Stability In Mean Square Sense Research Articles

Exponential stability for switched semi‐Markov jump systems with mode‐dependent average dwell time and generally uncertain transition rates

AbstractThis paper investigates the exponential stability in mean square sense for a class of stochastic switched semi‐Markov jump systems (ssSMJS). In this model, a hierarchical control switched regulation is designed, considering the external stochastic disturbances and generally uncertain transition rates (TRs) of the entire system and making the system more general and practical. By using appropriate Lyapunov–Krasovskii functional, linear matrix inequality, Jensen inequality, and mode‐dependent average dwell time (MDADT), sufficient conditions on the exponential stability in mean square sense for ssSMJS are derived. Furthermore, a nonfragile state feedback controller is designed. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed criterion.

Event-triggered dynamic output feedback control for networked Markovian jump systems with partly unknown transition rates

This paper addresses the problem of event-triggered dynamic output feedback control for uncertain Markovian jump systems with partly unknown transition rates. An event-triggered dynamic output feedback control scheme is proposed to stabilize a class of uncertain networked Markovian jump systems with partly unknown transition rates, and a suitable Lyapunov–Krasovskii functional is then introduced to guarantee the asymptotical stability in mean-square sense of the closed-loop controlled system. It is shown that the proposed control scheme is relatively convenient to choose the control gain matrices by the use of the Matlab LMIs Toolbox, and thus it might be convenient to application in practice. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methodology.

Weighted $$\mathbf{H}_\infty$$ Performance Analysis of Nonlinear Stochastic Switched Systems: A Mode-Dependent Average Dwell Time Method

In this paper, the issues about weighted $$H_\infty$$ performance analysis and $$H_\infty$$ control for the stochastic switched nonlinear systems (SSNSs) with multiplicative noise are investigated. The mode-dependent average dwell time (MDADT) method is used to deal with the switching in different modes. Firstly, we get a sufficient condition to show that the considered system with all stable subsystems achieves the exponential stability in mean square sense and a specified weighted $$H_\infty$$ performance, which is extended to the case that both stable and unstable subsystems coexist in terms of second-order Hamilton-Jacobi inequalities (HJIs). Then, by using Takagi-Sugeno (T-S) fuzzy approach, the $$H_\infty$$ controller for SSNSs is designed via solving a set of linear matrix inequalities (LMIs). Finally, an example is supplied to illustrate the effectiveness of our results.

L₂-l∞ State Estimation for Persistent Dwell-Time Switched Coupled Networks Subject to Round-Robin Protocol.

This article is concerned with the issue of l2 - l∞ state estimation for nonlinear coupled networks, where the variation of coupling mode is governed by a set of switching signals satisfying a persistent dwell-time property. To solve the problem of data collisions in a constrained communication network, the round-robin protocol, as an important scheduling strategy for orchestrating the transmission order of sensor nodes, is introduced. Redundant channels with signal quantization are used to improve the reliability of data transmission. The main purpose is to determine an estimator that can guarantee the exponential stability in mean square sense and an l2 - l∞ performance level of the estimation error system. Based on the Lyapunov method, sufficient conditions for the addressed problem are established. The desired estimator gains can be obtained by addressing a convex optimization case. The correctness and availability of the developed approach are finally explained via two illustrative examples.

Exact detectability and exact observability of discrete-time linear stochastic systems with periodic coefficients

This paper is devoted to the problem of detectability and observability of a class of time-varying stochastic systems. The main contribution of this note is to propose a PBH-type test which is equivalent to the detectability of the considered stochastic systems in the sense that all unstable modes produce some non-zero output. Then, we apply the obtained results in order to derive a criterion of exponential stability in mean square sense as well as existence conditions for the stabilizing solution of a class of discrete-time time-varying Riccati equations.

Exponential Stabilization of Markov Jump Systems with Mode-Dependent Mixed Time-Varying Delays and Unknown Transition Rates

This paper investigates the problem of exponential stability in mean square sense for stochastic Markov jump systems with mixed time-varying delays and partly unknown transition rates. By employing a class of appropriate stochastic Lyapunov functionals, the analysis process of stability for stochastic Markov jump systems can be effectively carried out. Based on the linear matrix inequalities technique, the mean square exponential stability criteria are presented for stochastic Markov jump systems with partly unknown transition rates. Furthermore, by expanding this case to uncertain Markov jump systems, we derive the sufficient conditions for guaranteeing the stability of uncertain Markov jump systems. A numerical example is presented to illustrate the effectiveness of the proposed results.

Mean‐square stability of networked control systems with event driven state quantisation and packet loss

This study addresses mean-square stability of networked control system with state quantisation process and suffers random packet losses. A logarithmic quantiser is applied in order to decrease the possibility of the packet loss phenomenon. Also, the quantisation process is designed as event driven so that the quantisation is only performed when necessary. It is shown that stability in mean-square sense is achieved if a series of LMI conditions are valid. Simulations are finally performed to show the effectiveness of the results of the theoretical analysis.

Stochastic consistency and stochastic stability in mean square sense for Cauchy advection problem

This work deals with the construction of finite difference solutions of random advection Cauchy type partial differential equation containing uncertainty through the coefficient of the velocity. Under appropriate hypothesis on the velocity random variable, we establish that the constructed random finite difference solution is mean square consistent and mean square stable over the whole real line. In addition, the main statistical functions, such as the mean, of the approximate solution stochastic process generated by truncation of the exact finite difference solution are given. Finally, we apply the proposed technique to several illustrative examples which show our discussing for the mean square stability.

