Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Robust Control of Markovian Switching Stochastic Systems with Noise Dependent States and Inputs

A problem of state feedback stabilization of discrete-time stochastic processes under Markovian switching and random diffusion (noise) is considered. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. Sufficient conditions based on linear matrix inequalities (LMI's) for stochastic stability is obtained. The robustness results of such stability concept against all admissible uncertainties are also investigated. An example is given to demonstrate the obtained results.     

Optimal Guaranteed Cost Control of Stochastic Discrete-Time Systems with States and Input Dependent Noise Under Markovian Switching

A problem of robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered. The control is simultaneously applied to both the random and the deterministic components of the system. The noise (the random) term depends on both the states and the control input. The jump Markovian switching is modeled by a discrete-time Markov chain and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Using linear matrix inequalities (LMIs) approach, the robust quadratic stochastic stability is obtained. The proposed control law for this quadratic stochastic stabilization result depended on the mode of the system. This control law is developed such that the closed-loop system with a cost function has an upper bound under all admissible parameter uncertainties. The upper bound for the cost function is obtained as a minimization problem. Two numerical examples are given to demonstrate the potential of the proposed techniques and obtained results.     

Stabilization of stochastic systems under Markovian switching

The problem of state feedback stabilization of discrete-time stochastic processes under Markovian switching is considered. The jump Markovian switching is modeled by a discrete-time Markov chain, and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Necessary and sufficient conditions based on linear matrix inequalities (LMI’s) for stochastic stability is obtained. The proposed control law for this stochastic stabilization result depends on the mode of the system as well as the environmental disturbances. The robustness results of such stability concepts against all admissible uncertainties are also investigated. An example is given to demonstrate the obtained results.     

随机混合动态系统的状态反馈控制

针对一类以有限齐次马氏链δ(k)作为切换信号的随机混合系统,首先,通过构造随机混合Lyapunov函数,得到整个随机混合系统渐近稳定的充分条件.然后,引入可调转移概率等相关概念,通过对有限齐次马氏链δ(k)及各子系统加入控制,以实现状态反馈控制.进一步,得到随机混合闭环系统渐近稳定的充分条件.     

Stabilization for hybrid stochastic systems by aperiodically intermittent control

In this paper, the mean-square exponential stabilization for stochastic differential equations with Markovian switching is studied. Specifically, a new set of sufficient conditions is derived to obtain the aperiodically intermittent control design which exponentially stabilizes the addressed hybrid stochastic differential equations. Further, stabilization problem by periodically intermittent control can be deduced as a special case from the developed results. As an application, we consider the Hopfield neutral network model with simulations to illustrate the effectiveness of developed aperiodically intermittent control design.     

Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Finite-time stability and stabilization of nonlinear stochastic hybrid systems

This paper deals with the problem of finite-time stability and stabilization of nonlinear Markovian switching stochastic systems which exist impulses at the switching instants. Using multiple Lyapunov function theory, a sufficient condition is established for finite-time stability of the underlying systems. Furthermore, based on the state partition of continuous parts of systems, a feedback controller is designed such that the corresponding impulsive stochastic closed-loop systems are finite-time stochastically stable. A numerical example is presented to illustrate the effectiveness of the proposed method.     

Delay-Dependent Criteria for Robust Stabilization of Markovian Switching Networks with Time-Varying Delay

Abstract

A problem of feedback stabilization of hybrid systems with time-varying delay and Markovian switching is considered. Delay-dependent sufficient conditions for stability based on linear matrix inequalities (LMI's) for stochastic asymptotic stability is obtained. The stability result depended on the mode of the system and of delay-dependent. The robustness results of such stability concept against all admissible uncertainties are also investigated. This new delay-dependent stability criteria is less conservative than the existing delay-independent stability conditions. An example is given to demonstrate the obtained results.     

Necessary and Sufficient Condition for Robust Stability and Stabilizability of Continuous-Time Linear Systems with Markovian Jumps

In this paper, we investigate the quadratic stability and quadratic stabilizability of the class of continuous-time linear systems with Markovian jumps and norm-bound uncertainties in the parameters. Under some appropriate assumptions, a necessary and sufficient condition is established for mean-square quadratic stability and mean-square quadratic stabilizability of this class of systems. The quadratic guaranteed cost control problem is also addressed via a LMI optimization problem.     

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.     

Stability of singularly perturbed nonlinear stochastic hybrid systems

The problem of exponential mean-square stability of nonlinear singularly perturbed, stochastic hybrid systems is studied in this article. Two groups of nonlinear systems are considered separately. To obtain the sufficient conditions of stability, two basic approaches of stability analysis for hybrid systems with a given Markovian switching rule and any Markovian switching rule and singularly perturbed non–hybrid systems were combined. The Lyapunov techniques were used in both approaches. The obtained results are illustrated by examples.     

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.     

