Direct Lyapunov Approach Research Articles

Stabilization of a Class of 2-DOF Underactuated Mechanical Systems Via Lyapunov's Direct Approach

We study a class of 2-DOF underactuated mechanical systems(UAMS) where the objective is to obtain a design framework for stabilization. We propose a stabilization procedure via Lyapunov's direct approach to a class of UAMS employing partial feedback linearization and a change of coordinates. Asymptotic stability of unstable equilibrium point has been achieved in a large region of attraction and we make use of La Salle's theorem to prove it. Designed controller scheme has been successfully illustrated through simulations on two different UAMS namely the pendulum on a cart and the inertia wheel pendulum.

Decentralized Stabilization of Uncertain Systems With Interconnection and Feedback Delays: An LMI Approach

This note presents a broad LMI condition that can ascertain the stability of uncertain systems under decentralized feedback in the presence of interconnection and feedback delays. Based on the Lyapunov's direct approach with a four-term energy functional and a three-term quadratic formulation of the given state dynamics, this method has a larger search space than used so far. Numerical examples corroborate the superiority of this method vis-a-vis the existing ones for several subsets of the general problem.

Observer Design for a Class of Transport-Reaction Systems: a Direct Lyapunov Approach

The paper considers the observer design problem for a class of scalar distributed transport-reaction systems with boundary measurements, mixed Dirichlet-Neumann boundary conditions, linear (diffusive-convective) transport and nonlinear conically bounded source. A direct Lyapunov method is developed for the exponential stability analysis of the observation error. The Lyapunov functional is chosen with a constant kernel for the small values of the so-called Peclet number (i.e. the quotient of diffusion to convection time) and with a spatial-dependent kernel for high Peclet numbers. The derived Lyapunov-based stability conditions correspond to the ones based on the frequency-domain analysis. A numerical example illustrates the efficiency of the method.

Global stabilization for constrained robot motions with constraint uncertainties

In this paper, the global stability problem for constrained robot motions in the presence of constraint uncertainties is investigated. We focus on the uncertainties in the constraint functions and their effects on the global stability. PD type controllers are used and conditions for global stability are developed using Lyapunov's direct approach. In the presence of the constraint uncertainties under investigation, the desired position and constraint force can be guaranteed with global asymptotic convergence. The developed conditions for feedback gain selections clearly show the effects of the constraint uncertainties. For the case when the velocity measurements are not available, conditions for global stability regulation are also established and the robot controller uses only the measurements of the position angles. Finally, we consider the case where the robot joints are flexible and global stability conditions are given.

A Direct Lyapunov Approach to Volterra Integrodifferential Equations

We propose a different approach to the Lyapunov theory for Volterra integrodifferential equations. Instead of using Lyapunov functionals we use Lyapunov functions, which typically are the norm of the solution raised to some power. Our Lyapunov functions are not decreasing along solutions, so we use separate estimates to bound them from above. Our approach is direct and straightforward, and it appears to be applicable to most of the results that have been proved earlier by means of Lyapunov functionals.

Stationary-state solutions to the rotation of solid bodies with liquid-filled cavities and their stability

In this paper, we discuss all the possible equilibrium states of axi-symmetrical solid bodies with liquid-filled cavities rotating around fixed axes according to the extremum conditions on the potential energy, and conclude that there exists a unique stable final-state solution, for which the system uniformly rotates around its vertical symmetrical axis, for both the inverted and suspended ones. And then applying the Lyapunov direct approach for a continuous system, we investigate the stability of the rotating systems subject to large disturbances. In addition, we describe an interesting analogue between the rotation of a solid body with a liquid-filled cavity in the inverted case and the motion of a small ball in a spinning spherical bowl. The results obtained herein theoretically provide an evidence of the reality of the secular stability.