Fault isolation for multivariate nonlinear non-Gaussian systems using generalized entropy optimization principle

This paper is concerned with the fault isolation (FI) problem for multivariate nonlinear non-Gaussian systems by using a novel filtering method. The generalized entropy optimization principle (GEOP) is established for non-Gaussian systems with multiple faults and disturbances, where the statistic information including entropy and mean of the residual variable is maximized in the presence of the target fault as well as all the nuisance faults and disturbances, and is minimized in the absence of the target fault but in the presence of the nuisance faults and disturbances. Different from the existing results where the output is measurable for feedback, the fault isolation filter is designed and driven by the joint output stochastic distributions rather than its deterministic value. The error dynamics is represented by a multivariate nonlinear non-Gaussian system, for which new recursive relationships are proposed to formulate the joint probability density functions (JPDFs) of the residual variable in terms of the JPDFs of the noises and the faults. Finally, a simulation example is given to demonstrate the effectiveness of the proposed multivariate FI algorithms.     

Minimum entropy filtering for multivariate stochastic systems with non-Gaussian noises

In this note, a minimum entropy filtering algorithm is presented for a class of multivariate dynamic stochastic systems, which are represented by a set of time-varying difference equations and are subjected to the multivariate non-Gaussian stochastic inputs. Several new concepts including the hybrid random vector, hybrid probability and hybrid entropy are firstly established to describe the probabilistic property of the estimation errors. New relationships are provided between the probability density functions (PDFs) of the multivariate stochastic input and output for different mapping cases. Recursive algorithms are then proposed to design the real-time sub-optimal filter so that the hybrid entropy of the estimation error can be minimized. Finally, an improved algorithm is provided through the on-line tuning of the weighting matrices so as to guarantee the local stability of the error system.     

Fault diagnosis and fault-tolerant control for non-Gaussian nonlinear stochastic systems via entropy optimisation

ABSTRACT

In this paper, the fault diagnosis (FD) and fault tolerant control (FTC) problems are studied for non-linear stochastic systems with non-Gaussian disturbance and fault. Unlike classical FD algorithms, the minimum entropy FD is adopted to minimise the residual entropy and control the shape of the probability density function (PDF) of the residual signal. The observation error system can be proved to be locally and ultimately bounded in the mean square sense. Since entropy can be used to characteriSe the uncertainty of the tracking error for non-Gaussian stochastic systems, the FTC controller is obtained by minimising the performance function with regard to the entropy of the tracking error in this paper. The PDF of the output tracking error is approximated by the B-spline model. An illustrative example is utilised to demonstrate the effectiveness of the FD and FTC algorithm, and satisfactory results have been obtained.     

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 ...     

Robust fault diagnosis for non-Gaussian stochastic systems based on the rational square-root approximation model

The task of robust fault detection and diagnosis of stochastic distribution control （SDC） systems with uncertainties is to use the measured input and the system output PDFs to still obtain possible faults information of the system. Using the rational square-root B-spline model to represent the dynamics between the output PDF and the input, in this paper, a robust nonlinear adaptive observer-based fault diagnosis algorithm is presented to diagnose the fault in the dynamic part of such systems with model uncertainties. When certain conditions are satisfied, the weight vector of the rational square-root B-spline model proves to be bounded. Conver- gency analysis is performed for the error dynamic system raised from robust fault detection and fault diagnosis phase. Computer simulations are given to demon- strate the effectiveness of the proposed algorithm.     

状态估计中的采样型滤波器

基于采样方法的滤波器在现代非线性滤波领域内得到了广泛应用。其共同特点是利用抽样粒子点模拟系统状态的概率分布,从而不受状态先验分布假设(如高斯假设)的约束,拥有更高的滤波精度和更广的应用范围。论文在抽样意义下重新论述了确定性采样滤波器如高斯和型滤波器和UKF,以及基于随机模拟的粒子滤波器,并对这三种滤波器及其衍生方法在状态估计领域(滤波)的应用进行了精度和计算负荷分析。某Benchmark信号处理算例验证了该类方法的估计精度和扩展能力。给出基于实际系统需求一般性评价和选用原则。     

Detecting and diagnosing faults in dynamic stochastic distributions using a rational B-splines approximation to output PDFs

This paper presents a novel approach to detect and diagnose faults in the dynamic part of a class of stochastic systems . the Such a group of systems are subjected to a set of crisp inputs but the outputs considered are the measurable probability density functions (PDFs) of the system output, rather than the system output alone. A new approximation model is developed for the output probability density functions so that the dynamic part of the system is decoupled from the output probability density functions. A nonlinear adaptive observer is constructed to detect and diagnose the fault in the dynamic part of the system. Conver-gency analysis is performed for the error dynamics raised from the fault detection and diagnosis phase and an applicability study on the detection and diagnosis of the unexpected changes in the 2D grammage distributions in a paper forming process is included.     

