Stability analysis for stochastic jump systems with time-varying delay

This paper deals with the mean-square asymptotic stability of stochastic Markovian jump systems with time-varying delay. Based on a new stochastic inequality and convex analysis property, some novel stability conditions are presented. In the derivation, the information of the time-varying delay is retained and the estimation of it by the worst-case enlargement is not involved. Some special cases of the systems under consideration are also investigated. Illustrative examples are given to show the effectiveness of the proposed approach.     

Quantized stabilization of stochastic systems with multiplicative noise under Markovian switching

A problem of quantized state feedback quadratic mean-square stabilization of discrete-time stochastic processes under Markovian switching and multiplicative noise is considered. A static quantizer is used in the feedback channel and 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. It is shown that the coarsest quantization density that permits quadratic mean-square stabilization of this system is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special linear stochastic Markovian switching control system. Also, sufficient conditions for exponential mean-square stabilization of such systems are also explored. An example is given to demonstrate the obtained results.     

具有时变区间参数的不确定随机线性系统的均方鲁棒稳定性

研究了一类具有时变区间参数的不确定随机线性系统的均方鲁棒稳定性.利用时变区间矩阵的分解技术、矩阵的Kronecker积的性质和Lyapunov函数法,得到了该系统均方鲁棒稳定的几个充分性条件.通过一个数值例子说明了所得的这些充分性条件的有效性和实用性.     

Variance-constrained dissipative observer-based control for a class of nonlinear stochastic systems with degraded measurements

This paper is concerned with the variance-constrained dissipative control problem for a class of stochastic nonlinear systems with multiple degraded measurements, where the degraded probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over a given interval. The purpose of the problem is to design an observer-based controller such that, for all possible degraded measurements, the closed-loop system is exponentially mean-square stable and strictly dissipative, while the individual steady-state variance is not more than the pre-specified upper bound constraints. A general framework is established so that the required exponential mean-square stability, dissipativity as well as the variance constraints can be easily enforced. A sufficient condition is given for the solvability of the addressed multiobjective control problem, and the desired observer and controller gains are characterized in terms of the solution to a convex optimization problem that can be easily solved by using the semi-definite programming method. Finally, a numerical example is presented to show the effectiveness and applicability of the proposed algorithm.     

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.     

Discrete-time reduced-order estimator design for bilinear stochastic systems with error covariance assignment

This paper is concerned with the design of reduced-order stateestimators for bilinear stochastic discrete-time systems subjectedto estimation error covariance assignment. The purpose of theproblem addressed is to design the reduced-order state estimators for the bilinear stochastic discrete-time systems such thatthe steady-state estimation error covariances achieve the prespecified values. A simple, effective matrix inequality approach is developed to solve this problem. Specifically, (1) the parameterisationof estimation error covariances that certain bilinear errordynamic processes may possess is presented, (2) the characterisationof all reduced-order state estimators that assign such errorcovariances is explicitly derived, and (3) the solvabilityof the assignability conditions is discussed. Furthermore,an illustrative example is used to demonstrate the effectivenessof the proposed design procedure.     

The variance and covariance of fuzzy random variables and their applications

The concepts of the variance and covariance of fuzzy random variables and their properties are introduced. Examples show their computation and applications in statistical estimation of parameters when samples or prior information are fuzzy. As their further applications the correlation function and the criterions of mean-square calculus for fuzzy stochastic processes are established.     

The the uniform mean-square ergodic theorem for wide sense stationary processes

It is shown that the uniform mean-square ergodic theorem holds for the family of wide sense stationary sequences, as soon as the random process with orthogonal increments, which corresponds to the orthogonal stochastic measure generated by means of the spectral representation theorem, is of bounded variation and uniformly continuous at zero in a mean-square sense. The converse statement is also shown to be valid, whenever the process is sufficiently rich. The method of proof relies upon the spectral representation theorem, integration by parts formula, and estimation of the asymptotic behaviour of total variation of the underlying trigonometric functions. The result extends and generalizes to provide the uniform mean-square ergodic theorem for families of wide sense stationary processes     

Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations

This paper deals with the mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. It is shown that the stochastic theta methods inherit the mean-square exponential stability property of the underlying system. Moreover, the backward Euler method is mean-square exponentially stable with less restrictions on the step size. In addition, numerical experiments are presented to confirm 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.     

Memory feedback controller design for stochastic Markov jump distributed delay systems with input saturation and partially known transition rates

This paper considers the problem of stabilization for a class of stochastic Markov jump distributed delay systems with partially known transition rates subject to saturating actuators. By employing local sector conditions and an appropriate Lyapunov function, a state memory feedback controller is designed to guarantee that the resulted closed-loop constrained systems are mean-square stochastic asymptotically stable. Some sufficient conditions for the solution to this problem are derived in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

随机延迟微分方程平衡方法的均方收敛性与稳定性

本文讨论求解刚性随机延迟微分方程的平衡方法.证明了随机延迟微分方程平衡方法的均方收敛阶为1/2.给出了线性随机延迟微分方程平衡方法均方稳定的条件.     

Exponential mean-square stability of the improved split-step theta methods for non-autonomous stochastic differential equations

We consider the mean-square stability of the so-called improved split-step theta method for stochastic differential equations. First, we study the mean-square stability of the method for linear test equations with real parameters. When θ 3/2, the improved split-step theta methods can reproduce the mean-square stability of the linear test equations for any step sizes h 0. Then, under a coupled condition on the drift and diffusion coefficients, we consider exponential mean-square stability of the method for nonlinear non-autonomous stochastic differential equations. Finally, the obtained results are supported by numerical experiments.     

Asymptotic mean-square stability of linear multi-step methods for SODEs

In this article we present results of a linear stability analysis of stochastic linear multi-step methods for stochastic ordinary differential equations. As in deterministic numerical analysis we use a linear time-invariant test equation and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution of that test equation. Sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods are obtained with the aide of Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams-Bashforth- and Adams-Moulton-methods and the BDF method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

随机扰动下统一混沌系统的有限时间同步

本文研究了具有随机扰动的统一混沌系统的有限时间同步问题,其中随机扰动是一维标准的维纳随机过程.利用了有限时间随机李雅普诺夫稳定性理论、伊藤公式,本文分三个步骤设立了三个控制器获得了驱动–响应系统在有限时间内的均方渐近同步.最后进行的数值模拟验证了理论结果的正确性和方法的有效性.     

Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise

The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index-1 differential-algebraic equations (DAEs). Based on the idea of Defect Correction we developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index-1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper. The first author acknowledges support by the BMBF-project 03RONAVN and the second author support by the Austrian Science Fund Project P17253.     

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

The paper deals with the numerical treatment of stochastic differential-algebraic equations of index one with a scalar driving Wiener process. Therefore, a particularly customized stochastic Runge-Kutta method is introduced. Order conditions for convergence with order 1.0 in the mean-square sense are calculated and coefficients for some schemes are presented. The proposed schemes are stiffly accurate and applicable to nonlinear stochastic differential-algebraic equations. As an advantage they do not require the calculation of any pseudo-inverses or projectors. Further, the mean-square stability of the proposed schemes is analyzed and simulation results are presented bringing out their good performance.