Stabilization of a chain of integrators with nonlinear perturbations: application to the inverted pendulum

Abstract

A solution to the long standing problem of the pendulum on a cart is presented. This solution insolves a reformulation of the original sjstem model in order to show that the inverted pendulum system belongs to a particular class of nonlinear systems: a class consisting of four cascaded integrators and a nonlinear "perturbation" term. Our controller first brings the pendulum close to the vertical unstable equilibrium p i n t and then regulates the cart position around the origin. We prore that using the proposed control strategy, all state rariables converge to zero for a &en set of initial conditions ($(O), B(0)) belonging to a special domain of attraction.

Stabilization of the Inverted Pendulum via a Constructive Lyapunov Function

“…The purpose of the first experiment was to assess that the strategy effectively brings the pendulum to its unstable upright position if it is located somewhere in the upper half plane. Once we were sure that our strategy worked properly, we compared its performance with the performance of both the well-known traditional linear control (TLC) and the Saturation Basic Controller (SBC) proposed by [20]. Although an exhaustive comparative analysis between our controller and the other two is beyond the scope of this work, we did not want to miss the opportunity to present, at least, the results of the simulations of the three schemes under the same conditions to have an idea of where we can locate our method according to its performance.…”

“…Let us then compute the largest invariant set M in S. Clearly, on the set S, it follows that the auxiliary variable w, given by w = q + (k 1 + k d ) sin θ , is a fixed constant on the set S. From the time derivative of E 2 (20), it is clear that the controller was selected, so that…”

“…That is, the pendulum is brought to the unstable top position, with zero displacement of the cart, assuming that the initial angle position of the pendulum satisfies |θ(0)| < π/2. For this, first a suitable candidate Lyapunov function is shaped by means of the technique of added integration, as presented in [20]. Secondly, a control law is obtained with its respective domain of attraction.…”

“…The resulting closed-loop system is asymptotically stable without switching to a stabilizing local controller. A similar work, using nested saturation function approach, is presented in [20]. The authors first write an approximate model of the IPC model as a chain of integrators, by using a convenient transformation.…”

Lyapunov function-based non-linear control for two-wheeled mobile robots

“…The CLF is shaped by means of the technique of added integration, presented by Lozano and Dimogianopoulos (2003). First, we introduced a Lyapunov function that represents the energy of the IB of the TWRs with certain constraint.…”