Stochastic Time-delay Systems Research Articles

Delay-range-dependent robust H ∞ filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters

This paper investigates the problem of robust H ∞ filtering for uncertain stochastic time-delay systems with Markovian jump parameters. Both the state dynamics and measurement of the system are corrupted by Wiener processes. The time delay varies in an interval and depends on the mode of operation. A Markovian jump linear filter is designed to guarantee robust exponential mean-square stability and a prescribed disturbance attenuation level of the resulting filter error system. A novel approach is employed in showing the robust exponential mean-square stability. The exponential decay rate can be directly estimated using matrices of the Lyapunov–Krasovskii functional and its derivative. A delay-range-dependent condition in the form of LMIs is derived for the solvability of this H ∞ filtering problem, and the desired filter can be constructed with solutions of the LMIs. An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.

Robust [formula omitted] filtering for discrete nonlinear stochastic systems with time-varying delay

In this paper, we are concerned with the robust H ∞ filtering problem for a class of nonlinear discrete time-delay stochastic systems. The system under study involves parameter uncertainties, stochastic disturbances, time-varying delays and sector-like nonlinearities. The problem addressed is the design of a full-order filter such that, for all admissible uncertainties, nonlinearities and time delays, the dynamics of the filtering error is constrained to be robustly asymptotically stable in the mean square, and a prescribed H ∞ disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and some new techniques, sufficient conditions are first established to ensure the existence of the desired filtering parameters. These conditions are dependent on the lower and upper bounds of the time-varying delays. Then, the explicit expression of the desired filter gains is described in terms of the solution to a linear matrix inequality (LMI). Finally, a numerical example is exploited to show the usefulness of the results derived.

Robust L2 - L∞ Filtering for Uncertain Nonlinearly Parameterized Stochastic Systems with Time-Varying Delays

This paper is concerned with the problem of robust L2 - L∞ filtering for uncertain stochastic time-delay systems in the nonlinear fractional transformation (NFT) form. Attention is focused on the design of both full-order and reduced-order filters, guaranteeing robust stochastic stability and a prescribed level of L2 - L∞ performance for the filtering error system. A parameter-dependent Lyapunov functional is employed to solve the problem. The concerned filters are not only parameter dependent but also have the NFT form. Sufficient conditions for the existence of such filters are presented in terms of certain linear matrix inequalities (LMIs). Desired filters are explicitly expressed based on the solutions to the proposed LMIs. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Generalized robust H∞ fault diagnosis filtering based on conditional stochastic distributions of system outputs

The efficiency of fault detection and diagnosis by using output probability density functions (PDFs) for stochastic time-delayed systems has been shown in practical processes. Neural network modelling has been applied to characterize the output PDFs and to the dynamical weighting system. In this paper, the system perturbations and disturbance are considered and the robust fault diagnosis design is studied for the general stochastic system in the presence of time delays. The main objective is to design linear matrix inequality (LMI)-based fault diagnostic filtering (FDF) to estimate the fault and attenuate the disturbances. The modelling errors and system uncertainties resulting from both B-spline expansion and the weighting system are merged into system disturbance. It can be seen that the resulting weighting system comprises non-linearities, uncertainties, disturbances, and time delays, and includes the non-zero initial condition. The generalized H∞ optimization is presented and applied to the fault diagnosis problem of the weighting system with the non-zero initial condition and truncated norms. Simulations are given to demonstrate the efficiency of the proposed approach.

Delay-dependent fault detection and diagnosis using B-spline neural networks and nonlinear filters for time-delay stochastic systems

A new fault detection and diagnosis (FDD) scheme is studied in this paper for the continuous-time stochastic dynamic systems with time delays, where the available information for the FDD is the input and the measured output probability density functions (PDFs) of the system. The square-root B-spline neural networks is used to formulate the output PDFs with the dynamic weightings. As a result, the concerned FDD problem can be transformed into a robust FDD problem subjected to a continuous time uncertain nonlinear system with time delays. Delay-dependent criteria to detect and diagnose the system fault are provided by using linear matrix inequality (LMI) techniques. It is shown that this new criterion can provide higher sensitivity performance than the existing result. Simulations are given to demonstrate the efficiency of the proposed approach.

