General Nonlinear Stochastic Systems Research Articles

Dynamic Event-Triggered Impulsive Control for Stochastic Nonlinear Systems With Extension in Complex Networks

This study proposes a novel dynamic event-triggered impulsive control (ETIC) scheme to study the exponential stabilization of general stochastic nonlinear systems, where the impulsive sequence is determined by a dynamic event-triggered mechanism. The dynamic ETIC can effectively reduce controller updates and significantly save energy under the same decay rate compared to traditional static event-triggered impulsive generators. Additionally, there is a guaranteed positive minimum inter-event time for each sample path solution of systems. Furthermore, the proposed dynamic ETIC scheme is employed to stabilize stochastic complex networks based on the graph theory and the Lyapunov method. Finally, we provide two illustrative examples to verify the effectiveness and correctness of the proposed dynamic ETIC scheme.

Explicit expression of stationary response probability density for nonlinear stochastic systems

Identifying the exactly or approximately explicit expression of the stationary response probability density for general nonlinear stochastic dynamical systems is of great significance in the fields of stochastic dynamics and control. Almost all the existing methods are devoted to determine the exact or approximate solution for specific values of system and excitation parameters. Herein, aimed at stochastic systems with polynomial nonlinearity and excited by Gaussian white noises, a novel method is proposed to identify the stationary response probability density which explicitly includes system and excitation parameters. The stationary probability density is first written as an exponential function according to the maximum entropy principle, the power of the exponential function is then expressed as a linear combination of prescribed nondimensional parameter clusters constituted by system and excitation parameters, and state variables, with the coefficients to be determined. The undetermined coefficients are derived by minimizing the residual of the associated Fokker-Planck- Kolmogorov equation. The application and efficacy of the proposed method are illustrated by a typical numerical example.

Dynamic event-triggered output feedback control for stochastic nonlinear systems

Abstract In this paper, the event-triggered stabilization problem for stochastic nonlinear systems is addressed. A dynamic event-triggered transmission strategy is proposed for general stochastic nonlinear systems without the linear growth assumption and the proposed transmission strategy can be applied to the output-based dynamic controller. Zeno phenomenon is avoided in the transmission strategy. The resulted closed-loop system is modeled into a stochastic hybrid system. Uniformly globally asymptotic stability in probability is concluded by showing that the equilibrium set is uniformly globally stable in probability and arbitrarily small neighborhood of the equilibrium set is uniformly globally recurrent in probability.

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

This paper mainly discusses the H∞ filtering of general nonlinear discrete time-varying stochastic systems. A nonlinear discrete-time stochastic bounded real lemma is firstly obtained by means of the smoothness of the conditional mathematical expectation, and then, based on the given stochastic bounded real lemma and a stochastic LaSalle-type theorem, a sufficient condition for the existence of the H∞ filtering of general nonlinear discrete time-varying stochastic systems is presented via a newly introduced Hamilton–Jacobi inequality (HJI), which is easily verified. When the worst-case disturbance {vk∗}k∈N is considered, the suboptimal H2∕H∞ filtering is studied. Three examples including a practical engineering example show the effectiveness of our main results.

Event‐triggered H∞ filtering for nonlinear discrete‐time stochastic systems with application to vehicle roll stability control

SummaryIn this article, the problem of event‐triggered H∞ filtering for general discrete‐time nonlinear stochastic systems is investigated. An event‐triggered mechanism is introduced to reduce the communication burden by judging whether the measured data should be transmitted. First, two results about the event‐triggered H∞ filtering for general discrete nonlinear stochastic systems are presented. Second, when the linear discrete‐time stochastic system is considered, the corresponding event‐triggered H∞ filtering can be designed via linear matrix equality approach. Finally, a numerical example and a practical example about the vehicle roll angle estimation are presented to verify the effectiveness of the proposed new design method.

Distributed Maximum Correntropy Filtering for Stochastic Nonlinear Systems Under Deception Attacks.

