semi-Markov Switching Research Articles

Stability of stochastic functional differential systems with semi-Markovian switching and Lévy noise by functional Itô’s formula and its applications

This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Lévy noise (SFDEs-SMS-LN). Based on functional Itô’s formula, multiple degenerate Lyapunov functionals and nonnegative semi-martingale convergence theorem, new pth moment and almost surely stability criteria with general decay rate for SFDEs-SMS-LN are established. Meanwhile, the diffusion operators are allowed to be controlled by multiple auxiliary functions with time-varying coefficients, which can be more adaptable to the non-autonomous SFDEs-SMS-LN with high-order nonlinear coefficients. Furthermore, as applications of the presented stability criteria, new delay-dependent sufficient conditions for general decay stability of the stochastic delayed neural network with semi-Markovian switching and Lévy noise (SDNN-SMS-LN) and the scalar non-autonomous SFDE-SMS-LN with non-global Lipschitz condition are respectively obtained in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the effectiveness of the proposed results.

The novel sufficient conditions of almost sure exponential stability for semi-Markov jump linear systems

This paper discusses the almost sure exponential stability problem for semi-Markov jump linear systems. By comprehensively utilizing the coupled Lyapunov matrices and the ergodic property of semi-Markov switching process, the novel sufficient stability conditions for the considered systems are obtained, which are expressed in terms of linear matrix inequalities and the probability structure of semi-Markov switching process. Finally, an example is given to illustrate the effectiveness of our results.

Quantized Fuzzy Finite-Time Control for Nonlinear Semi-Markov Switching Systems

This brief considers quantized control for finite-time synthesis of nonlinear semi-Markov switching systems (S-MSSs) via T-S fuzzy strategy. The stochastic phenomena of structural and parametric changes are modeled by the semi-Markov process, in which the sojourn-time (ST) is deemed to obey a non-exponential distribution. Compared with previous works, the input quantization is firstly investigated for studying the finite-time control via a logarithmic quantizer. A key issue under the consideration is how to design a fuzzy-model-based finite-time control law in the presence of quantized error. For this purpose, by using the key point of Lyapunov function, the finite-time boundedness (FTBs) performance is analyzed via establishing sojourn-time-dependent sufficient conditions within a given finite-time level. Then, the existence of a quantized controller is given in standard LMIs. Finally, an example for an electric circuit shows the effectiveness of the finite-time control scheme.

Asynchronous adaptive quantized feedback sliding mode control for semi-markovian jump systems: An event-triggered approach

This paper is concerned with the problem of event-triggered based asynchronous adaptive quantized sliding mode control (SMC) for Itoˆ stochastic control systems with semi-markovian switching and actuator failures. In this design work, the jump modes between plants and controller are asynchronous, which is represented by a hidden Markov model. The dynamical logarithmic quantizer is introduced in signal quantization both in sensors-to-controller and controller-to-actuators sides, and a new event-triggered detector condition is developed based on the quantized state vector. Combined with SMC and adaptive control techniques, a new event-triggered based asynchronous adaptive quantized SMC law is designed to stabilize the Itoˆ stochastic systems with semi-markovian switching, actuator failures and signal quantization. The proposed control law has ability to guarantee the robust performance of the closed-loop system in the presence of the effects of actuator failures, time delay resulting by event-triggered and signal quantization error. Moreover, the developed event-triggered based asynchronous adaptive quantized feedback SMC scheme can ensure that the sliding surface converges the defined small domain in finite time. The lower bound for inter-execution time is analyzed and presented, and sufficient condition under which the zenor behaviors can be avoided is given. Finally, an example is provided to demonstrate the effectiveness of the proposed design techniques.

