Linear and Nonlinear State-Feedback Controls of HOPF Bifurcation

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1 IOSR Journal of Mathematis (IOSR-JM) e-issn: , p-issn: X. Volume 1, Issue Ver. III (Mar. - Apr. 016), PP Linear and Nonlinear State-Feedbak Controls of HOPF Bifuration Mohammed O.Mahgoub 1, Mohammed A.Gubara 1 (Mathematis Department, College of Siene / Majmaa Universit, Saudi Arabia) (Mathematis Department, College of tehnolog of mathematial sienes/ Alnelain Universit, Sudan) Abstrat : Bifuration ontrol has gained inreasing attention in the last deade. This paper makes an attempt to highlight a simple and unified state-feedbak methodolog for developing Hopf bifuration linear or nonlinear ontrols for ontinuous-time sstems. The ontrol task an be either shifting an existing bifuration or reating a new one. Some numerial simulations are inluded to illustrate the methodolog and verif the theoretial results. Kewords: Eigenvalues, Hopf bifuration, Control. I. Introdution Bifuration ontrol refers to the ontrol of bifurative properties of a nonlinear dnamial sstem, thereb resulting in some desired sstem output behaviors. Tpial examples of bifuration ontrol inlude delaing the onset of an inherent bifuration, reloating an existing bifuration point, modifing the shape or tpe of a bifuration hain, introduing a new bifuration at a preferable parameter value, stabilizing a bifurated periodi trajetor, hanging the multipliit, amplitude, and/or frequen of some limit les emerging from bifuration, optimizing the sstem performane near a bifuration point, or a ertain ombination of some of these [,5,9,10]. Bifuration ontrol is also important as an effetive strateg for haos ontrol sine period-doubling bifuration is a tpial route to haos in man nonlinear dnamial sstems. Bifuration ontrol has great potential in man engineering appliations. In some phsial sstems suh as the stressed sstem, dela of bifurations provides an opportunit for obtaining stable operating onditions for the mahine beond the margin of operabilit at the normal situation. It is often desirable to be able to modif the stabilit of bifurated sstem limit les, in appliations of some onventional ontrol problems suh as thermal onvetion experiments [11]. Other examples inlude stabilization via bifuration ontrol in tethered satellites [6], magneti bearing sstems [], voltage dnamis of eletri power sstems [4]; dela of bifuration in ompressor stall in gas turbine jet engines, and in rotating hains via external periodi foring [11]; and in various mehanial sstems suh as robotis and eletroni sstems suh as laser mahines and nonlinear iruits [7]. Bifurations an be ontrolled b means of linear delaed-feedbak [7, 8] or nonlinear feedbak [1], emploing harmoni balane approximation [6], and appling the quadrati invariants in the normal form [8]. In this paper, a unified nonlinear as well as simple linear state feedbak tehnique is developed for Hopf bifuration ontrol, whih b nature is different from the aforementioned existing approahes. Both problems of shifting and reating a Hopf bifuration are disussed, and Computer simulations are inluded to illustrate the methodolog and to verif the theoretial results. II. Hopf Bifuration: Fundamentals In this setion, the lassial fundamentals of ontinuous-time Hopf bifuration are briefl reviewed. These bakground onepts are needed in the development of the bifuration ontrol tehnique given in the subsequent setions. First, onsider a two-dimensional ontinuous time autonomous parameterized sstem x f ( x, ; ) g ( x, ; ) (1) Where R is a real variable parameter, and both point ( x, ), satisfing f g C R 1, ( ). Assume that the sstem has an equilibrium f ( x, ; ) g( x, ; ) 0 for all R. Let J( ) be its Jaobian at this equilibrium, and suppose that J has a pair of omplex onjugate eigenvalues, ( ) with 1, 1. Assume that 1 moves from the left-half plane to the right as varies, so does, and the ross the imaginar axis at the moment in suh a wa that DOI: / Page

