The State Following Approximation Method.

Area of Science:

• Computational Mathematics
• Machine Learning Theory
• Applied Mathematics

Background:

• Efficient approximation of nonlinear functions is crucial for real-time simulations and experiments.
• Existing methods may struggle with accuracy and computational cost in dynamic environments.
• Reproducing kernel Hilbert spaces offer a theoretical framework for function approximation.

Purpose of the Study:

• To develop a novel state following (StaF) method for efficient nonlinear function approximation.
• To provide theoretical guarantees on the approximation accuracy and computational requirements.
• To demonstrate the method's applicability in derivative estimation, function approximation, and adaptive dynamic programming.

Main Methods:

• Development of the state following (StaF) method based on universal reproducing kernel Hilbert spaces.
• Introduction of theorems bounding the number of kernel functions required for approximation accuracy.
• Implementation of a gradient descent-based weight update law for achieving arbitrary accuracy.
• An experience-based approximation approach using radial basis function interpolation of weight estimates.

Main Results:

• A bound on the number of kernel functions needed for accurate approximation within a compact set is established.
• Arbitrarily close accuracy can be achieved with a sufficiently frequent weight update law.
• The StaF method demonstrated stability maintenance with a reduced number of basis functions in adaptive dynamic programming.

Conclusions:

• The state following (StaF) method offers an efficient strategy for approximating nonlinear functions in dynamic settings.
• The theoretical framework ensures accuracy and provides insights into computational efficiency.
• The method's successful application highlights its potential for real-time simulations and advanced control problems.