This article focuses on the distributed maximum correntropy filtering issue for general stochastic nonlinear systems subject to deception attacks. The considered nonlinear functions consist of a determined one and a stochastic one, and the stochastic signals sent by deception attacks with identified statistic characteristics could be non-Gaussian. The corresponding calculation formulas of both the filter gains and the upper bound of the filter error covariance are proposed by means of the Taylor series expansion and the fixed-point iterative update rule, where the weighted maximum correntropy criterion is utilized to take the place of traditional minimum covariance indexes. Such an upper bound is only dependent on the local information, neighbor information, and the identified statistics of deception attacks and, therefore, the developed filtering scheme realizes the requirement of distributed calculation. Furthermore, a simplified version is obtained by removing weights in the correntropy criterion. Finally, an illustrative example is given to verify the effectiveness of developed distributed maximum correntropy filtering subject to deception attacks.

Pth moment exponential stability of general nonlinear discrete-time stochastic systems

pth moment exponential stability of general nonlinear discrete-time stochastic systems

Identifying explicit expression of response probability density of nonlinear stochastic system: Information-theoretic method

For a general nonlinear stochastic system, the response probability density possesses crucial significance for response evaluation and reliability design. The existing literatures derive the response probability density by accurately or approximately solving the Fokker-Planck- Kolmogorov equation, by searching for the equivalent system according to some criterion and approximating the probability density of the original system by that of the equivalent one, or by numerically solving the original equation of motion and giving discrete information statistically. Herein, a novel procedure is established to identify the explicit expression of the stationary response probability density of a general nonlinear system directly from the discrete data of samples. The stationary probability density is expressed as an exponent form with the exponential power being linear combination of base functions elaborately selected. The Shannon information entropy is then arranged as a nonlinear function of the statistical moments of base functions which are evaluated by the discrete data of samples. The undetermined coefficients of base functions are finally determined by minimizing the Shannon information entropy. Three typical examples, i.e., Duffing oscillator, van der Pol system and a frictional system (as representatives of smooth systems, electric systems and non-smooth systems, respectively), are investigated to illustrate the accuracy of this procedure, the robustness to data noise, the insensitivity to sampling interval and strength of nonlinearity, and the low requirement on the amount of data.

General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment

This paper focuses on a general class of systems of nonlinear stochastic differential equations, inspired by stochastic chemostat models. In the first part, the system is formulated as a hybrid switching diffusion. A complete characterization of the asymptotic behavior of the system under consideration is provided. It is shown that the long-term properties of the system can be classified by using a real-valued parameter λ. If λ≤0, the bacteria will die out, which means that the process does not operate. If λ>0, the system has an invariant probability measure and the transition probability of the solution process converges to that of the invariant measure. The rate of convergence is also obtained. One of the distinct features of this paper is that the critical case λ=0 is also considered. Moreover, numerical examples are given to illustrate our results. In the second part of the paper, controlled diffusions with a long-run average objective function are treated. The associated Hamilton–Jacobi–Bellman (HJB) equation is derived and the existence of an optimal Markov control is established. The techniques and methods of analysis in this paper can be applied to many other stochastic Kolmogorov systems.

A Generalized Decomposed State Estimation Approach for Uncertain Constrained Stochastic Nonlinear Systems

In this paper, a decomposed state estimator is developed for stochastic constrained nonlinear discrete-time dynamical systems with uncertain parameters. The proposed estimator can deal with general nonlinear uncertain stochastic systems without any pre-defined specifications on the system structure and/or the measurement model. Moreover, it can handle estimation problems for nonlinear stochastic systems subject to a set of imposed linear and/or nonlinear equality and/or inequality constraints. The mathematical structure of the proposed estimator is developed using multiple projection approach and its performance is analyzed and compared with the global (un-decomposed) structure. The main features of the proposed estimator are: it reduces the processing time, it can handle estimation problems with bad initial conditions of the estimator, it can be implemented on modern parallel processing facilities to further reduce the processing time, and last but not least, it minimizes the rounding off errors and hence improves the numerical stability of the algorithm. Simulations results on different real applications are presented to show the effectiveness of the developed approach, and its improved performance when compared with other techniques reported in the literature.

