On the concept of switching nonlinearity (a comment on "Switching control of linear systems for generating chaos" by X. Liu, K-L. Teo, H. Zhang and G. Chen)

E.Gluskin, On the concept of switching nonlinearity… 1 On the concept of sw itching nonlinearity (a comment on " Switchi ng control of linear syste m s for generating chaos " by X. Liu , K-L. Teo, H. Z hang and G. Chen) Emanuel Glus kin Holon Institute of Technology, Holon 58102, Israel, and Electrical Engineering . Israel , 84105 Beer Sheva , Gurion University of the Negev - Ben , Department il . ac . bgu . ee @ gluskin htm . gluskin / staff / lectronics e / departments / il . ac . hit . www :/ / http Abstract : It is explained and stressed that th e chaotic states in [1] are o btained by means of nonlinear switching . In the very interesting w ork [1] some chaotic states are obtained, but at no place nonlinearity of the system (or at all the term 'nonlinear') is mentioned, and the whole terminology of [1] can crea te impression that by means of switching in a li near system one ca n obtain chaos. However, it is well known that chaos can be obtained only in nonlinear systems, and it has to be clearly seen t hat in the method of [1] nonlinearity presents. As is s hown in [ 2] (see als o [3] and [4]) if the switching ins tants are defined by the unknowns to be f ound (i.e. the state-vari ables of the system under study), then the switched system is nonlinear . As i s seen from Section 2 of [1], the switching there, from one linear system to another, is done at t ime instants ass ociated with some constraints on the vector X( t ) of the state-variables of t he system. (We use nota tions of [1].) In other w ords, these instants depend on X, t 1(2) = t 1(2) (X). Thus, even though the switching is done in [ 1] between some per se linear systems, in view of [ 2] such a switching mea ns nonlinearity of the whole system . Since the switching instants that influence t he matrix via the elements being interplaced (or, e quivalently, one being c hanged), de pend on X, such whole system obeys equations of the type dX/d t = [A( t ,X)]X + … . (1) Thus, it should be spoken in [1] about nonlinear swi tching (control) of linear systems . It is im portant to see that if the switching instants were prescribed , -- the whole system would be linear ( time-variant be cause of the switchings), satisf ying the equations of the type dX/d t = [A( t )]X + …, (2) and no chaos would be obtained. E.Gluskin, On the concept of switching nonlinearity… 2 We do think that the point of terminology is a part of correct ac ademic outlook on the very complica ted field of switched circuits. There a re so many chaotic ci rcuits that, ot herwise, in a pa rticular research it indeed ca n be not easy for us "to see the forest behind the trees". Certainly, it has to b e clear that "switching between linear systems" can mean nonlinearity. Hoping t hat this comme nt will be useful for the R eaders of [1], we would like to stress that it is not purposed, in any sense, to decrease the importance of the method of chaos generation suggeste d in [1]. Moreover, the di rection of seeking complicated constraints on X( t ) (in [1], using norms), and not only the simple st concept of zero- crossing, or level -crossing of some x ( t ) ∈ X( t ), as i n [2], in order to obtain some nonlinearity needed f or chaos, seems to us t o be promising. In particular, such s tudies will better outline the borders of the concept of "switching nonli nearity" the usefulness of which is stressed in [2-4] and here. References : [1] X. Liu, K-L. Teo, H. Zhang and G. C hen, "Switching c ontrol of linear systems f or generating chaos", Chaos, Solitons and Fractals, 30 (2006), 725-733. [2] E. Gluskin, " A point of view on the line arity and nonlinearity of switched systems", Proceedings of 2006 IEEE 24th Conve ntion of Electrical and Electronics Engineers in Israel (15-17 Nov. Eilat), pp.110-114. (The work appears in the IEEE XPlore site.) [3] E. Gluskin, "A small theorem on the nonlinearity of switched systems", AEU -- Int. J. of Electronics and Communications, vol. 64 no. 4 (2008) pp. 260-265. [4] E. Gluskin, "The nonlinear -by-switching systems: a heuristic discussion of some singular systems", man uscript , 2007, [nlin.SI].