Converse Lyapunov Theorem Research Articles

New results on asymptotic stability of time-varying nonlinear systems with applications

Abstract. In this paper, we present a converse Lyapunov theorem for the new notion of global generalized practical uniform h-stability of nonlinear systems of differential equations. We derive some sufficient conditions which guarantee the global generalized practical uniform h-stability of time-varying perturbed systems. In addition, these results are used to study the practical h-stability of models of infectious diseases and vaccination. Mathematics Subject Classification (2010): 35B40, 37B55, 34D20, 93D15, 92D30. Keywords: Epidemic models, generalized practical uniform h-stability, Gronwall’s inequalities, Lyapunov functions.

On Converse Lyapunov Theorem for Fixed-Time Input-to-State Stability

On Converse Lyapunov Theorem for Fixed-Time Input-to-State Stability

Semi-Global Stabilization of Discrete-Time Linear Systems Subject to Infinite Distributed Input Delays and Actuator Saturations.

The semi-global stabilization problem of discrete-time systems subject to infinite distributed input delays and actuator saturations is investigated in this article. This article develops two low-gain feedback control laws for two types of systems, respectively. It is shown that the resulting system is semi-globally exponentially stabilized. Our results include those existing results on systems subject to only input saturations and systems subject to bounded delays and input saturations as special cases. Compared with existing results on infinite delays and actuator saturations, this article develops a more accurate scaling utilizing a more general framework. Furthermore, a novel converse Lyapunov theorem for discrete-time linear infinite-delayed systems and a novel stability analysis theorem for perturbed discrete-time linear infinite-delayed systems are developed to handle the nonlinearity induced by saturations. Finally, this article provides two numerical examples to illustrate the effectiveness of the developed theorems.

Practical semiglobal uniform exponential stability of nonlinear nonautonomous systems

UDC 517.9 We solve the following twofold problem: In the first part, we deduce Lyapunov sufficient conditions for practical uniform exponential stability of nonlinear perturbed systems under different conditions for the perturbed term. The second part presents a converse Lyapunov theorem for the notion of semiglobal uniform exponential stability for parametrized nonlinear time-varying systems. We establish the possibility of application of a perturbed parametrized system, by using Lyapunov theory, to the investigation of the robustness properties that may provide practical semiglobal uniform exponential stability with respect to perturbations.

Asymptotic and Finite-Time Semistability for Nonlinear Discrete-Time Systems With Application to Network Consensus

This article focuses on semistability and finite-time semistability analysis and synthesis of discrete-time 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 article, we build on the theories of semistability and finite-time semistability for continuous-time dynamical systems to develop a rigorous framework for discrete semistability and discrete finite-time semistability. Specifically, Lyapunov and converse Lyapunov theorems for semistability and finite-time semistability are developed, and the regularity properties of the Lyapunov function establishing finite-time semistability are shown to be related to the settling time function capturing the finite settling time behavior of the dynamical system. These results are then used to develop a general framework for designing semistable and finite-time semistable consensus protocols for discrete dynamical networks for achieving multiagent coordination tasks asymptotically and in finite time. The proposed controller architectures involve the exchange of generalized energy state information between agents guaranteeing that the closed-loop dynamical network is semistable to an equipartitioned equilibrium representing a state of consensus consistent with basic thermodynamic principles.

ISS and iISS for discontinuously time-varying time-delay systems

Novel sufficient conditions are given for input-to-state stability (ISS) to imply integral-ISS (iISS) of time-varying time-delay systems. These conditions are more general and less restrictive than existing ones. Outstanding points of generality are that neither continuity with respect to time nor Lipschitz continuity with respect to the input is required from the function defining the system dynamics. Increased generality is achieved via the analysis of properties of solutions, without requiring converse Lyapunov theorems.

