dalan Unversty o Technology Dalan, P. R. Chna mnhan@dlut.edu.cn Janchao Fan School o Electronc and Inormaton Engneerng Dalan Unversty o Technology Dalan, P. R. Chna chao@student.dlut.edu.cn Abstract Ths paper presents a novel dynamc neural network (D) predctve control strategy based on moded partcle swarm optmzaton (PSO) or long tme delay nonlnear process. The proposed dynamc structure could approxmate to the actual system model and obtan the pure delay tme exactly. An mproved verson o the orgnal PSO s put orward to tran the parameters o to enhance the convergence and accuracy. The eectveness o the proposed control scheme s demonstrated by smulaton as well as a test on an experment on the actual ph eutralzaton Process. Keywords delay system, model predctve control, dynamc, PSO I. ITRODUCTIO Most ndustral processes n realty are nonlnear and long tme delay. Proportonal Integral Dervatve (PID) controllers are the most popular ones n ndustry []. But the pure delay tme may cause the control sngle response to the output long tme ago, so that such controllers may produce large error or even be out o control []. T hus, n ths case, the Smth predctor s a well-known dead tme compensator or stable processes wth large tme delays. When the model s accurate, the closed-loop characterstc wll be delay ree, and so the Smth predctor structure has sgncantly acltated controller desgn n the conventonal eedback systems. However, the exact pure tme s hard to obtan. Another mportant predctve control strategy, Model Predctve Control (MPC), developed rapdly n the past two decades, has an ncreasng acceptance n engneerng eld recently [3]. A maor problem engneers have to ace that the desgn o a MPC system reles heavly upon an explct mathematcal model o the system under control, whch has been studed extensvely. However, the tradtonal approach s usually based on approxmate lnearzaton theory, whch s oten dcult to denty an accurate mathematcal system model and mposes serous restrctons on the structure o nonlnear system. Ths work s supported n part by supported by the proect (007AA04Z58) o the atonal Hgh Technology Research and Development Program o Chna (863 Program), the proect (60674073) o the atonal ature Scence Foundaton o Chna, the proect (006BAB4B05) o the atonal Key Technology R&D Program o Chna and the proect (006CB403405) o the atonal Basc Research Program o Chna (973 Program). Artcal neural networks (As), whch have been wdely studed and appled to varous research elds, have superorty as compared wth other conventonal modelng methods. The advantage o As s that t does not need any knowledge about the process, whch s a black box model. For nonlnear plants, the ablty o the MPC to make accurate predctons can be enhanced an s used to learn the plant dynamc nstead o standard nonlnear modelng technques. The based predctve control strateges have been ound to be eectve n controllng a wde class o nonlnear processes n the past [4]. In the predctve control (PC), the s used as the predcton model o the nonlnear plant and the system perormance s greatly dependent on the onlne optmzaton. Some network structures have been successully appled to the actual applcaton, such as eedorward neural networks, radal bass networks, and dynamc neural networks. Dynamc neural networks, due to the more complex tme sequence structure, could ully express the nonlnear relatonshp between the nput and output [5]. Furthermore, several algorthms are commonly mplemented n the tranng process o the, such as gradent descent and ewton methods. However, these gradent-based optmzaton methods usually all nto local optma, and also ths s a complex procedure or calculatng the Jacoban and Hessan matrx. Thereore, the convergence speed o neural networks s slow and senstve to the ntal parameter values [6]. A number o derent evolutonary algorthms have been proposed to mprove the capablty o the neural networks. In order to enhance the approxmaton capablty o the sngle neuron model, partcle swarm optmzaton s ntroduced to tran the model. PSO developed by Kennedy and Eberhart [7] n 995, s an evolutonary computaton technque or optmzng hard numercal unctons on metaphor o socal behavor o locks o brds and schools o sh. It s an evolutonary computaton technque based on swarm ntellgence, and s also the latest evolutonary optmzaton technology used or solvng a wde range o real world optmzaton problem as a substtute or genetc algorthm (GA) [8]. A swarm conssts o ndvduals, called partcles, whch change ther poston over tme. Each partcle represents a potental soluton to the problem. Due to the smple concept, easy mplementaton, and quck convergence, nowadays, PSO has ganed much attenton and wde applcatons n derent 978--444-794-9/09/$5.00 009 IEEE 449
