Identification of a Nonlinear System by Determining of Fuzzy Rules

Transcription

1 Identfcaton of a Nonlnear System by Determnng of Fuzzy Rules Hodjatollah Hamd* Department of Industral Engneerng, K. N. Toos Unversty of Technology, Tehran, Iran h_hamd@kntu.ac.r Atefeh Darae Department of Industral Engneerng, K. N. Toos Unversty of Technology, Tehran, Iran adarae@mal.kntu.ac.r Receved: /Apr/06 Revsed: 9/Jul/06 Accepted: 0/Aug/06 Abstract In ths artcle the hybrd optmzaton algorthm of dfferental evoluton and partcle swarm s ntroduced for desgnng the fuzzy rule base of a fuzzy controller. For a specfc number of rules, a hybrd algorthm for optmzng all open parameters was used to reach maxmum accuracy n tranng. The consdered hybrd computatonal approach ncludes: opposton-based dfferental evoluton algorthm and partcle swarm optmzaton algorthm. To tran a fuzzy system hch s employed for dentfcaton of a nonlnear system, the results show that the proposed hybrd algorthm approach demonstrates a better dentfcaton accuracy compared to other educatonal approaches n dentfcaton of the nonlnear system model. The example used n ths artcle s the Mackey-lass Chaotc System on whch the proposed method s fnally appled. Keywords: System Identfcaton; Combned Tranng; Fuzzy Rules; Database Desgn.. Introducton System dentfcaton s extensvely used n a number of programs ncludng control systems [], communcatons [], sgnal processng [3], controllng chemcal processes [4], bologcal processes [5] and etc. to be exact all problems of the real world are nonlnear per se. Anyhow, we face less computatonal problems n dentfyng a lnear system and normally we are not faced wth smple problems n dentfyng nonlnear systems. In [6] partcle swarm optmzaton wth dfferent partcle length s proposed for producton of structure and parameters of fuzzy rule database. In [7], the contnuous verson of ant colony optmzaton s used for desgnng of fuzzy rules database. In ths work the onlne method s used for determnng the number of fuzzy rules and all open parameters n each fuzzy rule n the contnuous space was contnually optmzed by ant colony optmzaton algorthm. In [8], system tranng s presented usng two-step swarm ntellgence algorthm. Ths algorthm ncludes two steps. In the frst part structure and ntal parameter dentfcaton s carred out usng onlne clusterng of ant colony optmzaton algorthm n dsrupted space. In the second part partcle swarm optmzaton s used for greater optmzaton of all open parameters n the contnuous space. Fuzzy systems are sutable for complex systems modelng, due to the good feature of general approxmaton especally for systems n whch mathematcal descrpton s dffcult. It s proven that any contnuous functon can approxmate a logcal degree of accuracy usng a fuzzy system whch s traned by metaheurstc algorthms. Ths approxmaton functon can act as a model for a number of functonal complex systems. Juang et al, 04, have shown that fuzzy systems that are traned by algorthm can be effectvely used n dentfcaton of nonlnear models. Recently, the dfferental evoluton algorthm (DE s consdered as a modern technque of evoluton calculatons [9,0] that are used for optmzaton ssues. DE s preferred over other evoluton methods lke genetc algorthm (A [,] and partcle swarm optmzaton (PSO and ths s due to ts notable characterstcs lke smple concept, easy executon and rapd convergence [3,4]. enerally all populaton-based optmzaton algorthms whch also nclude DE suffer the long calculaton perod due to ther evolutonary-accdental nature. The concept of opposton-based learnng (OBL s ntroduced by Tzhoosh [5]. In ths artcle, ths concept s used for acceleraton of learnng n fuzzy systems. The man dea n OBL concept s smultaneous consderaton of an estmate and ts correspondng opposng estmate. OBL leads to achevement of a better estmaton and acceleraton the rate of DE convergences. PSO s an algorthm wth local search pattern and can be used to fne-tune the present results and faster access to global mnmum. Therefore the proposed method n ths artcle s called hybrd opposton-based dfferental evoluton wth partcle swarm optmzaton (HODEPSO. ODE utlzes opposng numbers durng the start of populaton and also for producton of new populaton durng the evoluton process. Here, opposng numbers are used to accelerate the rate of convergences of DE optmzng algorthm. Pure random samplng or selecton of solutons from data * Correspondng Author

