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.     

Multi-agent compositional stability exploiting system symmetries

This paper considers nonlinear symmetric control systems. By exploiting the symmetric structure of the system, stability results are derived that are independent of the number of components in the system. This work contributes to the fields of research directed toward compositionality and composability of large-scale system in that a system can be “built-up” by adding components while maintaining system stability. The modeling framework developed in this paper is a generalization of many existing results which focus on interconnected systems with specific dynamics. The main utility of the stability result is one of scalability or compositionality. If the system is stable for a given number of components, under appropriate conditions stability is then guaranteed for a larger system composed of the same type of components which are interconnected in a manner consistent with the smaller system. The results are general and applicable to a wide class of problems. The examples in this paper focus on the formation control problems for multi-agent robotic systems.     

Novel delay-dependent stability condition for mixed delayed stochastic neural networks with leakage delay signals

In this paper, the problem of stability condition for mixed delayed stochastic neural networks with neutral delay and leakage delay is investigated. A novel Lyapunov functional is constructed with double and triple integral terms. New sufficient conditions are derived to guarantee the global asymptotic stability of the concerned neural network. This paper is more general than the paper by Zhu et al. [Robust stability of Markovian jump stochastic neural networks with time delays in the leakage terms, Neural Process. Lett. 41 (2015), pp. 1–27]. In our paper, we considered both the neutral delay and leakage delay, but the paper by Zhu et al. is not considering the neutral delay. Also we employed triple integrals in the Lyapunov functional which is not used in the paper by Zhu et al. Finally, two numerical examples are provided to show the effectiveness of the theoretical results.     

Robust stability of interval time-delay systems with delay-dependence

This paper proposes a sufficient robust stability condition for interval time-delay systems with delay-dependence. The properties of the comparison theorem and matrix measure are employed to investigate the problem. The stability criteria are delay-dependent and less conservative than delay-independent stability criteria [5] , [6] , [12] and [16] and delay-dependent stability criteria 1, [14] , [15] and [17] when delay is small. However, the results of this paper indeed give us one more choice for the stability examination of the interval time-delay systems. Simulation examples are given to demonstrate the application of our result.     

Stability analysis for uncertain switched neutral systems with discrete time-varying delay: A delay-dependent method

This paper considers the robust stability of a class of switched neutral systems with discrete time-varying delay and time-varying structure. Sufficient conditions for exponential stability criteria are developed for arbitrary switching signal with average dwell time. The results are obtained based on Lyapunov’s stability analysis via Krasovsky–Lyapunov’s functionals and the related stability study is performed to obtain delay-dependent results. It is proved that the stabilizing switching rule is arbitrary if all the switched subsystems are exponentially stabile. These conditions are delay-dependent and are given in the form of linear matrix inequalities (LMIs). Some examples are worked out to illustrate the effectiveness of the result.     

Stability and stabilization of stochastic systems with multiplicative noise

In this paper, stability and stabilization of linear stochastic time-invariant systems are studied based on spectrum technique. Firstly, the relationship among mean square exponential stability, asymptotical mean square stability, second-order moment exponential stability and the spectral location of the systems is revealed with the help of a spectrum operator L _A,C . Then, we focus on almost sure exponential stability and stochastic stabilization. A criterion on almost sure exponential stability based on spectrum technique is obtained. Sufficient conditions for mean square exponentially stability and asymptotic mean square stability are given via linear matrix inequality approach and some numerical examples to illustrate the effectiveness of our results are presented.     

Stability analysis of T–S fuzzy models for nonlinear multiple time-delay interconnected systems

In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the stability analysis for nonlinear interconnected systems with multiple time-delays using linear matrix inequality (LMI) theory. In terms of Lyapunov’s direct method for multiple time-delay fuzzy interconnected systems, a novel LMI-based stability criterion which can be solved numerically is proposed. Then, the common P matrix of the criterion is obtained by LMI optimization algorithms to guarantee the asymptotic stability of nonlinear interconnect systems with multiple time-delay. Finally, the proposed stability conditions are demonstrated with simulations throughout this paper.     

