线性时变系统有限时间稳定性分析

针对一般非线性时变系统的有限时间稳定性分析问题,考虑到系统初始时刻在有限时间区间内的变化,本文分别提出了一般非线性时变系统的一致有限时间稳定性,一致收缩稳定性和固定调节时间一致收缩稳定性定义.针对一类线性时变系统,基于计算系统的所有轨线的包线的思想,本文分别给出了判定该类系统的收缩稳定性、固定调节时间收缩稳定性、一致有限时间稳定性、一致收缩稳定性、固定调节时间一致收缩稳定性的充分必要条件,同时给出了判定该类系统一致收缩稳定性及固定调节时间一致收缩稳定性的3个充分条件.进一步,本文将所得定理结果推广到了周期线性时变系统,所得结论为判定周期线性时变系统关于任意初始时刻的一致有限时间稳定性,一致收缩稳定性及固定调节时间一致收缩稳定性提供了理论依据.最后,以4个数值算例和两航天器相对运动过程为例验证了本文结果的正确性.     

Robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays

In this paper, we consider delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed delays. Although many kinds of stability conditions for fuzzy systems with discrete delays have already been obtained, almost no stability condition for fuzzy systems with distributed delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed delays. Here we employ a generalized Lyapunov functional to obtain delay-dependent stability conditions of fuzzy systems with discrete and distributed delays. We introduce some free weighting matrices to such a Lyapunov functional in order to reduce the conservatism in stability conditions. These techniques lead to generalized and less conservative stability conditions. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we give two examples to illustrate our results. Delay-dependent stability conditions obtained here are shown to guarantee a wide stability region.     

Novel development of Lyapunov stability of motion

The paper presents a refined analysis of the influence of initial data on dynamic behaviour and stability properties of non-stationary systems. As a result, new stability properties are discovered and defined as well as the corresponding stability domains. Furthermore, general necessary conditions are established for asymptotic stability of the unperturbed motion of these systems, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the motion in such cases novel criteria are established. Under the criteria the Eulerian derivative of a system Lyapunov function may be non-positive only and still guarantee asymptotic stability of the unperturbed motion

The results are illustrated by examples.     

Estimate of total stability of delay difference systems

The concepts of total stability and total asymptotic stability for delay difference systems in the general form are defined. Moreover, the criteria for total stability and total asymptotic stability of linear delay difference systems are established. The obtained results are qualitative and quantitative in the sense that the estimates of the total stability region and the total asymptotic stability region are given.     

Biquadratic stability of uncertain linear systems

Deals with the problem of stability analysis for linear systems with uncertain real, possibly time-varying, parameters. A robust stability approach based on a Lyapunov function which depends quadratically on the uncertain parameters as well as in the system state is proposed. This robust stability approach, referred to as biquadratic stability, is suited to deal with uncertain real parameters with magnitude and rate of change which are confined to a given convex region. A linear matrix inequality based sufficient condition for biquadratic stability is developed. The proposed robust stability analysis method includes quadratic stability and affine quadratic stability as particular cases     

Directional stability radius: a stability analysis tool for uncertain polynomial systems

Coefficients of characteristic polynomials for stable parametrically uncertain systems are allowed to perturb to some extent for stability. Stability radius is a useful tool to assess the allowance of the stability for the systems. To enhance its usefulness, we modify stability radius so that it takes into account of given restricted perturbations, which we call directional stability radius. For an application, we show shifted-Hurwitz stability conditions and a stability analysis method for interval polynomial systems using the directional stability radii.     

On partial detectability of the nonlinear dynamic systems

Conditions were obtained under which the uniform stability (uniform asymptotic stability) in one part of the variables of the zero equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position relative to another, larger part of variables. Conditions were also obtained under which the uniform stability (uniform asymptotic stability) in one part of variables of the “partial” (zero) equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position. These conditions complement a number of the well-known results of the theory of partial stability and partial detectability of the nonlinear dynamic systems. Application of the results obtained to the problems of partial stabilization of the nonlinear control systems was considered.     

Stability analysis of Takagi-Sugeno stochastic fuzzy Hopfield neural networks with discrete and distributed time varying delays

In this paper, the global stability problem of Takagi-Sugeno (T-S) stochastic fuzzy Hopfield neural networks (TSSFHNNs) with discrete and distributed time varying delays is considered. A novel LMI-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of TSSFHNNs with discrete and distributed time varying delays. Here we choose a generalized Lyapunov functional and introduce a parameterized model transformation with free weighting matrices to it, in order to obtain stability region. In fact, these techniques lead to generalized and less conservative stability condition that guarantee the wide stability region. The proposed stability conditions are demonstrated with numerical examples. Comparison with other stability conditions in the literature shows that our conditions are the more powerful ones to guarantee the widest stability region.     

Remarks on three types of asymptotic stability

In this note, three types of asymptotic stability (Liapunov stability, Poincaré stability, and Zhukovkij stability) of motion in continuous autonomous dynamical systems are discussed. It is shown that each type of asymptotic stability can be geometrically characterized by omega limit set of an orbit of this type of stability.     

Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions

A generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and/or uncertainty. This paper will further enhance absolute stability analysis by using the composite quadratic Lyapunov function whose level set is the convex hull of a family of ellipsoids. The absolute stability analysis will be approached by characterizing absolutely contractively invariant (ACI) level sets of the composite quadratic Lyapunov functions. This objective will be achieved through three steps. The first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems. The second step establishes stability conditions for linear difference inclusions and then for saturated linear systems. The third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability. Based on the conditions for absolute stability, optimization problems are formulated for the estimation of the stability region. Numerical examples demonstrate that stability analysis results based on composite quadratic Lyapunov functions improve significantly on what can be achieved with quadratic Lyapunov functions.     

六足步行机的能量稳定性

本文详细讨论了六足步行机器人的能量稳定方法,给出并证明了六足步行机能量稳定裕量的一般计算公式,分析了横向及纵向行走的六足步行机采用三角步态时的能量稳定性,比较了能量稳定法和几何稳定法.     

Finite‐time stability and stabilization of semi‐Markovian jump systems with time delay

Semi‐Markovian jump systems are more general than Markovian jump systems in modeling practical systems. On the other hand, the finite‐time stochastic stability is also more effective than stochastic stability in practical systems. This paper focuses on the finite‐time stochastic stability, exponential stochastic stability, and stabilization of semi‐Markovian jump systems with time‐varying delay. First, a new stability condition is presented to guarantee the finite‐time stochastic stability of the system by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. Second, the stability criterion is further proved to guarantee the exponential stochastic stability of the system. Moreover, a controller design method is also presented according to the stability criterion. Finally, an example is provided to illustrate that the proposed stability condition is less conservative than other existing results. Additionally, we use the proposed method to design a controller for a load frequency control system to illustrate the effectiveness of the method in a practical system of the proposed method.     

Observers for Lipschitz nonlinear systems

This paper presents some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The stability of the nonlinear observer for such systems is not determined purely by the eigenvalues of the linear stability matrix. The correct necessary and sufficient conditions on the stability matrix that ensure asymptotic stability of the observer are presented. These conditions are then reformulated to obtain a sufficient condition for stability in terms of the eigenvalues and the eigenvectors of the linear stability matrix. The eigenvalues have to be located sufficiently far out into the left half-plane, and the eigenvectors also have to be sufficiently well-conditioned for ensuring asymptotic stability. Based on these results, a systematic computational algorithm is then presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability     

On the gait stability of biped machines

The control and stability properties of legged machines in locomotion are considered. The general stability of all such machines is separated into three categories: stability of the body's orientation with respect to earth; stability of the body's trajectory; stability of the gait. The control laws for a fully dynamic system with six degrees of freedom are presented. Their stability properties are studied by simulation with the introduction of disturbances.     

Conditions for the equivalence of internal and external stabilityfor distributed parameter systems

Conditions are given on a controlled, observed distributed parameter system under which external stability, also known as input-output stability, implies internal stability, which is the exponential stability of the underlying semigroup generator. It is shown that when the system satisfies a general definition of stabilizability and detectability, external stability is equivalent to internal stability. Unlike the case previous definitions of stabilizability and detectability, a system which satisfies these does not necessarily have a spectrum decomposition. Therefore, in order to check internal stability, it is often sufficient to check the boundedness of the transfer function in the right-half complex plane. This is illustrated with a simple example     

Stability preserving mappings for stochastic dynamical systems

We first formulate a general model for stochastic dynamical systems that is suitable in the stability analysis of invariant sets. This model is sufficiently general to include as special cases most of the stochastic systems considered in the literature. We then adapt several existing stability concepts to this model and we introduce the notion of stability preserving mapping of stochastic dynamical systems. Next, we establish a result which ensures that a function is a stability preserving mapping, and we use this result in proving a comparison stability theorem for general stochastic dynamical systems. We apply the comparison stability theorem in the stability analysis of dynamical systems determined by Ito differential equations     

不确定多重状态时滞离散系统的LMI鲁棒稳定条件

研究了具有多重状态时滞的凸多面体不确定离散系统的鲁棒稳定性分析问题．基于参数依赖的李亚普诺夫稳定性和线性矩阵不等式推导出使得时滞鲁棒稳定系统鲁棒稳定的充分条件．应用此条件，通过测试一组线性矩阵不等式的可解性即可达到判定系统的鲁棒稳定性的目的．因为使用了参数依赖的李亚普诺夫稳定性思想，此鲁棒稳定条件比基于二次稳定概念的稳定条件的保守性更小．算例验证了结果．     

Stability and stabilization of stochastic systems with multiplicative noise

In this paper, stability and stabilization of linear stochastic time-invariant systems are studied based on spectrum technique. Firstly, the relationship among mean square exponential stability, asymptotical mean square stability, second-order moment exponential stability and the spectral location of the systems is revealed with the help of a spectrum operator L _A,C . Then, we focus on almost sure exponential stability and stochastic stabilization. A criterion on almost sure exponential stability based on spectrum technique is obtained. Sufficient conditions for mean square exponentially stability and asymptotic mean square stability are given via linear matrix inequality approach and some numerical examples to illustrate the effectiveness of our results are presented.     

Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag–Leffler stability and generalized Mittag–Leffler stability notions. With the definitions of Mittag–Leffler stability and generalized Mittag–Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.