elds. However, the conventonal PSO may easly get trapped n a local optmum when tacklng complex problems as well as other evolutonary methods [9]. Great eorts could be ound n the lterature. At present, u [0] ntroduces a bo-nspred mechansm to make the basc PSO more ntellgent by separatng the populaton n PSO nto two parts, a master swarm and several slave swarms. A hybrd PSO model has been reported by Lovberg [], n whch the velocty and poston update relatons employ the concepts o breedng and subpopulaton. However, due to the lmtaton o dynamc response capablty or common structure and the drawback o PSO algorthm lackng o enough ablty to sustanable development, a smple hybrd o these two optmzaton algorthms cannot obtan approvng generalzaton eect [,3]. Thereore, ths paper proposes an adaptve dynamc eedorward neural network based on moded partcle swarm optmzaton algorthm to the long tme delay nonlnear system predctve control. Between nput layer and rst hdden layer, and also the last hdden layer and output layer, the dynamc tme delay operators are adopted, whch could buld the relatonshp between the current output and the prevous tme nputs. Those are dynamc adusted wth the teratve system dentcaton process usng the PSO algorthm. The delay operators n the output part can accurately denty the pure tme delay, and correspondng operators n the nput part can enhance the s dynamc characters. Ater these operators n output part removed, the actual system model wthout any delay s bult exactly, so some classc predct control strateges, such as model predctve control, could acheve satsactory control eect. Consequently, the predctor and compensator n MPC are solved exactly by our moded structure. Otherwse, use the weght dgresson method to mprove the PSO algorthm s global search capablty. Then, the parameters n the PC are traned by moded PSO method, avodng the complexty o gradent calculaton and trappng n the local optmal ponts. That makes the neural networks ollow the system change as ast as possble and perorm realtme response. The remander o ths paper s organzed as ollows. Secton II presents the moded PSO algorthm (MPSO). Secton III descrbes the dynamc delay evolutonal eedorward neural networks. Furthermore, n Secton IV, we dscuss the combnaton o the above two moded parts or the nonlnear delay system control. Secton V hghlghts the potental o the proposed approach through expermental examples. Concludng remarks are presented n Secton VI. II. MODIFIED PSO ALGORITHM A. Standard PSO PSO s an evolutonary algorthm whch mantans a swarm o canddate solutons, reerred to as partcles. The best soluton each partcle has acheved so ar s P best, whch s denoted as P = ( p, p,, pd). The best poston P gbest among all the partcles n the populaton s represented P = ( p, p,, p ). PSO reles on the exchange o as g g g gd normaton between partcles, n other words ths s a hypothess that socal sharng o normaton among ndvduals oers an evolutonary advantage. PSO s smlar to other evolutonary algorthms n that the system s ntalzed wth a populaton o random solutons. The velocty or each partcle n D dmensonal problem space s dynamcally adusted accordng to the lyng experences o ts own and ts ellows. There are our man parameters n the whole PSO algorthm. The locaton o partcle or s represented as x = { x, x,, xd}, where xd [ xmn, d, x, d ], d =,, D. x mn,d and x,d are the lower and upper bounds or the dmenson d, respectvely. Let V = ( v, v,, vd) be the current velocty or partcle, whch s lmted to a mum veloctyv = ( v,,, v, D ).At each step, the velocty and poston o each partcle are updated toward ts P best and Pgbest locatons accordng to vd ( t+ ) = wvd t + crand [ pd t xd ] t + crand [ pgd ( t) pd ( t)] xt ( + ) = xt ( ) + vt ( + ) where w [0,] s the nerta weght, determnng how much o the prevous velocty o the partcle s preserved. And rand, rand [0,] denote two unorm random