Optimal Design of LQR Weighting Matrices based on Intelligent Optimization Methods

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1 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods S.Amir Ghoreishi, Mohammad Ali Nekoui and S. Omid Basiri Souh ehran Branch, Islamic Azad Universiy, ehran, Iran. doi:.56/ijiip.vol. issue.7 Absrac In his paper, considering some imporan indices such as closed-loop pole locaions, speed of response, and maximum level of conrol effor, and combining hem ino an objecive funcion, an opimizaion problem is defined o find he opimal weighing marices in LQR conroller. o solve his opimizaion problem four inelligen opimizaion mehods are uilized: Geneic Algorihm (GA, Paricle Swarm Opimizaion (PSO, Differenial Evoluion (DE, and Imperialis Compeiive Algorihm (ICA. he proposed mehod is applied o a nonlinear flexible robo manipulaor model, and obained resuls from he algorihms are compared. Keywords: Opimal Conrol, Geneic Algorihm (GA, Differenial Evoluion (DE. Inroducion Opimal conrol uses some mehods and mahemaical ools o design conrollers for dynamical sysems such ha a crierion o be opimized. Usually he crierion considers performance, energy consumpion, response ime and final sae siuaions. For example designing a conroller o ransfer he sae of a dynamical sysem o a desired sae in minimum ime can be caegorized in an opimal conrol problem. he Linear Quadraic Regulaor (LQR is a mehod used abundanly o design linear conrollers for linear sysems. he LQR conroller possesses suiable robusness wih minimum gain margin -6db and maximum gain margin o infiniy and a he same ime i can reach o 6 degree of phase margin. he design parameers for LQR are he weighing marices in he objecive funcion and should be seleced by he designer. Since hese marices direcly affec he opimal conrol performance many discussions have been done o selec hese marices called eigen-srucure assignmen [-3]. Besides he classical mehods, he algorihms based on inelligen opimizaion and sof compuing have been used gradually for selecing hese marices. For example, Geneic Algorihm (GA [-], combinaion of GA and Simulaed Annealing (SA [] and An Colony Opimizaion (ACO [] have been used for solving his problem. In his paper, by considering some imporan crieria like speed of response, he closed-loop pole locaions, and maximum level of conrol effor, and combining hem ino an objecive funcion, give us an opimizaion problem, soluion of which leads us o he bes weighing marices in LQR conroller. Since we have a nonlinear, complicaed opimizaion problem GA [3], Paricle Swarm Opimizaion (PSO [,5,6], Differenial Evoluion (DE [7,8], and Imperialis Compeiive Algorihm (ICA [9,] are used o solve his problem. he proposed mehod is applied o flexible robo manipulaor. his paper is organized as follows: Secion briefly describes he LQR conroller. he proposed mehod and is relaed crierion are described in Secion 3. Secion, briefly explains he inelligen opimizaion ools, GA, PSO, DE, and ICA, which are used o solve he opimizaion problem. In Secion 5 he flexible robo manipulaor model is described, which in Secion 6 he proposed mehod is applied o his model. Finally, Secion 7 concludes he paper.