Stochastic stability of Positive Markov Jump Linear Systems

This paper investigates on the stability properties of Positive Markov Jump Linear Systems (PMJLS’s), i.e. Markov Jump Linear Systems with nonnegative state variables. Specific features of these systems are highlighted. In particular, a new notion of stability (Exponential Mean stability) is introduced and is shown to be equivalent to the standard notion of 1-moment stability. Moreover, various sufficient conditions for Exponential Almost-Sure stability are worked out, with different levels of conservatism. The implications among the different stability notions are discussed. It is remarkable that, thanks to the positivity assumption, some conditions can be checked by solving Linear Programming feasibility problems.     

Stability and persistent disturbance attenuation properties for a class of networked control systems: switched system approach

In this paper, both the asymptotic stability and l ^∞ persistent disturbance attenuation issues are investigated for a class of networked control systems (NCSs) under bounded uncertain access delay and packet dropout effects. The basic idea is to formulate such NCSs as discrete-time switched systems with arbitrary switching. Then the NCSs' stability and performance problems can be reduced to the corresponding problems of such switched systems. The asymptotic stability problem is considered first, and a necessary and sufficient condition is derived for the NCSs' asymptotic stability based on robust stability techniques. Secondly, the NCSs' l ^∞ persistent disturbance attenuation properties are studied and an algorithm is introduced to calculate the l ^∞ induced gain of the NCSs. The decidability issue of the algorithm is also discussed. A network controlled integrator system is used throughout the paper for illustration.     

A delay-range-partition approach to analyse stability of linear systems with time-varying delays

In this paper, the stability analysis of linear systems with an interval time-varying delay is investigated. First, augmented Lyapunov–Krasovskii functionals are constructed, which include more information of the delay's range and the delay's derivative. Second, two improved integral inequalities, which are less conservative than Jensen's integral inequalities, and delay-range-partition approach are utilised to estimate the upper bounds of the derivatives of the augmented Lyapunov–Krasovskii functionals. Then, less conservative stability criteria are proposed no matter whether the lower bound of delay is zero or not. Finally, to illustrate the effectiveness of the stability criteria proposed in this paper, two numerical examples are given and their results are compared with the existing results.     

Stability of a vector integer quadratic programming problem with respect to vector criterion and constraints

The paper analyzes five types of stability against perturbations of initial data in criterion functions and constraints of vector integer quadratic optimization problems. Necessary and sufficient conditions are proved for all the types of stability. The relationship among stability with respect to changes of vector criterion coefficients, stability with respect to changes of initial data in constraints, and stability with respect to vector criterion and constraints is established. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 63–72, September–October 2006.     

On exponential stability of linear networked control systems

This paper addresses exponential stability of linear networked control systems. More specifically, the paper considers a continuous‐time linear plant in feedback with a linear sampled‐data controller with an unknown time varying sampling rate, the possibility of data packet dropout, and an uncertain time varying delay. The main contribution of this paper is the derivation of new sufficient stability conditions for linear networked control systems taking into account all of these factors. The stability conditions are based on a modified Lyapunov–Krasovskii functional. The stability results are also applied to the case where limited information on the delay bounds is available. The case of linear sampled‐data systems is studied as a corollary of the networked control case. Furthermore, the paper also formulates the problem of finding a lower bound on the maximum network‐induced delay that preserves exponential stability as a convex optimization program in terms of linear matrix inequalities. This problem can be solved efficiently from both practical and theoretical points of view. Finally, as a comparison, we show that the stability conditions proposed in this paper compare favorably with the ones available in the open literature for different benchmark problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability analysis for linear delayed systems via an optimally dividing delay interval approach

This paper addresses the problem of stability for linear systems with time-varying delay. A novel augmented Lyapunov–Krasovskii functional is constructed by using the idea of optimally dividing the delay interval [0,τ(t)] into some variable sub-intervals and line integral technology. Using the novel augmented functional, the new delay-dependent stability criteria are proposed for linear systems with time-varying delay. The gain is that this stability criterion can lead to much less conservative stability results compared to other methods for linear systems with delay. Two numerical examples are provided to verify the effectiveness of the proposed criteria.     