numbers samples,, c c are acceleraton constants. From the teratve, t s shown that the second part represents the prvate thnkng by tsel, and the thrd part s the socal part, whch represents the cooperaton among the ndvduals. In order to use the control theory to analyze the partcle swarm optmzaton algorthm, change the ormula to matrx ormat vt ( + ) vt ( ) ϕ t ϕ t p t At xt ( ) = xt ( ) + ϕ t ϕ t pg t + ω ϕ t where the varable At = ω ϕ( t), ϕ t = cr t, ϕ t = cr t, ϕ t = ϕ t + ϕ t. B. Moded Partcle Swarm Optmzaton (MPSO) The nertal weght w o the PSO algorthm coordnates the partcles global and local search capabltes. When the parameter w s larger, the change rate or each partcle s larger to keep the global search capablty, whereas ownng a better local approxmate capacty. There has been a lot o research n determnng the nerta weght. The lnear decreasng weght was presented by Sh [4]and descrbed as ( w wmn ) w= w ter (3) Maxstep where w mn = 0.4, w = 0.9, whch are commonly accepted mnmum and mum change range or nertal weght [0,]. ter and Maxstep denote the current teratve tme and the mum teratve tme, respectvely. Eberhart [5] proposed the PSO wth a random nerta weght actor, where the nerta weght w changes accordng to 440
w= 0.5 rand / (4) where rand s a unormly dstrbuted random number wthn the range [0,]. However, t s ound that the nerta weght n these moded methods could not combne wth the neural networks. In Equaton (3), the parameter w s n the process o the transton at the most teratve tmes. So the global and local search capablty o PSO algorthm s weakened. And also the Equaton (4) s random method, whch could not keep the contnual approxmate ablty and acheve the stable predctve control. Thereore, ths artcle adopts nonlnear ways to gradually reduce the weght o nerta, n accordance wth cosne curve. Furthermore, the system error shrnks wth the weght w, whch ensures that the neural networks contnuously approach the controlled system n the latter perod. The update ormula s shown as (5) w= wmn + ( w wmn ) + cos[( ter ) π / ( Maxstep )] (5) Wth the ncrease o varable ter and the decrease or system dentcaton error, nertal weght w dmnshes gradually. Thereore, the eedorward neural network wth moded PSO algorthm could hold local approxmate capacty n the later perod o system control process. The pseudocode o the whole moded partcle swarm optmzaton algorthm s shown as ollow: Start ntalze the populaton evaluate the tness: = ( x) update P and Pg whle ( termnaton condton = alse ) do update nerta weght w usng the ormula (5) or ( = to number o partcles ) or ( d = to number o dmensons ) calculate new velocty v d and update the poston x d usng the ormula ncrease d evaluate the tness: = ( x) update P and Pg ncrease end do end III. DYAMIC FEEDFORWARD EURAL ETWORKS Choose a mult-nput, sngle-output eedorward neural network as an example to llustrate proposed dynamc eedorward eural etworks (D), the structure s shown n Fg.. There s only one connecton between each o the two neurons. The delayed o dynamc operators and are accessoned n the rst hdden layer and nput layer, the last hdden layer and output one, respectvely. These two dynamc parameters are together adaptve adusted wth the connecton weghts w and the threshold b. Impled n the hdden layers between the nput and output layer relatonshps can be expressed, as (6) and (7), respectvely ( l ) ( l ) ( l ) net t = w O t + bl (6) = O t = [ net ] t (7) where net t and O t denote the nput and output o the th neural n the lth layer at tme t, respectvely. The weght connectng the th neuron n the lth layer to the th neuron n the (l-)th layer are represented by w. ote that vares l l rom to, vares rom to. bl denotes the threshold parameter o the th neural n the lth layer. [ ] s a nonlnear actvaton uncton, wrtten as cx e ( x) = K (8) cx + e b b( M ) u t ( M ) w z w z b M b yt b( M ) n u wn z n t Joned the dynamc delay operators o the neural network connecton, the descrpton o the rst hdden nodes to nput and output neurons can be expressed respectvely as net t = w O ( t τ ) + b (9) here τ and = ( M ) ( M) ( M ) ( M ) = ( τ ) + M = net t w O t b (0) τ denote the assocated delay connectng the hdden layers wth nput and output layers, respectvely. In the proposed dynamc eedorward neural networks structure as shown n Fg., consder the eect o nput u ( ) t to output yt ( ), whch s shown n ( M ) ( M ) yt = [ wpq ( ( ( w u ( t τ τ ) + b ) ) + b ] The parameters τ pq M and ( M ) Fg.. The structure o proposed dynamc neural networks. τ pq are adaptve adusted by MPSO w 44