2 6 Hamd & Darae, Identfcaton of a Nonlnear System by Determnng of Fuzzy Rules populaton provdes for a chance to vst or even nspecton of undscovered regons of the search space. It has been proven that the probablty of ths ncdent s less for opposng numbers than purely random numbers. In fact mathematcal proof has been used to show that the probablty of opposng numbers beng closer to desred solutons s hgher than completely pure numbers [6-9]. In [7], the beneft of opposng numbers s nvestgated by replacng them wth random numbers and ths method has been utlzed for ntalzng the populaton and generaton skppng for dfferent DE versons. Ths artcle presents a new educatonal sample of fuzzy systems whch are combned wth meta-heurstc evolutonary algorthm, meanng: Use of Opposton-Based Learnng concept as smultaneous consderng of an estmate and ts opposng correspondng estmate whch would lead to better estmaton and acceleraton of the rate of convergence of dfferental evoluton algorthm (DE. Combnaton of ODE and PSO to prevent the probablty of gettng caught up n local optmum and qucker and more accurate achevement of general optmum. Performance of the traned fuzzy system usng HODEPSO s shown by comparng the results of some of the present methods n the un-lnear system dentfcaton. Results of stmulaton, shows the sutable performance of the proposed method compared to other methods of dentfcaton.. Meta- Heurstc Optmzaton (MHO Algorthms Optmzaton methods are search methods that am at fndng answers to the optmzaton problem so that the evaluated quantty s optmzed. Accordng to evdence and records of results, the best qualty and tme opposton for fuzzy system optmzaton s provded usng meta-heurstc algorthms.. Dfferental Evoluton (DE Algorthm Dfferental evoluton algorthm (DE s one of the effectve search-based methods [0-33]. Lke other evoluton algorthms, ths one also starts by ntalzng a populaton. Then through mplementaton of agents lke combnaton, mutaton and generaton convergence, the new-born s formed and n the next step whch s called selecton, new born generaton s compared to parent generaton to determne the rate of apttude whch s evaluated by the goal functon. Then the best members enter the next round as the next generaton. Ths trend contnues untl desred results are reached. Dfferent levels of ths algorthm are stated here n sequence. Populaton Intalzaton: The number of varables n ths algorthm are shown wth D. each of these varables hold a hgh and low lmt. Intal populaton wth the sze of n D s randomly formed accordng to equaton (. o mn round (.( max mn (,,,..., N P Where s a random number n the (0,] doman, max and mn are the hgh and low lmts of the varables and s the number of members. Mutaton and Intersecton: n ths algorthm fve strateges can be utlzed for combnaton and producton of new-borns [8]. In ths artcle best person-random person- random person s used for mutaton as follows: Z, r3, best, r 4, F.( r, r, Where F s called standard factor, s are randomly selected members and best s the best member of the present populaton. For every varable of each member of the populaton a random number, K, n the [, D] doman and a random number, u, n the [0, ] doman s selected. Intersecton s carred out accordng to the equaton below: f u CR or j k then Z F( else Z, j r, j r3, j r, j, j, j Where j s the number of any varable from th member of the populaton and CR s the ntersecton constant and s chosen as a number between 0 and. Estmaton and selecton: at ths stage the new-borns and parents are valuated accordng to the goal functon and f the newborn has a hgher value than the parent, t replaces the parent. Otherwse the parent moves on to the next level wth the next generaton. ( (3 Z, g arg max( f ( z, g, z, g (4 In ths equaton g stands for generaton, s the new generaton populaton (new-borns and s the prevous generaton populaton (parents. F s the goal functon of the problem. Repetton: repeatng steps and 3 untl maxmum repetton or the whole populaton convergence s reached.. Opposton-Based Dfferentaton Evoluton (ODE Algorthm In optmzaton approaches of evoluton algorthm, a unfed random guess for the ntal populaton s consdered. In each generaton the goal ncludes movement towards the desred soluton and the research trend ends when some of the pre-determned crteron are satsfactory. Calculaton tme usually depends on ntal guess, meanng that the greater the dstance between ntal guess and desred soluton, the more tme t takes to reach the end and vce versa. Opposton-based learnng ncreases the chance to start wth a better ntal populaton through revson of opposng solutons.

