Forwarding control of cart-pendulum system by following homoclinic orbit and stabilizing cart

In this paper, we design a controller that attains a homoclinic motion of the pendulum and the asymptotic stability of the cart by using a kind of forwarding control design. First, we derive a controller that converges to a homoclinic orbit via a Lyapunov function of the pendulum subsystem. Next, we give a nonlinear stabilizing controller via another Lyapunov function of the cart subsystem. Moreover, using the third Lyapunov function and adding a complementary control input, we guarantee that the pendulum converges to the homoclinic orbit and the cart is stabilized. Finally, the simulation with MATLAB/Simulink is performed to demonstrate the validity of the proposed control law.