Stability analysis of complex dynamical systems

We present two new algorithms to estimate the domain of attraction of the equilibriumx=0 of a nonlinear systemx=f(x). One of these algorithms utilizes quadratic Lyapunov functions while the second algorithm makes use of norm Lyapunov functions. Both of these procedures yield estimates for the domain of attraction which are comparable to those obtained by existing methods; however, the present algorithms appear to be significantly more efficient than existing algorithms.We also show how sometimes the applicability of the above results can be extended to high order systems by invoking the comparison principle. In doing so, we establish some results for the comparison principle which are of interest in their own right. Specifically, we relate the domain of attraction of a low order comparison system to the domain of attraction of a higher order system and we give an interpretation of the comparison principle in terms of stability preserving maps.In order to demonstrate the applicability of the present results, and in order to compare the present results with existing results, several specific examples are presented.Supported in part by the National Science Foundation under grants ENG77-28446 and ECS-81-00690.     

基于观测器的一类输出耦合网络参数辨识

诸多复杂网络控制方法一般都是在网络节点动力学的参数已知的前提下提出的.然而实际工程中,由于节点系统的复杂性,它的某些参数难以测量或者确定,参数辨识问题一直是系统控制和同步领域中的重要课题.本文针对一类输出耦合复杂动态网络模型,提出一种基于观测器的网络节点未知参数辨识方法.基于网络外部同步思想构造响应网络,利用网络中可测...     

基于滑模控制的一类节点互异复杂动态网络同步研究

复杂动态网络同步问题是复杂动态网络研究领域的一个十分重要的研究内容.本文针对一类各个节点都不相同的复杂动态网络的情况,应用滑模控制理论,设计出一种比例积分滑模面,并最终提出滑模控制准则,得到系统同步控制器的参数选择方法,使得一类复杂动态网络在有限时间内迅速达到同步.通过选取统一混沌系统作为复杂网络中的节点,对网络中各个...     

Synchronization problem of 2-D coupled dynamical networks with communication delays and missing measurements

Multidimensional Systems and Signal Processing - This study addresses a synchronization problem for an array of discrete-time two-dimensional (2-D) coupled dynamical networks with time-varying...     

Stability analysis of linear 2-D systems

The present paper is concerned with stability analysis of linear two-dimensional systems described by Fornasini–Marchesini state-space model. Necessary and sufficient conditions for asymptotic stability of the systems are obtained first. Several simple stability criteria are derived via the nonnegative matrix theory, which are sharper than those in literature. When all the parameter matrices are nonnegative, the criteria are necessary and sufficient for stability of the system. Illustrative examples are provided.     

Synchronization in general complex delayed dynamical networks

This paper investigates synchronization dynamics of a general model of complex delayed networks as well as the effects of time delays. Some simple yet generic criteria ensuring delay-independent and delay-dependent synchronization are derived, which are less conservative than those reported so far in the literature. Moreover, a scale "/spl nu/" denoted by a function of the smallest and the second largest eigenvalues of coupling matrix is presented to analyze the effects of time delays on synchronization of the networks. Furthermore, various kinds of coupling schemes, including small-world networks and scale-free networks, are studied. It is shown that, if the coupling delays are less than a positive threshold, then the network will be synchronized. On the other hand, with the increase of the coupling delays, the synchronizability of the network will be restrained and even eventually desynchronized. The results are illustrated by a prototype composed of the chaotic Duffing oscillators. Numerical simulations are also given to verify theoretical results.     

A Method for Computing the 2-D Stability Margin Based on a New Stability Test for 2-D Systems

In this paper, a new method to compute the stability margin of 2-D (two-dimensional) discrete systems is considered. The method is based on a recently proposed 2-D stability test by the author. Illustrative examples are also included.     