Non-extensive minimal entropy martingale measures and semi-Markov regime switching interest rate modeling

A minimal entropy martingale measure problem is studied to investigate risk-neutral densities and interest rate modelling. Hunt & Devolder focused on the method of Shannon minimal entropy martingale measure to select the best measure among all the equivalent martingale measures and, proposed a generalization of the Ho & Lee model in the semi-Markov regime-switching framework [ 1 ]. We formulate and solve the optimization problem of Hunt & Devolder for deriving risk-neutral densities using a new non-extensive entropy measure [ 2 ]. We use the Lambert function and a new type of approach to obtain results without depending on stochastic calculus techniques.

H∞ leader-following consensus of nonlinear multi-agent systems under semi-Markovian switching topologies with partially unknown transition rates

This paper investigates the H∞ leader-following consensus problem for nonlinear multi-agent systems under semi-Markovian switching topologies. The switching of the topologies is governed by a semi-Markovian jump process, which covers a Markovian jump process as a special case. In many practical systems, it is difficult to obtain the transition rate matrix, thus the transition rates are considered to be not completely known in the paper. External perturbations are considered in the paper, and H∞ control theory is applied. By utilising stochastic technique, sufficient conditions expressed in terms of linear matrix inequalities are derived to ensure that the H∞ leader-following consensus can be reached with a prescribed performance index. Finally, a numerical example is given to show the effectiveness of the theoretical results.

Scaled consensus of multi-agent systems with switching topologies and communication noises

This paper concerns with the mean square scaled consensus for multi-agent systems(MAS) with Markovian switching topologies and communication noises. A time-homogeneous Markov process governs the switching and each topology corresponds to a state of the process. Meanwhile, the communication noises are also considered for practical applications. We introduce a time-varying gain in control protocol which can reduce the effect of communication noises. It is shown that the effect of Markovian switching topologies mainly depends on the union of topologies associated with the positive recurrent states of Markov process. Then necessary and sufficient criteria can be established to guarantee the mean square scaled consensus under the action of distributed protocols. Moreover, the LV operator for semi-Markovian switching topologies case is derived and we can extend our result to the cases that the elements of transition rates matrix are partly unknown and the topological structure is semi-Markovian switching. Finally, an example is conducted to demonstrate the validity of our results.

Finite-Time Observer-Based Sliding Mode Control for Quantized Semi-Markov Switching Systems With Application

This article investigates the problem of sliding mode control (SMC) for semi-Markov switching systems (S-MSSs) with quantized measurement in finite-time level. The transition between different subsystems obeys a stochastic semi-Markov process related to nonexponential distribution. Additionally, due to the sensor information constraints, the state vectors are not always measurable in practice. Moreover, compared with existing results in literature, the output quantization is first considered for finite-time SMC problem via a logarithmic quantizer. Our attention is to design an appropriate finite-time SMC law to attenuate the influences of parametrical uncertainty and external disturbance onto the overall performance of the system under consideration. First, by the key points of stochastic semi-Markov Lyapunov function and observer design theory, a desired SMC law is constructed to guarantee that the system trajectories can arrive at the specified sliding surface (SSS) within an assigned finite-time level. Then, ST-dependent sufficient conditions are established to ensure the required finite-time boundedness performance including both reaching phase and sliding motion phase. Finally, the applicability of the proposed results is demonstrated by a single-link robot arm model.

Stability of Stochastic Functional Differential Systems with Semi-Markovian Switching and Lévy Noise and Its Application

This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Levy noise (SFDEs-sMS-LN). Based on generalized multidimensional Ito’s formula and multiple Lyapunov functions, a new pth moment stability criterion with general decay rate is established. Meanwhile, as an applications of the presented stability criterion, we consider the stabilization problem of stochastic delayed neural networks with semi-Markovian switching and Levy noise (SDNN-sMS-LN). A vertex approach is proposed to design the controller in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, a numerical example is presented to demonstrate the effectiveness of the proposed results.

