Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

Abstract

Abstract-We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability.In this paper we use kinetic shaping to preserve symmetry and only stabilize systems modulo the symmetry group. In the sequel to this paper (Part II), we extend the technique to include potential shaping and we achieve stabilization in the full phase space.The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane.

Kinetic energy shaping in the inverted pendulum

“…Nevertheless, due to the fact that the first subsystem is not GAS, the usual forwarding stability result does not apply and a proof of asymptotical stability has been included. The resultant control law has strong similarities with the one proposed in (Bloch et al, 2000) but it shows some differences. Among them, it has different tuning parameters that allows us to improve the performance of the closed-loop system as is shown by means of simulations.…”

“…which is the same as (1.18) of (Bloch et al, 2000), because it is easy to show that −(kγ + β) = κβ, where κ is a parameter introduced in (Bloch et al, 2000).…”

“…This notation is very similar to the one in (Bloch et al, 2000) in order to facilitate the comparison. Based on Eq.…”

“…It is worth noting that control law (13) is the same as the one obtained in (Bloch et al, 2000) by the method of the controlled Lagrangians. This can be easily checked by substituting Eq.…”

“…Thus, in closed loop we look for a system with negative kinetic energy. It should be realized that the same happens in (Bloch et al, 2000). The condition α + f < 0 limits the domain of attraction of the desired equilibrium in the closedloop system 1 .…”