2 . LQR Conroller Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March One of he sae space based opimal conrol mehod is Linear Quadraic Regulaor (LQR. In his secion we briefly describe his mehod. Consider he following linear, coninuous-ime and conrollable sysem: x = Ax + Bu ( he following objecive funcion is defined: J = x Qx + u Ru d ò [ ] ( where, Q and R are weighing marices and should be posiive-semi-definie and posiivedefinie, respecively. Since sysem ( is conrollable, he mehod which is able o minimize ( is called LQR. Considering he funcional ( in LQR he following Riccai equaion should be solved: - PA + A P - PBR B P + Q = (3 By solving he above Riccai equaion he posiive-definie marix P is obained, hus he opimal gain and conroller are calculaed as: ( - K = R B P u =-Kx herefore, he closed-loop poles are he eigenvalues of A- BK. In his paper he eigenvalues of A- BK are shown as l = { l, l,, ln }. In LQR problem, he weighing marices Q and R have profound effec on conroller performance. On he oher hand, finding he bes Q and R needs many compuer simulaion and rial and errors, which are very ime-consuming. hus using inelligen opimizaion mehods for finding Q and R is more effecive. In his paper, we use Geneic Algorihm (GA, Paricle Swarm Opimizaion (PSO, Differenial Evoluion (DE, and Imperialis Compeiive Algorihm (ICA as opimizaion ools. Firsly, in he nex secion we are going o describe he opimizaion problem for finding Q and R. 3. Proposed Approach An opimizaion problem has a leas wo disinc soluions. he process of opimizaion is o search for he bes soluion of he opimizaion problem. In his paper, we are searching for bes weighing marices of LQR approach. So i is necessary o define erms beer and bes. A good LQR conroller, mus make he closed-loop sysem (i as fas as possible and (ii as sable as possible, using (iii as lowes as possible conrol effor. We define hree indices for he menioned requiremens of a good LQR. Definiions of he indices are lised below: Sabiliy Index. his index is relaed o real pars of closed-loop poles and defined as follows: SI =- (6 max Re{ l } i i (5

3 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March Obviously he beer Q and R give smaller sabiliy index. Seling-ime Index. For a ime response of a sysem his index is he minimum ime ha he response reaches o absolue error of.5. In mahemaical form: S ì y (-yd( ü = min ï í.5ï ý ïî yd( ïþ (7 where y ( is he ime response and y ( is desired oupu rajecory. If he sysem has more han d one oupu, he larges seling ime among oupus, is he seling ime of sysem. If desired value of he oupu is zero, hen seing ime is defined as follows, convenionally. ì max y ( ü S = min ï í y( ï ý ïî ïþ (8 Maximum Conrol Effor Index. Because of he limiaions of acuaors, he objecives should be saisfied wih minimum conrol effor, usually. Using he acuaors wih higher sauraion level needs more expendiure. Maximum conrol effor o he sysem can be defined as: umax = max u( (9 Finding he bes Q and R marices, using above indices, is a muli-objecive opimizaion problem. hus o avoid complicaion we define he following objecive funcion: J = w SI + w S + w u ( oal 3 max where, w, w and w 3 should be seleced by he designer. Remark : he mos imporan poin in Q and R marices is he consrains on hem. As menioned before, he Q and R marices should be posiive-semi-definie and posiive-definie, respecively. However, hese consrains canno be easily saisfied. I is hard o define a simple relaion beween marix elemens and is posiive definieness. hus, in his paper, a mehod is used o make posiivesemi-definie marices. I is known ha for any real marixa, B = A A is nonnegaive, i.e., i can be said ha marix B is posiive-semi-definie. Usually marix A is known as square roo of he marix B. he above mehod is used in his paper o make iniial responses and o code he responses of he problem. In oher words, insead of using Q and R marices as unknown variables he marices W and V ha saisfy Q = W W and R = V V are used as unknown variables. hus opimizaion algorihms firs find W and V, hen Q and R marices are calculaed by above equaions.. Opimizaion ools In his secion, he uilized inelligen opimizaion mehods are briefly reviewed.

