Stabilization of interconnected nonlinear stochastic Markovian jump systems via dissipativity approach

This paper considers the stabilization problems for interconnected nonlinear stochastic Markovian jump systems from the viewpoint of dissipativity theory. Based on the strongly stochastic passivity theory, the feedback equivalence and global stabilization problems are studied for interconnected nonlinear stochastic Markovian jump systems. The strongly stochastic γ-dissipativity sustains a direct H_∞ control for this class of systems instead of solving coupled Hamilton–Jacobi inequalities.     

网络化非线性系统的非脆弱H∞控制

具有对数量化、网络诱导时延和数据包丢失的网络化Lipschitz非线性系统控制器,存在参数摄动问题。为此,设计一种加性非脆弱状态反馈H∞控制器。将数据量化和网络诱导时延对被控系统的影响,转化为系统的不确定参数,网络化控制系统建模为马尔可夫跳变系统。采用Lyapunov稳定性理论和线性矩阵不等式方法,给出网络化Lipschitz非线性系统的加性非脆弱状态反馈H∞控制器存在的充分条件,该非脆弱H∞控制器可通过解线性矩阵不等式求出。仿真结果表明,当控制器存在参数摄动时,与传统控制器相比,非脆弱控制器不仅能使被控系统稳定,而且满足设定的H∞性能指标。     

A geometric approach to H∞ control of nonlinear Markovian jump systems

This paper first discusses the H_∞ control problem for a class of general nonlinear Markovian jump systems from the viewpoint of geometric control theory. Following with the updating of the Markovian jump mode, the appropriate diffeomorphism can be adopted to transform the system into special structures, which establishes the basis for the geometric control of nonlinear Markovian jump systems. Through discussing the strongly minimum-phase property or the strongly γ-dissipativity of the zero-output dynamics, the H_∞ control can be designed directly without solving the traditional coupled Hamilton–Jacobi inequalities. A numerical example is presented to illustrate the effectiveness of our results.     

网络化系统的状态反馈保性能控制

研究了具有随机网络诱导时延且数据包丢失服从马尔可夫链的网络化系统的保性能控制问题．将网络化控制系统建模为具有两种运行模式的马尔可夫跳变线性系统．根据马尔可夫跳变线性系统理论，给出了网络化系统状态反馈保性能控制器存在的充分条件；该保性能控制器为一组线性矩阵不等式的解．通过一个仿真示例说明了本文所提方法的有效性．     

Nonsmooth Stabilization of a Class of Markovian Jump Stochastic Nonlinear Systems with Parametric and Dynamic Uncertainties

In this paper, we investigate the finite‐time stabilization of Markovian jump stochastic nonlinear (SNL) systems with dynamic uncertainties. Firstly, a proper criterion on finite‐time globally asymptotically stability in probability (FGSP) and some useful lemmas are introduced. Then, overcoming the influence of coupled item which determined by Markovian switching, by adding a power integrator technique and induction method, a state‐feedback finite‐time controller is explicitly constructed. It is proven that, the system state of the closed‐loop systems is FGSP. Simulation examples illustrate the effectiveness of our method.     

具有Markov跳跃参数的一类随机非线性系统逆最优增益设计

研究了一类随机非线性系统的逆最优增益设计问题,系统中除了方差未知的Wiener噪声之外,还含有Markov跳跃参数. 首先,给出此类系统逆最优增益设计问题可解的一个充分条件. 其次,针对一类具有严格反馈形式的随机非线性系统,利用积分反推法,给出了依概率全局渐近稳定和逆最优控制策略的设计方法. 其中,所设计的Lyapunov函数和控制策略与模态显式无关,克服了由于Markov跳跃模态引起的耦合项所带来的设计困难. 最后,通过仿真验证了控制策略的有效性.     

Observer-based Non-fragile Mixed Passivity and H∞ Feedback Control for Fuzzy Stochastic Jump System Subject to Quantized Measurements

This paper considers the topic of the observer-based non-fragile mixed passivity and H_∞ state feedback control for time-varying delay and norm-bounded parameter uncertainties fuzzy stochastic Markovian jump system subject to quantized measurements. The aim of this paper is to provide a suitable observer-based non-fragile state feedback controller to enhance stochastic stabilization of the closed-loop system and satisfy mixed passivity and H_∞ performance. By constructing mode-dependent Lyapunov-Krasovskii functional, novel conditions are achieved in virtue of linear matrix inequalities. Two examples including truck-trailer model are given to illustrate the effectiveness of the developed method.

