Maximum Correntropy Kalman Filtering for Non-Gaussian Systems With State Saturations and Stochastic Nonlinearities

This paper tackles the maximum correntropy Kalman filtering problem for discrete time-varying non-Gaussian systems subject to state saturations and stochastic nonlinearities. The stochastic nonlinearities, which take the form of statemultiplicative noises, are introduced in systems to describe the phenomenon of nonlinear disturbances. To resist non-Gaussian noises, we consider a new performance index called maximum correntropy criterion(MCC) which describes the similarity between two stochastic variables. To enhance the “robustness” of the kernel parameter selection on the resultant filtering performance, the Cauchy kernel function is adopted to calculate the corresponding correntropy. The goal of this paper is to design a Kalman-type filter for the underlying systems via maximizing the correntropy between the system state and its estimate. By taking advantage of an upper bound on the one-step prediction error covariance, a modified MCC-based performance index is constructed. Subsequently, with the assistance of a fixed-point theorem, the filter gain is obtained by maximizing the proposed cost function. In addition, a sufficient condition is deduced to ensure the uniqueness of the fixed point. Finally, the validity of the filtering method is tested by simulating a numerical example.

Event-Triggered Finite-Time H∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H∞ filter gain matrices can be expressed in an explicit form.

A neural network solution of first-passage problems

This paper proposes a novel method for solving the first-passage time probability problem of nonlinear stochastic dynamic systems.The safe domain boundary is exactly imposed into the radial basis function neural network(RBF-NN)architecture such that the solution is an admissible function of the boundary-value problem.In this way,the neural network solution can automatically satisfy the safe domain boundaries and no longer requires adding the corresponding loss terms,thus efficiently handling structure failure problems defined by various safe domain boundaries.The effectiveness of the proposed method is demonstrated through three nonlinear stochastic examples defined by different safe domains,and the results are validated against the extensive Monte Carlo simulations(MCSs).

Stochastic stability of the derivative unscented Kalman filter

This is the second of two consecutive papers focusing on the filtering algorithm for a nonlinear stochastic discretetime system with linear system state equation. The first paper established a derivative unscented Kalman filter（DUKF） to eliminate the redundant computational load of the unscented Kalman filter（UKF） due to the use of unscented transformation（UT） in the prediction process. The present paper studies the error behavior of the DUKF using the boundedness property of stochastic processes. It is proved that the estimation error of the DUKF remains bounded if the system satisfies certain conditions. Furthermore, it is shown that the design of the measurement noise covariance matrix plays an important role in improvement of the algorithm stability. The DUKF can be significantly stabilized by adding small quantities to the measurement noise covariance matrix in the presence of large initial error. Simulation results demonstrate the effectiveness of the proposed technique.

Fault tolerant control based on stochastic distribution via RBF neural networks

A new fault tolerant control（FTC） via a controller reconfiguration approach for general stochastic nonlinear systems is studied.Different from the formulation of classical FTC methods,it is supposed that the measured information for the FTC is the probability density functions（PDFs） of the system output rather than its measured value.A radial basis functions（RBFs） neural network technique is proposed so that the output PDFs can be formulated in terms of the dynamic weighings of the RBFs neural network.As a result,the nonlinear FTC problem subject to dynamic relation between the input and the output PDFs can be transformed into a nonlinear FTC problem subject to dynamic relation between the control input and the weights of the RBFs neural network approximation to the output PDFs.The FTC design consists of two steps.The first step is fault detection and diagnosis（FDD）,which can produce an alarm when there is a fault in the system and also locate which component has a fault.The second step is to adapt the controller to the faulty case so that the system is able to achieve its target.A linear matrix inequality（LMI） based feasible FTC method is applied such that the fault can be detected and diagnosed.An illustrated example is included to demonstrate the efficiency of the proposed algorithm,and satisfactory results have been obtained.