Sliding Mode Control for Nonlinear Stochastic Semi-Markov Switching Systems With Application to SRMM

In this paper, the sliding mode control (SMC) design for nonlinear stochastic semi-Markov switching systems (S-MSSs) is studied via the bound of time-varying transition rate matrix, in which semi-Markov switching parameters, stochastic disturbance, uncertainty, and nonlinearity are all considered in a unified framework. The system under consideration is more general, which covers the Markov switching system with sojourn-time-independent transition rate matrix as a special case. Many practical systems subject to unpredictable structural variations can be characterized by nonlinear stochastic S-MSSs with sojourn-time-dependent transition rate matrix. The specific information about the bound of time-varying transition rate matrix is known for the sliding mode controller design. First, by using the stochastic semi-Markov Lyapunov function, sojourn-time-dependent sufficient conditions are developed to guarantee the closed-loop sliding mode dynamics stochastically stable. Then, the SMC law is constructed to ensure the reachability of the sliding mode dynamics in a finite-time level. Finally, one joint of space robot manipulator model is described as nonlinear stochastic S-MSSs to illustrate the validity of the proposed SMC design method.

Exponential synchronization of semi-Markovian coupled neural networks with mixed delays via tracker information and quantized output controller

In this paper, exponential synchronization of semi-Markovian coupled neural networks (NNs) with bounded time-varying delay and infinite-time distributed delay (mixed delays) is investigated. Since semi-Markov switching occurs by time-varying probability, it is difficult to capture its precise switching signal. To overcome this difficulty, a tracker is used to track the switching information with some accuracy. Then a quantized output controller (QOC) is designed by using the tracked information. Novel Lyapunov–Krasovskii functionals (LKFs) with negative terms and delay-partitioning approach, which reduce the conservativeness of the obtained results, are utilized to obtain LMI conditions ensuring the exponential synchronization. Moreover, an algorithm is proposed to design the control gains. Our results include both those derived by mode-dependent and mode-independent control schemes as special cases. Finally, numerical simulations validate the effectiveness of the methodology.

Finite-time stabilization of T–S fuzzy semi-Markov switching systems: A coupling memory sampled-data control approach

The present draft investigates the finite-time stabilization of T–S fuzzy semi-Markov switching systems by using a coupling memory sampled-data control approach. The main study concerned is how to effectively design the sampled-data control such that closed-loop T–S fuzzy semi-Markov switching systems is finite-time boundedness. By utilizing a Bernoulli distributed sequence, a more general coupling memory fuzzy sampled-data control strategy that involving time delay effect is derived. By virtue of fuzzy-basis-dependent membership functions, an asynchronous approach is proposed for T–S fuzzy semi-Markov switching system. Moreover, the inverted pendulum model is presented to demonstrate the feasibility and validity of the proposed strategies.

Anti-Windup Design for Saturated Semi-Markovian Switching Systems With Stochastic Disturbance

In this brief, we consider the anti-windup design for stochastic semi-Markovian switching systems (S-MSSs) with saturation nonlinearity and stochastic disturbance. In the system under consideration, stochastic S-MSSs with the semi-Markovian process can describe more complex systems in a practical control process. The main motivation of this brief is that the practical system described by stochastic S-MSSs always needs to consider the actuator saturation. To deal with this kind of complex problem, sufficient conditions for stochastic stability (SSY) analysis is developed by the Lyapunov function method. Then, the anti-windup strategy is constructed to make the dynamical system achieves SSY and maximize the domain of attraction. Finally, the simulation results of the practical system show the validity of the proposed method.

Necessary and Sufficient Conditions for Consensus of Continuous-Time Multiagent Systems With Markovian Switching Topologies and Communication Noises.

This paper investigates the mean square consensus problem for continuous-time multiagent systems with randomly switching topologies and noises. The switching is governed by a time-homogeneous Markov process, and each topology corresponds to a state of the process. Meanwhile, the communication noises are also considered for practical applications. We introduce a time-varying gain which can reduce the effect of communication noises. It is shown that the effect of Markovian switching topologies mainly depends on the union of topologies associated with the positive recurrent states of the Markov process. Then, necessary and sufficient conditions can be obtained under a control protocol with time-varying gain. Moreover, we extend our result to the cases where the topological structure is semi-Markovian switching and the elements of transition rate matrix are partly unknown. Finally, we give an example to illustrate the validity of our results.

