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.     

一类随机人口发展系统的指数稳定性

对人口系统的讨论 ,通常的数学模型没有考虑外界环境对系统的影响 .在假设随机的外界环境对迁移产生扰动的条件下 ,给出Hilbert空间中一类随机时变人口发展系统 .对随机时变人口发展系统的均方稳定性和指数稳定性进行了讨论 .利用Burkholder_Davis_Gundy不等式 ,Gronwall引理和Kolmogorov不等式得到了均方稳定和指数稳定的充分条件 .最后提出如果生育率选作控制变量 ,系统仍然是均方和指数稳定的 ,并可进一步讨论它的最优控制问题     

Parametric continuity of solutions to stochastic functional differential equations with poisson perturbations

The small-parameter method and the notion of averaged system are used to analyze the asymptotic stability in the mean square of the original system of stochastic differential equations. The stability of a system with continuous perturbations is considered. It is proved that the small-parameter method can be applied to stochastic differential equations with discontinuous trajectories, i.e., that stochastic differential depends on the Poisson integral.     

Stability of Solutions for Stochastic Impulsive Systems via Comparison Approach

This note studies stability problem of solutions for stochastic impulsive systems. By employing Lyapunov-like function method and It's formula, comparison principles of existence and uniqueness and stability of solutions for stochastic impulsive systems are established. Based on these comparison principles, the stability properties of stochastic impulsive systems are derived by the corresponding stability properties of a deterministic impulsive system. As the application, the stability results are used to design impulsive control for the stabilization of unstable stochastic systems. Finally, one example is given to illustrate the obtained results.     

Robustness of discrete-time systems for unstructured stochasticperturbations

Several stochastic stability robustness measures are presented for nominally exponentially stable linear discrete-time systems with unstructured perturbations having second-moment bounds. Dependence of these measures on the stability degree of the nominal system and other parameters used in the procedure is illustrated. By using the time evolution of the second moment of the system state and stochastic Lyapunov stability results (positive super-martingale convergence theorems), the ability of nominally exponentially stable systems to maintain stability in the presence of unstructured stochastic (linear and nonlinear) perturbations is demonstrated. Quantitative results are given to determine the maximum modeling uncertainty which can be tolerated in design. Upper bounds on the second moments of stochastic perturbations to maintain the mean-square and almost sure stability of these systems in the presence of unstructured perturbations are obtained     

Asymptotical stability of 2-D linear discrete stochastic systems

The main goal of the present paper is to find computable stability criteria for two-dimensional stochastic systems based on Kronecker product and nonnegative matrices theory. First, 2-D discrete stochastic system model is established by extending system matrices of the well-known Fornasini–Marchesini?s second model into stochastic matrices. The elements of these stochastic matrices are second-order, weakly stationary white-noise sequences. Second, a necessary and sufficient condition for 2-D stochastic systems is presented, this is the first time that has been proposed. Third, computable mean-square asymptotic stability criteria are derived via Kronecker product and the nonnegative matrix theory. The criteria are only sufficient conditions. Finally, illustrative examples are provided.     

New results on stability analysis and stabilization of time‐delay continuous Markovian jump systems with partially known rates matrix

In this note, the problems of stability analysis and controller synthesis of Markovian jump systems with time‐varying delay and partially known transition rates are investigated via an input–output approach. First, the system under consideration is transformed into an interconnected system, and new results on stochastic scaled small‐gain condition for stochastic interconnected systems are established, which are crucial for the problems considered in this paper. Based on the system transformation and the stochastic scaled small‐gain theorem, stochastic stability of the original system is examined via the stochastic version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time‐varying delay and the new result on the stochastic scaled small‐gain theorem. The proposed stability condition is demonstrated to be much less conservative than most existing results. Moreover, the problem of stabilization is further solved with an admissible controller designed via convex optimizations, whose effectiveness is also illustrated via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Markov切换系统的随机稳定性分析

