Multiplicative Stochastic Disturbances Research Articles

On the convergence of tracking differentiator with multiple stochastic disturbances

On the convergence of tracking differentiator with multiple stochastic disturbances

Global stabilization of low-order stochastic nonlinear systems with multiple time-varying delays by a continuous feedback control

The global stabilization problem is investigated in the present paper for a class of stochastic nonlinear systems, whose structure is in a p-normal form (0<p<1) and states are perturbed by multiple time-varying delays. By means of an extended stability theory of stochastic nonlinear time-delay systems, where the existence and uniqueness of strong solution of an initial value problem is not required, a nonsmooth but continuous delay-dependent global stabilizer is systematically constructed to achieve the global strong asymptotic stability in probability of the closed-loop system under some conditions. In order to overcome the obstacles caused by low-order nonlinear terms, multiple time-varying delays and stochastic disturbances, a novel nonlinear Lyapunov-Krasovskii functional is skillfully introduced, which plays an important role in the controller design. The effectiveness of the obtained results is verified by an illustrative example in the end.

Parameter Identification and the Finite-Time Combination–Combination Synchronization of Fractional-Order Chaotic Systems with Different Structures under Multiple Stochastic Disturbances

This paper researches the issue of the finite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and the parameters are unknown. The stochastic disturbances are considered parameter uncertainties, nonlinear uncertainties and external disturbances. The bounds of the uncertainties and disturbances are unknown. Firstly, we are going to put forward a new FO sliding surface in terms of fractional calculus. Secondly, some suitable adaptive control laws (ACL) are found to assess the unknown parameters and examine the upper bound of stochastic disturbances. Finally, combining the finite-time Lyapunov stability theory and the sliding mode control (SMC) technique, we propose a fractional-order adaptive combination controller that can achieve the finite-time synchronization of drive-response (D-R) systems. In this paper, some of the synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization, and so forth, have become special cases of combination-combination synchronization. Examples are presented to verify the usefulness and validity of the proposed scheme via MATLAB.

Global pinning synchronization of stochastic delayed complex networks

In this paper, the global pinning synchronization problem of a class of complex dynamical networks with hybrid couplings, time delays, random data packet dropouts and multiple stochastic disturbances is investigated. The hybrid couplings are portrayed in three forms: constant couplings, discrete-delay couplings and distributed-delay couplings. The phenomenon of random packet dropouts is described as a binary random variable which obeys the Bernoulli distribution taking values of 0 and 1 with a certain probability. Multiple noisy processes herein are characterized by Brownian motions, which act on all coupling terms as well as the overall dynamics equation. By applying the Lyapunov method and stochastic analysis technique, sufficient conditions are established to ensure the considered stochastic delayed complex network to globally synchronize to a reference trajectory in the mean-square sense via the developed randomly occurring pinning control strategy. The obtained criteria are within the framework of linear matrix inequations. Finally, a simulation example is presented to verify the feasibility and effectiveness of the derived theoretical results.

Hybrid Kalman Filtering Algorithm With Stochastic Nonlinearities and Multiple Missing Measurements

In this paper, the hybrid Kalman filter is designed for a class of special nonlinear systems where the state equation is nonlinear and the measurement equation is linear. The stochastic nonlinearities, which are described by statistical means, are considered in the system model to reflect the multiplicative stochastic disturbances. The phenomenon of multiple missing measurements is depicted by a set of the Bernoulli distributed random variables with known conditional probabilities and the missing rates of every sensor are different. We need to compute the parameters to reduce the effects of the stochastic nonlinearities and the phenomenon of multiple missing measurements. In addition then, based on the recursive projection formula and the unscented transformation approach, a new hybrid Kalman filtering algorithm is proposed such that, for the stochastic nonlinearities and multiple missing measurements, the filtering error is minimized. By solving the recursive matrix equation, the filter gain matrices and the error covariance matrices can be obtained and the proposed results can be easily verified by using the standard numerical software. We finally provide a numerical example to show the performance of the proposed approach.

