Lyapunov stability theory of nonsmooth systems

This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand side such as in nonsmooth dynamic systems or variable structure control     

Region stability and stabilisation of switched linear systems with multiple equilibria

This paper studies stability and stabilisation issues of switched linear time-invariant systems with stable/unstable multiple equilibria. Investigation of such switched systems is motivated by a switching economic system. The well-known common Lyapunov function method is shown to be ineffecctive in analysing region stability of switched systems with multiple equilibria via a counterexample. When every subsystem has an equilibrium point and all multiple equilibria pairwise differ, this paper proposes some sufficient conditons for region stability/instability of such switched systems with respect to a region containing all multiple equilibria under arbitrary quasi-periodical switchings. These novel results imply that there may exist stable limit cycles of such switched systems. Based on the stability results, a global asymptotic region-stabilising controller, quasi-periodical switching path, and corresponding algorithm are all designed for such switched control systems. Several illustrative examples demonstrate the effectiveness and practicality of our new results.     

Stability regions of nonlinear autonomous dynamical systems

A topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented. The stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points (and/or closed orbits) on the stability boundary. Several necessary and sufficient conditions are derived to determine whether a given equilibrium point (or closed orbit) is on the stability boundary. A method for finding the stability region on the basis of these results is proposed. The method, when feasible, will find the exact stability region, rather than a subset of it as in the Lyapunov theory approach. Several examples are given to illustrate the theoretical prediction     

Improved stability criteria for linear time-varying systems on time scales

ABSTRACT

In this paper, asymptotic stability problems of linear time-varying (LTV) systems on time scales are considered based on a less conservative Lyapunov inequality, whose right side is not required to be necessarily negative. It is shown that the Lyapunov inequality covers not only the corresponding trivial (continuous and discrete) ones but also nontrivial ones. Based on this inequality, some necessary and sufficient conditions for asymptotic stability, exponential stability, uniformly exponential stability of LTV systems on time scales are obtained. An example about nontrivial systems is given for illustrating the effectiveness of the proposed results.     

On convergence properties of piecewise affine systems

In this paper convergence properties of piecewise affine (PWA) systems are studied. In general, a system is called convergent if all its solutions converge to some bounded globally asymptotically stable steady-state solution. The notions of exponential, uniform and quadratic convergence are introduced and studied. It is shown that for non-linear systems with discontinuous right-hand sides, quadratic convergence, i.e., convergence with a quadratic Lyapunov function, implies exponential convergence. For PWA systems with continuous right-hand sides it is shown that quadratic convergence is equivalent to the existence of a common quadratic Lyapunov function for the linear parts of the system dynamics in every mode. For discontinuous bimodal PWA systems it is proved that quadratic convergence is equivalent to the requirements that the system has some special structure and that certain passivity-like condition is satisfied. For a general multimodal PWA system these conditions become sufficient for quadratic convergence. An example illustrating the application of the obtained results to a mechanical system with a one-sided restoring characteristic, which is equivalent to an electric circuit with a switching capacitor, is provided. The obtained results facilitate bifurcation analysis of PWA systems excited by periodic inputs, substantiate numerical methods for computing the corresponding periodic responses and help in controller design for PWA systems.     

Exponential stability of globally projected dynamic systems

In this paper, we further analyze and prove the stability and convergence of the dynamic system proposed by Friesz et al.(1994), whose equilibria solve the associated variational inequality problems. Two sufficient conditions are provided to ensure the asymptotic stability of this system with a monotone and asymmetric mapping by means of an energy function. Meanwhile this system with a monotone and gradient mapping is also proved to be asymptotically stable using another energy function. Furthermore, the exponential stability of this system is also shown under strongly monotone condition. Some obtained results improve the existing ones and the given conditions can be easily checked in practice. Since this dynamic system has wide applications, the obtained results are significant in both theory and applications.     

