Output Probability Density Function Research Articles

Fault Detection and Diagnosis for Non-Gaussian Singular Stochastic Distribution Systems via Output PDFs

This paper investigates the problem of fault detection and diagnosis (FDD) problem for non-gausian singulat stochastic distribution control (SDC) systems via the output probability density functions(PDFs). The PDFs can be approximated by using square-root B-spline expansion,via this expansions to represent the dynamics weighting systems between the system input and the weights related to the output PDFs. In this works ,an optimal fault detection and diagnosis algorith is present by introducing the parameter-updating .when the fault occurs, an adaptive network parameter-updating law is designed to approximated the fault. Finally, the simulation result are given to show that the approach can detect faults and estimate the size of fault.

Comparison of Surrogate-Based Uncertainty Quantification Methods for Computationally Expensive Simulators

Polynomial chaos and Gaussian process emulation are methods for surrogate-based uncertainty quantification and have been developed independently in their respective communities over the last 25 years. Despite tackling similar problems in the field, to our knowledge there has yet to be a critical comparison of the two approaches in the literature. We begin by providing a detailed description of polynomial chaos and Gaussian process approaches for building a surrogate model of a black-box function. The accuracy of each surrogate method is then tested and compared for two simulators used in industry: a land-surface model (adJULES) and a launch vehicle controller (VEGACONTROL). We analyze surrogates built on experimental designs of various size and type to investigate their performance in a range of modeling scenarios. Specifically, polynomial chaos and Gaussian process surrogates are built on Sobol sequence and tensor grid designs. Their accuracy is measured by their ability to estimate the mean, standard deviation, exceedance probabilities, and probability density function of the simulator output, as well as a root mean square error metric, based on an independent validation design. We find that one method does not unanimously outperform the other, but advantages can be gained in some cases, such that the preferred method depends on the modeling goals of the practitioner. Our conclusions are likely to depend somewhat on the modeling choices for the surrogates as well as the design strategy. We hope that this work will spark future comparisons of the two methods in their more advanced formulations and for different sampling strategies.

Robust fault diagnosis and fault‐tolerant control for non‐Gaussian uncertain stochastic distribution control systems

SummaryThe purpose of fault diagnosis of stochastic distribution control systems is to use the measured input and the system output probability density function to obtain the fault estimation information. A fault diagnosis and sliding mode fault‐tolerant control algorithms are proposed for non‐Gaussian uncertain stochastic distribution control systems with probability density function approximation error. The unknown input caused by model uncertainty can be considered as an exogenous disturbance, and the augmented observation error dynamic system is constructed using the thought of unknown input observer. Stability analysis is performed for the observation error dynamic system, and the H∞ performance is guaranteed. Based on the information of fault estimation and the desired output probability density function, the sliding mode fault‐tolerant controller is designed to make the post‐fault output probability density function still track the desired distribution. This method avoids the difficulties of design of fault diagnosis observer caused by the uncertain input, and fault diagnosis and fault‐tolerant control are integrated. Two different illustrated examples are given to demonstrate the effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

A surrogate method for density-based global sensitivity analysis

This paper describes an accurate and computationally efficient surrogate method, known as the polynomial dimensional decomposition (PDD) method, for estimating a general class of density-based f-sensitivity indices. Unlike the variance-based Sobol index, the f-sensitivity index is applicable to random input following dependent as well as independent probability distributions. The proposed method involves PDD approximation of a high-dimensional stochastic response of interest, forming a surrogate input–output data set; kernel density estimations of output probability density functions from the surrogate data set; and subsequent Monte Carlo integration for estimating the f-sensitivity index. Developed for an arbitrary convex function f and an arbitrary probability distribution of input variables, the method is capable of calculating a wide variety of sensitivity or importance measures, including the mutual information, squared-loss mutual information, and L1-distance-based importance measure. Three numerical examples illustrate the accuracy, efficiency, and convergence properties of the proposed method in computing sensitivity indices derived from three prominent divergence or distance measures. A finite-element-based global sensitivity analysis of a leverarm was performed, demonstrating the ability of the method in solving industrial-scale engineering problems.

Fault Detection Based On Online Probability Density Function Estimation

AbstractThis study presents a new fault detection scheme based on the probability density function (PDF) of system output. Unlike the classical fault detection and diagnosis methods, in the proposed method, distribution of the system output is estimated online. To achieve this goal, an algorithm is introduced to estimate PDF online using fuzzy logic. Furthermore, convergence of this algorithm is investigated. Then, a residual is constructed that can show the existence of a fault in the system. The main advantages of the proposed method are robustness against measurement noise, even though it does not need the exact model and measured data of inputs and states. Simulation results show that this scheme can detect abrupt faults very well.

DOB Fuzzy Controller Design for Non-Gaussian Stochastic Distribution Systems Using Two-Step Fuzzy Identification

This paper presents a novel non-Gaussian stochastic control framework for the problem of disturbance estimation and rejection by combining fuzzy identification technology with disturbance observer design. First, fuzzy logic models are used to approximate the output probability density functions (PDFs) of non-Gaussian processes such that the task of PDF shape control can be reduced to a fuzzy weight dynamics modeling and control problem. Next, Takagi–Sugeno fuzzy models with multiple disturbances are employed to describe the nonlinear relations between fuzzy weight dynamics and the control input, in which a novel disturbance-observer-based PI-type fuzzy feedback controller is designed to ensure the system stability and convergence of the tracking error to zero. Meanwhile, the disturbance estimation and attenuation performance as well as the state constrained requirement can also be guaranteed. Moreover, the novel composite observer is constructed by augmenting the disturbance estimation into the full-state estimation. The satisfactory tracking performance and full-state observation effect can be achieved by the designed optimization algorithm. Finally, simulations for paper-making process are given to show the efficiency of the proposed approach.

