On the stability of a class of slowly varying systems

Consider a system _ x = f(x; t; t ) with a timevarying vector eld which contains a regular and a slow time scale ( large). Assume there exists ( ) such that kx (t; t 0 ; x 0 )k K( )kx 0 ke ( )(t?t0) where x (t; t 0 ; x 0 ) is the solution of the system _ x = f(x; t; ) with initial state x 0 at t 0 . We show that for su ciently large, _ x = f(x; t; t ) is exponentially stable when the average of ( ) is negative. This result can be used to extend the circle criterion i.e. to obtain a su cient condition for exponential stability of a feedback interconnection of a slowly time-varying linear system and a sector nonlinearity. An example is included which shows that the technique can be used to obtain an exponential stability result for a pendulum with a nonlinear partially slowly time-varying friction attaining positive and negative values.

Automatica, 2005

We study the stability properties of a class of time-varying nonlinear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.

2016

This manuscript presents a recent overview of time-delay systems as described in the proposal of the ANR project SCIDIS. After a brief description of the various problems arising in the study of such a class of infinite dimensional systems, we recall the main tools employed in the literature to address stability using a time domain approach, mainly the Lyapunov-Krasovskii theorem. The report lists, for instance, the class of delay systems that have been mainly dealt with in the literature, possible selections for the Lyapunov-Krasovskii functional candidate, and some integral and matrix inequalities that have been mainly provided by the participant to the SCIDIS project. Finally, several examples of time-delay systems are considered and demonstrate the potentials of the recent advances in this field.

IEEE Transactions on Automatic Control, 2006

A new stability criterion for time-varying systems consisting of linear and norm bounded nonlinear terms with uncertain time-varying delays is formulated. An explicit delay-independent sufficient stability condition is formulated in the terms of the transition matrix of the given linear part without delay and the bounds for the uncertain terms. The obtained condition turns out to be also necessary if the matrix of the linear part is time-invariant and symmetric; it is shown that these systems satisfy the well-known Aizerman's conjecture. The obtained criterion is contrasted by some of stability estimates available in the literature for these kinds of systems; in all cases the proposed criterion provides less conservative stability bounds.

This paper overviews the research investigations pertaining to stability and stabilization of control systems with time-delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay-independent and delay-dependent stability results for time-delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay-dependent stability results are also discussed.

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems

The stability behavior of time varying systems can be studied using the concept of Lyapunov exponents and their corresponding Lyapunov subspaces. For linear time varying systems the entire Lyapunov spectrum can be approximated by the Floquet exponents of periodic systems. This leads to a variety of stability results, including the characterization of stability radii. Furthermore, a structural stability type theorem shows that stability features of time varying hyperbolic systems persist under small perturbations. For nonlinear time varying systems a stable manifold theorem allows us to interpret the linear results for the nonlinear system locally around an equilibrium point.

IEEE transactions on circuits and systems, 1994

with This paper has three contributions. The first K = {k = [kt, ..., km]' : 67 5 ki 5 kf). (1.2) involves polytopes of matrices whose characteristic polynomials also lie in a polytopic set (e.g. companion matrices). We show that this set is Hurwitz or Schur invariant iff there exist multiaffinely parameterized positive definite, Lyapunov matrices which solve an augmented Lyapunov equation. The second result concerns uncertain transfer functions with denomina-tor and numerator belonging to a polytopic set. We show all members of this set are Strictly Positive Real iff the Lyapunov matrices solving the equations featuring the Kalman-Yakubovic-Popov Lemma are multiaffinely parameterized. Moreover, under an alternative characterization of the underlying polytopie sets, the Lyapunov matrices for both of these resnlts admit affine parameterizations. Finally, we apply the Lyapunov equation results to derive stability conditions for a class of Linear Time Varying Systems.

Systems & Control Letters, 2007

In this paper, global and local uniform asymptotic stability of perturbed dynamical systems is studied by using Lyapunov techniques. The restriction about the perturbed term is that the perturbation is bounded by an integrable function under the assumption that the nominal system is globally uniformly asymptotically stable. We use a new Lyapunov function to obtain a global uniform asymptotical stability of some perturbed systems.

Springer eBooks, 2021

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Hacettepe Journal of Mathematics and …, 2011

This paper investigates the initial time difference equi-boundedness criteria in terms of two measures, initial time difference boundedness and Lagrange Stability in terms of two measures. These are unified with Lyapunov-like functions to establish a variational ...