Conditions For Mean Square Stability Research Articles

Stabilisation of wireless networked control systems with packet loss

This study is concerned with the stabilisation problem for a basic wireless networked control system (WNCS) with packet loss in the discrete-time domain. Utilising the relation between packet loss probability and signal-to-noise power ratio (SNR) of the wireless channel, the wireless link model is simplified as a Bernoulli process. Moreover, the WNCS can be expressed as a Markov jump linear system. The necessary and sufficient condition of mean-square stability for the WNCS is established for given packet loss probability. The stabilisation condition is developed for the existence of a stabilising state feedback controller and the state-feedback gain can be obtained by using the condition. A numerical example is provided to illustrate the main results. Furthermore, an upper bound on the packet loss probability and a lower bound on the SNR guaranteeing the existence of stabilising controller are estimated by exploiting the stabilisation condition in the example.

Stabilization of a Class of Switched Stochastic Systems with Time Delays Under Asynchronous Switching

This paper addresses stabilization problem for a class of switched stochastic systems with time delays by using piecewise Lyapunov–Krasovskii functional method. Asynchronous switching means the switching of the controller has a time delay to the switching of the system. We provide a set of Lyapunov–Krasovskii-type sufficient conditions for establishing the mean-square exponential stability. The mean-square exponential stability condition for the resulting closed-loop system is firstly derived by further allowing the Lyapunov–Krasovskii functional to increase during the running time of the active subsystem with the mismatched controller. Then, the corresponding solvability condition for stabilizing controllers is established. Finally, we present an example to show the effectiveness of the developed theory.

Combinational parametric resonance under imperfectly periodic excitation

Conditions for mean square stability of a two-degree-of-freedom system have been obtained for the case of sum combinational resonance due to sinusoidal parametric excitation with constant amplitude and white-noise random temporal variations in the instantaneous frequency. Analysis is based on the asymptotic Krylov–Bogoliubov averaging method. The resulting set of ten deterministic differential equations for second-order moments of the four new state variables (inphase and quadrature modal responses) was solved analytically for its neutral stability boundary. The imperfections in periodicity may lead to either stabilization or destabilization of the system and the corresponding conditions have been clearly established from the solution obtained. Furthermore, conditions for almost sure stability have been obtained for a special symmetric case of identical modal damping factors and identical modal excitation amplitudes.

Passivity Analysis and Synthesis for a Class of Discrete-Time Switched Stochastic Systems with Time-Varying Delay

This paper deals with the problems of passivity and passification for a class of discrete-time switched stochastic systems with time-varying delay. Based on the average dwell time approach, the piecewise Lyapunov function technique, and the free-weighting matrix method, a new Lyapunov functional is proposed and sufficient conditions for mean-square exponential stability and stochastic passivity are developed under average dwell time switching. Moreover, an estimate of state decay can be calculated in terms of linear matrix inequalities (LMIs). Then, the solvability condition for passification is established and the corresponding controller is designed. Two numerical examples are given to show the effectiveness of the proposed methods.

Exponential mean square stability of numerical methods for systems of stochastic differential equations

This paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split-step theta method for stochastic systems. First, we consider linear autonomous systems. Under a sufficient and necessary condition for exponential mean square stability of the exact solution, it is proved that the two classes of theta methods with θ≥0.5 are exponentially mean square stable for all positive step sizes and the methods with θ<0.5 are stable for some small step sizes. Then, we study the stability of the methods for nonlinear non-autonomous systems. Under a coupled condition on the drift and diffusion coefficients, it is proved that the split-step theta method with θ>0.5 still unconditionally preserves the exponential mean square stability of the underlying systems, but the stochastic theta method does not have this property. Finally, we consider stochastic differential equations with jumps. Some similar results are derived.

Volterra Integral Approach to Impulsive Renewal Systems: Application to Networked Control

We analyze impulsive systems with independent and identically distributed intervals between transitions. Our approach involves the derivation of novel results for Volterra integral equations with positive kernel. We highlight several applications of these results, and show that when applied to the analysis of impulsive systems they allow us to (i) provide necessary and sufficient conditions for mean square stability, stochastic stability and mean exponential stability, which can be equivalently tested in terms of a matrix eigenvalue computation, an LMI feasibility problem, and a Nyquist criterion condition; (ii) assess performance of the impulsive system by computing a second moment Lyapunov exponent. The applicability of our results is illustrated in a benchmark problem considering networked control systems with stochastically spaced transmissions, for which we can guarantee stability for inter-sampling times roughly twice as large as in previous papers.