4 .. Geneic Algorihm Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March Geneic Algorihm (GA [3] is one of he evoluionary algorihms inspired by naural evoluion. In GA, he proposed responses for an opimizaion problem are considered as living creaures and GA provides a virual environmen for growh and aciviy of hese proposed responses. In his virual environmen he beer responses are increased easier and more han ohers. In his virual environmen, similar o any phenomenon in naure some mechanisms are considered. he mos imporan operaors and mechanisms used in GA wih heir equivalencies in naure are: Selecion operaor which is equivalen o naural selecion phenomenon Crossover operaor which is equivalen o reproducion phenomenon Muaion operaor which is equivalen o geneic muaion phenomenon By using hese operaors on curren populaion, new populaion emerges which he average of hem are no worse han he curren populaion, neverheless he average quie ofen is beer. hus by ime passing he GA gives beer response for opimizaion problem... Paricle Swarm Opimizaion he PSO mehod was firs inroduced by James Kennedy and Rassel Eberhar [] in 995. hey were essenially aimed a producing compuaional inelligence by exploiing simple analogues of social ineracion, raher han purely individual cogniive abiliies. heir work developed ino a powerful opimizaion mehod called Paricle Swarm Opimizaion (PSO [,5]. In PSO, a number of simple eniies, namely he paricles, are placed in he search space of some problem or funcion, and each evaluaes he objecive funcion a is curren locaion. Each paricle hen deermines is movemen hrough he search space by combining some aspec of he hisory of is own curren and bes (bes-finess locaions wih hose of one or more members of he swarm, wih some random perurbaions. he algorihm, searches a space by adjusing he rajecories of paricles as hey are concepualized as moving poins in mulidimensional space. he individual paricles are drawn sochasically oward he posiions of heir own previous bes performance and he bes previous performance of heir neighbors. If he search space is considered as n-dimensional space, he j posiion and velociy of a paricle can be shown by n-dimensional vecors. Consider xi [] and v j [] o be he j -h elemen of posiion and velociy of he i -h paricle in -h ieraion, i respecively. he posiion and velociy of i -h paricle in ( + [,5]: j j j i i i -h ieraion are defined as x [ + ] = x [ ] + v [ ] ( ( ( j j j j j j i i i, bes i gbes i v [ + ] = wv [ ] + rc x [ ] - x [ ] + rc x [ ] - x [ ] ( where, w is ineria coefficien and can be consan, variable or random. his coefficien guaranees ha he paricles which give he bes response are no haled and coninue heir pervious rajecories [,5]. he consans c and c are learning coefficiens and hey are seleced in he inerval [,] and usually c + c = [5]. r and r are random numbers wih uniform disribuion in he inerval [,]. xibes, [] is he bes response ha is found by he i -h paricle unil -h ieraion and x [] gbes is he bes response of oal populaion unil -h ieraion.

5 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March According o [6], i is possible o define consricion coefficiens, which guaranee he sabiliy and good performance for PSO. For wo scalars f and f, where f + f >, he consricion coefficiens are defined by: w = ( f + f - + ( f + f - ( f + f (3 c ( = fw (5 c = f w A good choice is f = f =.5, which implies ha w».798 and c = c».96 [5,6]..3. Differenial Evoluion Price and Sorn proposed Differenial Evoluion (DE in mid 99s [7,8], o deal wih opimizaion problems, defined in coninuous domains. DE has similariies wih boh GA and PSO. DE uses informaion of all individuals, and differences beween hem, o creae new soluions for opimizaion problem. he new soluions are creaed using difference and rial vecors. o creae a new soluion y, an old soluion a is perurbed using he following rule: y = a + F Ä( b - c (6 where b and c are wo individuals, randomly seleced from populaion, and a ¹ b ¹ c. Vecor F is he scaling facor, and is elemens are uniformly disribued random numbers in [ Fmin, F max]. Operaor Ä is he elemen-wise muliplicaion operaor. o creae final soluion z, crossover operaor is applied o y and anoher randomly seleced individual x. here are various mehods of crossover. Simples case is formulaed as follows: z i ì yi, r CR or i = i = ï í ïï xi, oherwise î (7 where x i indicaes i -h elemen of vecor x, scalar r is a uniformly disribued random number in [,], CR is he Crossover Rae parameer, and i is a random ineger index in he se {,, 3,..., n }. I is assumed ha number of search space dimensions (also number of elemens of soluion vecors are equal o n. In his way, DE uses informaion of curren populaion, o creae individuals of he nex ieraion (generaion. his process is carried ou, unil erminaion condiions are saisfied... Imperialis Compeiive Algorihm Imperialis Compeiive Algorihm (ICA is proposed by Aashpaz-Gargari e al. [9,], and i is inspired by imperialis compeiion. Similar o GA, which simulaes he naural evoluion process o solve an opimizaion problem, ICA simulaes he socio-poliical evoluion o deal wih opimizaion problems. In ICA, individual soluions are referred as (virual counries. Some of good counries in he iniializaion phase, which are named imperialiss, form heir own imperial. hey capure heir colonies