2 Re{ ( )} 1 Re{ 1 ( )} 0 and 0 () Where, Re {.} denotes the real part of a omplex number. Here, the seond ondition in () refers to as the transversalit ondition for the rossing of the eigenlous at the imaginar axis, whih means that the eigenlous is not tangent to the imaginar axis. Then, the sstem undergoes a Hopf bifuration at the bifuration point ( x,, ) [4,18]. More preisel, in an small left-neighborhood of small right-neighborhood of amplitude O( ). (i.e., (i.e., ), ( x, ) is a stable fous; and in an ), this fous hanges to be unstable, surrounded b a limit le of III. Controlling Hopf Bifuration The Hopf bifuration ontrol problem is the following: Design a (simple) ontroller, u( t; ) u( x, ; ) () That an move the existing Hopf bifuration point ( x,, ) to a new position ( x,, ), whih does not hange the original equilibrium point at ( x, ). It is lear that this ontroller must satisf u( x, ; ) 0 (4) We Can Add The Controller () To An Of Two Equations Of Sstem (1). (a) For the sake of alulation let us add the ontroller () to the first equation, namel: x f ( x, ; ) u ( x, ; ) g ( x, ; ) (5) This ontrolled sstem has a Jaobian at ( x, ) given b fx ux f u J ( ) gx g (6) With eigenvalues 1 ( ) ( ) ( ) 4[ ( ) ( )] 1, x x x x x x x u g f u g g f u g f u (7) To have a Hopf bifuration at ( x,, ) as required, the lassial Hopf bifuration theor sas that the following onditions must be hold: (i) ( x, ) is an equilibrium point of the ontrolled sstem (5), namel, f ( x, ; ) u ( x, ; ) 0 g ( x, ; ) 0 (8) (ii) The eigenvalues omplex onjugate: ( fx ux g ) 0 (9) ( ) of the ontrolled sstem (5) are purel imaginar at the point ( x,, ) and are 1, g ( fx ux ) gx ( f u ) 0 (10) ( fx ux g ) 4[ g ( fx ux ) gx ( f u ) ] 0 (11) (iii) The rossing of the eigenlous at the imaginar axis is not tangential ( transversal), namel Re{ ( ) 1 ( )} f x u g x 0 (1) (b) Now let us add the ontroller () to the seond equation, namel: x f ( x, ; ) g ( x, ; ) u ( x, ; ) (1) This ontrolled sstem has a Jaobian at ( x, ) given b DOI: / Page

3 fx f J ( ) g x ux g u (14) With eigenvalues 1 ( ) ( ) ( ) 4[ ( ) ( )] 1, x x x x x u g f u g f g u f g u (15) To have a Hopf bifuration at ( x,, ) as required, the lassial Hopf bifuration theor sas that the following onditions must be hold: (i) ( x, ) is an equilibrium point of the ontrolled sstem (1), namel, f ( x, ; ) 0 g ( x, ; ) u ( x, ; ) 0 (16) (ii) The eigenvalues ( ) of the ontrolled sstem (1) are purel imaginar at the point ( x,, ) and are 1, omplex onjugate: ( fx u g ) 0 (17) fx ( g u ) f ( gx ux ) 0 (18) ( fx u g ) 4[ fx ( g u ) f ( gx ux )] 0 (19) (iii) The rossing of the eigenlous at the imaginar axis is not tangential (transversal), namel Re{ ( ) 1 ( )} f x u g 0 (0) Now we present some examples to illustrate these results: Example I: Van Der Pol Osillator x x x (a) (Nonlinear ontroller) (1 ) (1) Clearl the sstem has a Hopf Bifuration at the point the bifuration value of from ( x,, ) (0, 0, 0). If the aim of ontrol is to move 0 to 0while preserving the equilibrium point of the sstem to be unaltered: ( x, ) = ( x, ) = (0,0), then the ondition (9) ields a nonlinear ontroller of the form x u x(1 ) () Whih satisfied all onditions required b equations (4) and (8) (1) stated in (i) (iii) above. More preisel, ( x ) sine ( x, ) (0, 0) and u ( x, ; ) x (1 ) 0, onditions (4) and (8) are satisfied. For ondition (10) to be satisfied we must show that g ( f u ) g ( f u ). Now g ( f u ) x x and gx ( f u ) 1 x x x 1, so ondition (10) must be satisfied merel for 1 1, so ondition (10) sas that 1 or, but as mentioned before we hoose onl for values 0. Thus the nonlinear ontroller (14) is valid onl for 0 1. Similarl ondition (11) is satisfied. Re{ 1 ( )} ( x x ) And finall, sine f u g 1 0 ondition (1) is satisfied. DOI: / Page