Robust Fault-Tolerant Control for Uncertain Networked Control Systems with State-Delay and Random Data Packet Dropout

A robust fault-tolerant controller design problem for networked control system (NCS) with random packet dropout in both sensor-to-controller link and controller-to-actuator link is investigated. A novel stochastic NCS model with state-delay, model uncertainty, disturbance, probabilistic sensor failure, and actuator failure is proposed. The random packet dropout, sensor failures, and actuator failures are characterized by a binary random variable. The sufficient condition for asymptotical mean-square stability of NCS is derived and the closed-loop NCS satisfiesH∞performance constraints caused by the random packet dropout and disturbance. The fault-tolerant controller is designed by solving a linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the proposed method.

Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean‐square exponential stability of the exact solution is derived. Then, mean‐square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean‐square stable for any stepsize Δt = τ/m when 1/2 ≤ θ ≤ 1, and they are exponentially mean‐square stable if the stepsize Δt ∈ (0, Δt0) when 0 ≤ θ &lt; 1. Finally, some numerical experiments are given to illustrate the theoretical results.

H∞Filtering over Networks for a Class of Discrete-Time Stochastic System with Randomly Occurred Sensor Nonlinearity

TheH∞filtering problem for a class of discrete-time stochastic system with randomly occurred nonlinearity (RON) suffering from network packet loss is considered. Based on the Lyapunov-Krasovskii functional method, the asymptotical mean-square stability condition of the filtering system with a prescribedH∞level is derived. Besides, the filter parameters can be obtained simultaneously by solving the matrix inequalities we achieve. It is worth noting that no slack variable is introduced in the proposed conditions. The effectiveness of the theory developed is verified through a numerical example.

Diffusion Bias-Compensated RLS Estimation Over Adaptive Networks

We study the problem of distributed least-squares estimation over ad hoc adaptive networks, where the nodes have a common objective to estimate and track a parameter vector. We consider the case where there is stationary additive colored noise on both the regressors and the output response, which results in biased local least-squares estimators. Assuming that the noise covariance can be estimated (or is known a priori), we first propose a bias-compensated recursive least-squares algorithm (BC-RLS). However, this bias compensation increases the variance or the mean-square deviation (MSD) of the local estimators, and errors in the noise covariance estimates may still result in residual bias. We demonstrate that the MSD and residual bias can then be significantly reduced by applying diffusion adaptation, i.e., by letting nodes combine their local estimates with those of their neighbors. We derive a necessary and sufficient condition for mean-square stability of the algorithm, under some mild assumptions. Furthermore, we derive closed-form expressions for its steady-state mean and mean-square performance. Simulation results are provided, which agree well with the theoretical results. We also consider some special cases where the mean-square stability improvement of diffusion BC-RLS over BC-RLS can be mathematically verified.

Reliable H ∞ filter design for T–S fuzzy model‐based networked control systems with random sensor failure

SUMMARYThis paper investigates robust and reliable H ∞ filter design for a class of nonlinear networked control systems: (i) a T‐S fuzzy model with its own uncertainties is used to approximate the nonlinear dynamics of the plant, (ii) a new sensor failure model with uncertainties is proposed, and (iii) the signal transfer of the closed‐loop system is under a networked communication scheme and therefore is subject to time delay, packet loss, and/or packet out of order. Four new theorems are proved to cover the conditions for the robust mean square stability of the systems under study in terms of LMIs, and a decoupling method for the filter design is developed. Two examples, one of them is based on a model of an inverted pendulum, are provided to demonstrate the design method. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Mean Square Stability of Impulsive Stochastic Delay Differential Equations with Markovian Switching and Poisson Jumps

In the paper, based on stochastic analysis theory and Lyapunov functional method, we discuss the mean square stability of impulsive stochastic delay differential equations with markovian switching and poisson jumps, and the sufficient conditions of mean square stability have been obtained. One example illustrates the main results. Furthermore, some well-known results are improved and generalized in the remarks. Keywords—impulsive, stochastic, delay, Markovian switching, Poisson jumps, mean square stability.

