Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems

The Bogoliubov–Mitropolsky small parameter method is used to study the behavior of stochastic differential systems in the analysis of the corresponding properties of solutions of averaged systems.     

Stability in impulsive systems with Markov perturbations in averaging scheme. 2. Averaging principle for impulsive Markov systems and stability analysis based on averaged equations

The Bogoliubov–Mitropolsky small parameter method is used to study the behavior of stochastic differential systems in the analysis of the corresponding properties of solutions of averaged systems.     

Robust stability of impulsive periodic hybrid systems with dependent perturbations

This paper derives sufficient conditions for the stability of impulsive periodic hybrid systems with dependent perturbations. Bounds on the size of the polytope of uncertainties which still guarantees stability are derived.     

带有时滞摄动的线性切换系统的稳定性

研究一类带有时滞摄动的线性切换系统的稳定性，分别利用单李亚普诺夫函数方法和多李亚普诺夫函数方法给出了使切换系统渐近稳定的条件和切换律的设计方法。仿真结果验证了所提出设计方法的有效性。     

Stability of neutral delay-differential systems with nonlinear perturbations

In this paper, the problem of the stability analysis for neutral delay-differential systems with nonlinear perturbations is investigated. Using the Lyapunov method, we present new sufficient conditions for the stability, in terms of the linear matrix inequality (LMI), which can be easily solved by various convex optimization algorithms. Numerical examples are given to illustrate the application of the proposed method.     

The persistence of nonoscillatory solutions of difference equations under impulsive perturbations

We obtain a sufficient condition for the persistence of nonoscillatory solutions of the difference equation with continuous variable, ,under the impulsive perturbations, x(t_k+τ)−x(t_k)=I_k(x(t_k)),kN(1),     

Analysis of stability of stochastic dynamic systems under Poisson perturbations. II. Stability of solutions in quadratic mean of systems of linear stochastic differential equations under Poisson perturbations

Necessary and sufficient conditions of asymptotic stability in quadratic mean are obtained for trivial solutions of systems of linear stochastic differential equations under Poisson perturbations. Model problems are analyzed. Part I of this article is published in No. 4 (2005). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–66, November–December 2005.     

带有非线性扰动的时变时滞系统的稳定性准则

研究了带有非线性扰动的时变时滞系统的稳定性问题.基于时滞分割方法,提出了保守性更小的系统稳定性分析准则.利用一个自由参数将时滞区间分割为2个子区间,进而构造了含有时滞分割点状态信息和3重积分项的Lyapunov-Krasovskii泛函,并采用自由矩阵积分不等式界定泛函导数中的交叉项.基于Lyapunov稳定性定理,得到了以线性矩阵不等式描述的时滞相关型稳定性准则.数值算例表明该稳定性准则能够得到更大的时滞上界,与已有结果相比具有更小的保守性.     

Stability and estimates for the convergence of solutions for systems involving quadratic terms with constant deviating arguments

In this paper, we shall consider systems of differential equations involving quadratic terms with constant deviating arguments. Stability of the systems is investigated by employing the second Lyapunov's method with a special rational function. This function provides convergence rate estimates as well as offers explicit sufficient conditions for the asymptotic stability of solutions.     

Decentralized stabilization of singular and time-delay large-scalecontrol systems with impulsive solutions

Decentralized stabilization of singular and time-delay large-scale control systems with impulsive solutions which has not been studied up to now is introduced and studied. The control problem considered is the construction of a decentralized controller which ensures that a certain singular, impulsive, and time-delay closed-loop large-scale system is exponentially stable. By means of lumped Picard and Gauss-Seidel iteration methods which enable the authors to avoid the difficulties of constructing Lyapunov functions, explicit algebraic criteria of stabilization for such systems are established. An example is given to illustrate the effectiveness of results obtained     

Non-linear impulsive dynamical systems. Part I: Stability and dissipativity

In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems. Furthermore, we generalize dissipativity theory to non-linear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics and dissipated energy over the resetting instants. Furthermore, extended Kalman‐Yakubovich‐Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness via system storage functions are derived. Finally, the framework is specialized to passive and non-expansive impulsive systems to provide a generalization of the classical notions of passivity and non-expansivity for non-linear impulsive systems. These results are used in the second part of this paper to develop extensions of the small gain and positivity theorems for feedback impulsive systems as well as to develop optimal hybrid feedback controllers.     

