Continuous Lyapunov Function Research Articles

On the Global Existence of Solution of the Comparison System and Vector Lyapunov Asymptotic Eventual Stability for Nonlinear Impulsive Differential Systems

In this paper, the existence of maximal solution of the comparison system as well as the asymptotic eventual stability of nonlinear impulsive differential equations with fixed moments of impulse is examined using the vector Lyapunov functions which is generalized by a class of piecewise continuous Lyapunov functions. It was distinctly established that the maximal solution of the comparison system majorizes the vector form of the Lyapunov functions. The novelty in the use of the vector Lypunov functions lies in the fact that the "restrictions" encountered by the scalar Lyapunov function is safely handled especially for large scale dynamical systems, since it involves splitting the Lyapunov functions into components so that each of the components can easily describe the behavior of the solution state. Together with comparison results, sufficient conditions for eventual stability and asymptotic are presented.

A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost

Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function W, we say that the system is globally asymptotically controllable to the target withW-regulated cost, whenever, starting from any point z, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than W(z). In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a Minimum Restraint Function. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.

On a Class of Piecewise Continuous Lyapunov Functions and Uniform Eventual Stability of Nonlinear Impulsive Caputo Fractional Differential Equations via New Generalized Dini Derivative

On a Class of Piecewise Continuous Lyapunov Functions and Uniform Eventual Stability of Nonlinear Impulsive Caputo Fractional Differential Equations via New Generalized Dini Derivative

Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).

Predefined-time fault-tolerant attitude control for tailless aircraft considering actuator input saturation

This paper proposes a predefined-time fault-tolerant attitude control methodology to tailless aircraft in the presence of external disturbance, input saturation, and possibly the infinite number of time-varying actuator faults. First, a novel predefined-time performance function is designed, subsequently, a smooth function and bound-based adaptation laws are developed to compensate for the adverse influence caused by actuator fault and input saturation simultaneously. Such a design is less constrained to faults, requires less computation burden, and can be applied to a larger scope of fault scenarios. Besides, a piecewise continuous Lyapunov function is introduced to proceed with the stability analysis of the closed-loop system, and it has been analytically proved that the attitude tracking errors can converge to a predefined residual set within a predefined time while guaranteeing that actuator fault and input saturation are well handled. Finally, comparative simulations are carried out to validate the effectiveness and superiority of the developed control scheme.

Resilient Sampled-Data Control for Stabilization of T--S Fuzzy Systems via Interval-Dependent Function Method: Handling DoS Attacks

This paper is concerned with the resilient sampled-data control for stabilization of T-S fuzzy systems in the presence of denial-of-service (DoS) attacks. To describe the effects of DoS attacks, an appropriate mathematical model is established to characterize the complicated dynamical behaviors of DoS attacks and periodic sampling mechanism. On this basis, a resilient sampled-data control scheme including communication protocols and security controller gains is proposed to account for the effects of DoS attacks. Meanwhile, a novel index is developed to measure the anti-attack ability and security level. By utilizing the information of DoS attacks and resilient sampling intervals, a continuous interval-dependent Lyapunov function is constructed which does not increase at the resilient sampling instants. Then, according to the designed interval-dependent function, several inequalities estimation techniques, convex combination approach, and together with discrete-time Lyapunov theory, two sufficient criteria are derived to guarantee that the closed-loop T-S fuzzy systems are globally asymptotically stable with prespecified security levels subject to DoS attacks. The advantages of designed function are well analyzed in contrast with the previous discontinuous Lyapunov functionals. At last, two simulation examples are given to demonstrate the validity and effectiveness of the obtained results.

Designing Distributed Impulsive Controller for Networked Singularly Perturbed Systems

This paper develops a novel synthesis approach for synchronization of a network of singularly perturbed systems (SPSs) with a small singular perturbation parameter (SPP) <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> via distributed impulsive control. First, a decoupling method in the setting of directed networks is employed to decompose networked SPSs related to complex eigenvalues of the Laplacian matrix. Then, based on an improved piecewise continuous Lyapunov function, an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent synchronization criterion is established. The relationship among the impulse interval, the impulse gain matrix and <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> is revealed. By employing the newly-obtained synchronization criterion, sufficient conditions on the existence of an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent impulse gain matrix are derived. Finally, an example is simulated to verify the effectiveness of the theoretical results.

Worst-case error bounds for the super-twisting differentiator in presence of measurement noise

The super-twisting differentiator, also known as the first-order robust exact differentiator, is a well known sliding mode differentiator. In the absence of measurement noise, it achieves exact reconstruction of the time derivative of a function with bounded second derivative. This note proposes an upper bound for its worst-case differentiation error in the presence of bounded measurement noise, based on a novel Lipschitz continuous Lyapunov function. It is shown that the bound can be made arbitrarily tight and never exceeds the true worst-case differentiation error by more than a factor of two. A numerical simulation illustrates the results and also demonstrates the non-conservativeness of the proposed bound.