6 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March from oher non-imperialis (normal counries. In every ieraion (decade of ICA, he following operaions are carried ou: Assimilaion of Colonies. Colonies of each imperialis are assimilaed o heir respecive imperialis. Assimilaion is formulaed as following: new old old col col imp col x = x + br Ä( x -x (8 where b is assimilaion facor, and r is a vecor, and is elemens are uniformly disribued random old numbers in [,]. x imp, x col, and x new col are posiion of imperialis, old posiion of colony, new posiion of colony, respecively. In [9,], he new posiion of colony is angularly deviaed. For more informaion abou deviaion, refer o [9] and []. Revoluion of Colonies. Similar o muaion operaor in GA, seleced colonies of every imperialis are changed randomly, or revolved. Revoluion is applied o a colony, wih a probabiliy of p r. Exchange wih Bes Colony. If afer assimilaion and revoluion seps, here are colonies which are beer han heir respecive imperialiss, he imperialis is exchanged wih is bes colony. In oher words, imperialis will be colony, and he bes colony will be he new imperialis. Imperialis Compeiion. Weakes imperialis among ohers, loses is weakes colony. One of oher imperialiss will capure he los colony, randomly. he beer he imperial, he more probable i will possess he colony. An imperialis wihou colony will collapse. I will become a colony, and capured by oher imperialiss. he menioned seps are carried ou, while sop condiions are no saisfied. 5. Flexible Robo Manipulaor A single-link flexible robo manipulaor is used in his paper, o implemen and check he proposed mehod. Nonlinear differenial equaions which are governing he single-link flexible robo manipulaor sysem are lised below: Iq + MgL sin q + k( q - q = Jq -k( q - q = u (9 where q and q are angular posiions, I and J are momen ineria, M is he mass of link, L is he lengh of link, k is he siffness, g is he graviaional acceleraion, and u is he conrol inpu. Values of parameers of he sysem are given in able. able. Value of Parameers of Sysem ( Parameer Value Uni I.3 Kg.m J. Kg.m k 3 N.m/rad L.55 m M.3 Kg Defining sae vecor as = 3 = x ( éx ( x ( x ( x ( ù éq ( q ( q ( q ( ù êë úû êë úû (

7 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March sae space model of he flexible robo manipulaor is as follows: x = x MgL k x =- sin x - ( x -x3 I I x = x 3 k x = ( x - x3 + u J J ( If he model ( is linearized around he origin (he equilibrium poin, he obained linear model is: é ù é ù MgL k k + - ( I I x = x( + u( k k ê - ú ê J J ëj ú ë û û ( Oupus of sysem are defined by: é ù éq ( ù y( = x ( = ë û ë û q( ê ú ê ú (3 Desired se poin of boh oupus is zero. In all of simulaions, iniial sae of sysem is defined as x = é.3. ù ê ë ú û ( 6. Simulaion Resuls In his secion he proposed mehod is applied o roaional invered pendulum inroduced in Secion 5. he simulaion ime is considered 5 seconds, wih sep size of. seconds. Weighing facors in cos funcion J oal, defined by Eq. (, are se o w =, w = 5 and w 3 =. Firs of all, GA is used o solve he opimizaion problem. GA resuls are shown in Figs. and. In Fig., oal cos funcional, Sabiliy Index, Seling ime Index and Maximum conrol effor are ploed versus ieraion. Afer ieraions (generaions, GA reaches o for cos funcional. Fig. shows he ime response of y ( = q (, y ( = q ( and he conrol signal u ( for differen ieraions. I can be seen ha he performance of he conroller is modified by increasing he number of ieraions.

