Abstract
Recently, many nonlinear control design methods have been developed to handle the nonlinearities of the systems to achieve better control performance than linear ones. In this paper, we consider a nonlinear optimal control design method based on stable manifold theory(1) . In nonlinear optimal control problem, we have to solve a partial differential equation called Hamilton-Jacobi equation (HJE). However, in general, it is difficult to solve the HJE analytically. In order to obtain its solution, Sakamoto et al proposed the stable manifold method to solve the HJE iteratively with high accuracy, and then based on the approximated solution, a nonlinear optimal controller is constructed. In this research, we consider a two degree of freedom acrobot. This is an underactuated system with strong nonlinearities, therefore it is often used to evaluate the control performance of nonlinear controller design methods. The control objective is to swing up and stabilize the acrobot around an equilibrium position using a single controller. In the acrobot system, there are four equilibrium points corresponding to different postures of the links. We consider six swing-up motion patterns starting from different initial conditions. For each motion, a nonlinear optimal controller is designed via stable manifold method. The simulation results confirm the effectiveness of the designed controllers.
