Study of the Variance of Parametric Estimates of the Best Linear Approximation of Nonlinear Systems

Abstract

Abstract-This paper analyzes the variance of the estimated parametric best linear approximation (BLA) Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ) of a nonlinear system that is driven by random excitations. The estimated model Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q; θ) varies not only due to the disturbing measurement and process noise but also over different realizations of the random excitation because the nonlinear distortions depend on the input realization. For the nonparametric frequency response function (FRF) estimate, it has been shown that the variance expression is still valid in the presence of nonlinear distortions, and the same formulas can be used in the linear as in the nonlinear case. This result does not hold for the variance σ(Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> on the parametric estimate Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ). It is shown in this paper that it is still possible to upperbound the variance σ(Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> using linear expression by introducing an additional scaling factor that depends upon the maximal degree of nonlinearity.