8 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March oal Cos 5 Bes oal Cos Ieraion Sabiliy Index..3. Bes Sabiliy Index Seling ime ( s Fig. GA: Cos funcional J oal Maximum Inpu (u max Ieraion.5.5 Bes Seling ime Ieraion Ieraion, Sabiliy Index, Seling ime Index and Maximum conrol effor Bes u max y.3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion y..3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion u 6 - Ieraion Ieraion 5 Ieraion Ieraion Ieraion Fig. GA: Oupus y and y, and he conrol signal he above opimizaion problem is solved by PSO, as second approach. he simulaion resuls are depiced in Figs. 3 and. In Fig. 3, he oal cos funcional, Sabiliy Index, Seling ime Index and Maximum conrol effor are shown, versus ieraion coun. PSO reaches o.657 for he cos funcional afer ieraions. A he same ime Fig. shows he ime response of y ( = q (, y ( = q ( and he conrol signal u ( for differen ieraions. oal Cos 5 5 Bes oal Cos Ieraion Sabiliy Index..3.. Bes Sabiliy Index Seling ime ( s Maximum Inpu (u max.8.6. Fig 3. PSO: Cos funcional J oal Ieraion Bes Seling ime Ieraion Ieraion, Sabiliy Index, Seling ime Index and Maximum conrol effor Bes u max

9 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March y.3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion y..3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion u - Ieraion Ieraion 5 Ieraion Ieraion Ieraion Fig. PSO: Oupus y and y, and he conrol signal As hird approach, DE is uilized o solve he opimizaion problem and find he opimal values of weighing marices. he simulaion resuls are skeched in Figs. 5 and 6. In Fig. 5, he oal cos funcional, Sabiliy Index, Seling ime Index and Maximum conrol effor are shown, versus ieraion coun. DE finally reaches o.6967 for he cos funcional afer ieraions. ime response of y ( = q (, y ( = q ( and he conrol signal u ( for differen ieraions, are shown in Fig. 6. oal Cos 8 6 Bes oal Cos Ieraion Sabiliy Index Bes Sabiliy Index Seling ime ( s Maximum Inpu (u max Ieraion.5.5 Fig 5. DE: Cos funcional J oal Bes Seling ime Ieraion Ieraion, Sabiliy Index, Seling ime Index and Maximum conrol effor Bes u max y.3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion y..3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion u - Ieraion Ieraion 5 Ieraion Ieraion Ieraion Fig 6. DE: Oupus y and y, and he conrol signal

10 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March Finally ICA is used o solve he opimizaion problem, as fourh mehod. he simulaion resuls are shown in Figs. 7 and 8. oal cos funcional, Sabiliy Index, Seling ime Index and Maximum conrol effor are shown, versus ieraion coun, in Fig. 7. ICA reaches o.53 for he cos funcional afer ieraions. ime response of y ( = q (, y ( = q ( and he conrol signal u ( for differen ieraions, are shown in Fig. 8. oal Cos 5 5 Bes oal Cos Ieraion Sabiliy Index..3.. Bes Sabiliy Index Seling ime ( s Ieraion Bes Seling ime Ieraion Maximum Inpu (u max Ieraion Bes u max Fig 7. ICA: Cos funcional J oal, Sabiliy Index, Seling ime Index and Maximum conrol effor y.3.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion y.6.. Ieraion Ieraion 5 Ieraion Ieraion Ieraion u - Ieraion Ieraion 5 Ieraion Ieraion Ieraion Fig 8. ICA: Oupus y and y, and he conrol signal For all of algorihms, populaion size is se o 5 and maximum number of ieraions is se o. Resuls obained from used algorihms are summarized in able. he rank of each algorihm is presened in parenhesis, for each of indices. Considering Sabiliy Index, DE has bes performance among ohers. Sabiliy Index for DE is.8, which implies ha greaes real par of he closed-loop poles is ICA wih Seling ime of.37 sec, resuls he fases closed-loop sysem among ohers. PSO needs leas conrol effor among ohers and uses Maximum Conrol Effor of.7738 o sabilize and conrol he sysem. Considering oal cos value, ICA has bes performance. PSO, DE, and GA has second, hird, and fourh rank, respecively.