Minimum rational entropy fault tolerant control for non-Gaussian singular stochastic distribution control systems using T-S fuzzy modelling

In this paper, a new fault diagnosis (FD) and fault tolerant control (FTC) algorithm for a non-Gaussian nonlinear singular stochastic distribution control (SDC) system is studied. The rational square-root fuzzy logic model is used to approximate the output probability density function of non-Gaussian processes and a Takagi-Sugeno (T-S) fuzzy model is employed to transform the non-Gaussian nonlinear SDC system into a fuzzy SDC system. An adaptive fuzzy fault diagnosis observer is constructed to achieve reconstruction of system state and fault. Based on the estimated fault information, the controller is reconfigured by minimising the performance index with regard to the rational entropy subjected to mean constraint. Minimum rational entropy fault tolerant control is introduced to make the output of the past-fault SDC system still have the minimum uncertainty. Simulation results are provided to demonstrate the validity of the FD and minimum rational entropy FTC algorithm.     

Fuzzy Set-Membership Filtering for Discrete-Time Nonlinear Systems

In this article, the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering. First, an improved T-S fuzzy model is introduced to achieve highly accurate approximation via an affine model under each fuzzy rule. Then, compared to traditional prediction-based ones, two types of fuzzy set-membership filters are proposed to effectively improve filtering performance, where the structure of both filters consists of two parts: prediction and filtering. Under the locally Lipschitz continuous condition of membership functions, unknown membership values in the estimation error system can be treated as multiplicative noises with respect to the estimation error. Real-time recursive algorithms are given to find the minimal ellipsoid containing the true state. Finally, the proposed optimization approaches are validated via numerical simulations of a one-dimensional and a three-dimensional discrete-time nonlinear systems.      

Distribution function tracking filter design using hybrid characteristic functions

A new tracking filtering algorithm for a class of multivariate dynamic stochastic systems is presented. The system is expressed by a set of time-varying discrete systems with non-Gaussian stochastic input and nonlinear output. A new concept, such as hybrid characteristic function, is introduced to describe the stochastic nature of the dynamic conditional estimation errors, where the key idea is to ensure the distribution of the conditional estimation error to follow a target distribution. For this purpose, the relationships between the hybrid characteristic functions of the multivariate stochastic input and the outputs, and the properties of the hybrid characteristic function, are established. A new performance index of the tracking filter is then constructed based on the form of the hybrid characteristic function of the conditional estimation error. An analytical solution, which guarantees the filter gain matrix to be an optimal one, is then obtained. A simulation case study is included to show the effectiveness of the proposed algorithm, and encouraging results have been obtained.     

Cubature Kalman Filter Under Minimum Error Entropy With Fiducial Points for INS/GPS Integration

Traditional cubature Kalman filter(CKF)is a preferable tool for the inertial navigation system(INS)/global positioning system(GPS)integration under Gaussian noises.The CKF,however,may provide a significantly biased estimate when the INS/GPS system suffers from complex non-Gaussian disturbances.To address this issue,a robust nonlinear Kalman filter referred to as cubature Kalman filter under minimum error entropy with fiducial points(MEEF-CKF)is proposed.The MEEF-CKF behaves a strong robustness against complex nonGaussian noises by operating several major steps,i.e.,regression model construction,robust state estimation and free parameters optimization.More concretely,a regression model is constructed with the consideration of residual error caused by linearizing a nonlinear function at the first step.The MEEF-CKF is then developed by solving an optimization problem based on minimum error entropy with fiducial points(MEEF)under the framework of the regression model.In the MEEF-CKF,a novel optimization approach is provided for the purpose of determining free parameters adaptively.In addition,the computational complexity and convergence analyses of the MEEF-CKF are conducted for demonstrating the calculational burden and convergence characteristic.The enhanced robustness of the MEEF-CKF is demonstrated by Monte Carlo simulations on the application of a target tracking with INS/GPS integration under complex nonGaussian noises.     

