Lyapunov Characterization Research Articles

A Lyapunov Characterization of Prescribed Performance Control Systems

A Lyapunov Characterization of Prescribed Performance Control Systems

Lyapunov Characterization of Input-to-State Stability for a Class of Impulsive Systems

Lyapunov Characterization of Input-to-State Stability for a Class of Impulsive Systems

A Lyapunov characterization of robust policy optimization

A Lyapunov characterization of robust policy optimization

Lyapunov Characterizations of Exponential Mean Square Input-to-State Stability for Discrete-Time Markovian Switching Nonlinear Systems

Lyapunov Characterizations of Exponential Mean Square Input-to-State Stability for Discrete-Time Markovian Switching Nonlinear Systems

A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

<p style='text-indent:20px;'>Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.</p>

A unified Lyapunov characterization for finite time control and prescribed time control

AbstractThis work presents a novel approach to unifying finite time control and prescribed time control. The main idea is to establish a new Lyapunov differential inequality that allows finite time and prescribed time stability to be addressed uniformly. It is shown that the derived control algorithm is able to achieve prescribed time convergence with bounded control gain and bounded control action, completely solving for the first time the long‐lasting issue of infinite gain associated with current notion of prescribed time control. Numerical simulations also confirmed the effectiveness of the developed approach.

Reduction theorems for stability of compact sets in time-varying systems

Reduction theorems provide a framework for stability analysis that consists in breaking down a complex problem into a hierarchical list of subproblems that are simpler to address. This paper investigates the following reduction problem for time-varying ordinary differential equations on Rn. Let Γ1 be a compact set and Γ2 be a closed set, both positively invariant and such that Γ1⊂Γ2⊂Rn. Suppose that Γ1 is uniformly asymptotically stable relative to Γ2. Find conditions under which Γ1 is uniformly asymptotically stable. We present a reduction theorem for uniform asymptotic stability that completely addresses the local and global version of this problem, as well as two reduction theorems for uniform stability and either local or global uniform attractivity. These theorems generalize well-known equilibrium stability results for cascade-connected systems as well as previous reduction theorems for time-invariant systems. We also present Lyapunov characterizations of the stability properties required in the reduction theorems that to date have not been investigated in the stability theory literature.

Further Geometric and Lyapunov Characterizations of Incrementally Stable Systems on Finsler Manifolds

In this paper, we report several new geometric and Lyapunov characterizations of incrementally stable systems on Finsler and Riemannian manifolds. A new and intrinsic proof of an important theorem in contraction analysis is given via the complete lift of the system. Based on this, two Lyapunov characterizations of incrementally stable systems are derived, namely, converse contraction theorems, and revelation of the connection between incremental stability and stability of an equilibrium point, in which the second result recovers and extends the classical Krasovskii's method.

Lyapunov characterizations on input-to-state stability of nonlinear systems with infinite delays

This paper addresses Lyapunov characterizations on input-to-state stability (ISS) of time-varying nonlinear systems with infinite delays. With novel ISS definitions in the case of nonlinear systems with infinite delays, we present several results on their ISS Lyapunov characterizations in the form of both ISS Lyapunov theorems and converse ISS Lyapunov theorems. It is shown that an infinite-delayed system is (locally) ISS if it has a (local) ISS Lyapunov functional, and conversely, there exists a (local) ISS Lyapunov functional if it is (locally) ISS. To prove the converse ISS Lyapunov theorems, we establish a key technical lemma bridging ISS/LISS and robust asymptotic stability of systems with infinite delays and two converse Lyapunov theorems concerning robust asymptotic stability of systems with infinite delays. Two distinctive advantages of this work are that a large class of infinite dimensional spaces are allowed and the results are established based on a more general Lipschitz condition, i.e., the right hand side Lipschitz (RS-L) condition. An example is provided for illustration of the obtained results.

