Robust stabilization of Markovian delay systems with delay-dependent exponential estimates

A sufficient condition for the exponential estimates of a class of Markovian systems with time delay is established for the first time in terms of the linear matrix inequality (LMI) technique. The estimating procedure is implemented by solving an LMI. The proposed condition is also extended to the uncertain case. Moreover, a state feedback stabilizing controller which makes the closed-loop system exponentially stable with a prescribed lower bound of decay rate is designed. Numerical examples are provided to illustrate the effectiveness of the theoretical results.     

WDOP‐based Summation Inequality and its Application to Exponential Stability of Linear Delay Difference Systems

This paper is concerned with the exponential stability analysis of linear delay difference systems. Firstly, a set of weighted discrete orthogonal polynomials (WDOPs) is established by using the Gram‐Schmidt orthogonalization process, and then two WDOPs‐based summation inequalities, including some existing summation inequalities as special cases, are developed. Secondly, these WDOPs‐based summation inequalities are applied to investigate the exponential stability criteria and explicit exponential estimates of solutions of linear delay difference systems. Finally, two numerical examples indicate that the proposed WDOPs‐based approach can derive the exponential stability condition with larger decay rate than the existing ones.     

New delay-range-dependent exponential estimates for singular systems with time-varying delay

This paper considers the problem of exponential stability for continuous-time singular systems with interval time-varying delay. By defining a novel Lyapunov-Krasovskii function and giving a tighter upper bound of its derivative, a new delay-range-dependent exponential admissibility criterion, which not only guarantees the regularity, absence of impulses and exponential stability of the system but also gives the estimates of decay rate and decay coefficient, is established in terms of linear matrix inequality (LMI). The resulting criterion has advantages over the result previously reported by Haidar et al. [17] in that it involves fewer matrix variables but has less conservatism, which is established theoretically. Examples are provided to demonstrate the advantage of the proposed criterion.     

滞后动态系统广义指数稳定的充要条件

本语文针对一般的滞后动态系统建立了广义指数稳定性的充分和必要条件，并由所建立的充要条件得出了系统过渡过程的时变指数衰减估计，作为所建结果的应用，基于矩阵工给出了一类滞后动态系统过渡过程的相应指数衰减估计，本文揭示了所建衰减估计可好于那些仅基于充分稳定性条件所得的估计。     

Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin–Voigt damping

We consider the beam equation coupled by a transmission condition with a wave equation in an elastic beam. The beam has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is locally distributed acts through one of the equations only; its effect is transmitted to the other equation through the coupling. First, we consider the case where the dissipation acts through the beam equation. We show that in this case the coupled system is polynomially stable by using a recent result of Borichev and Tomilov on polynomial decay characterisation of bounded semigroups, we provide precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this system is also polynomially stable and we provide precise polynomial decay estimates for its energy. Finally, we show the lack of uniform exponential decay of solutions for both models.     

带马尔科夫跳的模态相关时变时滞随机系统的状态反馈控制器设计

针对带马尔科夫跳的模态相关时变时滞系统得到了一个改进的均方指数稳定结果并设计了状态反馈控制器. 首先, 通过构造一个改进的Lyapunov-Krasovskii泛函, 以线性矩阵不等式的形式给出一个均方指数稳定性条件; 这里, 衰减率可以是一个在区间内取值的有限常数, 同时, 时变时滞的导数上界不要求小于1; 基于得到的稳定性条件, 设计了状态反馈的控制器. 最后, 通过两个仿真算例验证了所得理论的结果的有效性, 并与已有结果相比较, 保守性较弱.     

Decay Rates for a Beam with Pointwise Force and Moment Feedback

We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator. Date received: October 30, 2000. Date revised: December 20, 2001.     

New Exponential Estimates for Time-Delay Systems

This note considers the problem of exponential stability for time-delay systems. In terms of linear matrix inequalities, a new sufficient condition for exponential stability is obtained. Based on this, an improved upper bound of the decay rate can be easily calculated. When time-varying norm-bounded parameter uncertainties appear, a new sufficient condition for robust exponential stability of uncertain time-delay systems is also provided. The reduced conservatism of the proposed conditions is illustrated via two numerical examples.     

Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters

This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method.     

An integral function approach to the exponential stability of linear time-varying systems

This paper studies the exponential stability of linear time-varying (LTV) systems using the recent proposed integral function. By showing the properties of the integral function and applying the Bellman-Gronwall Lemma, a sufficient and necessary condition for the exponential stability of LTV systems is derived. Furthermore, the exponential decay rate of the system trajectories can be obtained by computing the radii of convergence of integral function. The algorithm for computing the integral function is also developed and two classical examples are given to illustrate the proposed approach.     

