Second‐order sliding mode control of underactuated mechanical systems I: Local stabilization with application to an inverted pendulum

Abstract

. Second order sliding mode control of underactuated Mechanical systems I: Local stabilization with application to an inverted pendulum. International Journal of Robust and Nonlinear Control, Wiley, 2008, 18 (4-5) AbstractSecond order sliding mode control synthesis is developed for underactuated mechanical systems, operating under uncertainty conditions. In order to locally stabilize an underactuated system around an unstable equilibrium, an output is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Then, the desired stability property of the closed-loop system is provided by applying a quasihomogeneous second order sliding mode controller, driving the system to the zero dynamics manifold in finite time. Although the present synthesis exhibits an infinite number of switches on a finite time interval, it does not rely on the generation of first order sliding modes, while providing robustness features similar to those possessed by their standard sliding mode counterparts. A second order sliding mode appears on the zero dynamics manifold which is of co-dimension greater than the control space dimension. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of a cart-Pendulum system.

Stabilization of the cart pole system: by sliding mode control

“…Then, we proposed the strategy as a combination of a linear PD controller and a modified version of the twisting algorithm [16,28,[34][35][36] ,where the first acts over the pendulum position and velocity, while the second brings the whole system state to the origin. The corresponding stability analysis was carried out by using several Lyapunov functions.…”

“…Convincing numerical simulations were done to assesses the performance of the proposed control strategy. Finally, we mention that this work was inspired by [35,36]. However, our strategy, when the controller gains are adequately selected, may guarantee global convergence, as long as the position pendulum is initialized into the upper-half plane, though it has the disadvantage of being less robust in the presence of unmodeled perturbations.…”

“…To provide an idea of how good the proposed control strategy OC is, we compared it with the control technique proposed by Riachy et al in [35], here referred to as RC. The control parameters of RC were tuning heuristically, but to be fair, we tried to find the values that enable the best transient response.…”

“…1 The inequities in (14) are referred to in the sequel as assumption A1. propose v s as a discontinuous injection based on the twisting control algorithm [28,[34][35][36] .That is,…”

Second‐order sliding mode control of underactuated mechanical systems II: Orbital stabilization of an inverted pendulum with application to swing up/balancing control

“…We demonstrate that the quasihomogeneous synthesis, developed in the first part of the paper [17], is applicable to the cart-pendulum system to design a Second Order Sliding Mode (SOSM) controller that drives an actuated part of the system to a periodic reference orbit in finite time in spite of the presence of friction forces and external disturbances with an a priori known magnitude bound. The resulting controller, that presents a contribution of the paper, exhibits an infinite number of switches on a finite time interval, however, in contrast to first order sliding mode controllers, it does not rely on the generation of sliding motions on the switching manifolds but on their intersections.…”