     

Passivity-based non-fragile control for Markovian jump systems with aperiodic sampling

In this paper, the problem of non-fragile passive control for Markovian jump systems with aperiodic sampling is investigated. The considered controller is assumed to have either additive or multiplicative norm-bounded uncertainties. A time-dependent Lyapunov functional capturing the available information of the sampling pattern is constructed to derive a sufficient condition for non-fragile stochastic passivity of the resultant closed-loop system. Based on the condition, a mode-independent state feedback sampled-data controller is designed such that for all admissible uncertainties the closed-loop system is robustly stochastically passive. Two illustrative examples are included to demonstrate the effectiveness and merits of the proposed techniques.     

Delay‐dependent passivity and passification for uncertain Markovian jump systems with time‐varying delays

This paper deals with the problems of passivity analysis and passivity‐based controller design for Markovian jump systems with both time‐varying delays and norm‐bounded parametric uncertainties. Firstly, new delay‐dependent conditions for the considered system to be passive are obtained by using a mode‐dependent Lyapunov functional and by introducing some slack variables. These conditions are expressed by means of LMIs that are easy to check. It is shown through a numerical example that the obtained passivity conditions are less conservative than the existing ones in the literature. Secondly, the passification problem is investigated. On the basis of the obtained passivity conditions, dynamic output‐feedback controllers are designed, which ensure that the resulting closed‐loop system is passive. The effectiveness of the proposed design method is demonstrated by a numerical example. Copyright © 2011 John Wiley & Sons, Ltd.     

Robust stability and H-infinity control for uncertain discrete-time Markovian jump singular systems

The robust stability and stabilization, and H-infinity control problems for discrete-time Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent （r.s.e.） transformation and by introducing new state vectors, the singular system is transformed into a discrete-time Markovian jump standard linear system, and the linear matrix inequality （LMI） conditions for the discrete-time Markovian jump singular systems to be regular, causal, stochastically stable, and stochastically stable with 7- disturbance attenuation are obtained, respectively. With these conditions, the robust state feedback stochastic stabilization problem and H-infinity control problem are solved, and the LMI conditions are obtained. A numerical example illustrates the effectiveness of the method given in the oaoer.     

Dissipativity of diffusion Itô processes with Markovain switching and problems of robust stabilization

Consideration is given to a class of systems described by a finite set of controlled diffusion Itô processes that are control-affine, with jump transitions between them, and are defined by the evolution of a uniform Markovian chain (Markovian switching). Each state of this chain corresponds to a certain system mode. A stochastic version of the notion dissipativity by Willems is introduced, and properties of diffusion processes with Markovian switching are studied. The relationship between passivity and stabilizability in the process of output-feedback control is established. The obtained results are applied to the problem of robust simultaneous stabilization for the set of nonlinear systems with undetermined parameters. As a partial case, a problem of robust simultaneous stabilization for the set of linear systems where final results are obtained in terms of linear matrix inequalities.     

Robust stability and H-infinity control for uncertain discrete-time Markovian jump singular systems

The robust stability and stabilization, and H-infinity control problems for discrete-time Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a discrete-time Markovian jump standard linear system, and the linear matrix inequality (LMI) conditions for the discrete-time Markovian jump singular systems to be regular, causal, stochastically stable, and stochastically stable with °- disturbance attenuation are obtained, respectively. With these conditions, the robust state feedback stochastic stabilization problem and H-infinity control problem are solved, and the LMI conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper.     

Passive stochastic feedback stability-Part II. Applications

In Part I of this paper it was shown that the concept of passivity, used extensively in deterministic feedback stability, could be extended to stochastic feedback systems described through Markov theory to produce stochastic feedback stability theorems. To apply the theory developed in Part I, one is required to do a passivity analysis of the sub-systems of the feedback system. In this paper, six particular types of stochastic systems (sub-systems) are considered and a passivity analysis of each is performed. These systems have the appearance of standardnth-order linear differential systems with a random process entering at some point. The randomness is either white noise, impulsive noise or a Markov jump process. Once the passivity analysis has been made, feedback stability theorems can be derived for many feedback systems containing one or more of these six systems as sub-systems. Two such feedback systems are considered in the paper.     

