Trajectory tracking of inverted pendulums via Lie derivative of a scalar function

In this paper, we propose a nonlinear state feedback control law to satisfy a constraint on state variables. The constraint is expressed as an algebraic equation containing a scalar nonlinear function such as an energy function or a limit cycle. Computing the time-derivative of the scalar function that defines the constraint of the state variables, we derive a nonlinear controller to regulate the state variable satisfying the constraint. An inverted pendulum system is given to demonstrate the closed-loop performance of the proposed controller.