On the stability and convergence rate analysis for the nonlinear uncertain systems based upon active disturbance rejection control

Emerging as an effective control method, active disturbance rejection control technique (ADRC) can well deal with disturbances and uncertainties. Then focusing on the general nonlinear uncertain systems, this article illustrates the relation among stability, uncertainties, and parameters of linear ADRC controller when perturbation occurs in control input. On the one hand, if the reference and uncertainties satisfy particular conditions, the estimation and output errors are proved to be ultimately and globally bounded by Lyapunov function and Gronwall‐Bellman inequality, together with rigorous mathematical deducing and simulation verification. On the other hand, a globally and asymptotically stable result is obtained applying Lyapunov method and Cauchy inequality, and the convergence rate can be estimated when uncertainties exist. Besides, numerical simulations are carried out to fully display the correctness and dependability of results and proofs.     

An efficient neurodynamic model to solve nonconvex nonlinear optimization problems and its applications

This paper presents a recurrent neural network for solving nonconvex nonlinear optimization problems subject to nonlinear inequality constraints. First, the p-power transformation is exploited for local convexification of the Lagrangian function in nonconvex nonlinear optimization problem. Next, the proposed neural network is constructed based on the Karush–Kuhn–Tucker (KKT) optimality conditions and the projection function. An important property of this neural network is that its equilibrium point corresponds to the optimal solution of the original problem. By utilizing an appropriate Lyapunov function, it is shown that the proposed neural network is stable in the sense of Lyapunov and convergent to the global optimal solution of the original problem. Also, the sensitivity of the convergence is analysed by changing the scaling factors. Compared with other existing neural networks for such problem, the proposed neural network has more advantages such as high accuracy of the obtained solutions, fast convergence, and low complexity. Finally, simulation results are provided to show the benefits of the proposed model, which compare to or outperform existing models.     

线性定常大系统参数稳定域的一个新结论

本文应用标量李亚普诺夫函数分解法结合Balley不等式对线性定常大系统的参数稳定域进行了讨论,得到参数稳定域的一种新形式,并以具有两个子系统的n阶大系统为例,指出新的参数稳定域与用向量李亚普诺夫函数分解法得到的参数稳定域互不包含这一新的结论。     

延时系统输入状态稳定性的Lyapunov逆理论

研究了延时系统输入状态稳定性的局部Lipschitz连续的Lyapunov逆理论. 针对含有任意可测局部本质有界扰动的延时系统, 一个局部Lipschitz连续的Lyapunov泛函被证实是存在的, 如果该系统是鲁棒渐进稳定的. 根据该结论, 延时系统输入状态稳定性的Lyapunov特征被进一步得到.     

A Lyapunov approach to exponential stabilization of nonholonomicsystems in power form

In this paper a continuous feedback control law with time-periodic terms is derived for the control of nonholonomic systems in power form. The control law is derived by Lyapunov design from a homogeneous Lyapunov function. Global asymptotic stability is shown by applying the principle of invariance for time-periodic systems. Exponential convergence follows since the vector fields are homogeneous of degree zero     

Stability and synchronization of oscillators: New Lyapunov functions

The analysis of asymptotic stability of nonlinear oscillator is one of the classical problems in the theory of oscillations. Usually, it is solved by exploiting Lyapunov functions having the meaning of the full energy of the system. However, the Barbashin–Krasovskii theorem has to be used along this way, and no estimates can be found for the rate of convergence of the trajectories to equilibria points. In this paper we propose a different Lyapunov function which lacks transparent physical meaning. With this function, both the rate of convergence and the domain of attraction of equilibria points can be estimated. This result also enables an efficient analysis of another problem, synchronization of oscillations of two oscillators. We formulate conditions that guarantee frequency synchronization and, on top of that, phase synchronization. Generalizations to the case of arbitrary number of oscillators are also discussed; solution of this problem is crucial in the analysis of power systems.     