algorthm untl reachng the mnmum value o tness uncton, and vary orm 0 toτ. The weght connectng the th neuron n the lth layer to the th neuron n the (l-)th layer are represented by w. Furthermore, b l denotes the threshold parameter o the th neural n the lth layer. As a result, each connecton between the neurons can be seen on a random tme delay so that the network has the tme sequence. The current moment output may be related to a ew moments o nputs, whch enhances the dynamc nature o D. Through the dentcaton o the controlled long tme-delay system, the pure tme delays are obtaned by the delay operators between hdden and output layers. Thereore, the proposed D could successully represent the nonlnear and delay characters o the system only wth ewer parameters. Ater tranng, the delay operators τ were removed, the exact controlled plant wthout any delay s obtaned, so predctve controller n MPC s obtaned exactly. Due to the precse system outputs are predcted, the satsactory control eect could be acheved. IV. MPC BASED O MPSO+D The MPSO method s adopted to adaptvely adust the value o the connecton weghts w, threshold b and delay operators n the D. Ths way avods the complex calculaton on repeated partal dervaton n error eedback gradent descent methods, and accelerates the pace o convergence to the global optmum. The varables wbτ,, consttute the locaton o the partcle k, as shown n Fg. ( M ) xk t w, t, w t b, t, bm t τ, t, τ t Fg.. The structure or the partcles. Then calculate the obectve uncton o the partcle tness, and update each partcle velocty, locaton and the optmal locaton o the groups. The obectve uncton ( x ) s dened based on users desred speccaton. Typcal output speccatons n the tme doman are peak over-shootng, rse tme, settlng tme, and steady-state error. The ntegral absolute error (IAE) perormance ndex descrbed n Equaton () s usually consdered as the obectve uncton, IAE = n e 0 t dt () = The purpose o MPC s to select sgnal ut ( ), such that the output o system yt ( ) s made as close as possble to be a expectaton set-pont rt ( ). A schematc o the MPC based on MPSO+D s llustrated n Fg. 3. The man controller adopts the classc PID methods, and the denter s our proposed MPSO+D ater the system dentcaton. On the other hand, the predctor s the same MPSO+D structure except the delay operators omtted between the last hdden layers and output layers, whch could predct the plant output wthout pure delay exactly. Hence, through the rst close-loop, the predctve sngle r ( t) s transerred to the PID Controller n tme. And then, the man controller reposes to change the sngle ut ( ) accordng to r. Otherwse, n the second close-loop, the dentcaton error e t s the derence between the actual plant output yt ( ) and the denter output y ( t). Then the error e t s transerred to PID controller to revse the delecton durng the dentcaton process. In addton, the eedback part o error e t could overcome the dsturbance n the place o controlled plant and the eects on the change o system model successully. V. THE SIMULATIO O PH EUTRALIZATIO PROCESS PH neutralzaton process wdely exsts n the chemcal ndustry, sewage treatment, botechnology, power generaton, and other ndustral processes. Acd-base neutralzaton process owns hgh nonlnear eatures, and other actors, such as the valve and sensors, results n ths process exstng long tme delay. The lterature [6] provdes Hammersten's mathematcal model to descrbe actual ph process, wrtten as (3) and (4) 3 vk ( ) = uk ( ).07 u( k) +.5 u( k) (3) yk ( ).588 yk ( ) + 0.597 yk ( ) = (4) 0.085 vk ( 0) + 0.07 vk ( ) + 0.597 vk ( ) Use the PID controller to control the above-mentoned system, and choose 800 group nput and output data as a teacher sngle to MPSO+D. Among these data, the rst 900 group sgnals are adopted to tran the neural networks, the rest part to test the generalzaton capablty. Fg. 4 depcts the results o the system dentcaton. And also the smulaton condton s shown n Table I. TABLE I SIMULATIO CODITIO ame Value structure -0- Functon x e x = D( ) x e + Ampltude D = 4 Iteratve Tmes 30000 Partcle umber 5 Accelerate Coecent c =, c = Dmenson 4 Research Range ( 5,5) n Maxmum velocty v = 0.5 Maxmum nerta weght w = 0.9 Mnmum nerta weght w mn = 0.4 Fg. 3. The system block dagram o model predctve control on MPSO+D..Fg. 4 shows the mproved algorthm can also acheve very good results to denty actual ndustral process wth the 44