根据混合系统离散状态的动态行为和Markov链的状态也是离散的特点,提出了一类离散状态的动态行为是Markov链的混合系统。与传统的混合系统相比,这类系统能够刻画出混合系统离散动态行为的随机性,可以用来描述系统受到外界环境因素制约和内部突发事件等随机因素影响而发生变化的动态行为。根据动态系统的稳定性定义以及随机过程理论,给出了Markov线性切换系统的随机稳定性定义,并且分析了Markov线性切换系统的随机稳定性问题,给出了判定随机稳定性的充分必要条件。     

Stability and control of uncertain ICPT system considering time‐varying delay and stochastic disturbance

This paper studies the problem of robust exponential stability for uncertain inductively coupled power transfer (ICPT) system considering time‐varying delay and stochastic disturbance. Firstly, the model of the system is set up via a time domain method. Secondly, based on the Newton‐Leibnitz formula and stability theory, a new stability analysis of the ICPT system with uncertain parameter and time‐varying delay or stochastic disturbance is presented respectively. The proposed approach can be applied for analyzing other similar systems, and the obtained results can be also used for estimating convergence rate and region of stability. Thirdly, based on the Lyapunov‐Krasovskii functional (LKF) approach and the stochastic stability theory, robust exponential stability criteria in the mean square are derived and the relevant controller is designed. The proposed method can further reduce conservatism. Finally, the correctness and effectiveness of the obtained results are verified by an example and simulations indicate that the larger time delay, the slower attenuation. Simultaneously, the uncertain parameter and the stochastic disturbance have a significant influence on the region of stability. In addition, in respect of reducing conservatism, the effectiveness of the proposed method is demonstrated by comparing with other papers. In addition, the designed controller shows better performance for the ICPT system with the parameter uncertainty, the time‐varying delay, and the stochastic disturbance.     

Asymptotic stability in probability and stabilization for a class of discrete‐time stochastic systems

This paper investigates asymptotic stability in probability and stabilization designs of discrete‐time stochastic systems with state‐dependent noise perturbations. Our work begins with a lemma on a special discrete‐time stochastic system for which almost all of its sample paths starting from a nonzero initial value will never reach the origin subsequently. This motivates us to deal with the asymptotic stability in probability of discrete‐time stochastic systems. A stochastic Lyapunov theorem on asymptotic stability in probability is proved by means of the convergence theorem of supermartingale. An example is given to show the difference between asymptotic stability in probability and almost surely asymptotic stability. Based on the stochastic Lyapunov theorem, the problem of asymptotic stabilization for discrete‐time stochastic control systems is considered. Some sufficient conditions are proposed and applied for constructing asymptotically stable feedback controllers. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust stabilization of load frequency control system under networked environment

The deregulation of the electricity market made the open communication infrastructure an exigent need for future power system. In this scenario dedicated communication links are replaced by shared networks. These shared networks are characterized by random time delay and data loss. The random time delay and data loss may lead to system instability if they are not considered during the controller design stage. Load frequency control systems used to rely on dedicated communication links. To meet future power system challenges these dedicated networks are replaced by open communication links which makes the system stochastic. In this paper, the stochastic stabilization of load frequency control system under networked environment is investigated. The shared network is represented by three states which are governed by Markov chains. A controller synthesis method based on the stochastic stability criteria is presented in the paper. A one-area load frequency control system is chosen as case study. The effectiveness of the proposed method for the controller synthesis is tested through simulation. The derived proportion integration (PI) controller proves to be optimum where it is a compromise between compensating the random time delay effects and degrading the system dynamic performance. The range of the PI controller gains that guarantee the stochastic stability is determined. Also the range of the PI controller gains that achieve the robust stochastic stability is determined where the decay rate is used to measure the robustness of the system.     

Control design for stochastic one-sided Lipschitz differential inclusion system with time delay

This paper is concerned with the control design problem for stochastic one-sided Lipschitz differential inclusion system with time delay. Different from the current works on the differential inclusion system, the nonlinear function with time delay is assumed to be one-sided Lipschitz. First, the theory of stochastic differential equation is extended to stochastic differential inclusion and the stability criterion is given. Next, convex hull Lyapunov function is used to construct the nonlinear controllers, and the sufficient conditions for stability are derived by a set of bilinear matrix inequalities. Finally, three simulation examples are provided to verify the validity of the designed controllers.     

