1,012 research outputs found
Lyapunov constraints and global asymptotic stabilization
In this paper, we develop a method for stabilizing underactuated mechanical systems by imposing kinematic constraints (more precisely Lyapunov constraints). If these constraints can be implemented by actuators, i.e., if there exists a related constraint force exerted by the actuators, then the existence of a Lyapunov function for the system under consideration is guaranteed. We establish necessary and sufficient conditions for the existence and uniqueness of constraint forces. These conditions give rise to a system of PDEs whose solution is the required Lyapunov function. To illustrate our results, we solve these PDEs for certain underactuated mechanical systems of interest such as the inertia wheel-pendulum, the inverted pendulum on a cart system and the ball and beam system
Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum
The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed-loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed-loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed-loop that converges to the inverted equilibrium manifold is identified. Copyright Š 2007 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/56146/1/1178_ftp.pd
Dynamics of Simple Balancing Models with State Dependent Switching Control
Time-delayed control in a balancing problem may be a nonsmooth function for a
variety of reasons. In this paper we study a simple model of the control of an
inverted pendulum by either a connected movable cart or an applied torque for
which the control is turned off when the pendulum is located within certain
regions of phase space. Without applying a small angle approximation for
deviations about the vertical position, we see structurally stable periodic
orbits which may be attracting or repelling. Due to the nonsmooth nature of the
control, these periodic orbits are born in various discontinuity-induced
bifurcations. Also we show that a coincidence of switching events can produce
complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure
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
3 sampled-data control of nonlinear systems
This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research
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