Nonsmooth Lyapunov Functions Research Articles

Fixed-time synchronization of multilayered complex dynamic networks via quantized variable-gain saturated control

This paper studies fixed-time (FxT) quantitative synchronization of multilayered complex dynamic networks (CDNs). First, a new FxT stability theorem is established and two new estimations of the settling time of stability are acquired, which are more accurate than the existing ones. Then, by using the improved theorem, several sufficient conditions ensuring FxT synchronization of a multilayered CDN are derived via two classes of innovative quantized variable-gain saturated controllers, and some high-precision estimates of the synchronous settling time (SST) are attained. In general, the saturation function used to replace the signum function is nonlinear and even includes the odd-even requirement on the exponent, and no quantization is involved in the developed FxT controllers. However, in our design, the classical linear saturation function is employed and the input signals of the controllers are also quantized. Therefore, our proposed strategies are more practical and easier to implement. Besides, different from the traditional 2-norm/1-norm-based approaches, a novel nonsmooth Lyapunov function is constructed so that the resultant estimates of the SST are unrelated to either the network size or the node's dimension, which are less conservative and closer to the actual synchronized time. Lastly, some numerical examples are given to validate the theoretical results.

Generalized homogeneous control with integral action

AbstractA generalized homogeneous control with integral action for a multiple‐input plant operating under uncertainty conditions is designed. The stability analysis is essentially based on a special version of the nonsmooth Lyapunov function theorem for differential equations with discontinuous right‐hand sides. A Lyapunov function for analysis of the closed‐loop system is presented. For negative homogeneity degree, this Lyapunov function becomes a strict Lyapunov function allowing an advanced analysis to be provided. In particular, the maximum control magnitude and the settling‐time of the closed‐loop system are estimated and a class of disturbances to be rejected by the control law is characterized. The control parameters are tuned by solving a system of Linear Matrix Inequalities (LMIs), whose feasibility is proved at least for small (close to zero) homogeneity degrees. The theoretical results are illustrated by numerical simulations.

Nonlinear placement for networked Euler‐Lagrange systems: A finite‐time hierarchical approach

AbstractThe finite‐time nonlinear placement problem of networked Euler‐Lagrange systems (NELSs) is discussed in this paper. The problem is reformulated into a finite‐time aggregate game under an undirected graph. Then, several novel practical gradient‐based finite‐time hierarchical (GFTH) algorithms composed of a game layer, a Nash equilibrium (NE) seeking layer, and a control layer are proposed. Specifically, the game layer employs an aggregate function to reach a consensus on the potential aggregate value which is adopted by a gradient‐based finite‐time method to tackle the finite‐time NE seeking problem in the NE seeking layer, and then, the tracking problem is realized in the control layer. The convergence results are established by a nonsmooth Lyapunov function. In addition, the versatility of the GFTH algorithms is shown by extending to address the task‐space control problem of NELSs. The effectiveness of the proposed algorithms is illustrated via simulations.

Global dynamics and bifurcation for a discontinuous oscillator with irrational nonlinearity

The objective is to carry out thorough analysis of global dynamics and bifurcation for a discontinuous oscillator, showing the transition of dynamics between the piecewise smooth system (with irrational nonlinearity) and the piecewise linear system. The existence of one or three equilibrium sets is proven by the approach of Filippov. Based on non-smooth Lyapunov functions and their set-valued derivatives, asymptotical stability especially finite-time asymptotical stability is shown for the equilibrium sets. In particular, some conditions are established guaranteeing the uniqueness of an equilibrium set and its global finite-time asymptotical stability. Making use of the parametric representation, codimension-one bifurcation curves and a codimension-two bifurcation point are obtained. We give numerical simulations and bifurcation diagram to illustrate the transition of dynamics and verify the results. We further clarify some previous misconceptions in Li et al. (2016).

Backstepping-Based Controller Design for Uncertain Switched High-Order Nonlinear Systems via PI Compensation

This article presents an effective method to address the tracking control problem arising in uncertain switched high-order nonlinear system in strict-feedback form. The system under consideration contains unknown functions, which causally make the asymptotic tracking performance difficult to be achieved. By adopting the adding a power integrator approach in the framework of backstepping, a novel tracking controller is developed to guarantee an asymptotic tracking performance in the presence of the approximation error cased by neural networks (NNs) under arbitrary switching. The main contributions lie in: 1) the article for the first time embeds the backstepping technique in designing a kind of discontinuous controller with proportional integral (PI) compensation and 2) with the help of Filippov’s theory, a new defined system described by differential inclusions can be first obtained by taking some transformations, and then a novel nonsmooth Lyapunov function approach along with its upper right Dini derivative technique is applied to complete the construction of the discontinuous controller. Finally, two simulation examples are exhibited to verify the validity of the proposed design techniques.

Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

On the Boundedness of Solutions of Discontinuous Differential Equations

By using the notion of Carathéodory solution to differential equations, the present work studies the boundedness of solutions of discontinuous differential equations. For these discontinuous systems determined by discontinuous differential equations, results are obtained that guarantee sufficient conditions to boundedness of solutions in terms of nonsmooth Lyapunov functions.

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.

Output Multiformation Tracking of Networked Heterogeneous Robotic Systems via Finite-Time Hierarchical Control.

This article investigates the finite-time output multiformation tracking (OMFT) problem of networked heterogeneous robotic systems (NHRSs), where each robot model involves external disturbances, parametric uncertainties, and possible kinematic redundancy. Besides, the interactions among robotic systems are described as a directed graph with an acyclic partition. Then, several novel practical finite-time hierarchical control (FTHC) algorithms are designed. The convergence analysis of the closed-loop dynamics is extremely difficult due to the lack of effective analysis methods. Based on the mathematics induction and reductio ad absurdum, a new nonsmooth Lyapunov function is proposed to derive the sufficient conditions and settling time functions. Finally, numerical simulations are performed on the NHRS to verify the main results.

Fixed-time synchronization for delayed inertial complex-valued neural networks

This paper researches the problem of p-norm fixed-time synchronization for a class of delayed inertial complex-valued neural networks (ICVNNs). By using reduced-order transformation and separating real and imaginary parts of complex-valued parameters, the second-order ICVNNs can be converted into the form of first-order real-valued differential equations. Then some new flexible and adjustable algebraic criteria to ensure the fixed-time synchronization of ICVNNs are established by means of the non-smooth Lyapunov function and inequality analytical techniques. Moreover, the settling time of fixed-time synchronization is theoretically estimated, which does not depend on the initial value of systems. Finally, simulation examples and applications are presented to illustrate the validity and availability of the obtained results.

Synchronization of chaotic dynamical systems

Chaotic dynamical systems produce signals with unpredictable complexity. Their essential characteristic is that it does not synchronize with any other system. However, they can synchronize provided the sufficiently large coupling is introduced between them. In the present work, the Lyapunov stability analysis has been carried out using non-smooth Lyapunov function to obtain sufficient coupling gain for synchronization. The Mathematical investigations have been done independently for all-to-all coupled Lorenz attractors and all-to-all coupled Rossler attractors. The results in each case are confirmed through numerical simulations.

Synchronization of coupled oscillators in presence of disturbance and heterogeneity

Disturbance and heterogeneity have a strong influence on coupled oscillator dynamics. In general, strong disturbances and heterogeneity break the robustness of the coupled system. However, the system can be protected against bounded disturbance and parameter mismatch by introducing coupling and achieving synchronization. The present work gives mathematical investigations of coupled oscillators for synchronization in the presence of bounded disturbance, which may be periodic, aperiodic, and chaotic. Also, the synchronization analysis has been carried out for second-order coupled heterogeneous oscillators. The oscillators considered are second-order Van der Pol oscillator and Fitzhugh Nagumo types of oscillators. The Lyapunov stability approach using a quadratic Lyapunov function and non-smooth Lyapunov functions have been used to obtain theoretical bound on the quality of synchronization or asymptotic synchronization depending on the type of oscillators. The numerical simulations supporting the analytical findings are reported.

Lyapunov stability for discontinuous systems

The present work studies the stability analysis of equilibrium of ordinary differential equations with the discontinuous right side, also called discontinuous differential equations, using the notion of Carathéodory solution for differential equations. This way, it is studied the stability of equilibrium in the Lyapunov sense for discontinuous systems through nonsmooth Lyapunov functions. Then two existing Lyapunov theorems are obtained. The results established refer to systems determined by nonautonomous differential equations.

