An appropriate notion for stability studies of dynamical systems with a continuum of equilibria is semistability. One approach to semistability investigation is based on the nontangency between the vector field and the set of equilibria. In this work, we introduce a novel way of verifying nontangency by computing an outer estimate of the direction cone and illustrate its application through two well-known second-order systems. The stable equilibria of both the systems are shown to possess the semistability property.©2014 by ASME.

Arun D. Mahindrakar

Department of Electrical Engineering

Anup K. Ekbote

Dynamical systems

ITS applications

Mechanical oscillators

Second-order systemss

Semistability

Stability study

Stable equilibrium

Vector fields

Oscillistors

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Semistability Analysis of the Chaplygin Sleigh and Nonsmooth Mechanical Oscillator

Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME

Semistability analysis of the chaplygin sleigh and nonsmooth mechanical oscillator

This paper focuses on semistability and finite-time semistability for discontinuous dynamical systems having a continuum of equilibria. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we extend the theory of semistability to discontinuous autonomous dynamical systems. In particular, Lyapunov-based tests for semistability and finite-time semistability for autonomous differential inclusions are established. © 2009 IEEE.

IEEE Transactions on Automatic Control

Semistability, finite-time stability, differential inclusions, and discontinuous dynamical systems having a continuum of equilibria

This paper focuses on the stability analysis of systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit points that are Lyapunov stable. We give new Lyapunov-function-based results for convergence and semistability of nonlinear systems. These results do not make assumptions of sign definiteness on the Lyapunov function. Instead, our results use a novel condition based on nontangency between the vector field and invariant or negatively invariant subsets of the level or sublevel sets of the Lyapunov function or its derivative and represent extensions of previously known stability results involving semidefinite Lyapunov functions. To illustrate our results we deduce convergence and semistability of the kinetics of the Michaelis-Menten chemical reaction and the closed-loop dynamics of a scalar system under a universal adaptive stabilizing feedback controller.

SIAM Journal on Control and Optimization

Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria

Nonholonomic Mechanics and Control,Interdisciplinary Applied Mathematics

Nonholonomic Mechanics

Journal of Dynamic Systems, Measurement, and Control

An Invariance Principle for Discontinuous Dynamic Systems With Application to a Coulomb Friction Oscillator

Journal	Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME
Publisher	American Society of Mechanical Engineers
ISSN	00220434
Open Access	No