CONTROL OF HINDMARSH-ROSE MODEL BY NONLINEAR-OPEN-PLUS-CLOSED-LOOP (NOPCL)

Transcription

1 Euroea Scietific Joural December 013 /SPECIAL/ editio vol.3 ISSN: (Pri e - ISSN CONTROL OF HINDMARSH-ROSE MODEL BY NONLINEAR-OPEN-PLUS-CLOSED-LOOP (NOPCL) Touia Bezekri, PhD Uiverity of Sciece ad Techology Houari Boumedie, Faculty of mathematic, EL ALIA, Bab Ezzouar, Algier, Algeria Abtract I thi work, we ue the Noliear-Oe-Plu-Cloed-Loo (NOPCL) method to cotrol a oliear model: the Hidmarh-Roe model i which we ca exhibit regular ad chaotic dyamic. The aim of the NOPCL method i to etrai comlex dyamic to arbitrary give goal dyamic, by addig a uitable cotrol term to the ytem. We ue thi method to ure chao, by etraiig chaotic dyamic to a eriodic oe for the Hidmarh-Roe model. Keyword: Cotrol chao, Hidmarh-Roe model, oliear-oe-lu-cloed-loo Itroductio: There have bee a great umber of tudie related to the cotrol of oliear dyamical ytem (For review ee Ref.(1,,3,4,5,6). Thee method have bee alied i a wide umber of domai icludig hyical ad biological ytem, robotic, avioic ad may other. Particularly there wa a great deal of reearch to modelig ad cotrol mechaim of excitable biological media uch a activity of euro which exhibit chaotic behavior (e.g. Ref.(7,8,9). The Hidmarh-Roe model (HRM), which model a euroal electrical activity, i a three-dimeioal model caable of comlex dyamic uch a burtig ocillatio ad chao. Neuro react o ijectio of a curret by a quick, hort deolarizatio of their membrae otetial, which i egative i ret. The activity of euro coit of erie of ule, alterated by log eriod of low activity aroud ret otetial. Thi i kow a a actio otetial, or ike. Burtig ocillatio i a time evolutio coitig of burt of raid ike, alterated by hae of relative quiecece. Thee erie of ule are coidered to carry the iformatio tramitted by euro. We ue the NOPCL method to how how the Hidmarh-Roe model ca be cotrolled by drivig it outut to the deired atter. The aim of thi method i to add a cotrol term, a drivig term, to the iitial ytem i order to drive it dyamic from oe trajectory ito aother oe. I articular, thi method i able to witch chaotic dyamic ito a eriodic oe ad vice vera. Etraimet Cotrol Let u recall the Etraimet Cotrol a exlaied i Ref.. We deote by u the additive cotrol term, the cotrolled dyamical ytem i the give by: dx = F( + u( (1) 314

2 Euroea Scietific Joural December 013 /SPECIAL/ editio vol.3 ISSN: (Pri e - ISSN The cotrol roblem i to fid a cotrol fuctio u R, uch that the ytem tate x i etraied to arbitrary give goal dyamic g for which the error betwee x ad g atifie: lim x( g( = lim e( = 0 t The bai of etraimet aociated with a aroriate time t ad g i defied by: BE( t ) = = (3) t { x( t ) / lim e( 0} The goal i to how that the bai of etraimet i ot a emty et, that i the error e = 0 i aymtotically table for the error equatio, ad i ideedet o the goal dyamic g. Oe-Plu-Cloed-Loo cotrol (OPCL) A (OPCL) trategy wa firt be rooed by Hubler ad Lucher Ref.3 ad exteded by Jacko ad Grou Ref. to cotrol the ytem (1). The rooed cotrol term u i of the followig form: dg u ( = S( C( e(, (4) where the firt term of u i called the Huble' oe-loo iteractio ad S ( i a uitable calar witchig fuctio o time t atifyig: S ( = 0 for t < t ; 0 < S ( 1 for t t. (5) The liear cloed-loo iteractio C ( i give by: df( C( = A, (6) dg where A i a arbitrary matrix whoe eigevalue all have egative real art. Jacko ad Grou Ref. roved that if the fuctio F i everywhere Lichitz, with reect of x, the for a arbitrary mooth goal fuctio g, the cotrol u i uch that oe of bai of etraimet aociated to g are emty et. Ideed, ubtitutig equatio (4) ito the cotrol ytem (1) ad lettig S ( = 1 yield to the give equatio: de df( t = F( e + ) A. (7) dg Exadig F ( e + for mall e, i the firt order, yield the liear aroximatio equatio: de = Ae. (8) Sice all eigevalue of the matrix A have egative real art, the aymtotic tability of equatio (8) i etablihed. However, it wa how by Y. Tia et al. Ref.4, that for a certai cla of ytem the bai of etraimet i rather comlicated; it i deedet o the goal dyamic g. Noliear Oe-Plu-Cloed-Loo cotrol (NOPCL) The NOPCL cotrol i baed o the OPCL cotrol. The cotrol term u i recoidered a follow: dg u ( = S( C( e( N(, (9) t () 315