Stochastic stability of a modified unscented Kalman filter with stochastic nonlinearities and multiple fading measurements

Stochastic stability of a modified unscented Kalman filter is investigated for a class of nonlinear systems with stochastic nonlinearities and multiple fading measurements. Note that the stochastic nonlinearities are represented by statistical means which indicate multiplicative stochastic disturbances. The fading probability of fading measurements is individual for each sensor. Convergence on the modified unscented Kalman filter is established if there exists a lower bound for the mean of fading probability. Sufficient conditions are obtained to ensure stochastic stability of the modified unscented Kalman filter. The effectiveness of the filtering algorithm is verified by an illustrative numerical example.

Pinning distributed synchronization of stochastic dynamical networks: a mixed optimization approach.

This paper is concerned with the problem of pinning synchronization of nonlinear dynamical networks with multiple stochastic disturbances. Two kinds of pinning schemes are considered: 1) pinned nodes are fixed along the time evolution and 2) pinned nodes are switched from time to time according to a set of Bernoulli stochastic variables. Using Lyapunov function methods and stochastic analysis techniques, several easily verifiable criteria are derived for the problem of pinning distributed synchronization. For the case of fixed pinned nodes, a novel mixed optimization method is developed to select the pinned nodes and find feasible solutions, which is composed of a traditional convex optimization method and a constraint optimization evolutionary algorithm. For the case of switching pinning scheme, upper bounds of the convergence rate and the mean control gain are obtained theoretically. Simulation examples are provided to show the advantages of our proposed optimization method over previous ones and verify the effectiveness of the obtained results.

Gain-Constrained Recursive Filtering With Stochastic Nonlinearities and Probabilistic Sensor Delays

This paper is concerned with the gain-constrained recursive filtering problem for a class of time-varying nonlinear stochastic systems with probabilistic sensor delays and correlated noises. The stochastic nonlinearities are described by statistical means that cover the multiplicative stochastic disturbances as a special case. The phenomenon of probabilistic sensor delays is modeled by introducing a diagonal matrix composed of Bernoulli distributed random variables taking values of 1 or 0, which means that the sensors may experience randomly occurring delays with individual delay characteristics. The process noise is finite-step autocorrelated. The purpose of the addressed gain-constrained filtering problem is to design a filter such that, for all probabilistic sensor delays, stochastic nonlinearities, gain constraint as well as correlated noises, the cost function concerning the filtering error is minimized at each sampling instant, where the filter gain satisfies a certain equality constraint. A new recursive filtering algorithm is developed that ensures both the local optimality and the unbiasedness of the designed filter at each sampling instant which achieving the pre-specified filter gain constraint. A simulation example is provided to illustrate the effectiveness of the proposed filter design approach.

Extended Kalman filtering with stochastic nonlinearities and multiple missing measurements

In this paper, the extended Kalman filtering problem is investigated for a class of nonlinear systems with multiple missing measurements over a finite horizon. Both deterministic and stochastic nonlinearities are included in the system model, where the stochastic nonlinearities are described by statistical means that could reflect the multiplicative stochastic disturbances. The phenomenon of measurement missing occurs in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probability distribution over the interval [0,1]. Such a probability distribution is allowed to be any commonly used distribution over the interval [0,1] with known conditional probability. The aim of the addressed filtering problem is to design a filter such that, in the presence of both the stochastic nonlinearities and multiple missing measurements, there exists an upper bound for the filtering error covariance. Subsequently, such an upper bound is minimized by properly designing the filter gain at each sampling instant. It is shown that the desired filter can be obtained in terms of the solutions to two Riccati-like difference equations that are of a form suitable for recursive computation in online applications. An illustrative example is given to demonstrate the effectiveness of the proposed filter design scheme.

Robust reliable control for discrete‐time‐delay systems with stochastic nonlinearities and multiplicative noises

AbstractThis paper is concerned with the reliable control problem against actuator failures for a class of uncertain discrete‐time stochastic nonlinear time‐delay systems. The failures of actuators are quantified by a variable varying in a given interval. The stochastic nonlinearities described by statistical means cover several well‐studied nonlinear functions as special cases. The time‐varying delay is unknown with given lower and upper bounds. The multiplicative stochastic disturbances are in the form of a scalar Gaussian white noise with unit variance. Attention is focused on the analysis and design of a stable controller such that, for all possible actuator failures, stochastic nonlinearities and disturbances, time delays and admissible parameter uncertainties, the closed‐loop system is exponentially mean‐square stable. A linear matrix inequality approach is developed to solve the addressed problem. A numerical example is given to demonstrate the effectiveness of the proposed design approach. Copyright © 2010 John Wiley &amp; Sons, Ltd.