3 Journal of Informaton Systems and Telecommuncaton, Vol. 4, No. 4, October-December 06 7 Smlar approaches to ths can be used not only n ntal solutons but also utlzed contnually n the present populaton for any soluton [9]. V,t w. v c. r.(, t best c. r.( P P, t best P, t (5.. Defnton of Opposng Number Suppose s a real number. The opposng number s whch s defned by. Defnton of opposng pont: suppose s a pont n a D-dmensonal space n whch and. Opposng pont s: ~ p ( ~ x, ~ x,..., ~ x where ~ x a b x d.. Opposton-Based Optmzaton (OBO Suppose s a pont n a d- dmensonal space, meanng suppose an electve soluton. F(0 s a proporton functon whch s used to measure the proporton of selectons. Accordng to defnton of opposng ponts, opposes. Now, f, then p can be replaced by otherwse we contnue wth p. therefore the pont s evaluated smultaneously wth ts opposng pont so that we contnue the algorthm wth the more sutable ones..3 Partcle Swarm Optmzaton (PSO Algorthm Partcle Swarm Optmzaton Algorthm (PSO works accordng to the socal behavour of brds [0]. For better understandng of ths technque, consder the below scenaro: a flock of brds are randomly lookng for food n a specfc regon. There s only one pece of food n ths regon whch the brds are not aware of but are aware of ther dstance wth the food all the tme. At ths state, a sutable strategy for fnng the exact locaton of food s followng the brd that s closer to the food. Actually PSO has been nspred by such a scenaro too and presents a soluton for optmzaton problems n PSO each brd s a soluton to the problem. All the present responses have a ftness value whch s calculated by the defned ftness functon for the problem. The am of ths technque s fndng a locaton wth the best ftness value n the problem settng. Ths ftness value has a drect effect on the drecton and speed of these brds movement (solutons to the problem towards the locaton of the food (optmal response. PSO starts wth a number of ntal response (partcles and looks for optmal response by movng these responses n contnuous repettons. In every repetton two values are determned: P Best and Best. P Best : Locaton of the best P Best ftness value where each [artcle has reached n ts movement, Best : Locaton of the best partcle ftness n the present populaton. After the above values are calculated, the partcles speed of movement s calculated by equaton (4 and each partcle s next locaton s calculated by equaton (5. P P (6 t t v t In these equatons r and r values are random numbers between zero and one and c and c coeffcents whch are called learnng coeffcents are usually equalled to two ntalzatons. In every repetton of algorthm, the speed of partcle movement (rate of change for each partcle n every dmenson can be lmted wth a pre-determned V max value. At ths state f the speed of each partcle n each dmenson exceeds ths lmt, we replace t by V max. 3. Hybrd Opposton-Based Dfferental Evoluton and Partcle Swarm Optmzaton Algorthms In ths artcle, the hybrd algorthm of oppostonbased dfferental evoluton and partcle swarm optmzaton (HODEPSO s effectvely developed. The exact detals of steps n hybrd algorthm of oppostonbased dfferental evoluton and partcle swarm optmzaton s explaned below: Step : Random populaton ntalzaton and by consderng smultaneous aussan of opposng ntal values. Step : enforcement of opposton-based dfferental evoluton algorthm agents on the ntal populaton. Step 3: Evaluaton of the cost functon (whch s as RMSE n the solved example n ths artcle for each partcle and updatng P Best and Best. Step 4: Selecton of parents and aussan ther opposton and enforcement of opposton-based dfferental evoluton algorthm agents on them. Step 5: Evaluaton of cost functon for the new-borns and updatng P Best and Best partcle speed. Step 6: After selecton of new-borns from the elected parents, the survvor selecton mechansm s performed. Step 7: updatng partcle speed and status usng equatons (5 and (6. Step 8: Evaluaton of cost functon for each partcle and updatng P Best and Best. Step 9: If the condtons for endng s establshed, hybrd algorthm can end otherwse go to step Fuzzy System In ths secton, the desgn of the hybrd algorthmbased fuzzy system s descrbed. The fuzzy system used f of TSK, zero-degree knd n whch the th fuzzy rule s specfed wth R and descrbed as follows: R : If x s A ( x and x s A ( x then y s B (7 Where R s the th fuzzy rule, x j s the nput varable, y the output varable, s the fuzzy set and B s a

4 8 Hamd & Darae, Identfcaton of a Nonlnear System by Determnng of Fuzzy Rules certan value. The fuzzy set s a aussan membershp functon descrbed by the equaton below: ( { ( } (8 Where m s the center and s the wdth of the aussan membershp functon. For x and x nputs, the merdan or effectve weght w s calculated as follows: If the fuzzy system has r rules, the fzzy system output s calculated wth the defuzzcaton weghted average as follows: (9 (0 The status of each partcle s stated wth the vector below n the search space: P [ m,, m,, a, m,, m,, a ] ( Where a s the tally value n each fuzzy law. For example f the fuzzy system has n nput varables and r s the rule, the number of vector member, p (number of optmzng varables would equal. 5. Results of Smulaton In ths example the desgned fuzzy system s used to predct future values of Mackey-lass chaotc tme seres. Ths tme seres s produced usng the Mackey-lass delay dfferental equaton as below: 0.x( t x ( t 0.x( t ( 0 x ( t The ssue of predctng tme seres based on Mackey- lass dfferental equaton s a famous crteron for comparson of capactes of dfferent fuzzy models. 000 pars of nput-output data were extracted from Mackey- lass chaotc tme seres. The frst 500 pars were used for tranngng of the fuzzy system whle the remanng 500 were used as test data to evaluate the performance of the fuzzy model n predcton. Input x( t 30 Input x( t 8 Input 3 x( t Input 4 x( t (3 To evaluate the performance of the desgned fuzzy model RMSE was used whch s defned as follows: RMSE n n k ( y k yˆ k (4 Where n s the number of data, y k s the real output and s the fuzzy model output. To show the effcency of hybrd algorthm compared to algorthms of opposton-based dfferental evoluton and partcle swarm optmzaton, each algorthm was used ndvdually. The results of the use of these algorthms are shown n table ( wth an average of 50 tmes use. Accordng to the results, the desgned fuzzy system wth the proposed method, n addton to smplcty (reducton n the number of fuzzy rules shows a better performance compared to fuzzy models presented n [] and [] and has acheved a lower RMSE compared to those. log RMSE PSO ODE HODEPSO Iteraton Fg.. RMSE values acheved n every repetton by PSO, ODE and HODEPSO. Accordng to Fg. t s observed that the hybrd algorthm converges more quckly to the optmal soluton and has better performance compared to ndvdual ODE and PSO algorthms. The same example s studed n references [] and []. Comparson of the results of these methods and the proposed method are shown n table (. Fg.. Comparson of real output and fuzzy model output for tran data