On the Converse Safety Problem for Differential Inclusions: Solutions, Regularity, and Time-Varying Barrier Functions

This article presents converse theorems for safety in terms of barrier functions for unconstrained continuous-time systems modeled as differential inclusions. Via a counterexample, we show the lack of existence of autonomous and continuous barrier functions certifying safety for a nonlinear system that is not only safe but also has a smooth right-hand side. Guided by converse Lyapunov theorems for (nonasymptotic) stability, time-varying barrier functions and appropriate infinitesimal conditions are shown to be both necessary as well as sufficient under mild regularity conditions on the right-hand side of the system. More precisely, we propose a general construction of a time-varying barrier function in terms of a marginal function involving the finite-horizon reachable set. Using techniques from set-valued and nonsmooth analysis, we show that such a function guarantees safety when the system is safe. Furthermore, we show that the proposed barrier function construction inherits the regularity properties of the proposed reachable set. In addition, when the system is safe and smooth, we build upon the constructed barrier function to show the existence of a smooth barrier function guaranteeing safety. Comparisons and relationships to results in the literature are also presented.

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.

Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications

Stability and safety are two important aspects in safety-critical control of dynamical systems. It has been a well established fact in control theory that stability properties can be characterized by Lyapunov functions. Reachability properties can also be naturally captured by Lyapunov functions for finite-time stability. Motivated by safety-critical control applications, such as in autonomous systems and robotics, there has been a recent surge of interests in characterizing safety properties using barrier functions. Lyapunov and barrier functions conditions, however, are sometimes viewed as competing objectives. In this paper, we provide a unified theoretical treatment of Lyapunov and barrier functions in terms of converse theorems for stability properties with safety guarantees and reach-avoid-stay type specifications. We show that if a system (modeled as a dynamical system with measurable perturbations) possesses a stability with safety property, then there exists a smooth Lyapunov function to certify such a property. This Lyapunov function is shown to be defined on the entire set of initial conditions from which solutions satisfy this property. A similar but slightly weaker statement is made for reach-avoid-stay specifications. We show by a simple example that the latter statement cannot be strengthened without additional assumptions. We further extend the results for systems with control inputs and prove existence of converse Lyapunov-barrier functions for reach-and-avoid specifications. One clear limitation of the results of this paper is that the converse Lyapunov-barrier theorems are not constructive, as with classical converse Lyapunov theorems. We believe, however, that the unified necessary and sufficient conditions with a single Lyapunov-barrier function are of theoretical interest and can hopefully shed some light on computational approaches.

Uniform Global Asymptotic Stabilization of Semilinear Periodic Discrete-Time Systems

Semilinear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic semilinear discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Moreover, the converse Lyapunov theorem on Lyapunov (nonasymptotic) stability is proved for complex and real linear periodic discrete-time systems. Finally, examples of using the obtained results are presented.

Lyapunov theorems for stability and semistability of discrete-time stochastic systems

This paper develops Lyapunov and converse Lyapunov theorems for discrete-time stochastic semistable nonlinear dynamical systems expressed by Itô-type difference equations possessing a continuum of equilibria. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose difference operator involves a discrete-time analog of the infinitesimal generator for continuous-time Itô dynamical systems and decreases along the dynamical system sample solution sequences satisfying an inequality involving the average distance to the set of the system equilibria.

Estimating the Region of Attraction for Power Systems Using Gaussian Process and Converse Lyapunov Function

The significant penetration of renewable energy resources into power systems introduces intermittent disturbances that drive the system away from its stable operating point (i.e., equilibrium). Nevertheless, as long as the system states are still contained inside a region of attraction (ROA), the system will finally return to the normal operating equilibrium. Thus, the ROA plays a crucial role in the stability assessment of dynamical power systems under uncertainties. This brief introduces a novel framework to construct the ROA of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytic Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need for an analytic Lyapunov function and learning unknown Lyapunov functions with the Gaussian process (GP) approach. In addition, a GP upper confidence bound (GP-UCB)-based sampling algorithm is designed to reconcile the tradeoff between the exploitation for enlarging the ROA and the exploration for enhancing the confidence level of the sample region. Within the constructed ROA, it is guaranteed in the probability that the system state will converge to the stable equilibrium with a confidence level. Extensive simulations and experimental validations are conducted to substantiate the assessment approach for the ROA of IEEE test systems and a real smartgrid using phasor measurement unit (PMU) data. It is demonstrated that the proposed approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Converse Lyapunov theorems for control systems with unbounded controls

In this paper we extend well-known relationships between global asymptotic controllability, sample stabilizability, and the existence of a control Lyapunov function to a wide class of control systems with unbounded controls, including control-polynomial systems. In particular, we consider open loop controls and discontinuous stabilizing feedbacks, which may be unbounded approaching the target, so that the corresponding trajectories may present a chattering behaviour. A key point of our results is to prove that global asymptotic controllability, sample stabilizability, and existence of a control Lyapunov function for these systems or for an impulsive extension of them are equivalent.

