Exponential Stability and Stabilization for Quadratic Discrete‐Time Systems with Time Delay

In this note, we deal with the exponential stability and stabilization problems for quadratic discrete‐time systems with time delay. By using the quadratic Lyapunov function and a so called ‘Finsler's lemma', delay‐independent sufficient conditions for local stability and stabilization for quadratic discrete‐time systems with time delay are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.     

Delay‐Dependent Exponential Stability for Discrete‐Time Singular Switched Systems with Time‐Varying Delay

In this paper, the exponential stability problem is investigated for a class of discrete‐time singular switched systems with time‐varying delay. By using a new Lyapunov functional and average dwell time scheme, a delay‐dependent sufficient condition is established in terms of linear matrix inequalities for the considered system to be regular, causal, and exponentially stable. Different from the existing results, in the considered systems the corresponding singular matrices do not need to have the same rank. A numerical example is given to demonstrate the effectiveness of the proposed result.     

离散时间扰动脉冲切换系统鲁棒指数稳定性

The robust exponential stability of a class of discrete time impulsive switched systems with structure perturbations is studied. Based on the average dwell time concept and by dividing the total activation time into the time with stable subsystems and the time with unstable subsystems, it is shown that if the average dwell time and the activation time ratio are properly large, the given switched system is robustly exponentially stable with a desired stability degree. Compared with the traditional Lyapunov methods, our layout is more clear and easy to carry out. Simulation results validate the correctness and effectiveness of the proposed algorithm.     

Delay‐Dependent Stability Criterion for Discrete‐Time Systems with Time‐Varying Delays

The stability analysis problem is considered for linear discrete‐time systems with time‐varying delays. A novel summation inequality is proposed, which takes the double summation information of the system state into consideration. The inequality relaxes the recently proposed discrete Wirtinger inequality and its improved version. Based on construction of a suitable Lyapunov‐Krasovskii functional and the novel summation inequality, an improved delay‐dependent stability criterion for asymptotic stability of the systems is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.     

具有时变不确定性的脉冲时滞大系统的鲁棒稳定性

提出并研究了具有时变不确定性的脉冲时滞大系统的鲁棒稳定性,借助于向量比较原理和矩阵不等式等方法,对该系统建立了若干大范围指数鲁棒稳定的判据,并举例说明了结论的有效性。     

具有参数扰动时滞系统的鲁棒镇定性

利用标量的Ｌｙａｐｕｎｏｖ函数，结合代数Ｒｉｃｃａｔｉ方程和Ｒａｙｌｅｉｇｈ’ｓ原理，讨论了具有参数扰动时滞系统的鲁棒镇定性，获得了一个充分条件。     

Global Exponential Stability Analysis of Discrete‐Time Genetic Regulatory Networks with Time Delays

This paper focuses on the study of global robust exponential stability of discrete‐time genetic regulatory networks (GRNs) with time‐invariant/time‐varying delays and parameter uncertainties. Many existing results on this problem are based on the linear matrix inequality (LMI) approach, which needs to verify whether there exists a feasible solution of a set of LMIs. Along with the increase in the number of genes, dimensions of LMIs will increase accordingly, which will lead to a large amount of calculation. Based on M‐matrix theory, sufficient conditions ensuring the global robust exponential stability of a class of discrete‐time GRNs with time‐invariant/time‐varying delays and parameter uncertainties are presented. These given conditions are to check whether a constructed constant matrix is a nonsingular M‐matrix. Simulation results of several examples are given to demonstrate the validity of the proposed method.     

State Feedback Control for Discrete‐time Markovian Jump Systems with Random Communication Time Delay

This paper concerns the stabilization problem for a class of networked control system with Markovian parameter using mode dependent state feed‐back controller through a wireless networked control system. The problem that the random network‐induced delays generated both by wireless connection contention and the movement of wireless nodes is taken into consideration. The mode dependent controller can be obtained by solving a set of parameterized bilinear matrix inequalities (BMIs). A numerical example is exhibited to show the effectiveness of the proposed design method.     