Observer-based optimal fault detection using PDFs for time-delay stochastic systems

This paper concerns the problem of fault detection (FD) for general stochastic systems with time delays. Different from the classical FD problem, the available information for the FD is the input and the measured output probability density functions (PDFs) of the system. Based on this, firstly, square root B-spline expansions technique is applied to model the PDFs so that the concerned FD problem is transformed into a nonlinear FD problem subject to the weight dynamical systems. Secondly, by using the FD observer as residual generator and guaranteed cost performance index as objective function, design of the FD observer is formulated as an optimization problem. A delay-dependent condition to solve this problem is established in terms of the feasibility of linear matrix inequality (LMI). Moreover, in order to improve FD performance, the guaranteed cost algorithm is applied to optimal observer design. Finally, two examples are given to demonstrate the applicability of the proposed approach.

Delay-dependent [formula omitted]– [formula omitted] filter design for stochastic time-delay systems

This paper is concerned with the problem of L 2 – L ∞ filter design for a class of stochastic time-delay systems. A delay-dependent sufficient condition is presented, which guarantees the existence of a linear filter ensuring that the filtering error system is stochastically asymptotically stable and its L 2 – L ∞ performance satisfies a prescribed level. A desired filter can be constructed by solving certain linear matrix inequalities. A numerical example is given to demonstrate the effectiveness of the proposed method.

A delay-dependent approach to H ∞ filtering for stochastic delayed jumping systems with sensor non-linearities

In this paper, a delay-dependent approach is developed to deal with the stochastic H ∞ filtering problem for a class of Itô type stochastic time-delay jumping systems subject to both the sensor non-linearities and the exogenous non-linear disturbances. The time delays enter into the system states, the sensor non-linearities and the external non-linear disturbances. The purpose of the addressed filtering problem is to seek an H ∞ filter such that, in the simultaneous presence of non-linear disturbances, sensor non-linearity as well as Markovian jumping parameters, the filtering error dynamics for the stochastic time-delay system is stochastically stable with a guaranteed disturbance rejection attenuation level γ. By using Itô's differential formula and the Lyapunov stability theory, we develop a linear matrix inequality approach to derive sufficient conditions under which the desired filters exist. These conditions are dependent on the length of the time delay. We then characterize the expression of the filter parameters, and use a simulation example to demonstrate the effectiveness of the proposed results.

Robust Fuzzy Design for Nonlinear Uncertain Stochastic Systems via Sliding-Mode Control

This paper deals with the sliding-mode control (SMC) problem for nonlinear stochastic time-delay systems by means of fuzzy approach. The Takagi-Sugeno (T-S) fuzzy stochastic time-delay model with parametric uncertainties and unknown nonlinearities is presented. A sufficient condition for the exponential stability in mean square of the sliding motion is also derived. Moreover, it is shown that when the linear matrix inequalities (LMIs) with equality constraint are feasible, the designs of both sliding surface and sliding-mode controller can be easily obtained via convex optimization. A simulation example illustrating the proposed method is given.

Delay-dependent stabilization for stochastic fuzzy systems with time delays

This paper is concerned with the delay-dependent stabilization problem for a class of time-delay stochastic fuzzy systems. The time delays are assumed to appear in both the state and the control input. The purpose is the design of a state-feedback fuzzy controller such that the resulting closed-loop system is asymptotically stable in the mean square. A delay-dependent condition for the solvability of this problem is obtained in terms of relaxed linear matrix inequalities (LMIs). By solving these LMIs, a desired controller can be obtained. Finally, a numerical example is given to demonstrate the effectiveness of the present results.

Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances

In this paper, a delay-dependent approach is developed to deal with the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. The system matrices are assumed to be uncertain within given intervals, the time delays appear in both the system states and the nonlinear disturbances, and the stochastic perturbation is in the form of a Brownian motion. The purpose of the addressed stochastic stabilization problem is to design a memoryless state feedback controller such that, for all admissible interval uncertainties and nonlinear disturbances, the closed-loop system is asymptotically stable in the mean square, where the stability criteria are dependent on the length of the time delay and therefore less conservative. By using Itô's differential formula and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the stability of the stochastic interval delay systems. Then, the controller gain is characterized in terms of the solution to a delay-dependent linear matrix inequality (LMI), which can be easily solved by using available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed design procedure.

A data analysis method for identifying deterministic components of stable and unstable time-delayed systems with colored noise

A method is proposed to identify deterministic components of stable and unstable time-delayed systems subjected to noise sources with finite correlation times (colored noise). Both neutral and retarded delay systems are considered. For vanishing correlation times it is shown how to determine their noise amplitudes by minimizing appropriately defined Kullback measures. The method is illustrated by applying it to simulated data from stochastic time-delayed systems representing delay-induced bifurcations, postural sway and ship rolling.

Some basic remarks on eigenmode expansions of time-delay dynamics

We describe in elementary terms how eigenmode expansions can be used to deal with differential-difference equations. As particular applications we present the full analytical solution of linear stochastic time-delay systems and the weakly nonlinear analysis of nonlinear differential-difference equations in the limit of large time delay. Our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the Lambert W-function and on the explicit formula for the eigenfunctions of the adjoint equation.