5 Journal of Informaton Systems and Telecommuncaton, Vol. 4, No. 4, October-December Concluson Fg. 3. comparson of real output and fuzzy model output for test data Table : Comparson of results of dfferent methods Method No. of rules RMSE trang RMSE Test PSO 0.00E-3.300E-3 ODE E E-5 HODEPO E E-6 [] [] [] In ths artcle the hybrd optmzaton algorthm of dfferental evoluton and partcle swarm s ntroduced for desgnng the fuzzy rule base of a fuzzy controller. For a specfc number of rules, a hybrd algorthm for optmzng all open parameters was used to reach maxmum accuracy n tranng. The am of usng ths algorthm was to set the parameters of the rule base n the zero-degree fuzzy system (TSK n order to mnmze the performance ndex (Root Mean Square Error (RMSE.usng the mentoned algorthm, the tme-consumng process of parameter adjustment became a smple and quck task. The results show the sutable performance of the proposed model compared to other methods. References [] S. Chen, S.A. Bllngs, Representaton of non-lnear systems: the NARMA model, Int. J. Control 49 ( [] H. Hujberts, H. Njmejer, R. Wllems, System dentfcaton n communcaton wth chaotc systems, IEEE Trans. Crcuts Syst. I 47 (6, 000, pp [3] M. Adjrad, A. Belouchran, Estmaton of mult component polynomal phase sgnals mpngng on a multsensor array usng state-space modelng, IEEE Trans. Sgnal Process. 55 (, 007.pp [4] K. Watanbe, I. Matsuura, M. Abe, M. Kebota, D.M. Hmelblau, Incpent fault dagnoss of chemcal processes va artfcal neural networks, AICHE J. 35 ( ( [5] Y. e, B. uo, L. u, J. L, P. Stoca, Multstatc adaptve mcrowave magng for early breast cancer detecton, IEEE Trans. Bomed. Eng. 53 (8, 006, pp [6] D. Chen, J. Wang, F. Zou, H. Zhang, W. Hou, Lngustc fuzzy model dentfcaton based on PSO wth dfferent length of partcles, Appled Soft Computng, pp , 0. [7] C. F. Juang, and P. H. Chang, Desgnng Fuzzy-Rule-Based Systems Usng Contnuous Ant-Colony Optmzaton, IEEE Trans on Fuzzy Syst., vol. 8, no., pp , Feb 00. [8] C. F. Juang, C. Lo, Zero-order TSK-type fuzzy system learnng usng a two-phase swarm ntellgence algorthm, Fuzzy Sets and Systems 59, 90 96, 008. [9] R. Storn, System desgn by constrant adaptaton and dfferental evoluton, IEEE Trans. Evol. Comput. 3 ( [0] J. Ilonen, J.K. Kamaranen, J. Lampnen, Dfferental evoluton tranng algorthm for feed forward neural networks, Neural Proc. Lett. 7 ( [] E. oldberg, J. Rchardson, enetc algorthms wth sharng for multmodal functon optmzaton, n: J. Rchardson (Ed., enetc Algorthms and ther Applcatons (ICA 87., 987, pp [] K. Krstnsson,.A. Dumont, System dentfcaton and control usng genetc algorthms, IEEE Trans. Syst. Man Cybernet. ( [3] J. Ilonen, J.K. Kamaranen, J. Lampnen, Dfferental evoluton tranng algorthm for feed forward neural networks, Neural Proc. Lett. 7, 003, pp [4] R. Storn, System desgn by constrant adaptaton and dfferental evoluton, IEEE Trans. Evol. Comput. 3, 999, pp. 34. [5] H.R. Tzhoosh, Opposton-based learnng: a new scheme for machne ntellgence, n: Proc. Int. Conf. Comput. Intell. Modelng Control and Autom, vol. I, Venna, Austra, 005, pp [6] S. Rahnamayan, H.R. Tzhoosh, M.M.A. Salama, Opposton Versus Randomness n Soft Computng Technques, Elsever J. Appl. Soft Comput. 8 (March (, 008, pp [7] S. Rahnamayan, H.R. Tzhoosh, M.M.A. Salama, Opposton-based dfferental evoluton, IEEE Trans. Evol. Comput. (, 008. [8] K. V. Prce, R. M. Storn, and J. A. Lampnen, "Dfferental Evoluton: A Practcal Approach to lobal Optmzaton", (Kndle Edton. Sprnger, 005. [9] H.R. Tzhoosh, Opposton-based learnng: a new scheme for machne ntellgence, n: Proc. Int. Conf. Comput. Intell. Modelng Control and Autom, vol. I, Venna, Austra, 005, pp