Robust Stability Analysis for Systems with Time‐Varying Delays: A Delay Function Approach

This paper is concerned with the robust stability of time‐varying delay systems with structured uncertainties. Stability conditions are provided through a Lyapunov‐Krasovskii functional (LKF) method. The proposed method introduces a linear function of the time‐varying delay to construct the LKF. With this function, two‐dimensional partition is conducted on the integral domain in the derivative of LKF. Quadratic convex combination then is employed to present stability criteria in the form of linear matrix inequalities (LMIs). The method not only exploits the information of delay at different time instants, but also enables the handling of its derivative to reduce conservatism. Numerical examples are given to show the effectiveness of our method.     

单滞后时变区间动力系统的指数稳定性

用矩阵测度和时滞微分不等式研究了单滞后时变区间动力系统 x(t) =N[P(t) ,Q(t) ]x(t) +N[C(t) ,D(t) ]x(t -τ) ,τ≥ 0 的指数稳定性 ,给出了其指数稳定的判别准则 ,推广和改进了文 [1～ 3]的工作     

Exponential Stabilization of Switched Discrete‐Time Systems with All Unstable Modes

This paper studies the exponential stabilization of switched discrete‐time systems whose subsystems are unstable. A new sufficient condition for the exponential stability of the class of systems is proposed. The result obtained is based on the determination of a lower bound of the maximum dwell time by virtue of the multiple Lyapunov functions method. The key feature is that the given stability condition does not need the value of the Lyapunov function to uniformly decrease at every switching instant. An example is provided to illustrate the effectiveness of the proposed result.     

Adaptive Finite‐Time Robust Control of Nonlinear Delay Hamiltonian Systems Via Lyapunov‐Krasovskii Method

This paper investigates the adaptive finite‐time robust control problem of a class of nonlinear time‐delay Hamiltonian systems via the Lyapunov‐Krasovskii (L‐K) method, and proposes some delay‐dependent results on the issue. Different from existing works, this paper first presents a time‐varying finite‐time stability (FTS) criterion via an L‐K functional approach, and obtains two FTS conditions by constructing specific L‐K functionals. Then, the adaptive finite‐time robust control problem is investigated for nonlinear time‐delay port‐controlled Hamiltonian (PCH) systems, and a control design procedure is presented. Finally, the effectiveness of the results is demonstrated by an illustrative example.     

Robust Absolute Stability Analysis of Multiple Time‐Delay Lur'e Systems With Parametric Uncertainties

The problem of robust absolute stability for time‐delay Lur'e systems with parametric uncertainties is investigated in this paper. The nonlinear part of the Lur'e system is assumed to be both time‐invariant and time‐varying. The structure of uncertainty is a general case that includes norm‐bounded uncertainty. Based on the Lyapunov–Krasovskii stability theory, some delay‐dependent sufficient conditions for the robust absolute stability of the Lur'e system will be derived and expressed in the form of linear matrix inequalities (LMIs). These conditions reduce the conservativeness in computing the upper bound of the maximum allowed delay in many cases. Numerical examples are given to show that the proposed stability criteria are less conservative than those reported in the established literatures.     

具有时变时滞不确定大系统的鲁棒稳定性判据

研究具有时变时滞的不确定大系统的鲁棒稳定性，利用Lyapunov泛函方法．给出该系统鲁棒稳定的充分性判别条件，并结合算例验证了所得结果的有效性和低保守性。     

Robust Exponential Stability and Stabilization of a Class of Nonlinear Stochastic Time‐Delay Systems