一类无限维随机关联大系统的全局指数稳定性

基于箱体理论,利用向量函数法,研究了一类无限维随机非线性关联大系统的全局指数稳定性.通过分析相应的随机微分不等式的稳定性,得到了该类大系统全局指数稳定的一个判据.该判据利用随机大系统的系数矩阵以及与大系统关联的Lyapunov矩阵方程的解构造判定条件来判定大系统的全局指数稳定性,计算简便,便于应用.     

On the stochastic stability of a parabolic type system

A stochastic distributed parameter system of a parabolic type is dealt with. The system is stochastic duo to a distributed multiplicative gain. The gain is a non-linear function of a Wiener process. By applying a finite-difference scheme on the spatial coordinate, and using a stochastic Linpunov type functional, sufficient conditions for n weak stability of the system's solutions, are derived.     

时延网络化控制系统的H2/H∞混合控制

针对存在多步随机传输时延的网络化控制系统模型 ,研究了其随机稳定性及H₂/H_∞混合控制问题 .在一定的系统通信控制模式下 ,网络传输时延可以建模为一个马尔可夫随机过程 ,通过增广系统状态的方法将原系统转化为一个具有随机跳变系数的离散系统 ,同时通过建立随机跳变Lyapunov函数 ,构建了满足系统随机稳定的H_∞次优和H₂/H_∞混合控制状态反馈控制器 .该控制器可通过求解一组耦合的矩阵不等式而得 .     

Robust stabilization of stochastic systems based on the LQ controller

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems, we have designed an optimal controller which guarantees the exponential stability of the system. Actually, we employed Lyapunov fimction approach and the stochastic algebraic Riccati equation (SARE) to have shown the robusmess of the linear quadratic(LQ) optimal control law.And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed-loop systems are given.     

Exponential stability of stochastic systems with hysteresis switching

For stochastic systems with state-dependent switching which are motivated by active regions of subsystems, the exponential stability is studied in this paper. Distinct from most of the existing references, the existence of the solution to stochastic switched systems is not given as a priori information but can be proved under some easily verified conditions. By the aid of Dynkin’s formula, Itô’s formula and exponential martingale inequality, the criteria on moment exponential stability and almost sure exponential stability of the stochastic switched system are established based on Lyapunov-like techniques. Simulation examples are presented to illustrate the validity of the results.     

Robust stabilization of stochastic systems based on the LQ cont roller

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems ,we have designed an optimal controller which guarantees the exponential stability of the system. Actually ,we employed Lyapunov function approach and the stochastic algebraic Riccati equation (SARE) to have shown the robustness of the linear quadratic (LQ) optimal control law. And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed- loop systems are given.     

Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances

In this paper, a delay-dependent approach is developed to deal with the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. The system matrices are assumed to be uncertain within given intervals, the time delays appear in both the system states and the nonlinear disturbances, and the stochastic perturbation is in the form of a Brownian motion. The purpose of the addressed stochastic stabilization problem is to design a memoryless state feedback controller such that, for all admissible interval uncertainties and nonlinear disturbances, the closed-loop system is asymptotically stable in the mean square, where the stability criteria are dependent on the length of the time delay and therefore less conservative. By using Itô's differential formula and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the stability of the stochastic interval delay systems. Then, the controller gain is characterized in terms of the solution to a delay-dependent linear matrix inequality (LMI), which can be easily solved by using available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed design procedure.     

Mean-square exponential stability of impulsive stochastic time-delay systems with delayed impulse effects

This paper is concerned with the mean-square exponential stability problem for a class of impulsive stochastic systems with delayed impulses. The delays exhibit in both continuous subsystem and discrete subsystem. By constructing piecewise time-varying Lyapunov functions and Razumikhin technique, sufficient conditions are derived which guarantee the mean-square exponential stability for impulsive stochastic delay system. It is shown that the obtained stability conditions depend both on the lower bound and the upper bound of impulsive intervals, and the stability of system is robust with regard to sufficiently small impulse input delays. Finally, two examples are proposed to verify the efficiency of the proposed results.