6 0 Hamd & Darae, Identfcaton of a Nonlnear System by Determnng of Fuzzy Rules [0] J. Kennedy, R. Eberhart, Partcle swarm optmzaton, n: Proc. IEEE Internat. Conf. Neural Networks, Perth, Australa, pp , 995. [] C. F. Juang, C. W. Hung and C. H. Hsu, Rule-Based Cooperatve Contnuous Ant Colony Optmzaton to Improve the Accuracy of Fuzzy System Desgn, IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO. 4, AUUST 04 [] W. Zhao, Q. Nu, K. L and. W. Irwn, A Hybrd Learnng Method for Constructng Compact Rule-Based Fuzzy Models, IEEE TRANSACTIONS ON CYBERNETICS, VOL. 43, NO. 6, DECEMBER 03. [3] H.Hamd, A Model for Impact of Organzatonal Project Benefts Management and ts Impact on End User, JOEUC, Volume 9, Issue, 07, pp [4] K.Mohammad, H.Hamd., Modelng and Evoluton of Fault-Tolerant Moble Agents n Dstrbuted System.The Second IEEE and IFIP Internatonal Conference on wreless and Optcal Communcatons Networks (WOCN 005, March 6 8, 005. [5] S. A.Monadjem, H.Hamd, A.Vafae. Analyss and Evaluaton of a New Algorthm Based Fault Tolerance for Computng Systems. Internatonal Journal of rd and Hgh Performance Computng (IJHPC, 4(, 0, pp [6] S. A.Monadjem, H..Hamd., A.Vafae. ANALYSIS AND DESIN OF AN ABFT AND PARITY- CHECKIN TECHNIQUE IN HIH PERFORMANCE COMPUTIN SYSTEMS Journal of Crcuts, Systems, and Computers (JCSC, JCSC Volume Number 3, 0. [7] A.Vafae., S. A.Monadjem, H..Hamd., Evaluaton of Fault Tolerant Moble Agents n Dstrbuted Systems. Internatonal Journal of Intellgent Informaton Technologes (IJIIT, 5(, 009, pp [8] A.Vafae, S. A.Monadjem, H.Hamd Evaluaton and Check pontng of Fault Tolerant Moble Agents Executon n Dstrbuted Systems, Journal of Networks, VOL. 5, NO [9] H.Hamd, A New Method for Transformaton Technques n Secure Informaton Systems Journal of Informaton Systems and Telecommuncaton, Vol. 4, No., January- March 06, pp [30]. Ye, T.Sakura, Robust Smlarty Measure for Spectral Clusterng Based on Shared Neghbors, ETRI Journal, vol. 38, no. 3, June. 06, pp [3] J.Wu, F. Dng, M. u, Z. Mo, A..Jn. Investgatng the Determnants of Decson-Makng on Adopton of Publc Cloud Computng n E-government. JIM, 4(3, 06, pp [3] S. KUMAR, "Performance Evaluaton of Novel AMDF- Based Ptch Detecton Scheme," ETRI Journal, vol. 38, no. 3, June. 06, pp [33] B. Shadloo, A. Motevalan, V. Rahm-movaghar, M.A. Esmael, V. Sharf, A. Hajeb, R. Radgoodarz, M. Hefaz, A. Rahm- Movaghar, Psychatrc Dsorders Are Assocated wth an Increased Rsk of Injures: Data from the Iranan Mental Health Survey, Iranan Journal of Publc Health 45(5, 06, pp Hodjatollah Hamd born 978, n shazand Arak, Iran, He got hs Ph.D n Computer Engneerng. Hs man research nterest areas are Informaton Technology, Fault-Tolerant systems (fault-tolerant computng, error control n dgtal desgns and applcatons and relable and secure dstrbuted systems, Machne learnng, Knowledge Dscovery and Data Mnng. Snce 03 he has been a faculty member at the IT group of K. N. Toos Unversty of Technology, Tehran Iran. Informaton Technology Engneerng roup, Department of Industral Engneerng, K. N.Toos Unversty of Technology. Atefeh Darae born n Khorram Abad, Lorestan Iran. She receved her B.Sc n Informaton Technology n 0 from Unversty College of Nab Akram, Tabrz, Iran. She s currently an M.Sc student n Informaton Technology (E-Commerce, at K. N. Toos Unversty of Technology, Tehran Iran. Her research nterests nclude Machne learnng, Knowledge Dscovery and Data Mnng and Customer Relatonshp Management.