3 Euroea Scietific Joural December 013 /SPECIAL/ editio vol.3 ISSN: (Pri e - ISSN where C i a defied i (6) ad A defied a reviouly. The oliear term N R i the cloed-loo cotrol actio whoe ith elemet N i (, i give for ufficietly mooth F, by: 3 m 1 Fi ( 1 Fi ( 1 Fi ( N i ( = e jek + e jekel e jek... e! 3! m!... g k = 1 j k k, l= 1 j k l k,..., = 1 j k m, i = 1,,..., (10) where m i the order of derivative of F called the order of arameter of the fuctio N. I thi cae, exadig F ( e +, for mall e, oe obtai: de i 1 = Aei + ( m + 1)! k,..., = 1 ( m + 1) Fi ( e... j k j e... e k +..., i = 1,,..., It i eaily rove Ref. that the bai of etraimet are the whole hae ace for ytem for which the fuctio F i olyomial of degree m, m. Thi i due to the fact that i thi cae, (11) will be reduced to (8), ad e olved from thi lat equatio aroache zero for all iitial coditio e t ). ( Cotrol of Hidmarh-Roe Model The Hidmarh-Roe Model wa develoed by Hidmarh ad Roe (1984) to decribe a iolated triggered burt of actio otetial oberved i a brai cell of a od ail. The equatio are give by: dx 3 = y x + 3 x z dz dy = 1 5x y = ε ( 4x + K z) (1) where x i the membrae otetial, y ad z rereet emirical variable decribig the activatio ad iactivatio of the ioic coductace. They decribe reectively ome fat ad low gatig variable for ioic. Slow activatio of z i due to the mall arameter 0 < ε 1. Thee equatio model the electrical activity of the membrae otetial of a igle euro. The exteral curret K i viewed a a cotrol arameter delayig ad advacig the activatio of the low curret i the model. Notice that the ytem i autoome. Simulatio reult I order to illutrate the effect of the drivig term, we fix the arameter ε of the HRM to ε = For thi ytem all the fourth order artial derivative are equal to zero ice the fuctio F of HRM i a olyomial of degree 3. Notice that it i eay to ee that the fuctio F i everywhere Lichitz. The cotrol arameter m i thu take to be 3 i the NOPCL cotrol. It follow that the cloed-loo cotrol actio N ( i give by: 3 N( = [( 3g1( + 3) e1 ( e1 (, 5e1,0] (13) For coveiece, the matrix A i take diagoal ad the liear cloed loo iteractio i give by: C g) = 3g ( + 6g ( e ( e ( e (, (14) ( ) ) 1( t (11) 316