11 Opimal Design of LQR Weighing Marices based on Inelligen Opimizaion Mehods Inernaional Journal of Inelligen Informaion Processing, Volume, Number, March Algorihm 7. Conclusions able. Opimizaion Resuls Sabiliy Index Seling ime Maximum Conrol Effor oal Cos GA.63 (.5 (.6 ( ( PSO.88 (3.39 ( (.657 ( DE.8 (.38 (.993 ( (3 ICA.866 (.37 (.87 (.53 ( In his paper, he problem of finding weighing marices for an LQR conroller has been formulaed as an opimizaion problem. By considering some imporan indices in designing conrollers such as; closed-loop pole locaions, speed of response and maximum conrol effor, and combing hem ino an objecive funcional, he problem of finding he weighing marices for an LQR conroller has been convered o an opimizaion one. o solve his opimizaion problem four inelligen opimizaion mehods are uilized: Geneic Algorihm (GA, Paricle Swarm Opimizaion (PSO, Differenial Evoluion (DE, and Imperialis Compeiive Algorihm (ICA. he proposed mehod has been applied o a single-link flexible robo manipulaor sysem, and he obained resuls from hese four algorihms have been compared. Considering sabiliy index, seling ime, and maximum conrol effor, DE, ICA, and PSO have beer performance among ohers, respecively. Considering all of indices, ICA has he bes performance among all. 8. References [] F. L. Lewis and V. L. Syrmos, Opimal Conrol, nd Ediion, John Wiley & Sons, Inc., 995. [] G. P. Liu and R. J. Paon, Eigensrucure Assignmen for Conrol Sysem Design, John Wiley & Sons, Inc., 998. [3] J. W. Choi and Y. B. Seo, Conrol design mehodology using EALQR, in Proceedings of he IEEE 37h SICE Annual Conference, 998. [] J. V. da Fonseca Neo and C. P. Boura, Parallel Geneic Algorihm Finess Funcion eam for Eigensrucure Assignmenvia LQR Designs, in Proceedings of IEEE Congress on Evoluionary Compuaion, 999. [5] C. P. Boura and J. V. da Fonseca Neo, Parallel Eigensrucure Assignmen via LQR Design and Geneic Algorihms, in Proceedings of he American Conrol Conference, San Diego, California, 999. [6] J. V. da Fonseca Neo, e al., Modelos e Convergência de um AlgorimoGenéicoparaAlocação de Auo-esruura via RLQ, in IEEE Lain America ransacions, vol. 6, no., pp. -9, March 8. [7] R. Davis and. Clarke, A parallel implemenaion of he geneic algorihm applied o he fligh conrol problem, in IEE Colloquium on High Performance Compuing for Advanced Conrol, pp. 6/ - 6/3, 99. [8] C. Wongsahan and C. Sirima, Applicaion of GA o Design LQR Conroller for an Invered Pendulum Sysem, in Proceedings of he IEEE Inernaional Conference on Roboics and Biomimeics, 9. [9] A. H. Zaeri, M. BayaiPoodeh, and S. Eshehardiha, Improvemen of Cûk Converer Performance wih Opimum LQR Conroller Based on Geneic Algorihm, in Proceedings of Inernaional Conference on Inelligen and Advanced Sysems, 7. [] M. BayaiPoodeh, e al., Opimizing LQR and Pole placemen o Conrol Buck Converer by Geneic Algorihm, in Proceedings of Inernaional Conference on Conrol, Auomaion and Sysems, 7.

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