On stabilization for a class of nonlinear stochastic time-delay systems: A matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally (globally) robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also given.

Nonlinear H∞ control of stochastic time-delay systems with Markovian switching

In this paper, the stabilization and H ∞ control problems are investigated for a class of stochastic time-delay systems with both nonlinear disturbances and Markovian jumping parameters. The purpose of the stochastic stabilization problem is to design a memoryless state feedback controller such that, for the addressed nonlinear disturbances as well as Markovian jumping parameters, the closed-loop system is stochastically exponentially stable in the mean square, independent of the time delay. In the H ∞ control problem, in addition to the mean-square exponential stability requirement, a prescribed H ∞ performance index is required to be achieved. By using Itô’s differential formula and the Lyapunov stability theory, sufficient conditions for the solvability of these problems are derived in term of linear matrix inequalities, which can be easily checked by resorting to available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed results.

Robust H/sub /spl infin// filtering for stochastic time-delay systems with missing measurements

In this paper, the robust Hinfin filtering problem is studied for stochastic uncertain discrete time-delay systems with missing measurements. The missing measurements are described by a binary switching sequence satisfying a conditional probability distribution. We aim to design filters such that, for all possible missing observations and all admissible parameter uncertainties, the filtering error system is exponentially mean-square stable, and the prescribed Hinfin performance constraint is met. In terms of certain linear matrix inequalities (LMIs), sufficient conditions for the solvability of the addressed problem are obtained. When these LMIs are feasible, an explicit expression of a desired robust Hinfin filter is also given. An optimization problem is subsequently formulated by optimizing the Hinfin filtering performances. Finally, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach

STABILIZATION AND H∞ CONTROL FOR UNCERTAIN STOCHASTIC TIME‐DELAY SYSTEMS VIA NON‐FRAGILE CONTROLLERS

ABSTRACTThis paper considers the problems of robust non‐fragile stochastic stabilization and H∞ control for uncertain time‐delay stochastic systems with time‐varying norm‐bounded parameter uncertainties in both the state and input matrices. Attention is focused on the design of memoryless state feedback controllers which are subject to norm‐bounded uncertainties. For both the cases of additive and multiplicative controller uncertainties, delay‐independent sufficient conditions for the solvability of the above problems are obtained. The desired state feedback controller can be constructed by solving a certain linear matrix inequality.

Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control

A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.

RobustH∞ control of stochastic time-delay jumping systems with nonlinear disturbances

This paper deals with the problems of robust stabilization and H∞ control for a class of uncertain stochastic jumping systems with nonlinear disturbances and time delays. The uncertain parameters are assumed to be norm-bounded and mode dependent, and the time delays enter into the state matrix, the stochastic perturbation term, as well as the state feedback. The stochastic robust stabilization problem addressed in this paper is to design a state feedback controller with input delay such that, for all admissible uncertainties and the nonlinear disturbances, the closed-loop system is robustly, stochastically, exponentially stable in the mean square. Moreover, the purpose of the robust H∞ control problem is to guarantee a specified H∞ performance index, while still achieving the mean-square exponential stability requirement for the closed-loop system. By resorting to the Itô's differential formula and the Lyapunov stability theory, sufficient conditions are derived, respectively, for the robust stabilization and the robust H∞ control problems. It is shown that the addressed problems can be solved if a set of linear matrix inequalities (LMIs) are feasible. A numerical example is employed to illustrate the usefulness of the proposed LMI-based design methods. Copyright © 2006 John Wiley & Sons, Ltd.

FAULT DIAGNOSTIC FILTERING USING STOCHASTIC DISTRIBUTIONS IN NONLINEAR GENERALIZED H∞ SETTING

A fault diagnosis problem is considered by using output probability density functions (PDFs) for stochastic time-delayed systems in the continuous time domain. For such systems, a B-spline approximation is used to model the output PDFs and the approximation coefficients (i.e., the weights) are then dynamically linked with the control input in the form of a weighting system. The modelling errors and system uncertainties resulting from both the B-spline expansion and the weighting system are merged into the system disturbances and the established weighting system is also subjected to nonlinearities, uncertainties and time delays. The generalized H∞ optimization is applied to the fault diagnosis problem with the non-zero initial condition and the truncated norms. An LMI-based fault diagnostic filtering (FDF) method is presented such that the fault can be estimated and the disturbances can be attenuated. Simulations are given to demonstrate the efficiency of the proposed approach.