Robust stability of stochastic genetic regulatory networks with discrete and distributed delays

In this paper, the problem on asymptotical and robust stability of genetic regulatory networks with time-varying delays and stochastic disturbance is considered. The time-varying delays include not only discrete delays but also distributed delays. The parameter uncertainties are time-varying and norm-bounded. Based on the Lyapunov stability theory and Lur’s system approach, sufficient conditions are given to ensure the stability of genetic regulatory networks. All the stability conditions are given in terms of linear matrix inequalities, which are easy to be verified. Illustrative example is presented to show the effectiveness of the obtained results.     

Stability Analysis and Performance Design for Fuzzy-Model-Based Control System Under Imperfect Premise Matching

This paper investigates the stability analysis and performance design of nonlinear systems. To facilitate the stability analysis, the Takagi–Sugeno (T–S) fuzzy model is employed to represent the nonlinear plant. Under the imperfect premise matching in which T–S fuzzy model and fuzzy controller do not share the same membership functions, a fuzzy controller with enhanced design flexibility and robustness property is proposed to control the nonlinear plant. However, the nice characteristic given by the perfect premise matching, leading to conservative stability conditions, vanishes. In this paper, under the imperfect premise matching, information of membership functions of the fuzzy model and controller are considered in stability analysis. With the introduction of slack matrices, relaxed linear matrix inequality (LMI)-based stability conditions are derived using Lyapunov-based approach. Furthermore, LMI-based performance conditions are provided to guarantee system performance. Simulation examples are given to illustrate the effectiveness of the proposed approach.      

Stability analysis for stochastic neural network with infinite delay

This paper considers a stochastic neural network (SNN) with infinite delay. Some sufficient conditions for stochastic stability, stochastic asymptotical stability and global stochastic asymptotical stability, respectively, are derived by means of Lyapunov method, Itô formula and some inequalities. As a corollary, we show that if the neural network with infinite delay is stable under some conditions, then the stochastic stability is maintained provided the environmental noises are small. Estimates on the allowable sizes of environmental noises are also given. Finally, a three-dimensional SNN with infinite delay is analyzed and some numerical simulations are illustrated to show our results.     

Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable

This paper is concerned with the stability and robust stability of switched positive linear systems (SPLSs) whose subsystems are all unstable. By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively. The copositive Lyapunov functions constructed in this paper are timevarying during the dwell time and time-invariant afterwards. In addition, the above approach is extended to the switched interval positive systems. A numerical example is proposed to illustrate our approach.      

p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.     

An analysis of the exponential stability of linear stochastic neutral delay systems

This paper is concerned with the analysis of the mean square exponential stability and the almost sure exponential stability of linear stochastic neutral delay systems. A general stability result on the mean square and almost sure exponential stability of such systems is established. Based on this stability result, the delay partitioning technique is adopted to obtain a delay‐dependent stability condition in terms of linear matrix inequalities (LMIs). In obtaining these LMIs, some basic rules of the Ito calculus are also utilized to introduce slack matrices so as to further reduce conservatism. Some numerical examples borrowed from the literature are used to show that, as the number of the partitioning intervals increases, the allowable delay determined by the proposed LMI condition approaches h_max, the maximal allowable delay for the stability of the considered system, indicating the effectiveness of the proposed stability analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Design of decentralized PI control systems based on Nyquist stability analysis

This paper presents a simple, but effective, design method for decentralized PI control systems with guaranteed closed-loop stability. Nyquist stability conditions are used to derive the stability region for each PI controller in terms of the controller parameters. A detuning factor for each loop is specified based on a diagonal dominance index. Then appropriate controller settings are determined using this index and the stability region. Simulation results for a variety of 2 × 2, 3 × 3, and 4 × 4 systems demonstrate that the proposed design method guarantees closed-loop stability and provides good set-point and load responses.