非Lipschitz连续级联系统的稳定性分析及其应用

当考虑级联系统稳定性时,一般都需要系统满足局部或者全局Lipschitz连续性条件.与已有文献中的结果不同,本文给出了一种处理满足非Lipschitz连续条件下级联系统的稳定性分析方法.首先,基于积分输入状态稳定的定义,给出了级联系统全局稳定的Lyapunov形式条件.基于此,继续讨论了非Lipschitz连续情况下级联系统的有限时间稳定性.然后,利用上述稳定性分析结果,讨论了一类驱动子系统具有上三角结构的级联系统的控制设计问题.最后,给出几个例子验证了上述结果的有效性.     

非线性采样系统指数稳定的新条件

研究了非线性采样系统的稳定性问题. 对以采样周期为参数的离散时间系统族, 证明了全局指数稳定的Lyapunov定理和逆定理. 分别基于系统的一般近似模型和Euler近似模型, 给出了闭环系统全局指数稳定的新条件. 与现有结果相比, 取消了Lyapunov函数全局Lipschitz连续的假设, 减弱了闭环系统全局指数稳定的充分条件.     

一类模糊系统的分解性质和稳定性分析

探讨一类动态模糊模型的分解性质,通过一个简单的构造性的分解过程,将模糊系统的输入空间划分为一些子输入空间,从而把一般的模糊状态空间系统分解为一些定义在子输入空间上的子模糊系统.这些子模糊系统具有最简单的结构,从而可以简化模糊系统稳定性的分析及设计.进一步基于模糊系统的分解性质,引入分段连续的Lyapunov函数和S-procedure,以线性矩阵不等式的形式给出了模糊系统的稳定性条件的一个新结果.     

具有脉冲干扰和可变时滞的区间关联大系统的鲁棒指数稳定性

研究了一类具有脉冲干扰和可变时滞区间关联大系统的鲁棒指数稳定性.假设该系统的关联函数满足全局Lipschitz条件,基于矢量Lyapunov函数法和数学归纳法,给出确保该关联系统鲁棒指数稳定的充分条件.最后给出一个数值算例用以说明本文所得到结论的正确性和有效性.     

切换系统的不变性原理与不变集的状态反馈镇定

证明了一类切换系统的一个不变性原理,并将输入对状态稳定的概念推广到输入对系统某个非负能量函数稳定的情况.基于这个不变性原理以及输入对系统能量函数稳定的概念,利用多Lyapunov函数方法提出并证明了一类具有Lyapunov稳定子系统的切换系统的不变集可状态反馈镇定的条件.最后讨论了输入对系统能量函数稳定与输入对状态稳定的关系.仿真结果证明了该方法的可行性.     

带有不稳定子系统的切换非线性系统的积分输入状态稳定

本文针对带有不稳定子系统的切换非线性系统研究了系统的积分输入状态稳定性问题. 应用导数不定的 类Lyapunov函数得出切换非线性系统的积分输入状态稳定. 导数不定的类Lyapunov函数方法比传统的导数正定 的Lyapunov函数的方法更具有一般性. 文中包含两种情况: 当所有子系统为积分输入状态稳定时, 切换非线性系统 是积分输入状态稳定的; 当部分子系统为非积分输入状态稳定时, 本文证明了切换非线性系统的积分输入状态稳 定. 最后应用一个仿真例子描述了所提结果的有效性.     

一类T-S模糊控制系统的稳定性分析及设计

研究了一类输入采用双交叠模糊分划的T-S模糊控制系统稳定性分析及控制器设计问题.基于分段模糊Lyapunov函数,提出了一个新的判定开环T-S模糊系统稳定性的充分条件,该方法只需在各个模糊区间里满足模糊Lyapunov方法中的条件,其保守性比公共Lyapunov函数法和分段Lyapunov函数法的保守性更低.运用并行分布补偿法(PDC)进一步探讨了闭环T-S模糊控制系统的稳定性分析问题并设计了模糊控制器.最后,一个仿真示例说明了本文方法的有效性.     

Lipschitz非线性系统观测器设计新方法

考虑了非线性项满足Lipschitz条件的非线性系统观测器设计问题, 利用Lyapunov方法给出了新的判断观测误差稳定性的条件, 并由所给的条件通过求解线性矩阵不等式来设计观测器. 通过算例与其它方法进行的比较, 说明了所提方法的有效性.     

一类离散切换模糊系统的稳定性

针对离散模糊系统,提出一类离散切换模糊系统的稳定性问题.使用切换技术及单Lyapunov函数、多Lyapunov函数方法构造出连续状态反馈控制器,使得相应的闭环系统渐近稳定,同时设计可以实现系统全局渐近稳定的切换律.模型中的每个切换系统的子系统是离散模糊系统,取常用的平行分布补偿PDC控制器,主要条件以凸组合和矩阵不等式的形式给出,具有较强的可解性.计算机仿真结果表明设计方法的可行性与有效性.     