Stability, boundedness and controllability of solutions of measure functional differential equations

Generalized ordinary differential equations (we write generalized ODEs), introduced by J. Kurzweil in 1957, are known to encompass several other types of equations as measure functional differential equations, for instance. In this paper, we obtain converse Lyapunov theorems for generalized ODEs and, in particular, for measure functional differential equations which, in turn, encompass impulsive functionals differential equations as well as functional dynamic equations on time scales. We also relate uniform stability to boundedness of solutions. As an application, we establish necessary and sufficient conditions for a system of non–homogeneous nonlinear generalized ODEs defined in a Banach space and for a system of non–homogeneous measure functional differential equations to be asymptotically controllable. We include an example which illustrates the main results.

A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.

Input-to-State Stability of homogeneous infinite dimensional systems with locally Lipschitz nonlinearities

The Input-to-State Stability (ISS) of homogeneous evolution equations with unbounded linear operators and locally Lipschitz nonlinearities in Banach spaces is studied using a new homogeneous converse Lyapunov theorem. It is shown that, similarly to finite-dimensional models, uniform asymptotic stability of an unperturbed homogeneous system guarantees its ISS with respect to homogeneously involved exogenous inputs.

Converse Lyapunov theorems for measure functional differential equations

The aim of this paper is to obtain converse Lyapunov theorems for measure functional differential equations which are known to encompass dynamic equations on time scales, impulsive differential equations and functional differential equations (see [7,8]). Our main tool is to consider the framework of generalized ordinary differential equations, introduced by Jaroslav Kurzweil, and to manage all other equations in this setting which has the advantage to handle well both highly oscillating functions and functions with many discontinuities.

Converse theorem for practical $h$-stability of time-varying nonlinear systems

This paper treats the concept of practical uniform $h$-stability for such perturbed dynamical systems as an extension of practical uniform exponential stability. We present a converse Lyapunov theorem and we give sufficient conditions that guarantee practical uniform $h$-stability for a time-varying perturbed system using the Gronwall-Bellman inequality and Lyapunov's theory. Some examples are introduced to illustrate the applicability of the main results.

Converse Lyapunov Theorem for Nabla Asymptotic Stability Without Conservativeness

This article focuses on the conservativeness issue of the existing Lyapunov method for linear time-invariant (LTI) nabla fractional-order systems and proposes a converse Lyapunov theorem to overcome the conservative problem. It is shown that the LTI nabla fractional-order system is asymptotically stable if and only if there exist a positive-definite Lyapunov function whose first-order difference is negative definite. After developing a systematic scheme to construct such Lyapunov candidates, the Lyapunov indirect method is derived for the nonlinear system. Finally, the effectiveness and practicability of the proposed methods are substantiated with four examples.

Input Delay Tolerance of Nonlinear Systems Under Smooth Feedback: A Semiglobal Control Framework

This article studies the problem of when a globally asymptotically stabilizable nonlinear system by smooth feedback is tolerable with respect to input delay. We illustrate, by means of theoretic and practical examples, some fundamental limitations including: 1) A globally exponentially stabilizable (GES) nonlinear system may not guarantee global asymptotic stabilizability (GAS) of the nonlinear system even with an arbitrarily small input delay; 2) GAS cannot even ensure semiglobal asymptotic stabilizability of the nonlinear system with a small input delay. To overcome these obstacles, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">semiglobal input delay tolerance</i> (SGIDT) and present a semiglobal control framework for the asymptotic analysis/synthesis of input delay tolerance of multi-input–multi-output (MIMO) nonlinear systems under smooth feedback. With the aid of Razumikhin theorem and the converse Lyapunov theorem on global asymptotic local exponential stability (GALES), it is proved that in the case of state feedback, GALES does ensure the SGIDT of a MIMO nonlinear system. In the case of output feedback, it is further proved that GALES and uniform observability imply the SGIDT of the nonlinear system.