4 (b) (Linear ontroller) Again the sstem has a Hopf Bifuration at the point ( x,, ) (0,0,0). If the aim of ontrol is to move the bifuration value of from 0 to 0while preserving the equilibrium point of the sstem to be unaltered: ( x, ) = ( x, ) = (0,0), then the ondition (17) ields a linear ontroller of the form u (1 ( x ) ) () Whih satisfied all onditions required b equations (4) and (16) (0) stated in (i) (iii) above. We an hek these results b the same manner as we did in the nonlinear ontroller. Note that this linear feedbak ontroller ma signifiantl extend the operational range of the ke sstem parameter,, of the sstem as desired. Suh an extension an be ver important in some real appliations: it ma atuall enhane the stabilit and/or performane robustness of the dnamial sstems. It is ver important to mention that different bifuration ontrol tasks an be performed b means of different ontrollers for the Van Der Pol osillator. For example if the goal of ontrol is to shift the original equilibrium point of the sstem from ( x, ) =(0,0) to, sa ( x, ) == ( x, 0), with 0 x 1, while maintaining the bifuration parameter value at 0, then another linear feedbak ontroller an be designed. Atuall the simple onstant ontroller u x an perform the task. This fat an be easil heked b following the same steps disussed above. Example II: the above ontrol proedures an be applied to other dnamial sstems, for whih different ontrollers an be designed with satisfation of onditions (8) (1) or (16) (0). The following two sstems give tpial examples [1]: the first one is x x x x For whih linear ontroller (4) u ( ( ) ) 4 (0.1, 0.1, 0.9) and the nonlinear ontroller u (0.1, 0.1, 0.99), respetivel ; and the seond sstem is x x x x x For whih linear ontroller (5) hanges the original Hopf bifuration point (0, 0, 0) to u ( ( ) ) 1 6 (4 6 ) (,, ) and the nonlinear ontroller 4 8 (0.1, 0.1, 0.99), respetivel. DOI: / Page moves the original bifuration point to hanges the original Hopf bifuration point (0, 0, 0) to u x moves the original bifuration point to IV. Conlusion In this paper, a simple and unified state-feedbak ontrol strateg has been developed for Hopf bifurations, for ontinuous-time sstems and for both problems of shifting and reating a Hopf bifuration point in the ontrolled sstem. Other tpes of bifurations, for instane, pithfork, saddle node, transritial, and perioddoubling bifurations an also be ontrolled b similar linear or simple nonlinear ontrollers, whih are investigated in other topis and publiations. Referenes [1]. G. Chen, K. Yap and J. Lu, Feedbak Control of Hopf Bifuration IEEE Trans. on Cir. pp ,1998. []. S. Wiggins, Introdution to Applied Nonlinear Dnamial Sstems and Chaos, Springer-Verlag, New York, []. H. Yabuno, Bifuration ontrol of parametriall exited Duffing sstem b a ombined linear-plus nonlinear Feedbak ontrol, Nonlin. Dnazn., Vol. 1, pp. 6-74, [4]. M. E. Brandt and G. Chen, Bifuration ontrol of two nonlinear models of ardia ativit, IEEE Trans. on Cir. Ss. - I, Vol. 44, pp ,1997. [5]. G. Chen and X. Dong, From Chaos to Order: Perspetives - methodologies and appliations world sientifi Pub. Co., Singapore, [6]. R. Genesio, A. Tesi, H. 0. Wang, and E. H. Abed, Control of period doubling bifurations using harmoni balane, Pro. of Conf. on Deis. Contr., San Antonio, TX, 199, pp

5 [7]. G. Gu, X. Chen, A. G. Sparks, and S. S. Banda, Bifuration stabilization with loal output feedbak, 1998, preprint. [8]. W. Kang, Bifuration and normal form of nonlinear ontrol sstems, Parts I and 11, SIAM J. of Contr.Optjm., Vol. 6, [9]. E. H. Abed and J. H. Fu, Loal feedbak stabilization and bifuration ontrol, Ss. Contr. Lett., Part I: [10]. Hopf bifuration, , pp , 1986; Part 11: Stationar bifuration, Vol. 8, pp , [11]. E. H. Abed, H. 0. Wang, and R. C. Chen, Stabilization of period doubling bifurations and impliationsfor ontrol of haos, Pbsia D, Vol. 70, pp ,1994. [1]. S. Weibel and J. Baillieul, Osillator ontrol of bifurations in rotating hains, Pro. of Amer. Contr. Conf., Albuquerque, NM, June 1997, pp DOI: / Page