4 Euroea Scietific Joural December 013 /SPECIAL/ editio vol.3 ISSN: (Pri e - ISSN C ( g) = 10g1( e1 ( + (1 + a ) e (, (15) C 3( g) = ε (4e1 ( (1 + a33) e3(. (16) I our cae, the uroe of the cotrol actio i to teer the Hidmarh-Roe model from oe of it trajectory to aother oe. Hece g ( i uch that dg( g) = 0, t. (17) The cotrol term u i the the um of C (g) ad N (. The error equatio i the ame a i (8). Hece, for the Hidmarh-Roe Model, the bai of etraimet BE (g) i global for all value of g ad e i ( = ex( aii ) for i = 1,, 3 (18) For umerical aalyi, we chooe the matrix A a follow: A = diag( 1, 1, 0.01) (19) The goal trajectory g ( i a burtig eriodic motio. Our aim i to teer the HRM from chaotic trajectory to burtig ocillatig trajectory. Numerical imulatio how that the olutio tur out to be chaotic for the curret K = ad eriodic for K = We deict, i reectively Figure1 ad Figure, the eriodic burtig trajectory for K = 5.1 ad the chaotic trajectory for K = ad iitial coditio: x t ), y( t ), z( t ) ( , ,0. ( ).01644) ( = Figure1. Burtig ocillatio of the membrae otetial x ( for K = 5. 1 ad iitial coditio ( , , ). 317

5 Euroea Scietific Joural December 013 /SPECIAL/ editio vol.3 ISSN: (Pri e - ISSN Figure. Chaotic olutio of the membrae otetial x ( for K = ad iitial coditio ( , , ). Lyauov exoet are ued to decribe the eriodic ad chaotic dyamic of oliear dyamical ytem. The time varyig larget Liauov exoet, howi for K = the chaotic motio, i rereeted i Figure3. Figure3. The larget Lyauov exoet for chaotic olutio of HRM. I order to avoid the traitio hae of the trajectory, we tart cotrol of the chaotic motio at t = 1300 for the ame iitial coditio a above. We oberve i Figure4 that at thi time t, the trajectory i drive to the burtig trajectory, thu removig chao. 318

6 Euroea Scietific Joural December 013 /SPECIAL/ editio vol.3 ISSN: (Pri e - ISSN Figure4. The HRM drive from chaotic olutio to burtig olutio by addig the cotrol term u. Cocluio: We coidered the Hidmarh-Roe model. We have how by uig the NOPCL method that it i oible to witch from oe trajectory of the ytem ito aother oe ad therefore chagig the dyamic of the otetial actio. The aim of thi method i to add a uitable drivig term to the HRM, which force the cotrolled ytem to erform a motio which coicide with a target trajectory of the model. We howed that we ca ure the chaotic dyamic of the HRM. Referece: G.Che, X.Do From Chao to Order:Perective ad Methodologie i Cotrollig chaotic Noliear Dyamic Sytem, It.J.Bifurcatio Chao, vol. 3, o. 6, E.A. Jacko, I.Grou,A oe-lu-cloed-loo (OPCL) cotrol of comlex dyamic ytem, Phyica D, vol.85, 1993, A.Hubler, E. Lucher, Reoat timulatio ad cotrol of oliear ocillator, Naturwiechafte, vol. 76, Yu-Chu. Tia, M.O. Tade, J.Y. Ta Noliear oe-lu-cloed-loo 5NOPCL) cotrol of dyamic ytem, Chao Solito ad Fractal, vol. 11, o. 7, 1993, E.A. Jacko, OPCL migratio cotrol betwee five attractor of the Chua ytem, It.J.Bifurcatio Chao N. Nijmeijer, A.J. Va Der Schaft, Noliear Dyamical cotrol Sytem, Sriger-Verla A.Garfikel, ML.Sao, WL.Ditto,JN.Wei, Cotrollig cardiac chao, Sciece vol.57, ,199. S.Rajaekar, Cotrollig of chao by weak eriodic erturbatio i Duffig-va der Pol ocillator. Pramaa J. Phy. vol.41, , S.Rajaekar, M.Lakhmaa, Bifurcatio, chao ad ureio of chao i FitzHugh- Nagumo erve coductio model equatio, J Theor Biol., vol. 166, o.3,.75-88,