Approximating Linearizations for Nonlinear Systems

Given a nonlinear control system m x(t) = f(x(t)) + I u.(t)g.(x(t)J on H and a point x. in R, we want to approximate the system near x by a linear system. Of course, one approach is to use the usual Taylor series linearization. However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration. This suggests that we should construct a linear approximation based on Lie bracket matching at x_. IK general, the linearizations based on the Taylor method and the Lie , .'f bracket approach are different. However, under certain mild assumptions, we show that there is a coordinate system for K near x in which these two types of linearizations agree. We indicate the imf portance of this agreement by examining the time responses of the n<in..-; linear system and its linear approximation and comparing the lower ' order kernels in Volterra expansions of each. {Na'sa-TM-88772) APPBOXIMailNG N86-29578 LIBEABIZailOBS.IOE BOBLIH'EAB SYSTEMS semiannual Progress Report (KiSft) 23 p CSC1 12A Unclas G3/64 U3229 https://ntrs.nasa.gov/search.jsp?R=19860020106 2017-09-12T19:29:20+00:00Z