Polynomial Bases for Optimal Worst Case Approximation of Nonlinearities

Several identification and control problems present nonlinearities that cannot be neglected and are often approximated by polynomials. In some previous works optimal set of interpolation nodes that minimizes the uncertainties of the approximation have been derived for the Vandermonde base that, however, can lead to ill-conditioned numerical problems. In this paper the conditions under which polynomial bases, used for representing static nonlinear blocks, derived by linear transformation from the Vandermonde base preserve the optimal worst case design features of the Vandermonde base are investigated. Explicit meaningful geometrical and analytical conditions to which the transformation matrix must satisfy in order to allow the new base to maintain the optimal sampling schedule of the Vandermonde matrix are derived.