ROBUST SYNCHRONIZATION OF A CLASS OF COUPLED DELAYED NETWORKS WITH MULTIPLE STOCHASTIC DISTURBANCES: THE CONTINUOUS-TIME CASE

In this paper, the robust synchronization problem is investigated for a new class of continuous-time complex networks that involve parameter uncertainties, time-varying delays, constant and delayed couplings, as well as multiple stochastic disturbances. The norm-bounded uncertainties exist in all the network parameters after decoupling, and the stochastic disturbances are assumed to be Brownian motions that act on the constant coupling term, the delayed coupling term as well as the overall network dynamics. Such multiple stochastic disturbances could reflect more realistic dynamical behaviors of the coupled complex network presented within a noisy environment. By using a combination of the Lyapunov functional method, the robust analysis tool, the stochastic analysis techniques and the properties of Kronecker product, we derive several delay-dependent sufficient conditions that ensure the coupled complex network to be globally robustly synchronized in the mean square for all admissible parameter uncertainties. The criteria obtained in this paper are in the form of linear matrix inequalities whose solution can be easily calculated by using the standard numerical software. The main results are shown to be general enough to cover many existing ones reported in the literature. Simulation examples are presented to demonstrate the feasibility and applicability of the proposed results.

Global Synchronization for Discrete-Time Stochastic Complex Networks With Randomly Occurred Nonlinearities and Mixed Time Delays

In this paper, the problem of stochastic synchronization analysis is investigated for a new array of coupled discrete-time stochastic complex networks with randomly occurred nonlinearities (RONs) and time delays. The discrete-time complex networks under consideration are subject to: 1) stochastic nonlinearities that occur according to the Bernoulli distributed white noise sequences; 2) stochastic disturbances that enter the coupling term, the delayed coupling term as well as the overall network; and 3) time delays that include both the discrete and distributed ones. Note that the newly introduced RONs and the multiple stochastic disturbances can better reflect the dynamical behaviors of coupled complex networks whose information transmission process is affected by a noisy environment (e.g., internet-based control systems). By constructing a novel Lyapunov-like matrix functional, the idea of delay fractioning is applied to deal with the addressed synchronization analysis problem. By employing a combination of the linear matrix inequality (LMI) techniques, the free-weighting matrix method and stochastic analysis theories, several delay-dependent sufficient conditions are obtained which ensure the asymptotic synchronization in the mean square sense for the discrete-time stochastic complex networks with time delays. The criteria derived are characterized in terms of LMIs whose solution can be solved by utilizing the standard numerical software. A simulation example is presented to show the effectiveness and applicability of the proposed results.

Global synchronization for delayed complex networks with randomly occurring nonlinearities and multiple stochastic disturbances

This paper is concerned with the synchronization problem for a new class of continuous time delayed complex networks with stochastic nonlinearities (randomly occurring nonlinearities), interval time-varying delays, unbounded distributed delays as well as multiple stochastic disturbances. The stochastic nonlinearities and multiple stochastic disturbances are investigated here in order to reflect more realistic dynamical behaviors of the complex networks that are affected by the noisy environment. By utilizing a new matrix functional with the idea of partitioning the lower bound ℏ1 of the time-varying delay, we employ the stochastic analysis techniques and the properties of the Kronecker product to establish delay-dependent synchronization criteria that ensure the globally asymptotically mean-square synchronization of the addressed stochastic delayed complex networks. The sufficient conditions obtained are in the form of linear matrix inequalities (LMIs) whose solutions can be readily solved by using the standard numerical software. A numerical example is exploited to show the applicability of the proposed results.

