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.     

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

Passive Control for Stochastic Interval Systems with Interval Time‐Varying Delay

This paper is concerned with the problem of delay‐dependent passive analysis and control for stochastic interval systems with interval time‐varying delay. The system matrices are assumed to be uncertain within given intervals, and the time delay is a time‐varying continuous function belonging to a given range. By the transformation of the interval uncertainty into the norm‐bounded uncertainty, partitioning the delay into two segments of equal length, and constructing an appropriate Lyapunov–Krasovskii functional in each segment of the delay interval, delay‐dependent stochastic passive control criteria are proposed without ignoring any useful terms by considering the information of the lower bound and upper bound for the time delay. The main contribution of this paper is that a tighter upper bound of the stochastic differential of Lyapunov–Krasovskii functional is obtained via a newly‐proposed bounding condition. Based on the criteria obtained, a delay‐dependent passive controller is presented. The results are formulated in terms of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness of the method.     

State‐feedback stabilization for a class of stochastic time‐delay nonlinear systems

This paper investigates the problem of state‐feedback stabilization for a class of lower‐triangular stochastic time‐delay nonlinear systems without controllable linearization. By extending the adding‐a‐power‐integrator technique to the stochastic time‐delay systems, a state‐feedback controller is explicitly constructed such that the origin of closed‐loop system is globally asymptotically stable in probability. The main design difficulty is to deal with the uncontrollable linearization and the nonsmooth system perturbation, which, under some appropriate assumptions, can be solved by using the adding‐a‐power‐integrator technique. Two simulation examples are given to illustrate the effectiveness of the control algorithm proposed in this paper.Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic stability of semi‐Markovian jump systems with mode‐dependent delays

Semi‐Markovian jump systems, due to the relaxed conditions on the stochastic process, and its transition rates are time varying, can be used to describe a larger class of dynamical systems than conventional full Markovian jump systems. In this paper, the problem of stochastic stability for a class of semi‐Markovian systems with mode‐dependent time‐variant delays is investigated. By Lyapunov function approach, together with a piecewise analysis method, a sufficient condition is proposed to guarantee the stochastic stability of the underlying systems. As more time‐delay information is used, our results are much less conservative than some existing ones in literature. Finally, two examples are given to show the effectiveness and advantages of the proposed techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Network‐based H∞ control for stochastic systems

This paper investigates the problem of network‐based control for stochastic plants. A new model of stochastic time‐delay systems is presented where both network‐induced delays and packet dropouts are taken into consideration for a sampled‐data network‐based control system. This model consists of two successive delay components in the state, and we solve the network‐based H_∞ control problem based on this model by a new stochastic delay system approach. The controller design for the sampled‐data systems is carried out in terms of linear matrix inequalities. Finally, we illustrate the methodology by applying these results to an air vehicle control problem. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Delay‐range‐dependent control of nonlinear time‐delay systems under input saturation

This paper describes a delay‐range‐dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time‐delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay‐range‐dependent and delay‐dependent stabilization problems for linear and nonlinear time‐delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay‐rate‐independent condition for control of nonlinear systems in the presence of input saturation with unknown delay‐derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L₂ gain reduction from disturbance to output is devised to address the problem of the stabilization of input‐constrained nonlinear time‐delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time‐delay network and a large‐scale chemical reactor under input saturation and varying interval time‐delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Distributed delay‐dependent filtering for Markovian jump systems interconnected over an undirected graph with time‐delay

In this paper, the problems of delay‐dependent stochastic stability analysis and distributed filter synthesis are considered for Markovian jump systems interconnected over an undirected graph with state time‐invariant delay. A sufficient condition for the well‐posedness, delay‐dependent stochastic stability and contractiveness of the plant is developed in terms of linear matrix inequalities (LMIs). The distributed filter synthesis aims to design a distributed filter inheriting the structure of the plant such that the filtering error systems is well‐posed, delay‐dependent stochastically stable and contractive. Specifically, a corresponding sufficient condition to guarantee the filtering error system contractive is first presented by a set of nonlinear matrix inequalities. Next, for coupling these nonlinear matrix inequalities, a sufficient condition on the existence of such a distributed filter is proposed via a series of finite‐dimensional LMIs. Finally, a numerical simulation is presented to demonstrate the effectiveness of the proposed approach.     

Delay‐dependent robust stability for stochastic time‐delay systems with polytopic uncertainties

This paper considers a delay‐dependent and parameter‐dependent robust stability criterion for stochastic time‐delay systems with polytopic uncertainties. The delay‐dependent robust stability criterion, as expressed in terms of linear matrix inequalities (LMIs), is obtained by using parameter‐dependent Lyapunov functions. It is shown that the result derived by a parameter‐dependent Lyapunov functional is less conservative. Numerical examples are provided to illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Razumikhin‐type approach on state feedback of stochastic high‐order nonlinear systems with time‐varying delay

The Razumikhin‐type approach is introduced to solve the state feedback stabilization problem for a class of stochastic high‐order nonlinear systems with time‐varying delay. Based on the general Razumikhin‐type theorem on stochastic systems established in our paper and backstepping design method, a state feedback controller is constructed to ensure the origin of closed‐loop system is globally asymptotically stable in probability. Our methodology enables us to completely remove the limitations on the derivative of delay, which is the common assumption of stochastic high‐order nonlinear systems with time‐varying delay. The efficiency of the state feedback controller is demonstrated by simulation examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Generalized criteria on delay‐dependent stability of highly nonlinear hybrid stochastic systems

Our recent paper (Fei W, etal. Delay dependent stability of highly nonlinear hybrid stochastic systems. Automatica. 2017;82:165‐170) is the first to establish delay‐dependent criteria for highly nonlinear hybrid stochastic differential delay equations (SDDEs) (by highly nonlinear, we mean that the coefficients of the SDDEs do not have to satisfy the linear growth condition). This is an important breakthrough in the stability study as all existing delay stability criteria before could only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In this continuation, we will point out one restrictive condition imposed in our earlier paper. We will then develop our ideas and methods there to remove this restrictive condition so that our improved results cover a much wider class of hybrid SDDEs.     