A New Nonlinear Model-Based Fault Detection Method Using Mann–Whitney Test

A well-established theory of statistical inference enables the generation of statistical fault detection approaches. Previous works mainly focus on parametric tests that assume that probabilistic distributions of both healthy and faulty residuals can be parameterized. However, such assumptions may be quite limited for general nonlinear stochastic systems because those residuals are usually with unknown distribution. In this article, a new fault indicator is defined to replace the traditional residual, and a multistep fault detection method is developed via the standard Mann-Whitney (MW) test. Moreover, with weak assumptions on faults and systems, a one-step fault detection approach is proposed by means of the modified MW test. Finally, the effectiveness of the proposed fault detection method is verified by a simulation of three water tank system.

Stabilization and destabilization of nonlinear stochastic differential delay equations

ABSTRACTThis present paper deals with the problem of stabilization and destabilization of nonlinear stochastic differential delay equations. The key techniques used in this paper are the LaSalle-type stability theorems, the nonnegative semimartingale convergence theorem, and the law of large numbers for martingales. The proposed results can be applied to study the stabilization and destabilization of more general nonlinear stochastic dynamical time-delay systems. Computer simulations are presented to illustrate our theory.

Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects

This paper focuses on the problem of almost sure exponential stability and stochastic stabilization of nonlinear stochastic differential systems with impulsive effects. The moment stability analysis of impulsive stochastic differential systems has received considerable attention. But relatively little is known about the almost sure exponential stability and noise stabilization. In this paper, by using Lyapunov function, we shall not only establish the general criteria on almost sure exponential stability for general nonlinear impulsive stochastic differential systems but also discuss exact method to design a stochastic perturbation to stabilize a given unstable impulsive differential systems. The efficiency of the proposed results is illustrated by a numerical example.

Stochastic convergence analysis of cubature Kalman filter with intermittent observations

The stochastic convergence of the cubature Kalman filter with intermittent observations (CKFI) for general nonlinear stochastic systems is investigated. The Bernoulli distributed random variable is employed to describe the phenomenon of intermittent observations. According to the cubature sample principle, the estimation error and the error covariance matrix (ECM) of CKFI are derived by Taylor series expansion, respectively. Afterwards, it is theoretically proved that the ECM will be bounded if the observation arrival probability exceeds a critical minimum observation arrival probability. Meanwhile, under proper assumption corresponding with real engineering situations, the stochastic stability of the estimation error can be guaranteed when the initial estimation error and the stochastic noise terms are sufficiently small. The theoretical conclusions are verified by numerical simulations for two illustrative examples; also by evaluating the tracking performance of the optical-electric target tracking system implemented by CKFI and unscented Kalman filter with intermittent observations (UKFI) separately, it is demonstrated that the proposed CKFI slightly outperforms the UKFI with respect to tracking accuracy as well as real time performance.

NN-based output tracking for more general stochastic nonlinear systems with unknown control coefficients

This paper considers the output tracking problem for more general classes of stochastic nonlinear systems with unknown control coefficients and driven by noise of unknown covariance. By utilizing the radial basis function neural network approximation method and backstepping technique, we successfully construct a controller to guarantee the solution process to be bounded in probability. The tracking error signal is 4th-moment semi-globally uniformly ultimately bounded (SGUUB) and can be regulated into a small neighborhood of the origin in probability. A simulation example is given to demonstrate the effectiveness of the control scheme.

Adaptive DSC of stochastic non‐linear systems with input unmodelled dynamics

An adaptive dynamic surface control (DSC) method is proposed for a class of stochastic non-linear systems with input and state unmodelled dynamics in this paper. A normalisation signal is introduced to restrict the input unmodelled dynamics. By introducing a non-linear transformation, modified DSC is extended and applied to a class of more general stochastic non-linear systems. By theoretical analysis, it is shown that all the signals in the closed-loop system are bounded in probability. Two simulation examples further demonstrate the effectiveness of the proposed control approach.

