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Approximating linearizations for nonlinear systems

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Abstract

Given a nonlinear control system

$$\dot x(t) = f(x(t)) + \sum\limits_{i = 1}^m {u_i (t)g_i (x(t))}$$

on ℝn and a pointx 0 in ℝn, we want to approximate the system nearx 0 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 atx 0. In general, the linearizations based on the Taylor method and the Lie bracket approach are different. However, under certain mild assumptions, we show that there is a coordinate system for ℝn nearx 0 in which these two types of linearizations agree. We indicate the importance of this agreement by examining the time responses of the nonlinear system and its linear approximation and comparing the lower-order kernels in Volterra expansions of each.

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Authors and Affiliations

  1. Programs in Mathematical Sciences, The University of Texas at Dallas, Box 830688, 75083-0688, Richardson, Texas, USA

    L. R. Hunt

  2. Department of Electrical Engineering, University of Colorado, 80309, Boulder, Colorado, USA

    R. Su

  3. NASA Ames Research Center, MS 210-3, 94035, Moffett Field, California, USA

    G. Meyer

Authors
  1. L. R. Hunt
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  2. R. Su
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  3. G. Meyer
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Additional information

The research of L. R. Hunt was supported by NASA Ames Research Center under Grant Numbers NAG2-189 and NAG2-366 and the Joint Services Electronics Program under ONR Contract N00014-76-C1136. The research of R. Su was supported by NASA Ames Research Center under Grant Number NAG2-203 and the Joint Services Electronics Program under ONR Contract N00014-76-C1136.

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Hunt, L.R., Su, R. & Meyer, G. Approximating linearizations for nonlinear systems. Circuits Systems and Signal Process 5, 419–433 (1986). https://doi.org/10.1007/BF01599618

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  • DOI: https://doi.org/10.1007/BF01599618

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