di Bernardo, M. (1995). A purely adaptive controller to synchronize and control chaotic systems.

Transcription

1 di ernardo, M. (995). A purely adapive conroller o synchronize and conrol chaoic sysems. hps://doi.org/.6/375-96(96)8-x Early version, also known as pre-prin Link o published version (if available):.6/375-96(96)8-x Link o publicaion record in Explore risol Research PDF-documen Universiy of risol - Explore risol Research General righs This documen is made available in accordance wih publisher policies. Please cie only he published version using he reference above. Full erms of use are available: hp://

2 A pure adapive conroller o synchronize and conrol chaoic sysems Mario di ernardo Deparmen of Engineering Mahemaics, Universiy of risol, Queen's uilding, Universiy Walk, risol S8 TR, U.K. M.Diernardo@brisol.ac.uk Absrac Following he work presened by he auhor in a previous paper, a model reference adapive conroller, which requires minimal knowledge of he sysem srucures, is synhesized o conrol or synchronize chaoic sysems. This is achieved by exploiing he boundedness of chaoic evoluions. Moreover, when he linear erm of he error equaion is characerized by a Hurwiz marix, he conrol law is reduced o a pure disconinuous acion, whose ampliude is adapively esimaed. The mehod is applied o he conrol of a Lorenz sysem and he synchronizaion of wo idenical Chua circuis.

3 Inroducion The problem of conrolling chaoic evoluions of nonlinear dynamical sysems or of synchronizing wo or more equivalen sysems have been approached in several dieren ways [Chen & Dong 993]. Adapive conrol sraegies have also been applied o solve he problem [di ernardo 995, Mossayebi, Qammar & Harley 99, Huebler 985], showing ha even sandard or slighly modied conrol engineering mehods can be applied o achieve he desired goal. This paper is an exension of he work presened by he auhor in di ernardo [995], in which an adapive conroller for chaoic sysems was designed and esed. Here, he original adapive scheme is furher modied. Firs, he requiremen of knowing a coninuous funcion, which upper-bounds he nonlineariy of he sysem o conrol, is removed. Then, any linear feedback acion wih xed gains is omied, in case he linear par of he error equaion is characerized by an Hurwiz marix. Therefore he original conrol law is modied ino one in which he eor of achieving synchronizaion and conrol is lef only o a pure adapive erm. A he same ime knowledge of he sysems involved is minimized. Finally, he mehod is applied o conrol a Lorenz sysem and o synchronize wo Chua circuis. The numerical resuls conrm wha is forecased by he heoreical background developed in he paper. The original conroller The adapive sraegy, presened in he paper menioned above, was concerned wih he problem of conrolling and synchronizing chaos. Namely, given wo sysems _x = f(x; ) + u; x R n ; _y = g(y; ) y R n ;

4 wih u R m ; R nm, he problem consiss of choosing an appropriae conroller u = u() in such a way as o have lim jx()? y()j = :! The sraegy proposed in di ernardo [995] can be oulined as follows. Firs, he error equaion is formed, An orhogonal projecion operaor : R n rewrien as _e() = _x()? _y() = f(x; )? g(y; ) + u: () _e() = Le() + [h(x; )? l(y; ) + u];! Im() is found so ha () could be where Le() is he projecion of f(x; )? g(y; ) on he complemenary space of Im(), which is assumed o be linear, and h(x; ), l(y; ) are he projecion on Im() of f(x; ) and g(y; ) respecively. Then, given a gain marix K R n, such ha b L = L? K is an Hurwiz marix, ha is all is eigenvalues are in he lef half plane, we solve he Lyapunov equaion P b L + b L T P + I = : () Finally, exploiing he fac ha he reference model is evolving eiher on a chaoic aracor or a limi cycle or an equilibrium poin (hence is evoluion is bounded, i.e. jl(y; )j W; W R + ), we form he conroller u() =?Ke()? k()( + (x)) T P e? T P e (3) In (3) (x) is a coninuous funcion upper-bounding he nonlineariy of he sysem o conrol and k() is adapively esimaed according o he law _k() = ( + (x)) T P e : (4) Using an appropriae Lyapunov funcion, i is possible o prove ha he error asympoically decays o zero, while k() ends oward a bounded value [di ernardo 995]. 3

5 3 A modied adapive approach If we suppose, now, ha he sysem o conrol is evolving in a chaoic regime and ha is nonlineariy is upper bounded by a coninuous funcion (x), we can hen deduce ha j(x)j T; T R (5) In ha case, he adapive esimaion law (4) can be modied o _k() = T P e wihou losing eiher he global asympoic sabiliy of he origin of he error sysem () or he boundedness of k(). Assuming (5) o be valid, we obain he following resul. ; Theorem Le P R nn be he posiive denie soluion of () and le _k() = T P e : The conroller u() =?Ke()? k() T P e? T P e guaranees ha for every iniial condiion (e(); k()) = (e ; k );. lim! k() = k < +;. lim! e() = : Proof. Consider he funcion V (e; k) = e T P e + (W + T? k) (6) We have ha V (e; k) is greaer han zero for all (e; k) R n R: Moreover diereniaing (6) we ge _V (e; k) = _e T P e + e T P _e? (W + T? k) _ k? kek + k T P e? e T P T P e + e T P h(x; )? e T P l(y; )? (W + T? k) T P e? kek + (kl(y; )k + kh(x; )k) T P e? (W + T ) T P e? kek 4