Global synchronization and asymptotic stability of complex dynamical networks

Global synchronization and asymptotic stability of complex dynamical networks are investigated in this paper. Based on a reference state, a sufficient condition for global synchronization and stability is derived. Unlike other approaches where only local results were obtained, the complex network is not linearized in this paper. Instead, the sufficient condition for the global synchronization and asymptotical stability is obtained here by introducing a reference state with the Lyapunov stability theorem rather than the Lyapunov exponents, and this condition is simply given in terms of the network coupling matrix therefore is very convenient to use. Furthermore, the developed technique is applied to networks consisting of nodes with unknown but bounded nonlinear functions. A typical example of a complex network with chaotic nodes is finally used to verify the theoretical results and the effectiveness of the proposed synchronization scheme.     

Stability analysis for a class of discrete-time two-dimensional nonlinear systems

Stability analysis for a class of discrete-time two-dimensional nonlinear systems is addressed in this paper. A linear matrix inequality based sufficient condition for asymptotic stability is proposed. This condition is an analogous counterpart for two-dimensional linear systems which is shown to be easily verifiable and suitable for design problems. The result is illustrated by a numerical example.     

Design for 2-D error feedback networks with identical coefficient sets

In realization of recursive digital filters with fixed point arithmetic, an error caused by roundoff arises. It is known that the level of the roundoff noise of an IIR filter tends to be high when the poles are close to the unit circle. Error feedback (EF) is an effective method to reduce the roundoff noise. It is desirable to design an EF network using as few parameters as possible in order to keep computational costs low. In this paper, we propose a method for designing a 2D EF network with identical coefficient sets. That is, the EF coefficients are divided into several subsets such that all the elements within each set have the same absolute value. In order to optimize the coefficient sets, we propose an algorithm by using the genetic algorithm. In the numerical example, we demonstrate the effectiveness of the proposed method.     

一类2-D奇异系统的鲁棒稳定性分析

针对一类不确定2-D奇异系统第二类Fornasini-Marchesini(SFM-Ⅱ)模型,在闭环系统边界相容,且无跳跃模的条件下,基于Lyapunov稳定性分析方法,利用线性矩阵不等式(LMI)得到具有不确定参数的二维SFM-Ⅱ系统鲁棒稳定性充分条件,并给出了不确定系统的静态状态反馈控制律,为2-D奇异系统的分析与设计提供了一条有效途径。     

Closed-system approach for a class of dissipative dynamical systems

Using the ideas of Livsic on closed systems (with no input), a set of state equations with special properties for a class of dissipative time-invariant systems is derived.     

Lyapunov Equation for the Stability of 2-D Systems

An analogue of the characterization of asymptotic stability of the 1-D systems by the solvability of associated Lyapunov equation is proposed here for 2-D systems. It is shown that internal stability of Roesser model is equivalent to the feasibility of some linear matrix inequality (LMI), related to quadratic Lyapunov functions.     

Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions

This paper investigates the problem of stability analysis and stabilization for two-dimensional (2-D) discrete fuzzy systems. The 2-D fuzzy system model is established based on the Fornasini–Marchesini local state-space model, and a control design procedure is proposed based on a relaxed approach in which basis-dependent Lyapunov functions are used. First, nonquadratic stability conditions are derived by means of linear matrix inequality (LMI) technique. Then, by introducing an additional instrumental matrix variable, the stabilization problem for 2-D fuzzy systems is addressed, with LMI conditions obtained for the existence of stabilizing controllers. Finally, the effectiveness and advantages of the proposed design methods based on basis-dependent Lyapunov functions are shown via two examples.     

Stability analysis of a particular class of stochastic system

In the letter, a particular class of linear system with multiplicative coloured noise is considered. It is shown that the stability of the moments can be analysed by means of an equivalent lt? equation involving white noise. The necessary and sufficient condition for almost sure asymptotic stability is also given.     

一类无不动点的二维混沌映射的复杂动力学

当前,离散映射在经济、生物、工程等领域具有广泛研究,而离散映射中的隐藏吸引子的研究目前涉及较少。文中首先通过相图、吸引域研究该映射在参数固定的情况下的动力学特性,然后将分别通过分岔图等研究系统在单参数及双参数变化情况下的动力学演化过程,从而揭示该类映射的复杂动力学的产生与演化机理。