具有输入和状态延时网络系统的鲁棒拥塞控制

研究基于状态空间的主动队列管理算法,以状态变量的形式描述具有状态延时和输入延时的TCP／AQM模型,设计基于观测器的状态反馈控制器,观测器在线测量控制器的输出。应用线性矩阵不等式和Lyapunov—Krasovskii定理,给出AQM控制器的控制率和不依赖于延时的稳定条件。NS2仿真表明该控制算法在延时变化和突发业务流情况下,能够快速收敛于期望队列长度,动静态性能优于已有的P1控制算法。     

Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approach

In this paper, robust fuzzy model predictive control of a class of nonlinear discrete systems subjected to time delays and persistent disturbances is investigated. Based on the modeling method of delay difference inclusions, nonlinear discrete time-delay systems can be represented by T–S fuzzy systems comprised of piecewise linear delay difference equations. Moreover, Lyapunov–Razumikhin function (LRF), instead of Lyapunov–Krasovskii functional (LKF), is employed for time-delay systems due to its ability to reflect system original state space and its advantages in controller synthesis and computation. The robust positive invariance and input-to-state stability with respect to disturbance under such circumstances are investigated. A robust constraint set is adopted that the system state is converged to this set round the desired point. In addition, the controller synthesis conditions are derived by solving a set of matrix inequalities. Simulation results show that the proposed approach can be successfully applied to the well-known continuous stirred tank reactor (CSTR) systems subjected to time delay.     

Stability and control of nonlinear systems described by retarded functional equations: a review of recent results

This paper reports on recent results in a series of the work of the authors on the stability and nonlinear control for general dynamical systems described by retarded functional differential and difference equations. Both internal and external stability properties are studied. The corresponding Lyapunov and Razuminkhin characterizations for input-to-state and input-to-output stabilities are proposed. Necessary and sufficient Lyapunov-like conditions are derived for robust nonlinear stabilization. In particular, an explicit controller design procedure is developed for a new class of nonlinear time-delay systems. Lastly, sufficient assumptions, including a small-gain condition, are presented for guaranteeing the input-to-output stability of coupled systems comprised of retarded functional differential and difference equations.     

Input-to-State Stability of Time-Delay Systems: A Link With Exponential Stability

The main contribution of this technical note is to establish a link between the exponential stability of an unforced system and the input-to-state stability (ISS) via the Liapunov–Krasovskii methodology. It is proved that a system which is (globally, locally) exponentially stable in the unforced case is (globally, locally) input-to-state stable when it is forced by a measurable and locally essentially bounded input, provided that the functional describing the dynamics in the unforced case is (globally, on bounded sets) Lipschitz and the functional describing the dynamics in the forced case satisfies a Lipschitz-like hypothesis with respect to the input. Moreover, a new feedback control law is provided for delay-free linearizable and stabilizable time-delay systems, whose dynamics is described by locally Lipschitz functionals, by which the closed-loop system is ISS with respect to disturbances adding to the control law, a typical problem due to actuator errors.      

Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form

In this article, we show that the existence of a Lyapunov–Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability (GES) of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map on the right-hand side of the equation is Lipschitz on bounded sets. A link between GES and input-to-state stability is also provided.     

Input to state set stability for pulse width modulated control systems with disturbances

New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. The results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. The nonsmooth nature of PWM systems is accommodated by working with upper semicontinuous set-valued maps, locally Lipschitz inflations of these maps, and locally Lipschitz parameterizations of locally Lipschitz set-valued maps.     

Lyapunov stability and generalized invariance principle for nonconvex differential inclusions

This paper studies the system stability problems of a class of nonconvex differential inclusions. At first, a basic stability result is obtained by virtue of locally Lipschitz continuous Lyapunov functions. Moreover, a generalized invariance principle and related attraction conditions are proposed and proved to overcome the technical difficulties due to the absence of convexity. In the technical analysis, a novel set-valued derivative is proposed to deal with nonsmooth systems and nonsmooth Lyapunov functions. Additionally, the obtained results are consistent with the existing ones in the case of convex differential inclusions with regular Lyapunov functions. Finally, illustrative examples are given to show the effectiveness of the methods.     

Stability of the cell dynamics in acute myeloid leukemia

In this paper we analyze the global asymptotic stability of the trivial solution for a multi-stage maturity acute myeloid leukemia model. By employing the positivity of the corresponding nonlinear time-delay model, where the nonlinearity is locally Lipschitz, we establish the global exponential stability under the same conditions that are necessary for the local exponential stability. The result is derived for the multi-stage case via a novel construction of linear Lyapunov functionals. In a simpler model of hematopoiesis (without fast self-renewal) our conditions guarantee also global exponential stability with a given decay rate. Moreover, in this simpler case the analysis of the PDE model is presented via novel Lyapunov functionals for the transport equations.     

Direct and converse Lyapunov theorems for functional difference systems

New Lyapunov criteria for asymptotic stability and input-to-state stability of infinite dimensional systems described by functional difference equations are provided. Conditions in terms of both Lyapunov–Razumikhin functions defined on Euclidean spaces and of Lyapunov–Krasovskii functionals defined on infinite dimensional spaces are found. For the case of Lyapunov–Krasovskii functionals, necessary and sufficient conditions are provided for the asymptotic stability, in both the local and the global case, and for the input-to-state stability. This is the first time in the literature that converse Lyapunov theorems are provided for the class of nonlinear systems here studied.     

On the conservatism of the sum-of-squares method for analysis of time-delayed systems

In this paper, we present a converse Lyapunov result which shows that using polynomial Lyapunov functionals to prove the stability of linear time-delay systems is not conservative. This result motivates the sum-of-squares approach to stability analysis of linear time-delay systems. This main result is based on an extension of the Weierstrass approximation theorem to show that linear varieties of polynomials can be used to approximate linear varieties of the space of continuous functions.     

The analysis of global input-to-state stability for piecewise affine systems with time-delay

In this paper, global input-to-state stability (ISS) for discrete-time piecewise affine systems with time-delay are considered. Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhin and Lyapunov-Krasovskii methods are used. The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown, respectively.