The viability of switched nonlinear systems with piecewise smooth Lyapunov functions

In this paper, we focus on the viability and attraction for switched nonlinear systems with nonsmooth Lyapunov functions. We determine the viable set and region of attraction for switched systems in which Lyapunov functions are piecewise smooth. The switching law is constructed by using the directional derivatives of a piecewise smooth Lyapunov function along the trajectories of the subsystems. Sufficient conditions are derived to guarantee the viability and attraction of switched nonlinear systems on the level set of a piecewise smooth Lyapunov function. We further extend the method to switched systems involving possible sliding motions. The approach in the paper provides a unified framework for studying viability and attraction with a systematic consideration of sliding motions. Finally, considering two certain classes of piecewise smooth functions, the related conditions of the viability and attraction for the level set are developed.

Lyapunov-based Singular Perturbation Results in the Framework of Hybrid Systems

Abstract Stability properties of singularly perturbed hybrid systems are investigated via Lyapunov functions with assistance from the invariance principle. Both continuously differentiable Lyapunov functions and non-smooth Lyapunov functions are considered. In each case, under appropriate assumptions, uniform asymptotic stability and uniform global asymptotic stability are established. An estimate of the basin of attraction is given for the former property. Two examples are given to illustrate the proposed theoretical results based on continuously differentiable Lyapunov functions. In addition, one example for switched learning inclusions with unstable modes is given to show the effectiveness of the results obtained based on non-smooth Lyapunov functions.

Disturbance Induced Synchronization in Networked Oscillatory System

The problem of synchronization of oscillators that are not coupled directly but are connected to a common disturbance has been studied in this work. Mathematical formalisms to obtain sufficient coupling gain for synchronization of a complex network of oscillators forced by a common disturbance have been developed. The Lyapunov stability approach using quadratic Lyapunov function and non-smooth Lyapunov function has been used for the purpose. The networked oscillatory systems considered are formed by Van der Pol and Fitzhugh Nagumo oscillators independently. These oscillators are diffusively coupled to common disturbance. The comparison of analytical approaches has been made, and the results are validated through numerical simulations.

Fuzzy Approximation Based Asymptotic Tracking Control for a Class of Uncertain Switched Nonlinear Systems

The problem of asymptotic tracking control for a class of uncertain switched nonlinear systems under fuzzy approximation framework is solved in this paper. Superior to most existing results based on fuzzy adaptive control strategy that can only achieve bounded error tracking performance, our proposed control scheme can guarantee the local asymptotic tracking performance for the uncertain switched nonlinear systems under consideration. This is accomplished by constructing a nonsmooth Lyapunov function and introducing a novel discontinuous controller with dynamic feedback compensator in the design procedure. Meanwhile, some concepts, such as differential inclusion and set-valued map, are introduced to theoretically verify the local asymptotic tracking performance of the systems with our proposed controller. With the help of set-valued Lie derivative, the common virtual control functions, the desired controller, and the adaptive laws can be precisely constructed. Finally, simulation results are given to show the effectiveness of the proposed method.

Stochastic Source Seeking for Mobile Robots in Obstacle Environments Via the SPSA Method

This paper considers a class of stochastic source-seeking problems to drive a mobile robot to the minimizer of a source signal. Our approach is first analyzed in an obstacle-free scenario, where measurements of the signal at the robot location and information of a contact sensor are required. We extend our results to environments with obstacles under mild assumptions on the step size. Our approach builds on the simultaneous perturbation stochastic approximation idea to obtain information of the signal field. We prove the practical convergence of the algorithms to a ball whose size depends on the step size that contains the location of the source. The novelty relies in that we consider nondifferentiable convex functions, a fixed step size, and the environment may contain obstacles. Our proof methods employ nonsmooth Lyapunov function theory, tools from convex analysis, and stochastic difference inclusions. Finally, we illustrate the applicability of the proposed algorithms in a two-dimensional scenarios.

Delayed Feedback Controller based Finite Time Synchronization of Discontinuous Neural Networks with Mixed Time-Varying Delays

This paper addresses finite-time synchronization of artificial neural networks with discrete and distributed time-varying delays as well as discontinuous neuron activation functions which may be unbounded or non-monotonic. Under the framework of Filippov solution, delay feedback controller is studied by constructing nonsmooth Lyapunov function and differential inclusions theory based on finite time convergence theorem. Several effective new criteria are derived. Moreover, the estimation of settling time is established. The obtained results ensure that the synchronization in finite time is achieved. Finally, simulation results are presented to demonstrate the analytical results.

Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach

Finding the conditions that foster synchronization in networked oscillatory systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with neuronal spiking dynamics, we show that our new sufficient condition is tighter than those found in previous analyses which used nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network configuration is a significant step toward facilitating understanding and control of complex oscillatory systems.