Detectability, observability and Lyapunov-type theorems of linear discrete time-varying stochastic systems with multiplicative noise

ABSTRACTThe objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, -exact detectability (resp. -exact detectability, -exact detectability, -exact detectability) and -exact observability (resp. -exact detectability, -exact detectability, -exact detectability) are introduced, respectively, and nice properties associated with uniform detectability, exact detectability and exact observability are also obtained. Moreover, some Lyapunov-type theorems associated with generalised Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, -exact observability and -exact detectability, respectively.

Mean-square global exponential stability in Lagrange sense for delayed recurrent neural networks with Markovian switching

The paper discusses the mean-square global exponential stability in Lagrange sense for delayed recurrent neural networks (DRNNs) with Markovian switching. Two different types of activation functions are considered, which include both bounded and unbounded activation functions. By using the vector Lyapunov function and stochastic analysis technique, we establish two L-operator differential inequalities. By employing the two L-operator differential inequalities and vector Lyapunov functions methods, we provide easily verifiable criteria for Lagrange stability in mean-square sense of DRNNs with Markovian switching. Finally, two numerical examples are given to illustrate the efficiency of the derived results.

Existence and exponential stability for some stochastic neutral partial functional integrodifferential equations

Abstract. The aim of this work is to study the existence, uniqueness and exponential stability of mild solutions for some stochastic neutral partial functional integrodifferential equations. We suppose that the linear part has a resolvent operator in the sense given in Grimmer [Transactions of the American Mathematical Society 273 (1982), 333–349]. The nonlinear part is assumed to be continuous and lipschitzian with respect to the second argument. Firstly, we study the existence of mild solutions. Secondly we give some results on the exponential stability in mean square sense. An example is provided to illustrate the results of this work.

Exponential stability conditions for switched linear stochastic systems with time-varying delay

The authors investigate the stability in mean-square sense for switched stochastic systems with time-varying delay by using multiple Lyapunov functionals method. In order to characterise the joint effects of switching and time delay on stability, some practical techniques are applied in computing the Lyapunov functional constructed for individual subsystem. The first one among others is to make use of Ito calculus rules to reduce the conservatism produced by noise since it basically plays a negative role for preserving stability. Also, the authors present a necessary number of slack matrices to create convex conditions so as to accommodate the computation to time-varying delay. In the following, the authors show the stability of overall system via confining switching frequency within a certain level. An example is used to illustrate the method with describing the mutually constraint situation of switching and time delay.

Predictive Control of Constrained Linear Systems with Multiple Missing Measurements

This paper investigates the problem of predictive control for constrained control systems, in which the measurement signal may be multiply missing. An augmented stochastic model is firstly introduced to describe and compensate these missing measurements. By means of an infinite horizon quadratic performance objective, a state-feedback predictive control law involving missing probability is designed by minimizing the upper bound on performance objective at each sampling instant. It is shown that the on-line optimization problem subject to input and state constraints can effectively be solved in terms of linear matrix inequalities. The designed predictive controller can achieve the desired control performance and also guarantee the closed-loop stability in mean square sense. Finally, an example is given to illustrate the proposed results.

Delayed-State-Feedback Fault-Tolerant Control for Fuzzy Markovian Jumping Systems

This paper is concerned with the robust delayed-state-feedback fault-tolerant control problem for fuzzy markovian jump systems with time-varying delay. A new criterion of delayed-dependent robust fault-tolerant stabilization for such systems is established in terms of linear matrix inequalities (LMIs) by using Lyapunov stability theory and free-weighting matrix methods. Based on the obtained criterion, designed hybrid controller combining state feedback and delayed state feedback can guarantee robust asymptotic stability in mean square sense for the resulting closed-loop system not only when all control components are operating well, but also in the presence of some outages of actuators within a prespecified subset of actuators.

Stabilisation of singular Markovian jump systems with discontinuities and saturating inputs

In this study, the problems of stability and stabilisation of singular Markovian jump systems with external discontinuities and saturating inputs are addressed. A design procedure is proposed to determine simultaneously a state feedback controller and an associated domain of safe admissible states for which the regularity, the absence of impulsive behaviour and the stochastic stability in mean square sense of the closed-loop systems are guaranteed. This design procedure is based on linear matrix inequality technique (LMI) and the sequential linear programming matrix method (SLPMM). A numerical example is provided to demonstrate the effectiveness of the proposed methods.

Stochastic output feedback controller for singular Markovian jump systems with discontinuities

The class of continuous-time linear Markovian jump singular systems with discontinuities has been dealt with. Based on the stochastic stability in mean square sense of the system under study, a sufficient condition is proposed for the design of an output feedback controller which guarantees that the closed-loop system is piecewise regular, impulse-free and stochastically stable in mean square sense. First, the proposed approach is established in terms of bilinear and linear matrix inequalities. Then, the sequential linear programming matrix method is used to convert the nonlinear and non-convex output feedback problem on a convex optimisation one. A numerical example is presented to show the usefulness of the proposed results.