Fault-tolerant leader-following consensus for multi-agent systems subject to semi-Markov switching topologies: An event-triggered control scheme

For multi-agent systems with switching topologies, the work addresses the fault-tolerant leader-following consensus issue. A semi-Markov chain is applied to describe randomly topological switching which may be caused by the complexity of network communication. In addition, in order to improve the utilization of bandwidth and relieve the burden of the communication channels in the multi-agent networks, a fault-tolerant event-triggered control protocol is developed to obtain the leader-following consensus of the multi-agent systems. By constructing an appropriate Lyapunov–Krasovskii functional and using some novel inequalities, the sufficient conditions of leader-following consensus are derived. One numerical simulation example validates the effectiveness of the presented method at length.

Procedure of Stochastic Approximation for the Diffusion Process with Semi-Markov Switchings

We establish sufficient conditions for the convergence of the procedure of stochastic approximation for the diffusion process in the case of a uniformly ergodic semi-Markov process of switchings of the regression function with the use of a small parameter in the scheme of series.

Finite-time synchronization for complex dynamic networks with semi-Markov switching topologies: An H∞ event-triggered control scheme

For complex dynamic networks subject to switching topologies, an H∞ event-triggered control scheme is developed to deal with the finite-time synchronization problem. A semi-Markov process is employed to govern the switching of the networks topologies. The purpose focuses on designing a controller via the event-triggered scheme, which guarantees that the resulting error networks are finite-time bounded with a prescribed level of the H∞ performance. By utilizing the stochastic analysis theory and finite-time synchronization concept, some sufficient conditions are deduced to determine whether the desired controller will exist or not for the complex dynamic networks. Eventually, an illustrated example is proposed to explain the validity and practicability of this method.

Robust finite-time stabilization for positive delayed semi-Markovian switching systems

Robust finite-time stabilization is discussed for positive semi-Markovian switching systems (S-MSSs), in which semi-Markovian process, time-varying delay, external disturbance, and transient performance in finite-time level are all considered in a unified framework. In the system under consideration, finite-time problem can describe transient performance of practical control process. Firstly, some finite-time boundedness and L1 finite-time boundedness criteria for positive delayed S-MSSs are proposed by constructing the stochastic semi-Markovian Lyapunov–Krasovskii functional with mode-dependent integral term. Then, a developed L1 finite-time feedback controller design method is presented to reduce some constraints of input matrices, which guarantees the resulting closed-loop system achieves positivity, finite-time boundedness, and has a prescribed L1 noise attenuation performance index in a novel standard linear programming. Finally, a practical example is illustrated to validate the proposed results by applying a virus mutation treatment model.

Stabilization for switched stochastic systems with semi-Markovian switching signals and actuator saturation

In this paper, the asymptotic stabilization problem is investigated for a class of switched stochastic systems with semi-Markovian switching signals and actuator saturation. By using the stochastic analysis theory and multiple Lyapunov function method, sufficient conditions for the local asymptotic mean square stability of the related system are established based on the stationary distribution of the embedded Markov chain. Moreover, the mode-dependent state feedback controller and estimation of domain of attraction in mean square sense are proposed in terms of a family of decoupled linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the effectiveness of the proposed results.

$H_{\infty}$ Consensus of Linear Multi-Agent Systems With Semi-Markov Switching Network Topologies and Measurement Noises

This paper investigates the H∞ consensus of linear multi-agent systems with semi-Markov switching network topologies and measurement noises. The information that each agent measures its neighbors's has multiplicative noises. The switching of the network topologies are modeled by a semiMarkov process. Taking the external disturbance into account, H∞ consensus of multi-agent systems with multiplicative noises is achieved under semi-Markov switching network topologies by using semi-Markov jump theory, stochastic theory and algebraic graph theory. Finally, simulation example is provided to illustrate the effectiveness of the theoretical results.