6 Therefore, along he soluion (e(); k()) _V (e(); k())? kek ; for almos all. Hence (e(); k()) is bounded and he proof can be compleed as in di ernardo [995], wihou subsanial modicaions. 3. Example (Conrolling he Lorenz Sysem) Given wo Lorenz sysems wih dieren parameer values, associaed wih wo disinc aracors, an equilibrium poin and a chaoic aracor respecively, we wan o nd an appropriae conroller o make he chaoic Lorenz sysem o behave as he non-chaoic one. Firs, we noice ha he evoluion of boh he sysems are bounded. Then, by looking a he srucure of he Lorenz model? _x = r? C x x 3 x x we decided o add he conrol only o he second sae of he chaoic sysem, hence T = : Therefore, if we call ; r ; b he parameers of he reference model (a he equilibrium poin), he error equaion () becomes _e() = Le() + r(x(); y()) + u; C ; A where L r??b C A ; r(x; y) + )y + ( + )y (r + r )y? y? x x 3 + y y 3?(b + b )y 3 + x x + y y C ; = C : A A linear gain marix K is hen chosen in order o have a sable L b = (L? K): Hence, he Lyapunov equaion is solved and he conroller (3) is synhesized. Figs.,,3 show he evoluions of he hree componens of he error, he adapively esimaed gain k() and 5

7 he conrol inpu u() respecively. As we can see, conrol is achieved afer a relaively shor ransien, while he gain evolves owards a consan value and he conrol inpu decays o zero, afer only a few swichings. 4 A pure adapive acion The conroller synhesized above consiss of wo dieren conribuions: a linear feedback erm and a disconinuous acion whose ampliude is adapively esimaed. If we now suppose ha in he linear erm of he error equaion (), he linear marix L is an Hurwiz marix (all eigenvalues having negaive real pars), we can omi he linear feedback erm, considering he conroller 8 >< >: _k() = T P e ; u() =?k() T P e? T P e; which consiss of a pure adapive conribuion. If he marix L is Hurwiz, we can prove ha here exiss a posiive denie marix P which saises he Lyapunov equaion (7) P L + L T P + I = : (8) and ha P is is unique soluion [Khalil, 99 p.7]. Therefore here is no need for a linear feedback acion o sabilize he linear par of he error equaion and he conrol (7), under his hypohesis, guaranees he claim of Theorem. Remark. The conrol law (7) requires no specic knowledge of he nonlineariies of eiher he plan or he reference model excep he fac ha heir evoluions are bounded, for example as a consequence of evolving on a chaoic aracor. Preliminary numerical resuls by he auhor have shown ha he conroller (7) gives excellen resuls even when L is no Hurwiz. In ha case, however, he marix P canno be obained as he soluion of he Lyapunov equaion, bu has o be chosen following a rial and error procedure. 6

8 4. Example (Synchronizing wo Chua circuis) The problem of synchronizing wo idenical Chua circuis saring from dieren iniial condiions (see Chua, Ioh, Kocarev & Ecker [993]) is solved following he sraegy oulined above. Given wo idenical Chua circuis? _x C x + f(x) wih f(x) = bx + (a? b)[jx + j? jx + j], one is considered as he reference model, _y = g(y), and he oher as he sysem o conrol, _x = f(x) + u, he conrol being added only o he rs sae of he sysem. Hence he error equaion is? _e C e + [f(x)? f(y)] from which we see he linear par is a Hurwiz marix, and so we can apply he conrol (7). The dynamics of he hree componen of he error sysem are shown in g. 4, while gs. 5, 6 repor he evoluion of k() and u() respecively. Even in his case synchronizaion is obained via a dissipaive conrol acion; once he conrol goal has been achieved he conrol is swiched o, leaving he sysems evolving coherenly and synchronously as required. C A C A C : A 5 Conclusion y exploiing he boundedness of chaoic evoluions, we have been able o minimize he adapive conroller srucure, removing any knowledge of he nonlineariies of he sysems involved from he conroller. In addiion o his, under cerain condiions, he conrol law is furher minimized and he conroller is lef wih only a pure disconinuous acion, whose ampliude is esimaed adapively. This shows ha, o a cerain exen, we can exploi chaos o make simpler he conrol of cerain classes of nonlinear sysems. I is only 7

9 because of he bounded evoluion of he sysem o conrol, ha we were able o remove from he conroller direc knowledge of he nonlineariies appearing in i. Furhermore, boh he heoreical and numerical resuls seem o sugges ha he adapive scheme proposed in his paper can be successfully applied o a large number of chaoic sysems, showing is exibiliy and simple implemenaion. Neverheless, many deails need sill o be invesigaed, for insance is robusness agains parameer variaions and exernal disurbances. These issues are lef for furher sudy. References Chen, G. & Dong, X. [993], `From chaos o order{perspecives and mehodologies in conrolling chaoic nonlinear dynamical sysems', Inernaional Journal of ifurcaion and Chaos 3, 363{49. Chua, L. O., Ioh, M., Kocarev, L. & Ecker, K. [993], `Chaos synchronizaion in chua's circui', Journal of Circuis, Sysems and Compuers 3, 93{8. di ernardo, M. [995], An adapive approach o he conrol and synchronizaion of coninuous-ime chaoic sysems, submied o he Inernaional Journal of ifurcaion and Chaos. Huebler, A. [985], `Adapive conrol of chaoic sysems', Helveica Physica Aca 6, 343{ 346. Mossayebi, F., Qammar, H. K. & Harley, T. T. [99], `Adapive esimaion and synchronizaion of chaoic sysems', Phys. Le. A6, 55{6. 8

10 e() 5 5 e() 5 5 e3() 5 5 Figure : Error dynamics for he conrolled Lorenz sysem k() Figure : Evoluion of he gain 9

11 3 u() Figure 3: Conrol inpu evoluion 6 4 e() Figure 4: Error dynamics of he conrolled Chua circui e () solid line, e () dashed line and e 3 () do-dashed line

12 6 5 4 k() Figure 5: Evoluion of he gain 6 4 u() Figure 6: Conrol inpu evoluion