Event-triggered feedback stabilization of switched linear systems using dynamic quantized input

This paper is concerned with the co-design of event-triggered sampling, dynamic input quantization and constrained switching for a switched linear system. The mismatch between the plant and its corresponding controller is considered. This behavior is raised by switching within the event-triggered sampling interval. Accordingly, novel update laws of dynamic quantization parameter are designed separately for matched sampling intervals (without switching) and mismatched sampling intervals (with a switch). We technically transform the total variation (increment or decrement) of Lyapunov functions in one sampling interval into the discrete-time update of quantization parameter. Based on this transformation, a hybrid quantized control policy is developed. This policy, in conjunction with the average dwell-time switching law and the constructed event-triggered condition, can ensure the exponential stabilization of the switched system with finite-level quantized input. Besides, the event-triggered scheme is proved to be Zeno-free. The effectiveness of the developed method is verified by a simulation example.     

Guaranteed Cost Control for a Class of Uncertain Stochastic Impulsive Systems with Markovian Switching

Abstract

This article is concerned with the problem of guaranteed cost control for a class of uncertain stochastic impulsive systems with Markovian switching. To the best of our knowledge, it is the first time that such a problem is investigated for stochastic impulsive systems with Markovian switching. For an uncontrolled system, the conditions in terms of certain linear matrix inequalities (LMIs) are obtained for robust stochastical stability and an upper bound is given for the cost function. For the controlled systems, a set of LMIs is developed to design a linear state feedback controller which can stochastically stabilize the class of systems under study and guarantee the given cost function to have an upper bound. Further, an optimization problem with LMI constraints is formulated to minimize the guaranteed cost of the closed-loop system. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

Quantized stabilization of discrete-time systems in a networked environment

This paper is concerned with the problem of output feedback stabilization for a class of discrete-time systems with sector nonlinearities and imperfect measurements. A unified control law model is proposed to take the network-induced delay, random packet dropout and measurement quantization into consideration simultaneously. By choosing appropriate Lyapunov functional, a new stability condition, which is dependent on multiple network status, is established for the resulting closed-loop system. Based on the result, a design criterion for the static output feedback controller is formulated in the form of nonconvex matrix inequalities, and the cone complementary linearization (CCL) procedure is exploited to solve the nonconvex feasibility problem. Incidentally, a less conservative synthesis method is also developed for the state feedback stabilization purpose. Finally, two illustrative examples are provided to illustrate the effectiveness and applicability of the proposed design method.     

Quantized control for networked switched systems under denial-of-service attacks via a barrier event-triggered mechanism

This paper studies the quantized control problem for networked switched systems (NSSs) under denial-of-service (DoS) attacks. The quantized state information, together with the switching signal, is transmitted to the controller through a network. In order to reduce communication consumption and controller update frequency, a barrier event-triggered mechanism is utilized to monitor the state at discrete time. Because of the event-triggered mechanism and the DoS attacks on the network, the mismatch between the system mode and the controller mode is thus frequently encountered, which may lead to quantization saturation and system instability. To solve the problem, an update rule is presented for the dynamic quantizer by switching between zooming in and zooming out of the zooming variable, and a feedback controller is proposed with a jointly designed event-triggered mechanism and a dynamic quantizer. Sufficient conditions on the constraints of DoS frequency and duration are obtained to ensure the exponential stability of the switched system. The effectiveness of the obtained results is illustrated by simulation examples and comparative studies.     

Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations

In this paper we study the mean-square (MS) stability of the Milstein method for linear stochastic delay integro-differential equations (SDIDE) with Markovian switching by extending the techniques of [Z. Wang, C. Zhang, An analysis of stability of Milstein method for stochastic differential equations with delay, Computers and Mathematics with Applications 51 (2006) 1445–1452; L. Ronghua, H. Yingmin, Convergence and stability of numerical solutions to SDDEs with Markovian switching, Applied Mathematics and Computation 175 (2006) 1080–1091]. It is established that the Milstein method is MS-stable for linear stochastic delay differential equations (Wang and Zhang (2006); in the above reference). Here we prove that it is MS-stable for linear SDIDE with Markovian switching also under suitable conditions on the integral term. A numerical example is provided to illustrate the theoretical results.     

具时变时滞和 Markovian转换的不确定奇异随机系统的鲁棒 H_∞ 滤波(英文)

本文处理了一类具与模式有关的时变时滞和 Markovian转换的不确定奇异随机系统的鲁棒H∞滤波问题.所考虑的系统包含参数不确定性,Markovian参数,随机扰动和与模式有关的时变时滞.本文的目的是设计一个滤波器以保证滤波错误系统是正则的、无脉冲的、鲁棒指数均方稳定的和可达到一个给定的 H∞扰动衰减水平.文章首先得到所求鲁棒指数H∞滤波器存在的充分条件,然后给出所求滤波器参数的显示表示.     

Robust stability and stabilization of linear stochastic systems with Markovian switching and uncertain transition rates

This paper is concerned with the robust stabilization problem for a class of linear uncertain stochastic systems with Markovian switching. The uncertain stochastic system with Markovian switching under consideration involves parameter uncertainties both in the system matrices and in the mode transition rates matrix. New criteria for testing the robust stability of such systems are established in terms of bi-linear matrix inequalities (BLMIs), and sufficient conditions are proposed for the design of robust state-feedback controllers. A numerical example is given to illustrate the effectiveness of our results.