A NEW STOCHASTIC OPTIMAL CONTROL STRATEGY FOR HYSTERETIC MR DAMPERS

A new stochastic optimal control strategy for randomly excited quasi-integrable Hamiltonian systems using magneto-rheological (MR) dampers is proposed. The dynamic be- havior of an MR damper is characterized by the Bouc-Wen hysteretic model. The control force produced by the MR damper is separated into a passive part incorporated in the uncontrolled system and a semi-active part to be determined. The system combining the Bouc-Wen hysteretic force is converted into an equivalent non-hysteretic nonlinear stochastic control system. Then It?o stochastic di?erential equations are derived from the equivalent system by using the stochastic averaging method. A dynamical programming equation for the controlled di?usion processes is established based on the stochastic dynamical programming principle. The non-clipping nonlin- ear optimal control law is obtained for a certain performance index by minimizing the dynamical programming equation. Finally, an example is given to illustrate the application and e?ectiveness of the proposed control strategy.

Fixed-Time Lyapunov Criteria and State-Feedback Controller Design for Stochastic Nonlinear Systems

This paper investigates the fixed-time stability theorem and state-feedback controller design for stochastic nonlinear systems.We propose an improved fixed-time Lyapunov theorem with a more rigorous and reasonable proof procedure.In particular,an important corollary is obtained,which can give a less conservative upper-bound estimate of the settling time.Based on the backstepping technique and the addition of a power integrator method,a state-feedback controller is skillfully designed for a class of stochastic nonlinear systems.It is proved that the proposed controller can render the closed-loop system fixed-time stable in probability with the help of the proposed fixed-time stability criteria.Finally,the effectiveness of the proposed controller is demonstrated by simulation examples and comparisons.

Sampled-data Observer Design for a Class of Stochastic Nonlinear Systems Based on the Approximate Discrete-time Models

In this paper,we studied the approximate sampleddata observer design for a class of stochastic nonlinear systems.Euler-Maruyama approximation was investigated in this paper because it is the basis of other higher precision numerical methods,and it preserves important structures of the nonlinear systems.Also,the form of Euler-Maruyama model is simple and easy to be calculated.The results provide a reference for sampled-data observer design method for such stochastic nonlinear systems,and may be useful to many practical control applications,such as tracking control in mechanical systems.And the effectiveness of the approach is demonstrated by a simulation example.

Extended Riccati Equation Rational Expansion Method and Its Application to Nonlinear Stochastic Evolution Equations

In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons ＆ Fractals 25 （2005） 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions＇ soliton-like solution.s and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.

A Novel PDF Shape Control Approach for Nonlinear Stochastic Systems

In this work,a novel shape control approach of the probability density function(PDF)for nonlinear stochastic systems is presented.First,we provide the formula for the PDF shape controller without devising the control law of the controller.Then,based on the exact analytical solution of the Fokker-PlanckKolmogorov(FPK)equation,the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response.To validate the performance of the proposed control approach,we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments.The results show that the novel PDF shape control approach is effective and feasible.Using an equal number of parameters,our method can achieve a similar or better control effect as the exponential polynomial method.By comparison with the multiGaussian closure method,our method has clear advantages in PDF shape control performance.For all cases,the integral of squared error and the errors of first four moments of our proposed method were very small,indicating superior performance and promising good overall control effects of our method.The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

Controller design for stochastic nonlinear systems with matched conditions

This paper is concerned with the global boundedness problem for a class of stochastic nonlinear systems with matched conditions. The uncertainties in the systems are due to parameter variations and external stochastic disturbance. Only the matched conditions and the possible bound of the uncertainties are demanded. Based on the stochastic Lyapunov stability theory, an explicit controller is constructed in the gradient direction, which renders responses of the closed-loop systems be globally bounded in probability. When the systems degrade to linear systems, the controller becomes linear. Illustrative examples are given to show the effectiveness of the proposed method.

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.

Adaptive NN stabilization for stochastic systems with discrete and distributed time-varying delays

A new adaptive neural network（NN） output-feedback stabilization controller is investigated for a class of uncertain stochastic nonlinear strict-feedback systems with discrete and distributed time-varying delays and unknown nonlinear functions in both drift and diffusion terms.First,an extensional stability notion and the related criterion are introduced.Then,a nonlinear observer to estimate the unmeasurable states is designed,and a systematic backstepping procedure to design an adaptive NN output-feedback controller is proposed such that the closed-loop system is stable in probability.The effectiveness of the proposed control scheme is demonstrated via a numerical example.

