# A Direct Lyapunov Approach for Stabilization of Underactuated Mechanical Systems

@article{White2007ADL, title={A Direct Lyapunov Approach for Stabilization of Underactuated Mechanical Systems}, author={W. White and M. Foss and Xin Guo}, journal={2007 American Control Conference}, year={2007}, pages={4817-4822} }

Control of underactuated systems is treated from a Lyapunov direct method approach. The method results in a set of three matching conditions, the solution of which is easily accomplished. The method developed is capable of treating more complicated systems than that reported in an earlier publication. The suitability of the Lyapunov candidate function is demonstrated through mathematical proofs. An application of the method to the ball and beam is presented.

#### 18 Citations

Improvements in direct Lyapunov stabilization of underactuated, mechanical systems

- Engineering, Computer Science
- 2008 American Control Conference
- 2008

A Lyapunov direct method is presented for the stabilization of underactuated, mechanical systems and a method is demonstrated for assuring the positive definiteness of certain matrices associated with the formulation. Expand

Stabilisation of a class of 2-DOF underactuated mechanical systems via direct Lyapunov approach

- Computer Science, Engineering
- Int. J. Control
- 2013

This paper represents an alternative stabilisation procedure for a class of two degree-of-freedom underactuated mechanical systems based on a set of transformations and a Lyapunov function and asymptotic convergence for each system is proven by means of La Salle's invariance principle. Expand

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

- Mathematics
- 2010

Abstract 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… Expand

Enlarging the region of attraction for underactuated systems using impulsive inputs

- Computer Science, Mathematics
- 2013 American Control Conference
- 2013

An impulsive control algorithm is proposed to enlarge this region of attraction of the stabilized equilibrium and the algorithm uses impulsive inputs that are applied in conjunction with the stabilizing controller. Expand

The direct Lyapunov method for the stabilisation of the Furuta pendulum

- Mathematics, Computer Science
- Int. J. Control
- 2010

Using the Lyapunov direct method, a nonlinear controller for the stabilisation of the Furuta pendulum is presented and it is demonstrated that the closed-loop system has a large region of attraction. Expand

On the stabilization of the ball and beam system using a direct Lyapunov method

- Mathematics
- 2009

I presents the stabilization of the ball on the actuated beam, whether the damping force acts or not in the non-actuated coordinate. The control law is based on the application of the traditional… Expand

On the stabilization of the ball and beam system using a direct Lyapunov method

- 2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)
- 2009

I present the stabilization of the ball on the actuated beam, whether the damping force acts or not in the non-actuated coordinate. The control law is based on the application of the traditional… Expand

The Lyapunov direct method for the stabilisation of the ball on the actuated beam

- Mathematics
- 2009

This article deals with the stabilisation of a ball on an actuated beam independently of whether or not the force acts on the non-actuated coordinate. The feedback control law is deduced via… Expand

Distributed backstepping control for synchronization of networked class of underactuated systems: A passivity approach

- Engineering
- 2011 19th Mediterranean Conference on Control & Automation (MED)
- 2011

This paper considers the cooperative control problem of networked class of underactuated systems. We propose a new set up for synchronization of this type of systems based on the use of passivity as… Expand

Stabilization of uncoupled PVTOL aircraft based on a Lyapunov function

- Engineering
- 2012

In this paper we present a stabilization procedure for PVTOL (planar vertical take off and landing) aircraft with zero coupling coefficient that is defined as the effect of rolling moment on lateral… Expand

#### References

SHOWING 1-10 OF 11 REFERENCES

Control of nonlinear underactuated systems

- Computer Science, Mathematics
- 1999

In this paper we introduce a new method to design control laws for nonlinear, underactuated systems. Our method produces an infinite-dimensional family of control laws, whereas most control… Expand

Feedback stabilization of underactuated nonlinear pendulum cart system using matching conditions

- Mathematics
- Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301)
- 2002

The feedback stabilization problem of an underactuated nonlinear system is considered in this paper. The example system considered here is an inverted pendulum-cart system. The PDEs describing the… Expand

A direct Lyapunov approach for a class of underactuated mechanical systems

- Computer Science, Mathematics
- 2006 American Control Conference
- 2006

It is shown for this class of systems, that the solutions of these linear differential and partial differential equations necessary for assuring asymptotic stability can be evaluated numerically as part of the feedback process. Expand

Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment

- Mathematics, Computer Science
- IEEE Trans. Autom. Control.
- 2002

This work describes a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction, given in terms of solvability of certain partial differential equations. Expand

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

- Mathematics, Computer Science
- Proceedings of the 2004 American Control Conference
- 2004

The problem of (asymptotic) stabilization of mechanical systems with underactuation degree one is considered and a state-feedback design is derived applying the interconnection and damping assignment passivity-based control methodology that endows the closed-loop system with a Hamiltonian structure with desired potential and kinetic energy functions. Expand

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

- Mathematics, Computer Science
- IEEE Trans. Autom. Control.
- 2000

A method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians is developed, which uses kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. Expand

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

- Computer Science, Mathematics
- IEEE Trans. Autom. Control.
- 2001

The method of controlled Lagrangians is extended to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems and extends the method to include a class of mechanical system without symmetry such as the inverted pendulum on a cart that travels along an incline. Expand

Partial Differential Equations: An Introduction

- Mathematics
- 1992

Chapter 1: Where PDEs Come From 1.1 What is a Partial Differential Equation? 1.2 First-Order Linear Equations 1.3 Flows, Vibrations, and Diffusions 1.4 Initial and Boundary Conditions 1.5 Well-Posed… Expand

Nonlinear Systems, Upper Saddle River

- NJ: Prentice-Hall,
- 2002

Partial Differential Equations, An Introduction, Boston, Massachusetts: Allyn and Bacon, 1972

- FrA18.3
- 1972