Identification of look-up tables using gradient algorithm.

Area of Science:

• Embedded Systems Engineering
• Control Systems Theory
• Computational Mathematics

Background:

• Look-up tables (LUTs) are crucial for approximating nonlinear functions and system variables in embedded applications due to their low computational cost.
• Offline identification of LUT parameters is challenging, particularly for complex nonlinear systems.

Purpose of the Study:

• To develop an efficient gradient-based algorithm for offline estimation of look-up table parameters.
• To provide a theoretical framework for ensuring the convergence of parameter estimation.
• To validate the proposed method on both 1D and 2D look-up table identification tasks.

Main Methods:

• Approximation of nonlinear functions using piecewise linear interpolation with LUT parameters.
• Reformulation of the approximation as a dot product using membership functions.
• Application of a gradient algorithm to update LUT parameters based on approximation error.
• Derivation of conditions for persistent excitation of LUT input to guarantee convergence.

Main Results:

• An explicit derivation of the relationship between parameter estimation error and model approximation error.
• A condition for persistent excitation of the look-up table input was established, providing a theoretical basis for data requirements.
• Successful validation of the method through the identification of 1D and 2D look-up tables, including a throttle discharge coefficient for an engine simulation.

Conclusions:

• The proposed gradient-based method effectively estimates look-up table parameters for nonlinear function approximation in embedded systems.
• The derived persistent excitation condition ensures reliable parameter convergence for offline identification.
• The method's efficacy is demonstrated across various dimensions and a practical engine application.