随机激励的非线性Markov跳变系统的稳态响应

大量实际工程问题需要用同时包含连续和离散变量的Markov跳变系统来描述.本文介绍了一类随机激励的单自由度(强)非线性Markov跳变系统的稳态响应的研究方法.首先,基于随机平均法导出具有Markov跳变参数的平均It随机微分方程,原系统方程的维数得到降低.接着,根据跳变过程原理,建立Fokker-Planck-Kolmogorov(FPK)方程组,方程组中的方程与系统的结构状态一一对应且互相耦合.求解该FPK方程组,得到Markov跳变系统的稳态随机响应及其统计量.最后,以一个高斯白噪声激励的Markov跳变Duffing振子为例,计算得到不同跳变规律下系统的稳态响应.研究结果表明,Markov跳变系统的稳态响应可以看作是各结构状态子系统稳态响应的加权和,加权值由跳变规律决定.     

Reliable H∞ control for nonlinear discrete-time systems with multiple intermittent faults in sensors or actuators

This study proposes a fault-tolerant control method for stochastic systems with multiple intermittent faults (IFs) and nonlinear disturbances, and both sensor and actuator faults are considered. The occurrence and disappearance of IFs are governed by Markov chain, and its transition probabilities are partly known. Hence, the faulty system can be described by a Markovian jump system (MJS). In order to ensure that the MJS is stochastically stable and satisfies H_∞ performance index, mode-dependent output feedback controllers are modelled using linear matrix inequalities. Numerous sufficient conditions for stochastic stability are obtained on the basis of Lyapunov stability theory. Finally, the effectiveness of the developed method is evaluated on the three-tank system.     

Markov跳跃非线性系统逆最优增益设计

证明了一类严格反馈Markov跳跃系统是依概率输入–状态可稳定的.其次,证明了逆最优增益设计问题可解的一个充分条件是存在一组满足小控制量的依概率输入–状态稳定控制李雅普诺夫函数.最后,利用积分反推方法,给出了严格反馈Markov跳跃系统逆最优增益设计问题的一个构造性解.其中,为了克服由于Markov跳跃引起的耦合项所带来的困难,所设计的李雅普诺夫函数以及控制器是与模态无关的.     

Feedback predictive control for constrained fuzzy systems with Markovian jumps

In this paper, a feedback model predictive control method is presented to tackle control problems with constrained multivariables for uncertain discrete‐time nonlinear Markovian jump systems. An uncertain Markovian jump fuzzy system (MJFS) is obtained by employing the Takagi‐Sugeno (T‐S) fuzzy model to represent a discrete‐time nonlinear system with norm bounded uncertainties and Markovain jump parameters. To achieve more generality, the transition probabilities of the Markov chain are assumed to be partly unknown and partly accessible. The predictive formulation adopts an on‐line optimization paradigm that utilizes the closed‐loop state feedback controller and is solved using the standard semi‐definite programming (SDP). To reduce the on‐line computational burden, a mode independent control move is calculated at every sampling time based on a stochastic fuzzy Lyapunov function (FLF) and a parallel distributed compensation (PDC) scheme. The robust mean square stability, performance minimization and constraint satisfaction properties are guaranteed under the control move for all admissible uncertainties. A numerical example is given to show the efficiency of the developed approach. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Fault-tolerant Control for Nonlinear Systems with Multiple Intermittent Faults and Time-varying Delays

This study investigates a new fault-tolerant control method for uncertain nonlinear systems with multiple intermittent faults and time-varying delays. The considered intermittent faults appear in sensors and actuators simultaneously. A Markov chain is used to describe the random occurrence and disappearance of intermittent faults. The uncertain nonlinear system with intermittent faults is augmented as a Markovian jump system. By using H-infinity control theory and linear matrix inequality (LMI), we design fault tolerant controllers to make augmented Markovian jump system work steadily. Several sufficient conditions for stochastic stability with given H-infinity performance index and the existence of output-feedback controllers are derived. The effectiveness of the proposed fault-tolerant method is validated by a continuously stirred tank reactor (CSTR).     

网络控制系统随机稳定性研究

研究了具有随机网络诱导时延及数据包丢失的网络控制系统随机稳定性问题. 本文用一个具有两个状态的马尔可夫链来描述数据通过网络传输时随机数据包丢失过程, 利用马尔可夫跳变线性系统理论, 将网络控制系统建模为一个具有两种运行模式的马尔可夫跳变线性系统, 给出了在状态反馈控制下网络控制系统随机稳定的线性矩阵不等式形式的充分条件, 最后用一个仿真示例验证了该方法的有效性.