FP13 1450 Proceedings of the 37th IEEE Conference on Decision & Control Tampa, Florida USA December 1998 Differential representations of driftless discrete-time dynamics S. MONACO D. NORMAND-CYROT Dipartimento di Informatica e Sistemistica Universitj di Roma “La Sapienza” Via Eudossiana 18, 00184 Rome, Italy. Tel.: (39) 6 44 58 58 73 Fax: (39) 6 44 58 53 67 Laboratoire des Signaux et Systkmes CNRS ~ Supklec, Plateau de Moulon 91192 Gif-sur-Yvette, France Tel.: (33) 1 69 85 17 48 Fax: (33) 1 69 41 30 60 e-mail:monaco@dis.uniromal .it e-mail:cyrot@lss.supelec.fr Abstract The paper deals with nonlinear driftless discrete-time dy- namics. It is shown that such dynamics can always be represented by an exponential form which corresponds to the solution of a suitable differential equation. This enables us to characterize the associated Lie group struc- ture and to further understand invariance, accessibility and passivity properties. 1 Introduction Following the ideas of [12],[11] and [13], it will be shown in this paper that a nonlinear driftless discrete-time dy- namics assigned through a set of first-order difference equations admits a differential representation with re- spect to the control variables. As shown in [13],this new differential interpretation leads to considering discrete- time dynamics as mixed ones composed of a differential equation, enlightening the control dependency and a dis- crete one associated with the drift jump. In this context analysis and control properties can be revisited and may even lead to further understanding. This has been il- lustrated in the context of invariance and accessibility, passivity, feedback linearization, decoupling ([ll], [lo], Two well-known difficulties occur when dealing with nonlinear discrete-time dynamics: the nonlinearity in the control variables and the drift term evolution which gen- erates discrete jumps and renders a local study difficult. The current literature is thus generally restricted to the analysis around an equilibrium pair. These considera- tions motivate the present paper. Dealing with driftless [121). dynamics avoids the jump caused by the drift term and makes possible a complete differential geometric analy- sis. The canonical family of vector fields introduced in [8] with reference to general drift invertible discrete-time dynamics is the starting point of the present study. It is shown that the associated Lie algebra corresponds to the Lie algebra of the Lie group associated with the opera- tion of substitution of one function into another; such an operation being at the basis of the analysis of discrete- time dynamics over several steps. The paper is organized as follows. Section 2 introduces the differential representation of driftless dynamics and the group action associated with the operation of substi- tution is stressed thus characterizing the dynamics be- haviour over several steps. Section 3 discusses some geo- metric properties and particular cases. Section 4 discuss passivity and losslessness conditions for driftless dynam- ics. 2 Nonlinear driftless dynamics Let the driftless discrete-time dynamics be where x belongs to M, an analytic n-dimensional man- ifold, F : M x IR -+ M, is an analytic function of both arguments satisfying F(., 0) = 0. For U in a neighbour- hood U, of 0, let the expansion of Id + F( ., U) be Id + F(.,u) = Id + ‘11 22 1 0-7803-4394-8198 $10.00 0 1998 IEEE 4620 Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on April 23,2010 at 13:38:47 UTC from IEEE Xplore. Restrictions apply.