On the stability of motion

“…If H = {0}, then Definition 2 reduces to the standard definition of positive definiteness [10, p. 4] while Definitions 3 and 4 reduce to Definitions 1.1 and 1.5, respectively, of Matrosov [3]. If, in addition, f(t,0) = 0 for all t > 0, then Definition 5(i) reduces to Duboshin's definition of total stability of the zero solution of (N).…”

Section: Notation and Definitionsmentioning

confidence: 99%

Total Stability of Sets for Nonautonomous Differential Systems

Athanassov¹

1986

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

ABSTRACT. The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations. The concept of stability is one of those which reaches beyond the general domain in mathematics. In his well-known dissertation Problème général de la stabilité du mouvement, Liapunov [1] has given several criteria for stability and asymptotic stability of solutions of systems of differential equations with the help of certain auxiliary scalar functions which are now commonly called Liapunov functions. The problem of stability and relationships between stabilities and Liapunov functions has been discussed by many authors since Liapunov.The classical Liapunov theorem on asymptotic stability of the zero solution for a differential equation x = f(t, x) uses a positive definite function V (t, x) whose time derivative V(t, x) along solutions of the differential equation has to be negative definite. However, this stronger property is not always possessed by the natural candidate for the Liapunov function, namely, the energy. The damped harmonic oscillator is the standard example.An effective tool to overcome the difficulty was developed by LaSalle, and is

show abstract

“…For an arbitrary control input, an explicit function of time, we resort to one of the techniques called "invariance-like" methods by Khalil [21]. In particular, we use the theorem first presented by Matrosov [22] and extended to non-scalar auxiliary functions by Rouche [23]. We refer readers to the translated Matrosov paper and the Rouche paper for proofs of the original theorems.…”

Section: A Stability Analysismentioning

confidence: 99%

An almost global estimator on SO(3) with measurement on S<sup>2</sup>

Swensen

Cowan

2012

2012 American Control Conference (ACC)

View full text Add to dashboard Cite

Abstract-This paper presents an almost globally convergent state estimator for the orientation of a rotating rigid body. The estimator requires knowledge of the angular velocity of the body at each time instant and the measurement consists of a single unit vector on the body, which we take without loss of generality to be the first column of the rotation matrix. The stability proof involves a relatively simple Lyapunov and invariance-like analysis. A mild non-degeneracy constraint on the control input guarantees the fulfillment of the invariance criterion. We apply the result to needle-tip orientation estimation for tip-steerable needles.

show abstract

On the stability of motion

Cited by 152 publications

References 0 publications

Uniform stability of sets for difference inclusions under summability criteria

Uniform stability of sets for difference inclusions under summability criteria

Total Stability of Sets for Nonautonomous Differential Systems

An almost global estimator on SO(3) with measurement on S<sup>2</sup>

Contact Info

Product

Resources

About