Adaptive neural control for a class of uncertain stochastic nonlinear systems with dead-zone

The problem of adaptive stabilization is addressed for a class of uncertain stochastic nonlinear strict-feedback systems with both unknown dead-zone and unknown gain functions.By using the backstepping method and neural network（NN） parameterization,a novel adaptive neural control scheme which contains fewer learning parameters is developed to solve the stabilization problem of such systems.Meanwhile,stability analysis is presented to guarantee that all the error variables are semi-globally uniformly ultimately bounded with desired probability in a compact set.The effectiveness of the proposed design is illustrated by simulation results.

State feedback stabilization for high-order stochastic nonlinear systems with zero dynamics

In this paper, for a class of high-order stochastic nonlinear systems with zero dynamics which are neither necessarily feedback linearizable nor affine in the control input, the problem of state feedback stabilization is investigated for the first time. Under some weaker assumptions, a smooth state feedback controller is designed, which ensures that the closed-loop system has an almost surely unique solution on [0,∞）, the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability, and all the states can be regulated to the origin almost surely. A simulation example demonstrates the control scheme.

Output Feedback for Stochastic Nonlinear Systems with Unmeasurable Inverse Dynamics

This paper considers a concrete stochastic nonlinear system with stochastic unmeasurable inverse dynamics. Motivated by the concept of integral input-to-state stability （iISS） in deterministic systems and stochastic input-to-state stability （SISS） in stochastic systems, a concept of stochastic integral input-to-state stability （SiISS） using Lyapunov functions is first introduced. A constructive strategy is proposed to design a dynamic output feedback control law, which drives the state to the origin almost surely while keeping all other closed-loop signals almost surely bounded. At last, a simulation is given to verify the effectiveness of the control law.

Output-feedback adaptive stochastic nonlinear stabilization using neural networks

For the first time, an adaptive backstepping neural network control approach is extended to a class of stochastic non- linear output-feedback systems. Different from the existing results, the nonlinear terms are assumed to be completely unknown and only a neural network is employed to compensate for all unknown nonlinear functions so that the controller design is more simplified. Based on stochastic LaSalle theorem, the resulted closed-loop system is proved to be globally asymptotically stable in probability. The simulation results further verify the effectiveness of the control scheme.

Optimal and suboptimal white noise smoothers for nonlinear stochastic systems

A new approach of smoothing the white noise for nonlinear stochastic system was proposed. Through presenting the Gaussian approximation about the white noise posterior smoothing probability density fimction, an optimal and unifying white noise smoothing framework was firstly derived on the basis of the existing state smoother. The proposed framework was only formal in the sense that it rarely could be directly used in practice since the model nonlinearity resulted in the intractability and infeasibility of analytically computing the smoothing gain. For this reason, a suboptimal and practical white noise smoother, which is called the unscented white noise smoother （UWNS）, was further developed by applying unscented transformation to numerically approximate the smoothing gain. Simulation results show the superior performance of the proposed UWNS approach as compared to the existing extended white noise smoother （EWNS） based on the first-order linearization.

Bayesian estimation for nonlinear stochastic hybrid systems with state dependent transitions

The Bayesian approach is considered as the most general formulation of the state estimation for dynamic systems. However, most of the existing Bayesian estimators of stochastic hybrid systems only focus on the Markov jump system, few liter- ature is related to the estimation problem of nonlinear stochastic hybrid systems with state dependent transitions. According to this problem, a new methodology which relaxes quite a restrictive as- sumption that the mode transition process must satisfy Markov properties is proposed. In this method, a general approach is presented to model the state dependent transitions, the state and output spaces are discreted into cell space which handles the nonlinearities and computationally intensive problem offline. Then maximum a posterior estimation is obtained by using the Bayesian theory. The efficacy of the estimator is illustrated by a simulated example .

DECOMPOSITION OF NONLINEAR DISCRETE-TIME STOCHASTIC SYSTEMS

This paper gives a mathematical definition for the "caution" and "probing", and presents a decomposition theorem for nonlinear discrete-time stochastic systems. Under some assumptions, the problem of finding the closed-loop optimal control can be decomposed into three problems: the deterministic optimal feedback, cautious optimal and probing optimal control problems.