Mode Switching Control for Drag-Free Satellite Based on Region of Attraction

This paper aims to propose a switching rule to improve the efficiency and stability of the mode switching process for the drag-free satellite. The switching rules will ensure the stability of the different controllers during the switching process. Unlike traditional satellite switching control, the inner loop and the outer loop of the drag-free satellite are strongly coupled. The drag-free satellite not only needs to consider the controller design in the inner loop but also the controller design of the outer loop. In the outer loop, a Proportion Integration Differentiation control method is adopted to design the controller. In the inner loop, considering the release error effect of the test mass, a nonlinear sliding-mode control is employed as a controller before the mode switch. The H∞ mixed-sensitivity controller, to improve the robustness of the system and solve the problem of controller saturation, is designed after the mode switch. In the stability analysis of the switching system, the piecewise continuous Lyapunov function method is adopted. The region of attraction, which is used as the switching rule, is calculated based on the sum of squares. The obtained results demonstrate that the proposed switching rule satisfies the control accuracy and system stability.

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.

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.

Design of interval observer for continuous linear large-scale systems with disturbance attenuation

A design method of interval observers for linear continuous large-scale systems is proposed in this paper. The considered systems are subject to disturbances that are assumed to be unknown but bounded. With the aid of continuous Lyapunov functions, the error system of the original system and the interval observer system is asymptotically stable and its state is nonnegative, allowing sufficient conditions for interval observers of large-scale systems to be obtained and expressed as linear matrix inequalities. Furthermore, the L stability theory is used in the design of interval observers for large-scale systems to increase the accuracy of interval estimation. Interval observers design methods are proposed for large-scale systems with L1/L2 gain performance. Two numerical examples are provided to demonstrate our findings.

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.

A Conley-type Lyapunov function for the strong chain recurrent set

Let ϕ:X×R→X be a continuous flow on a compact metric space (X,d). In this article we constructively prove the existence of a continuous Lyapunov function for ϕ which is strictly decreasing outside SCRd(ϕ). Such a result generalizes Conley's Fundamental Theorem of Dynamical Systems for the strong chain recurrent set.

Transient stability domain estimation of AC/DC systems considering HVDC switching characteristics based on the polynomial Lyapunov function method

Transient stability analysis is a traditional yet significant topic in power systems. The switching control schemes of the HVDC converter station after fault bring new challenges to the direct method of transient stability analysis in AC/DC power systems. To apply the direct method to investigate the transient stability of power systems accommodating the switching characteristics of the line commutated converter (LCC) HVDC, continuous piecewise Lyapunov functions are constructed for a whole AC/DC system model including several classical LCC-HVDC control modes, and the continuity of Lyapunov functions is studied and guaranteed in transitions between control modes. Then, the Sum-of-Squares optimization is employed to find the optimal continuous piecewise Lyapunov functions. Finally, the transient stability domain is estimated for a sample system including the main control loops of LCC-HVDC, a three-order generator model, and a one-order exciter model. Tests in MATLAB environment are conducted to demonstrate the feasibility of this method.

Numerical design of Lyapunov functions for a class of homogeneous discontinuous systems

AbstractThis paper deals with the analytical and numerical design of a Lyapunov function for homogeneous and discontinuous systems. First, the presented converse theorems provide two analytic expressions of homogeneous and locally Lipschitz continuous Lyapunov functions for homogeneous discontinuous systems of negative homogeneity degree, generalizing classical results. Second, a methodology for the numerical construction of those Lyapunov functions is extended to the class of systems under consideration. Finally, the developed theory is applied to the numerical design of a Lyapunov function for some higher‐order sliding mode algorithms.

Block-coordinate and incremental aggregated proximal gradient methods for nonsmooth nonconvex problems

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope, which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka–Łojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.

Stability Analysis of Simple and Online User Association Policies for Millimeter Wave Networks

In a millimeter wave (mmWave) network, user association– the process of deciding as to which base station (BS) a given user should associate with– is a crucial process, which affects the throughput and delay performance seen by users in the network and the amount of load at each BS. In the existing research literature, the stability region of a user association policy, i.e. , the set of user arrival rates for which the user association policy stabilizes the network, has not been analytically characterized for any user association policy for mmWave networks. In this paper, we study the user association problem in mmWave networks and compare the performances of four user association policies: Signal to Noise Ratio (SNR) based, Throughput based, Load based and Mixed. All these policies are simple, easy to implement, distributed, and online. We use a Continuous Time Markov Chain (CTMC) model and Lyapunov function techniques to analytically characterize the stability region of each of the above four user association policies; our analysis provides several insights into the performances of the policies. We also evaluate the performances of the above four user association policies in a large mmWave network, in which link qualities fluctuate with time and users are mobile, via detailed simulations. Our results show that the Throughput based policy outperforms the other three user association policies in terms of stability region as well as average throughput, average delay, fairness and energy efficiency performance.

Stability Analysis and Stabilization of Switched Systems With Average Dwell Time: A Matrix Polynomial Approach

This paper focuses on stability analysis and stabilization of a switched system under average dwell time criteria in the continuous-time domain. The matrix polynomial approach is applied to the switched system to construct a continuous Lyapunov function and design a more effective controller, which can not only improve the system performance but also lessen the system conservativeness. The stability problem is studied with square matricial representation for the first time. The new method is more applicable than previous work under average dwell time switching with the time-varying controller gains. In addition, new sufficient conditions of stability and stabilization are derived to guarantee the global uniform exponential stability of the switched system. A numerical example is provided to show the advantages of the new method.

Homogeneous Lyapunov functions for homogeneous infinite dimensional systems with unbounded nonlinear operators

The existence of a locally Lipschitz continuous homogeneous Lyapunov function is proven for a class of asymptotically stable homogeneous infinite dimensional systems with unbounded nonlinear operators.