Input‐to‐state stability of min‐max MPC scheme for nonlinear time‐varying delay systems

This paper studies the robustness problem of the min–max model predictive control (MPC) scheme for constrained nonlinear time‐varying delay systems subject to bounded disturbances. The notion of the input‐to‐state stability (ISS) of nonlinear time‐delay systems is introduced. Then by using the Lyapunov–Krasovskii method, a delay‐dependent sufficient condition is derived to guarantee input‐to‐state practical stability (ISpS) of the closed‐loop system by way of nonlinear matrix inequalities (NLMI). In order to lessen the online computational demand, the non‐convex min‐max optimization problem is then converted to a minimization problem with linear matrix inequality (LMI) constraints and a suboptimal MPC algorithm is provided. Finally, an example of a truck‐trailer is used to illustrate the effectiveness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Finite‐time adaptive robust control of nonlinear time‐delay uncertain systems with disturbance

This paper investigates finite‐time adaptive robust control problem for a general class of nonlinear time‐delay systems with uncertain and external disturbance via the Lyapunov‐Krasovskii (L‐K) method and presents some delay‐independent and delay‐dependent results on the issue. First, by applying the orthogonal decomposition method, this paper presents an equivalent form. Based on which, we study the finite‐time adaptive robust control problem for the systems by constructing a specific L‐K functional and designing a suitable controller. Different from existing works, this paper studies finite‐time adaptive robust control problem for general nonlinear delay system and presents a delay‐dependent sufficient condition on the problem. Finally, an illustrative example is given to show the effectiveness of the result in this paper.     

Tracking Control for Stochastic Multi‐Agent Systems Based on Hybrid Event‐Triggered Mechanism

This paper deals with the leader‐following consensus for nonlinear stochastic multi‐agent systems. To save communication resources, a new centralized/distributed hybrid event‐triggered mechanism (HETM) is proposed for nonlinear multi‐agent systems. HETMs can be regarded as a synthesis of continuous event‐triggered mechanism and time‐driven mechanism, which can effectively avoid Zeno behavior. To model the multi‐agent systems under centralized HETM, the switched system method is applied. By utilizing the property of communication topology, low‐dimensional consensus conditions are obtained. For the distributed hybrid event‐triggered mechanism, due to the asynchronous event‐triggered instants, the time‐varying system method is applied. Meanwhile, the effect of network‐induced time‐delay on the consensus is also considered. To further reduce the computational resources by constantly testing whether the broadcast condition has been violated, self‐triggered implementations of the proposed event‐triggered communication protocols are also derived. A numerical example is given to show the effectiveness of the proposed method.     

New insight into delay‐dependent stability of time‐delay systems

This paper presents a new insight into the delay‐dependent stability for time‐delay systems. Because of the key observation that the positive definiteness of a chosen Lyapunov–Krasovskii functional does not necessarily require all the involved symmetric matrices in the Lyapunov–Krasovskii functional to be positive definite, an improved delay‐dependent asymptotic stability condition is presented in terms of a set of LMIs. This fact has been overlooked in the development of previous stability results. The importance of the present method is that a vast number of existing delay‐dependent results on analysis and synthesis of time‐delay systems derived by the Lyapunov–Krasovskii stability theorem can be improved by using this observation without introducing additional variables. The reduction of conservatism of the proposed result is both theoretically and numerically demonstrated. It is believed that the proposed method provides a new direction to improve delay‐dependent results on time‐delay systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Separation principle for quasi‐one‐sided Lipschitz nonlinear systems with time delay

In this article, we address the problem of output stabilization for a class of nonlinear time‐delay systems. First, an observer is designed for estimating the state of nonlinear time‐delay systems by means of quasi‐one‐sided Lipschitz condition, which is less conservative than the one‐sided Lipschitz condition. Then, a state feedback controller is designed to stabilize the nonlinear systems in terms of weak quasi‐one‐sided Lipschitz condition. Furthermore, it is shown that the separation principle holds for stabilization of the systems based on the observer‐based controller. Under the quasi‐one‐sided Lipschitz condition, state observer and feedback controller can be designed separately even though the parameter (A,C) of nonlinear time‐delay systems is not detectable and parameter (A,B) is not stabilizable. Finally, a numerical example is provided to verify the efficiency of the main results.     

On the Controllability of Nonlocal Second‐Order Impulsive Neutral Stochastic Integro‐Differential Equations with Infinite Delay

In this paper, we investigate the controllability for a class of nonlocal second‐order impulsive neutral stochastic integro‐differential equations with infinite delay in Hilbert spaces. More precisely, a set of sufficient conditions for the controllability results of nonlocal second‐order impulsive neutral stochastic integro‐differential equations with infinite delay are derived by means of the Banach fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. As an application, an example is provided to illustrate the obtained theory.     

Output feedback stabilization for delayed large‐scale stochastic systems with markovian jumping parameters

This paper deals with the problem of decentralized output feedback stabilization for a class of large‐scale stochastic time‐delay systems with Markovian jumping parameters. Attention is focused on the design of a decentralized dynamic output feedback controller, which is also with Markovian jumping parameters, such that the closed‐loop system is exponentially mean‐square stable. A sufficient condition for the solvability of this problem is proposed in terms of linear matrix inequalities. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society