41 research outputs found
Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem
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. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. 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
Matching and stabilization of discrete mechanical systems
Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems
with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in
the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for
matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers
Physical Dissipation and the Method of Controlled Lagrangians
We describe the effect of physical dissipation on stability of
equilibria which have been stabilized, in the absence of damping,
using the method of controlled Lagrangians. This method
applies to a class of underactuated mechanical systems including
“balance” systems such as the pendulum on a cart. Since
the method involves modifying a system’s kinetic energy metric
through feedback, the effect of dissipation is obscured.
In particular, it is not generally true that damping makes a
feedback-stabilized equilibrium asymptotically stable. Damping
in the unactuated directions does tend to enhance stability,
however damping in the controlled directions must be “reversed”
through feedback. In this paper, we suggest a choice
of feedback dissipation to locally exponentially stabilize a class
of controlled Lagrangian systems
Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems
The method of controlled Lagrangians for discrete mechanical systems is
extended to include potential shaping in order to achieve complete state-space
asymptotic stabilization. New terms in the controlled shape equation that are
necessary for matching in the discrete context are introduced. The theory is
illustrated with the problem of stabilization of the cart-pendulum system on an
incline. We also discuss digital and model predictive control.Comment: IEEE Conference on Decision and Control, 2006 6 pages, 4 figure
Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System
Matching techniques are developed for discrete
mechanical systems with symmetry. We describe new phenomena
that arise in the controlled Lagrangian approach for mechanical
systems in the discrete context. In particular, one needs
to either make an appropriate selection of momentum levels or
introduce a new parameter into the controlled Lagrangian to
complete the matching procedure. We also discuss digital and
model predictive control