A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.

A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.

We propose a Lyapunov-function-based control for the stabilization of the under-actuated Furuta pendulum. Firstly, by a suitable partial feedback linearization that allows to linearize only the actuated coordinate of the system, we proceed to find a candidate Lyapunov function. Based on this candidate function, we derive a stabilizing controller, in such away that the closed-loop system is locally and asymptotically stable around the unstable vertical equilibrium rest, with a computable domain of attraction.

A simplified model of the hovercraft system, used in the literature to illustrate nonlinear control options in underactuated systems, is shown to be diflerentially flat. The flat outputs are given by the position coordinates with respect to the fixed earth frame. This fact is here exploited for the design of a dynamic feedback controller for the global asymptotic stabilization of the system's trajectory tracking error with respect to off-line planned position trajectories.

A simple way to determine the parameters of Rössler's system based on a suitable output is presented in this paper. The fact that the nonlinear system is observable and algebraically identifiable with respect to the selected output, allows us to propose, in a first stage, a high-gain observer to estimate the output time derivatives. And then, based on these facts two suitable schemes to recover the parameters are presented.