Optimal stochastic fault detection filter

A fault detection and identification algorithm, called optimal stochastic fault detection filter, is determined. The objective of the filter is to detect a single fault, called the target fault, and block other faults, called the nuisance faults, in the presence of the process and sensor noises. The filter is derived by maximizing the transmission from the target fault to the projected output error while minimizing the transmission from the nuisance faults. Therefore, the residual is affected primarily by the target fault and minimally by the nuisance faults. The transmission from the process and sensor noises is also minimized so that the filter is robust with respect to these disturbances. It is shown that the filter recovers the geometric structure of the unknown input observer in the limit where the weighting on the nuisance fault transmission goes to infinity. Further, the asymptotic behavior of the filter near the limit is determined by using a perturbation method. Filter designs can be obtained for both time-invariant and time-varying systems.     

Adaptive control of linear systems with Markov perturbations

The stochastic model considered is a linear jump diffusion process X for which the coefficients and the jump processes depend on a Markov chain Z with finite state space. First, we study the optimal filtering and control problem for these systems with non-Gaussian initial conditions, given noisy observations of the state X and perfect measurements of Z. We derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter. We study a quadratic control problem and show that the separation principle holds. Next, we investigate an adaptive control problem for a state process X defined by a linear diffusion for which the coefficients depend on a Markov chain, the processes X and Z being observed in independent white noises. Suboptimal estimates for the process X, Z and approximate control law are investigated for a large class of probability distributions of the initial state. Asymptotic properties of these filters and this control law are obtained. Upper bounds for the corresponding error are given     

一种非线性非高斯随机系统的故障诊断方法

针对非线性非高斯随机系统在线故障诊断的问题,运用粒子滤波器提出了一种基于方差自适应粒子滤波器的非线性非高斯随机系统的故障诊断方法,可以用来解决系统的参数偏差型故障诊断问题。通过对连续搅拌釜式反应器(CSTR)的仿真研究,可以看出,该算法实现简单,易于对系统进行在线估计,对于发生缓变和突变的参数偏差型故障的检测与估计均较为有效。     

Stochastic system identification with noisy input using cumulantstatistics

Addresses the problem of estimating the parameters of stochastic linear systems when the measurements of the system input as well as the system output are noise contaminated. It is assumed that the input is non-Gaussian and the noises are Gaussian. The square root of the magnitude of the fourth cumulant of a generalized error signal is proposed as a performance criterion for parameter estimation. An optimization algorithm is presented. Strong consistency of the proposed parameter estimators is proved under certain sufficient conditions. Both single-input single-output and multiple-input multiple-output cases are investigated. Finally, simulation results are presented to illustrate the proposed approach     

Event-triggered filtering and fault estimation for nonlinear systems with stochastic sensor saturations

This paper is concerned with the filtering problem for a class of nonlinear systems with stochastic sensor saturations and event-triggered measurement transmissions. An event-triggered transmission scheme is proposed with hope to ease the traffic burden and improve the energy efficiency. The measurements are subject to randomly occurring sensor saturations governed by Bernoulli-distributed sequences. Special effort is made to obtain an upper bound of the filtering error covariance in the presence of linearisation errors, stochastic sensor saturations as well as event-triggered transmissions. A filter is designed to minimise the obtained upper bound at each time step by solving two sets of Riccati-like matrix equations, and thus the recursive algorithm is suitable for online computation. Sufficient conditions are established under which the filtering error is exponentially bounded in mean square. The applicability of the presented method is demonstrated by dealing with the fault estimation problem. An illustrative example is exploited to show the effectiveness of the proposed algorithm.     

Constrained discrete-time PI controller design for output PDFs of stochastic systems with time delays

This article presents a new proportional-integral (PI) tracking control strategy for non-Gaussian stochastic systems based on a square root B-spline model for the output probability density functions (PDFs). Following the square root B-spline approximation to the measured output PDF, a non-linear discrete-time dynamical model can be established between the control input and the weights related to the PDFs. It is noted that the PDF tracking is transformed to a constrained dynamical tracking control problem for weight dynamics. For the non-linear discrete-time weight model including time-delay terms and exogenous disturbances, convex linear matrix inequality optimisation algorithms are used to design a generalised PI controller such that stabilisation, state constraint and tracking performance can be guaranteed simultaneously. Furthermore, in order to enhance the robustness, the peak-to-peak measure index is applied to optimise the tracking performance. 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.     

Entropy-robust randomized forecasting under small sets of retrospective data

This paper suggests a new randomized forecasting method based on entropy-robust estimation for the probability density functions (PDFs) of random parameters in dynamic models described by the systems of linear ordinary differential equations. The structure of the PDFs of the parameters and measurement noises with the maximal entropy is studied. We generate the sequence of random vectors with the entropy-optimal PDFs using a modification of the Ulam–von Neumann method. The developed method of randomized forecasting is applied to the world population forecasting problem.