Stability of stochastic θ-methods for stochastic delay integro-differential equations

In this paper, we are concerned with the numerical stability of linear stochastic delay integro-differential equations (SDIDEs). A sufficient condition for mean square stability of the exact solution of a linear SDIDE with multiplicative noise is derived. Then the mean square stability of stochastic θ-methods is investigated, and it is shown that the numerical solution can reproduce the mean square stability of the exact solution under appropriate conditions. At last, we present some numerical experiments to support our conclusions.

Deterministic and stochastic stability of a mathematical model of smoking

Our aim in this paper, is first constructing a Lyapunov function to prove the global stability of the unique smoking-present equilibrium state of a mathematical model of smoking. Next we incorporate random noise into the deterministic model. We show that the stochastic model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. Then a stochastic Lyapunov method is performed to obtain the sufficient conditions for mean square and asymptotic stability in probability of the stochastic model. Our analysis reveals that the stochastic stability of the smoking-present equilibrium state, depends on the magnitude of the intensities of noise as well as the parameters involved within the model system.

Closed-Loop Stabilization Over Gaussian Interference Channel

AbstractThe problem of closed-loop stabilization of two scalar linear time invariant systems over symmetric white Gaussian interference channel with correlated noise components and arbitrarily distributed initial states is addressed. We propose to use linear and memoryless communication and control scheme based on the coding schemes introduced for interference networks which are an extension of the Schalkwijk-Kailath coding scheme. By employing the proposed communication and control scheme over the Gaussian interference channel, we derive sufficient conditions for mean square stability of the two linearly controlled LTI systems.

Convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay

In this paper, we study the convergence and stability of the stochastic theta method (STM) for a class of index 1 stochastic delay differential algebraic equations. First, in the case of constrained mesh, i.e., the stepsize is a submultiple of the delay, it is proved that the method is strongly consistent and convergent with order 1/2 in the mean-square sense. Then, the result is further extended to the case of non-constrained mesh where we employ linear interpolation to approximate the delay argument. Later, under a sufficient condition for mean-square stability of the analytical solution, it is proved that, when the stepsizes are sufficiently small, the STM approximations reproduce the stability of the analytical solution. Finally, some numerical experiments are presented to illustrate the theoretical findings.

Dynamic instabilities in a single-axis gyroscope subjected to stochastic angular rate perturbations

Instabilities in a vibratory MEMS gyroscope that is subject to stochastic fluctuations in input angular rates are investigated. The vibratory-type gyroscope considered in the present study is of the spring-mass type. An asymptotic approach based on the method of stochastic averaging has been employed for this purpose, and closed-form conditions for mean square stability of dynamic response are obtained for the case of exponentially correlated noise. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. The presented results remain valid if the stochastic parametric excitation has a small correlation time compared with the system relaxation time. Stability predictions have been illustrated via stability diagrams in the power-spectral-density–damping-ratio space.

The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points

In the paper, the asymptotic mean square stability of the zero solution for neutral stochastic delay differential equations with Poisson jumps is studied by fixed points theory without Lyapunov functions. The coefficient functions have not been asked for a fixed sign, and the sufficient condition for mean square stability has been obtained. Therefore, some well-known results are improved and generalized.

Delay‐distribution‐dependent robust stability of uncertain systems with time‐varying delay

AbstractBy employing the information of the probability distribution of the time delay, this paper investigates the problem of robust stability for uncertain systems with time‐varying delay satisfying some probabilistic properties. Different from the common assumptions on the time delay in the existing literatures, it is assumed in this paper that the delay is random and its probability distribution is known a priori. In terms of the probability distribution of the delay, a new type of system model with stochastic parameter matrices is proposed. Based on the new system model, sufficient conditions for the exponential mean square stability of the original system are derived by using the Lyapunov functional method and the linear matrix inequality (LMI) technique. The derived criteria, which are expressed in terms of a set of LMIs, are delay‐distribution‐dependent, that is, the solvability of the criteria depends on not only the variation range of the delay but also the probability distribution of it. Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed method. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.