Identification of look-up tables using gradient algorithm

In view of the relatively low computational load, look-up tables (or maps) are usually used to approximate nonlinear function or characterize operating-point-dependent system variables in typical embedded applications. Aiming at the problem of off-line identifying the look-up tables, a method based on the gradient algorithm is presented to estimate the look-up table parameters in this paper. The nonlinear function is approximated in terms of the piecewise linear interpolation model with the look-up table parameters, which can be rewritten as a dot product between the regression vector and unknown parameter vector using membership function. With the approximation error of the nonlinear function, a method for updating look-up tables using the gradient algorithm is given, and the relationship between the parameter estimation error and model approximation error is explicitly derived. To guarantee the convergence of the look-up table parameters estimation, a condition for the persistent excitation of the look-up table input is derived, which also provides a theoretical basis for the data characteristics of the look-up table input required to identify look-up table parameters offline using dynamic data. The validity of the proposed method is verified respectively by updating a one-dimensional (1D) look-up table, and the identification of the two-dimensional (2D) look-up table for the throttle discharge coefficient of a spark ignition gasoline engine form engine simulation tool enDYNA.