Approximation of Nonlinear Systems by Bilinear Ones

  • C. Hespel
  • G. Jacob
Part of the Mathematics and Its Applications book series (MAIA, volume 29)


Given an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k).

For that purpose, we use a noncommutative formal power series called the generating series of the system. Let s be the series, and y the output of the analytic system: we notice that the Taylor expansion of y up to order k in t=0 can be obtained from the coefficients of the words of length not greater than k in the series s. As rational series are characteristic of finite dimensional bilinear systems, the problem is reduced to the following: build a rational series g, which is an approximation of s up to order k (i.e. the coefficients of the words of length not greater than k in g and s are identical), and which is of minimal rank. Then with g we associate a bilinear system, which is a solution to our problem.


Taylor Expansion Rational Series Generate Series Bilinear System Volterra Series 
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Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • C. Hespel
    • 1
  • G. Jacob
    • 2
  1. 1.I.N.S.ARennes CedexFrance
  2. 2.Laboratoire de Recherche en Informatique Fondamentale (LA 369)Université de Lille IVilleneuve D’Ascq CedexFrance

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