Weak convergence results for sequential regression in memoryless systems†

Weak convergence results are obtained for a sequential regression algorithm that arises in the identification of nonlinear, memoryless systems and the adaptive design of moving average filters. The algorithm is shown to be weakly consistent if the system input is a wide-sense stationary sequence of order four that satisfies certain covariance and fourth-cumulant conditions. The conditions are essentially asymptotic independence requirements that permit one to relax the (usually required) strict independence requirements on the input data.     

Stability of dynamic systems with aftereffect with due regard for Markov perturbations

Lyapunov-Krasovskii functionals and infinitesimal operators are employed to analyze a system for asymptotic stochastic stability on the whole and asymptotic stability on the whole. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 134–146, November–December 2007.     

一类含扰动的非线性切换系统稳定性分析

首先在非零扰动情况下,利用平均滞留时间的方法给出保证切换系统一致有界或一致终极有界的条件以及最终边界.当切换系统中含有不稳定子系统时,证明了当切换规则满足一定条件时,仍可保证切换系统是有界的.然后在零扰动情况下,给出了扰动非线性切换系统稳定的充分条件.最后通过仿真算例说明了所得结果的有效性.     

Stability of networked control systems with asynchronous renewal links: An impulsive systems approach

We consider networked control systems in which sensors, actuators, and controller transmit through asynchronous communication links, each introducing independent and identically distributed intervals between transmissions. We model these scenarios through impulsive systems with several reset maps triggered by independent renewal processes, i.e., the intervals between jumps associated with a given reset map are identically distributed and independent of the other jump intervals. For linear dynamic and reset maps, we establish that mean exponential stability is equivalent to the spectral radius of an integral operator being less than one. We also prove that the origin of a non-linear impulsive system is (locally) stable with probability one if its local linearization about the zero equilibrium is mean exponentially stable, which justifies the importance of studying the linear case. The applicability of the results is illustrated through an example using a linearized model of a batch-reactor.     

Stability of solutions of differential equations under generalized pulse perturbations

Consideration was given to stability of the solution of the system of differential equations to the function of a locally bounded variation added to the right-hand side of the derivative (understood in the general sense). Simple attributes of stability and asymptotic stability were established for different classes of these equations.     

Stability analysis and control synthesis of switched impulsive systems

In this paper, we investigate the stability analysis problem of switched impulsive nonlinear systems and several stabilization problems of switched discrete‐time linear systems are studied. First, sufficient conditions ensuring globally uniformly asymptotically stability of switched nonlinear impulsive system under arbitrary and DDT (dynamical dwell time which defines the length of the time interval between two successive switchings) switching are derived, respectively. In the DDT switching case, we first consider the switched system composed by stable subsystems, then we extend the results to the case where not all subsystems are stable. The stabilizations of switched discrete‐time linear system under arbitrary switching, DDT switching and asynchronous switching are investigated respectively. Based on the stability analysis results, the control synthesis consists of controller design for each subsystem and state impulsive jumping generators design at switching instant. With the aid of the state impulsive jumping generators at switching instant, the ‘energy’ produced by switching can be minimized, which leads to less conservative results. Several numerical examples are given to illustrate the proposed results within this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability criterion of linear delayed impulsive differential systems with impulse time windows

In this paper, we study the uniform stability of linear delayed differential equationswith impulse time windows. By means of Lyapunov functions and Razumikhin technique combined with classification discussion technique, the criterion of uniform stability is obtained, which may be used to discuss others stability of delayed differential equations with impulse time win-dows. Two examples are given to illustrate the effectiveness of the theoretic result.