Controller design for nonlinear and non-Gaussian multivariable systems based on survival information potential criterion

In this paper, a new model-free control strategy for general nonlinear and non-Gaussian multivariable stochastic systems has been proposed. The strategy applies minimum survival information potential control (MSIPC) scheme to decrease the closed-loop randomness of the output in the information theory framework. Compared with traditional entropy measures, survival information potential (SIP) has many advantages, such as validity in a wide range of distributions, robustness, the simplicity in computation, and so on. In order to calculate the SIP, the empirical SIP formulation under scalar data case is derived directly based on ordered error sample data. By minimizing the performance index mainly consists of SIP, a new model-free control algorithm is obtained for the considered multivariable nonlinear and non-Gaussian stochastic systems. The analysis on the proposed MSIPC convergence is made and a numerical example is provided to show the effectiveness of the obtained control algorithm, where encouraging results have been obtained.

Adaptive neural control for a class of stochastic nonlinear systems by backstepping approach

This paper addresses adaptive neural control for a class of stochastic nonlinear systems which are not in strict-feedback form. Based on the structural characteristics of radial basis function (RBF) neural networks (NNs), a backstepping design approach is extended from stochastic strict-feedback systems to a class of more general stochastic nonlinear systems. In the control design procedure, RBF NNs are used to approximate unknown nonlinear functions and the backstepping technique is utilized to construct the desired controller. The proposed adaptive neural controller guarantees that all the closed-loop signals are bounded and the tracking error converges to a sufficiently small neighborhood of the origin. Two simulation examples are used to illustrate the effectiveness of the proposed approach.

Robust H 2/H ∞ global linearization filter design for nonlinear stochastic time-varying delay systems

One can design a robust H ∞ filter for a general nonlinear stochastic system with external disturbance by solving a second-order nonlinear stochastic partial Hamilton-Jacobi inequality (HJI), which is difficult to be solved. In this paper, the robust mixed H 2/H ∞ globally linearized filter design problem is investigated for a general nonlinear stochastic time-varying delay system with external disturbance, where the state is governed by a stochastic Ito-type equation. Based on a globally linearized model, a stochastic bounded real lemma is established by the Lyapunov–Krasovskii functional theory, and the robust H ∞ globally linearized filter is designed by solving the simultaneous linear matrix inequalities instead of solving an HJI. For a given attenuation level, the H 2 globally linearized filtering problem with the worst case disturbance in the H ∞ filter case is known as the mixed H 2/H ∞ globally linearized filtering problem, which can be formulated as a linear programming problem with simultaneous LMI constraints. Therefore, this method is applicable for state estimation in nonlinear stochastic time-varying delay systems with unknown exogenous disturbance when state variables are unavailable. A simulation example is provided to illustrate the effectiveness of the proposed method.

Event-triggered consensus control for discrete-time stochastic multi-agent systems: The input-to-state stability in probability

This paper is concerned with the event-triggered consensus control problem for a class of discrete-time stochastic multi-agent systems with state-dependent noises. A novel definition of consensus in probability is proposed to better describe the dynamics of the consensus process of the addressed stochastic multi-agent systems. The measurement output available for the controller is not only from the individual agent but also from its neighboring ones according to the given topology. An event-triggered mechanism is adopted with hope to reduce the communication burden, where the control input on each agent is updated only when a certain triggering condition is violated. The purpose of the problem under consideration is to design both the output feedback controller and the threshold of the triggering condition such that the closed-loop system achieves the desired consensus in probability. First of all, a theoretical framework is established for analyzing the so-called input-to-state stability in probability (ISSiP) for general discrete-time nonlinear stochastic systems. Within such a theoretical framework, some sufficient conditions on event-triggered control protocol are derived under which the consensus in probability is reached. Furthermore, both the controller parameter and the triggering threshold are obtained in terms of the solution to certain matrix inequalities involving the topology information and the desired consensus probability. Finally, a simulation example is utilized to illustrate the usefulness of the proposed control protocol.