Improved multiple Lyapunov functions of input–output-to-state stability for nonlinear switched systems

This study is concerned with the Lyapunov characterization of input–output-to-state stability (IOSS) for time-varying nonlinear switched systems. Considering the restriction in current methods that each subsystem must be IOSS, and the Lyapunov function of each subsystem is strictly descended in the definition domain, in this work, by extending multiple Lyapunov functions to an indefinite form, the restriction on the negativity of the time derivative of Lyapunov functions is relaxed; additionally, all subsystems are permitted to not be IOSS in the state space. The results obtained are then extended to verifying the integral input–output-to-state stability (iIOSS) and output-to-state stability (OSS); and a state-norm estimator is proposed based on the indefinite Lyapunov function as a by-product. Finally, a numerical example illustrates the effectiveness of the results.

Improved Multiple Lyapunov Functions of input-to-output Stability and input-to-state Stability for Switched Systems

This paper investigates the Lyapunov characterizations of input-to-output stability (IOS) and input-to-state stability (ISS) for nonlinear switched systems. The restriction on the negativity of derivatives of Lyapunov functions is relaxed; additionally, subsystems are not restricted to be ISS/IOS throughout the state space. These problems are overcome by a newly proposed method, which extends multiple Lyapunov functions (MLFs) to an indefinite form. Indefinite multiple Lyapunov functions (iMLFs) are proposed on the basis of an inequality to estimate the upper boundary of the system state. Moreover, we apply the iMLFs method to verify the extended notions of IOS in a switched system. More relaxed sufficient conditions for ISS and IOS are proposed, a simulation experiment is carried out on a numerical example of the system, which demonstrates their effectiveness and superiority.

Lyapunov Characterization of Uniform Exponential Stability for Nonlinear Infinite-Dimensional Systems

In this article, we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to uncertainties. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to uncertainties) local, semi-global, and global exponential stability, through the existence of coercive and non-coercive Lyapunov functionals. The importance of the obtained results is underlined through some applications concerning 1) exponential stability of nonlinear retarded systems with piecewise constant delays, 2) exponential stability preservation under sampling for semilinear control switching systems, and 3) the link between input-to-state stability and exponential stability of semilinear switching systems.

Generalized Mittag-Leffler Input Stability of the Fractional-Order Electrical Circuits

This article addresses new applications of the generalized Mittag-Leffler input stability to the fractional-order electrical circuits. We consider the fractional-order electrical circuits in the context of the generalized Caputo-Liouville derivative. We propose the Lyapunov characterizations of the fractional differential equations. A new numerical discretization, including the fractional differential equations represented by the generalized Caputo derivative, has been successfully applied to the fractional electrical circuits. To support the results, we have proposed the graphics generated by our numerical discretization. The graphics of the solutions have been analyzed and interpreted in the context of generalized Mittag-Leffler input stability and the generalized Mittag-Leffler stability. The generalized Mittag-Leffler input stability is a new stability notion for the fractional differential equations recently introduced in the literature.

Lyapunov characterizations of input-to-state stability for discrete-time switched systems via finite-step Lyapunov functions

Abstract This paper addresses Lyapunov characterizations of input-to-state stability for nonlinear switched discrete-time systems via finite-step Lyapunov functions with respect to closed sets. The use of finite-step Lyapunov functions permits not-necessarily input-to-state stable systems in the systems family, while input-to-state stability of the resulting switched system is ensured. The result is generally presented for systems under arbitrary switching. It additionally covers the case of constrained switchings. We illustrate the effectiveness of our results by application to networked control systems with periodic scheduling policies under a priori known and dwell time-based switching mechanism.

Lyapunov Characterization and Analysis of the Operating Modes of the AC–DC Ćuk Converter

In this paper, all operating modes of the ac–dc Cuk converter have been modeled as linear-time invariant systems where some of the modes were not only modeled but also detected for the first time. Another feature provided in this paper is that the converter can be modeled and analyzed as a switching dynamical system controlled by an external constant duty-ratio switching and the internal switchings shown to be related to the minimum among the time derivatives of a common quadratic Lyapunov function. Some representative simulations are provided to illustrate and validate the technical results of this paper.

Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations

The behavior of the analytical solutions of the fractional differential equation described by the fractional order derivative operators is the main subject in many stability problems. In this paper, we present a new stability notion of the fractional differential equations with exogenous input. Motivated by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, we present our work here. Applications of Mittag-Leffler functions in certain areas of physical and applied sciences are also very common. During the last two decades, this class of functions has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, to name just a few. Moreover, we propose the generalized Mittag-Leffler input stability conditions. The left generalized fractional differential equation has been used to help create this new notion. We investigate in depth here the Lyapunov characterizations of the generalized Mittag-Leffler input stability of the fractional differential equation with input.

ISS implies iISS even for switched and time-varying systems (if you are careful enough)

For time-invariant systems, the property of input-to-state stability (ISS) is known to be strictly stronger than integral-ISS (iISS). Known proofs of the fact that ISS implies iISS employ Lyapunov characterizations of both properties. For time-varying and switched systems, such Lyapunov characterizations may not exist, and hence establishing the exact relationship between ISS and iISS remained an open problem, until now. In this paper, we solve this problem by providing a direct proof, i.e. without requiring Lyapunov characterizations, of the fact that ISS implies iISS, in a very general time-varying and switched-system context. In addition, we show how to construct suitable iISS gains based on the comparison functions that characterize the ISS property, and on bounds on the function f defining the system dynamics. When particularized to time-invariant systems, our assumptions are even weaker than existing ones. Another contribution is to show that for time-varying systems, local Lipschitz continuity of f in all variables is not sufficient to guarantee that ISS implies iISS. We illustrate application of our results on an example that does not admit an iISS-Lyapunov function.

Mittag-Leffler input stability of fractional differential equations and its applications

This paper addresses the Mittag-Leffler input stability of the fractional differential equations with exogenous inputs. We continuous the first note. We discuss three properties of the Mittag-Leffler input stability: converging-input converging-state, bounded-input bounded-state, and Mittag-Leffler stability of the unforced fractional differential equation. We present the Lyapunov characterization of the Mittag-Leffler input stability, and conclude by introducing the fractional input stability for delay fractional differential equations, and we provide its Lyapunov-Krasovskii characterization. Several examples are treated to highlight the Mittag-Leffler input stability.

Fractional input stability and its application to neural network

This paper deals with fractional input stability, and contributes to introducing a new stability notion in the stability analysis of fractional differential equations (FDEs) with exogenous inputs using the Caputo fractional derivative. In particular, we study the fractional input stability of FDEs with exogenous inputs. A Lyapunov characterization of this notion is proposed and several examples are provided to explain the fractional input stability of FDEs with exogenous inputs. The applicability and simulation of this method are illustrated by studying the particular class of fractional neutral networks.

Integral ISS for Switched Nonlinear Time-Varying Systems Using Indefinite Multiple Lyapunov Functions

This paper is concerned with studying Lyapunov characterization of integral input-to-state stability (iISS) for switched nonlinear time-varying systems. Sufficient conditions are given to verify iISS for switched nonlinear time-varying systems under a time-varying state-dependent switching law designed, which allow all subsystems to be not integral input-to-state stable (iISS) and the time derivative of Lyapunov functions of individual subsystems to be indefinite. An indefinite multiple Lyapunov functions (iMLFs) method for analyzing the dynamic behavior of switched nonlinear time-varying systems is provided. Also, an iMLFs-based small-gain theorem for switched interconnected nonlinear time-varying systems is presented, where each lower dimensional subsystem is allowed to be not iISS, which extends the small-gain technique from its original nonswitched nonlinear time-invariant version to a switched nonlinear time-varying version. Finally, an illustrative example is used to demonstrate the feasibility of the theoretical results.