ON SYNCHRONIZATION OF DISCRETE-TIME MARKOVIAN JUMPING STOCHASTIC COMPLEX NETWORKS WITH MODE-DEPENDENT MIXED TIME-DELAYS

In this paper, the synchronization problem is investigated for a new class of discrete-time complex networks. Such complex networks involve the Markovian jumping parameters, mode-dependent discrete and distributed time-delays, constant and delayed couplings, as well as multiple stochastic disturbances. The stochastic disturbances influence the constant coupling term, the delayed coupling term, as well as the overall network dynamics, which could better describe the dynamical behavior of a coupled complex network presented within a noisy environment. With help from the Lyapunov functional method and the properties of Kronecker product, we employ the stochastic analysis techniques to derive several delay-dependent sufficient conditions under which the coupled complex network is asymptotically synchronized in the mean square. The criteria obtained in this paper are in the form of LMIs whose solution can be easily calculated using the standard numerical software. It is shown that our main results can cover many existing ones reported in the literature. A numerical example is presented to illustrate the usefulness of our results.

Robust filtering with stochastic nonlinearities and multiple missing measurements

This paper is concerned with the filtering problem for a class of discrete-time uncertain stochastic nonlinear time-delay systems with both the probabilistic missing measurements and external stochastic disturbances. The measurement missing phenomenon is assumed to occur in a random way, and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over the interval [ 0 1 ] . Such a probabilistic distribution could be any commonly used discrete distribution over the interval [ 0 1 ] . The multiplicative stochastic disturbances are in the form of a scalar Gaussian white noise with unit variance. The purpose of the addressed filtering problem is to design a filter such that, for the admissible random measurement missing, stochastic disturbances, norm-bounded uncertainties as well as stochastic nonlinearities, the error dynamics of the filtering process is exponentially mean-square stable. By using the linear matrix inequality (LMI) method, sufficient conditions are established that ensure the exponential mean-square stability of the filtering error, and then the filter parameters are characterized by the solution to a set of LMIs. Illustrative examples are exploited to show the effectiveness of the proposed design procedures.

Structural properties of gradient recurrent high-order neural networks

The structural properties of Recurrent High-Order Neural Networks (RHONN) whose weights are restricted to satisfy the symmetry property, are investigated. First, it is shown that these networks are gradient and stable dynamical systems and moreover, they remain stable when either bounded deterministic or multiplicative stochastic disturbances concatenate their dynamics. Then, we prove that such networks are capable of approximating arbitrarily close, a large class of dynamical systems of the form /spl chi//spl dot/=F(/spl chi/). Appropriate learning laws, that make these neural networks able to approximate (identify) unknown dynamical systems are also proposed. The learning laws are based on Lyapunov stability theory, and they ensure error stability and robustness. >

Filtering, prediction, and learning properties of ECE neural networks

The capabilities of recurrent high order neural networks (RHONNs), whose synapses are adjusted according to the learning law proposed in Kosmatopoulos and Christodoulou (1992), and Koostmatopoulos, Christodoulou, and Ioannou (1993) are examined in 1) spatiotemporal pattern learning, recognition, and reproduction and 2) stochastic dynamical system identification problems. The mathematical model describing the stochastic disturbances that affect the spatiotemporal patterns or the system dynamics is quite general, and includes both additive and multiplicative stochastic disturbances. Under an extensive mathematical analysis, the authors show that, for any selection of the neural network's high order terms, the prediction error converges to zero exponentially fast. Extensions are also made to the case where the energy coordinate equivalent (ECE) RHONN's are used.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A computationally efficient technique for state estimation of nonlinear systems

An estimation technique is presented for the class of nonlinear systems consisting of memoryless nonlinearities embedded in a dynamic linear system. The approach is based on a useful sampled-data nonlinear system simulation method, which involves the addition of an extra state variable for each nonlinear element. The nonlinear estimator is developed along the lines of the basic Kalman state estimation, using quasilinearization instead of the Taylor series linearization used in extended Kalman filters. It is demonstrated that this new method out performs the extended Kalman filter in terms of the mean-square error of the state estimate. This estimator was used effectively for state estimation in cases where the extended Kalman filter does not converge. Moreover the new method is directly applicable to feedback systems with multiple nonlinearities and stochastic disturbances.