nonlnear and long tme delay. The tranng error s 0.00000, the generalzaton error s 0.00044, and the consumng tme s 7.74 mnute. The moded could acheve a satsactory accuracy and better generalzaton capablty. What s more, the consumng tme s also greatly reduced, and the whole convergence process s speed up sgncantly. As shown n the Equaton (4), the delay tme s 0 sample perod. Ater the system dentcaton process, the delay operators between the last hdden and output layer are the nteger n the range o [8, ]. Thereore, our proposed MPSO+D structure could obtan the pure tme parameters. Equaton (5) 3 vk ( ) = uk ( ).07 u( k) +.5 u( k) yk ( ).588 yk ( ) + 0.597 yk ( ) = (5) 0.0585 vk ( 5) + 0.07 vk ( 6) + 0.597 vk ( 7) The perormance s shown n Fg. 6. It s ound that the novel model predctve control wth MPSO+D could get back to the stable state wth small transton tme compared wth the Smth method n the msmatch stuaton. Thus our proposed predctve control s less dependent on the system model and the robust character s better. Fg. 4. The system dentcaton results. In order to very the eectveness o our proposed predctve control strategy, three control methods are compared wth the example o PH neutralzaton process. The rst one s classc PID sngle-closed-loop control, the second s PID control wth the Smth predctor, and the last one s our proposed model predctve control wth MPSO+D. The nput sngle s the square wave sgnal, whose samplng perod s 0.s, and three thousand ponts n all. Fg. 5 presents the control results obtaned by these control strateges or square sgnal and t shows a good trackng result. Fg. 5. The comparson o control perormance eect wth three methods. From the gure, t s shown that our proposed control system could comes to a steady state rapdly wthout the any overshoot, and the perormance s very close to the Smth predctve control. However, the man advantage o our predctve controller s desgned n the case o system model unknown. Furthermore, n order to analyze the dsturbance stuaton, or the operaton process and envronment causng the change o system model, the experment on robust characters s desgned. At the 600 th step, both the model coecent and pure tme delay parameter have vared as the ollowng Fg. 6. The robust character analyss n the msmatch stuaton VI. COCLUSIO In ths paper, a moded PSO algorthm s ntroduced to tran the dynamc neural networks. Through the nerta weghts decreasng, the PSO method could be t or the neural network tranng process. What s more, the dynamc structure could represent the system nonlnear and the pure tme delay exactly, and overcomes the drawback o gradent descent way easly allng nto local optma ponts. The ntroduced dynamc delay operators make that the pure delay tme could be obtan precsely or the model predct control. The smulaton results show that the proposed model predct control wth MPSO+D s superor to other predctve control methods. Moreover, the method s o strong robustcty on background dsturbance and msmatch stuaton. REFERECES [] D. W. Clarke, C. Mohtad, and P. S. Tus, Generalzed predctve control Part I and Part II, Automatca, vol. 3, no., pp. 37-60, 987. [] Z. Song and A. Kusak, Optmzaton o Temporal Processes: A Model Predctve Control Approach, IEEE Transactons on Evolutonary Computaton, vol. 3, no., pp. 69-79, 009. [3] B. Potocnk, G. Musc, I. Skranc, and B. Zupancc, Model-based predctve control o hybrd systems: A probablstc neural-network approach to real-tme control, Journal o Intellgent & Robotc Systems, vol. 5,no. 7, pp. 45-63, 008. [4] X. J. Wu, X. J. Zhu, G. Y. Cao, and H. Y. Tu, Predctve control o SOFC based on a GA-RBF neural network model, Journal o Power Sources, vol. 79, no., pp. 3-39, 008. [5] M. Jall, S. Atashbar, S. Momenbellah, and F. H. Roudsar, eural networks as a tool or nonlnear predctve control: Applcaton to some benchmark systems, Internatonal Journal o Wavelets Multresoluton and Inormaton Processng, vol. 5,no., pp. 69-99, 007. 443
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