Case studies on the application of the stable manifold approach for nonlinear optimal control design

This paper presents application results of a recently developed method for approximately solving the Hamilton–Jacobi equation in nonlinear control theory. The method is based on stable manifold theory and consists of a successive approximation algorithm which is suitable 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. First, the stable manifold approach for approximately solving the Hamilton–Jacobi equation is reviewed from the computational viewpoint and next, the detailed applications are reported for the problems such as 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).