Robust state observer and control design using command-to-state mapping

In this paper, by introducing the concept of command-to-state/output mapping, it is shown that the state of an uncertain nonlinear system can robustly be estimated if command-to-state mapping of the system and that of an uncertainty-free observer converge to each other. Then, a global Jacobian system is defined to capture this convergence property for the dynamics of estimation error, and a set of general stability and convergence conditions are derived using Lyapunov direct method. It is also shown that the conditions are constructive and can be reduced to an algebraic Lyapunov matrix equation by which nonlinear feedback in the observer and its corresponding Lyapunov function can be searched in a way parallel to those of nonlinear control design. Case studies and examples are used to illustrate the proposed observer design method. Finally, observer-based control is designed for systems whose uncertainties are generated by unknown exogenous dynamics.     

New conditions for global exponential stability of continuous-time neural networks with delays

In this paper, we investigate the global exponential stability of delayed neural network systems. For this purpose, the activation functions are assumed to be globally Lipschitz continuous. The properties of norms and the relationship of homeomorphism are adjusted to ensure the existence as well as the uniqueness of the equilibrium point. Then by employing suitable Lyapunov functional, some delay-independent sufficient conditions are derived for exponential convergence toward global equilibrium state associated with different input sources. The obtained results are shown to be more general and less restrictive than the previous results derived in the literature. Lastly, a number of examples are provided to demonstrate the validity of the results proposed.     

Adaptive robust control of uncertain systems with state and input delay

In this paper, adaptive robust control of uncertain systems with multiple time delays in states and input is considered. It is assumed that the parameter uncertainties are time varying norm-bounded whose bounds are unknown but their functional properties are known. To overcome the effect of input delay on the closed loop system stability, new Lyapunov Krasovskii functional will be introduced. It is shown that the proposed adaptive robust controller guarantees globally uniformly exponentially convergence of all system solutions to a ball with any certain convergence rate. Moreover, if there is no disturbance in the system, asymptotic stability of the closed loop system will be established. The proposed design condition is formulated in terms of linear matrix inequality (LMI) which can be easily solved by LMI Toolbox in Matlab. Finally, an illustrative example is included to show the effectiveness of results developed in this paper.     

Lyapunov and converse Lyapunov theorems for stochastic semistability

This paper develops Lyapunov and converse Lyapunov theorems for stochastic semistable nonlinear dynamical systems. Semistability is the property whereby the solutions of a stochastic dynamical system almost surely converge to (not necessarily isolated) Lyapunov stable in probability equilibrium points determined by the system initial conditions. 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 infinitesimal generator decreases along the dynamical system trajectories and is such that the Lyapunov function satisfies inequalities involving the average distance to the set of equilibria.     

On global exponential stability for impulsive cellular neural networks with time-varying delays

In this paper, the problem of global exponential stability for cellular neural networks (CNNs) with time-varying delays and fixed moments of impulsive effect is studied. A new sufficient condition has been presented ensuring the global exponential stability of the equilibrium points by using piecewise continuous Lyapunov functions and the Razumikhin technique combined with Young’s inequality. The results established here extend those given previously in the literature. Compared with the method of Lyapunov functionals as in most previous studies, our method is simpler and more effective for stability analysis.     

Global exponential stability of a class of Hopfield neural networks with delays

This paper investigates global exponential stability of a class of Hopfield neural networks with delays based on contraction mapping principle, Lyapunov function and inequality technique. Some sufficient conditions are derived that ensure the existence, uniqueness, global exponential stability of equilibrium point of the neural networks. Finally, an illustrative numerical example is given to demonstrate the effectiveness of our results.     

Impulsive effects on stability of fuzzy Cohen-Grossberg neural networks with time-varying delays.

In this correspondence, the impulsive effects on the stability of fuzzy Cohen-Grossberg neural networks (FCGNNs) with time-varying delays are considered. Several sufficient conditions are obtained ensuring global exponential stability of equilibrium point for the neural networks by the idea of vector Lyapunov function, M-matrix theory, and analytic methods. Moreover, the estimation for exponential convergence rate index is proposed. The obtained results not only show that the stability still remains under certain impulsive perturbations for the continuous stable FCGNNs with time-varying delays, but also present an approach to stabilize the unstable FCGNNs with time-varying delays by utilizing impulsive effects. An example with simulations is given to show the effectiveness of the obtained results.