时滞切换系统的时滞相关性稳定判据

考虑由时滞子系统构成的切换系统，分析其在任意切换作用下保持指数稳定性的条件．利用较为一般的Lyapunov—Krasovskii泛函构造方式及相关的解析技巧给出了以线性矩阵不等式表述的时滞相关性稳定判据；进而证明了其指数衰减率对于所有切换序列一致成立，即其完全取决于子系统的结构特征．仿真算例证实了方法的有效性．     

区间化时变时延的网络化切换系统建模与控制

研究了具有从传感器到控制器和控制器到执行器存在双边时变时延的网络控制系统指数稳定性的问题.首先将时延变化范围划分为多个等分区间,然后采用增广矩阵的方法建立了参数不确定的离散时间切换闭环系统模型.同时基于平均驻留时间分析方法,给出了系统满足指数稳定的条件,接着进一步的建立了时延区间划分个数与系统状态指数衰减率的定量关系.该方法有效降低了系统设计的保守性,一定程度上减少了系统状态收敛的时间.最后通过数值仿真验证了所提方法的有效性.     

Delay-dependent exponential stability of stochastic delay differential system whose coefficients obey the polynomial growth condition

This article discusses the exponential stability of nonlinear stochastic delay differential systems (SDDSs) whose coefficients obey the polynomial growth condition. Delay-dependent criteria on almost sure exponential stability and pth moment exponential stability of such SDDSs have been established. By applying some novel techniques, our criteria work for many SDDSs including some cases in which the ?V operator has a complicated form, which seemingly prevents the existing results from being directly used. The range of order of moment exponential stability and the decay rate can be estimated through the coefficients of the system.     

Razumikhin‐type theorems on general decay stability and robustness for stochastic functional differential equations

This paper establishes Razumikhin‐type theorems on general decay stability for stochastic functional differential equations. This improves existing stochastic Razumikhin‐type theorems and can make us examine the stability with general decay rate in the sense of the pth moment and almost sure. These stabilities may be specialized as the exponential stability and the polynomial stability. When the almost sure stability is examined, the conditions of this paper may defy the linear growth condition for the drift term, which implies that the theorems of this paper can work for some cases to which the existing results cannot be applied. This paper also examines some sufficient criteria under which this stability is robust. To illustrate applications of our results clearly, this paper also gives two examples and examines the exponential stability and the polynomial stability, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

Global exponential stability of 2D switched positive nonlinear systems described by the Roesser model

In this paper, we aim to investigate the stability of 2D switched positive nonlinear systems with time‐varying delays in the Roesser model, which includes 2D switched positive linear systems as a special case. By using the average dwell time approach, we give a sufficient condition for the exponential stability of 2D switched positive nonlinear systems. The difficulty caused by the delays is overcome by introducing a model transform and the method used in this paper is different from conventional Lyapunov‐Krasovskii functional method. An explicit exponential bound on the decay rate is presented. We also extend the result to the general 2D switched linear systems, not necessarily positive. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.     

Exponential stability of nonlinear delay equation with constant decay rate via perturbed system method

This paper studies the exponential stability of nonlinear differential equations with constant decay rate under the assumption that the corresponding crisp equation (without delay, simply, nondelay equation) is exponentially stable. Different from most publications dealing with delay systems by applying Lyapunov-type methods, the perturbed system method is used in this paper. It shall be shown that the considered equations will remain exponentially stable provided the time lag is small enough. Moreover, we formulate and estimate the threshold of delay ensuring exponential stability when a constant decay rate appears explicitly in system model, which is better than the existing results.     

Detecting systematic deviations from single exponentiality,using numerical integration and difference plots

Non-exponential behavior due to, e.g., a second exponential, a second-order process or a poor base line, is made more obvious with enhanced S/N when the data are numerically integrated. Both model calculations and decay data show ≈ 10-fold enhancement. It is shown that ≈ 2% of a second exponential (with twice the rate constant) is detected in a decay that is almost single exponential.     

Stabilization of Sampled-Data Control Systems With Control Inputs Missing

This note is concerned with the stabilization problem for sampled-data control systems with control inputs missing. A sufficient condition is derived for the existence of exponentially stabilizing state feedback controllers via the switched system approach. The obtained condition establishes several quantitative relations among some system parameters (such as the sampling period and the exponential decay rate), the actual data missing rate, and the admissible data missing rate bound. Two illustrative examples are given to show the effectiveness of the proposed results.      

New criteria on asymptotic behavior of neutral stochastic functional differential equations

This paper investigates the asymptotic behavior of neutral stochastic functional differential equations (NSFDEs) under both the local Lipschitz condition and the one dependent on the diffusion operator and on a coercivity term, which is more general than the classical growth condition. Some sufficient conditions for stability with general decay rate and boundedness of NSFDEs are derived via the Lyapunov analysis method and some stochastic analysis techniques. Our results not only cover a wide class of highly nonlinear NSFDEs but they can also deal with general stability issues including the polynomial stability and the exponential stability. Finally, an illustrative example is provided to show the effectiveness of our theoretical results.     

网络控制系统时变采样周期的建模与切换控制*

针对具有双边随机时延和丢包的网络控制系统,首先采用了主动时变采样周期的方法,利用事件和时间驱动相结合方式,传感器的采样周期可实时地跟随网络延时和丢包的变化而改变,克服了长时延和数据包错序的问题。然后将系统建立为统一的切换系统模型,结合基于平均驻留时间的方法,给出了系统状态满足指数稳定的条件,并且描述了其指数衰减率和丢包率之间的定量关系。最后通过数值例仿真验证了本文所提方法的有效性。