Approximate solution method for the Hamilton-Jacobi equation based on stable manifold theory with applications

Abstract

A recently developed method for approximately solving the Hamilton-Jacobi equation in nonlinear control theory is introduced. The method utilizes the fact that a stabilizing solution of a Hamilton-Jacobi equation corresponds to a stable manifold of an associated Hamiltonian system. The contraction mapping argument, which is a standard tool in stable manifold theory, is converted into a successive approximation algorithm that constitutes the primary part of the computation for the stabilizing solution. The algorithm is written in a suitable way for computer calculations. Numerical approach for this algorithm is advantageous in that the computational complexity does not increase with respect to the accuracy of approximation and non-analytic nonlinearities such as saturation can be handled. Several applications will be reported, for example, swing up and stabilization of a 2-dimensional inverted pendulum (simulation), stabilization of systems with input saturation (simulation) and a (sub)optimal servo system design for magnetic levitation system (experiment).