This paper presents new exponential stability and delayed‐state‐feedback stabilization criteria for a class of nonlinear uncertain stochastic time‐delay systems. By choosing the delay fraction number as two, applying the Jensen inequality to every sub‐interval of the time delay interval and avoiding using any free weighting matrix, the method proposed can reduce the computational complexity and conservativeness of results. Based on Lyapunov stability theory, exponential stability and delayed‐state‐feedback stabilization conditions of nonlinear uncertain stochastic systems with the state delay are obtained. In the sequence, the delayed‐state‐feedback stabilization problem for a nonlinear uncertain stochastic time‐delay system is investigated and some sufficient conditions are given in the form of nonlinear inequalities. In order to solve the nonlinear problem, a cone complementarity linearization algorithm is offered. Mathematical and/or numerical comparisons between the proposed method and existing ones are demonstrated, which show the effectiveness and less conservativeness of the proposed method.     

Exponential Stability and Delayed Impulsive Stabilization of Hybrid Impulsive Stochastic Functional Differential Systems

This paper is concerned with the stability and impulsive stabilization of hybrid impulsive stochastic functional differential systems with delayed impulses. Using the Razumikhin techniques and Lyapunov functions, some sufficient conditions for the pth moment exponential stability of the systems under consideration are established. Based on the derived stability results, impulsive controllers are designed to stabilize a given unstable linear or nonlinear hybrid stochastic delayed differential system. Different from the existing stability and impulsive stabilization results in the literature, the results obtained in this paper shown that the delayed part of impulses can make a contribution to the stability of systems. Three examples are provided to present the effectiveness and advantages of the proposed results.     

Stability of discrete‐time systems with time‐varying delay via a novel Lyapunov‐Krasovskii functional

This article is concerned with the problem of stability analysis of discrete‐time systems with time‐varying delay. Unlike previous articles concentrating on the improvement of summation inequalities, this article constructs a novel augmented Lyapunov‐Krasovskii (L‐K) functional by fully considering the coupling information among state‐related vectors. Based on the newly developed L‐K functional, a more relaxed stability criterion is derived by exploring the interest of existing inequalities. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

Robust Exponential Stability Analysis of Uncertain Discrete Time‐Varying Linear Systems

This paper studies the robust exponential stability of uncertain discrete linear time‐varying (UDLTV) systems. The key tool is the recently proposed generating functions. It can be found that a class of improved generating functions (IGFS) can fully characterize the robust exponential stability of UDLTV systems, and the maximum exponential decay rate of system trajectories can be computed by the radius of convergence of the IGFS. Moreover, the application of convex optimization technique and dynamic programming method provides an effective algorithm for the computation of the IGFS. Finally, the numerical example illustrates the efficacy and advantage of the proposed approach.     

Robust Finite‐Time Stability and Stabilization of Linear Uncertain Time‐Delay Systems

Robust finite‐time stability and stabilization problems for a class of linear uncertain time‐delay systems are studied. The concept of finite‐time stability is extended to linear uncertain time‐delay systems. Based on the Lyapunov method and properties of matrix inequalities, a sufficient condition that ensures finite‐time stability of linear uncertain time‐delay systems is given. By virtue of the results on finite‐time stability, a memoryless state feedback controller that guarantees that the closed‐loop system is finite time stable, is proposed. The controller design problem is solved by using the linear matrix inequalities and the cone complementarity linearization iterative algorithm. Numerical examples verify the efficiency of the proposed methods.     

线性不确定时滞系统的鲁棒指数稳定性

针对带有时变有界的不确定参数和状态时滞的线性不确定时滞系统,研究其时滞依赖型具有指定收敛率的鲁棒指数稳定性问题。通过应用Lyapunov-Krasovskii稳定性定理方法,合理建立Krasovskii泛函,利用线性矩阵不等式方法导出了系统鲁棒指数稳定且具有指定收敛率的判据。所得结论采用线性矩阵不等式表示,与时滞参数的导数无关,克服了通常所得结论与时滞导数有关,且要求时滞参数导数小于常数1的限制。应用实例证明了设计方案切实可行。