Fault diagnosis and fault tolerant control for non-Gaussian time-delayed singular stochastic distribution systems

Stochastic distribution control (SDC) is a new branch of stochastic system control that the system output is the probability density function (PDF) of the output. In practice, some algebraic relations exist between the input and the weights of SDC systems, leading to a singular state space model between the weights and the control input which increases the complexity of the system. The ignorance of time delay in practical systems will make the effectiveness of the fault diagnosis (FD) and fault tolerant control (FTC) be reduced. In this paper, the linear B-spline basis functions are used to approximate the output PDF. A FD approach based on the adaptive observer is established to diagnose the size of fault in the singular time-delayed SDC system. With the fault diagnosis information, a fault tolerant controller based on PI tracking control scheme is constructed to make the post-fault PDF still track the given distribution. The post-fault closed-loop stability analysis with the practical fault tolerant controller is carried out based on the Lyapunov stability theorem. Finally, a numerical simulation is provided to demonstrate the effectiveness of the proposed approach.

New Lagrange Multipliers for the Blind Adaptive Deconvolution Problem Applicable for the Noisy Case

Recently, a new blind adaptive deconvolution algorithm was proposed based on a new closed-form approximated expression for the conditional expectation (the expectation of the source input given the equalized or deconvolutional output) where the output and input probability density function (pdf) of the deconvolutional process were approximated with the maximum entropy density approximation technique. The Lagrange multipliers for the output pdf were set to those used for the input pdf. Although this new blind adaptive deconvolution method has been shown to have improved equalization performance compared to the maximum entropy blind adaptive deconvolution algorithm recently proposed by the same author, it is not applicable for the very noisy case. In this paper, we derive new Lagrange multipliers for the output and input pdfs, where the Lagrange multipliers related to the output pdf are a function of the channel noise power. Simulation results indicate that the newly obtained blind adaptive deconvolution algorithm using these new Lagrange multipliers is robust to signal-to-noise ratios (SNR), unlike the previously proposed method, and is applicable for the whole range of SNR down to 7 dB. In addition, we also obtain new closed-form approximated expressions for the conditional expectation and mean square error (MSE).

A New Approach to Analysis and Design of Chaos-Based Random Number Generators Using Algorithmic Converter

This paper presents a new approach to analysis and design of ADC-based random number generators. To this end, different full-bit and half-bit redundant stages of algorithmic converter are used to design chaotic maps. It is shown that, in the redundant and nonredundant structures, output probability density function of the converter stages and their related chaotic functions always converge to uniformity. It is demonstrated that residues become independent and uniformly distributed. This fact leads to the randomness and uniformity of distribution of the random number generator output bits. Moreover, it is shown that some common chaotic maps that are employed in chaotic random number generators can be implemented using nonredundant and half-bit redundant stages of algorithmic converter. In this way, the capability of ADC-based generators in designing chaotic maps and producing random number sequences is illustrated. The validity of the proposed chaos-based random number generator is confirmed using NIST statistical tests even in the presence of nonidealities in algorithmic converter. Since the ADCs are mixed-signal integrated circuits and can be used in high-speed applications, the ADC-based random number generator has high throughput and is easily embeddable in all analog and digital circuits.

Fault Diagnosis and Fault Tolerant Control for Non-Gaussian Singular Time-Delayed Stochastic Distribution Systems with Disturbance Based on the Rational Square-Root Model

For the non-Gaussian singular time-delayed stochastic distribution control (SDC) system with unknown external disturbance where the output probability density function (PDF) is approximated by the rational square-root B-spline basis function, a robust fault diagnosis and fault tolerant control algorithm is presented. A full-order observer is constructed to estimate the exogenous disturbance and an adaptive observer is used to estimate the fault size. A fault tolerant tracking controller is designed using the feedback of distribution tracking error, fault, and disturbance estimation to let the postfault output PDF still track desired distribution. Finally, a simulation example is included to illustrate the effectiveness of the proposed algorithms and encouraging results have been obtained.

Adaptive Fault-Tolerant Stochastic Shape Control With Application to Particle Distribution Control

This paper investigates the fault-tolerant shape control (FTSC) problem for stochastic distribution systems. For this problem, in addition to measurable input signals, it is assumed that the distribution function of the system output can be evaluated so that it is available. It is also assumed that the system is subject to actuator faults. In this case, the main control objective is for the output of the stochastic distribution system to track a given target distribution even in the presence of actuator faults. By estimating these actuator faults, an effective FTSC strategy is proposed, which consists of a normal control law and an adaptive compensation control law. The former can track the given output distribution with an optimized performance index in the fault-free case, while the latter can automatically reduce (or even eliminate) the adverse effects caused by the actuator faults. The proposed method can be applied to tracking control of output probability density functions. To demonstrate the effectiveness of the proposed scheme, simulation is performed on one numerical example with satisfactory results obtained. In addition, a practical example of soil particle gradation control in geotechnical applications is given in this paper, and the results show that the proposed fault-tolerant scheme is applicable to practical particle size distribution control.

Fault diagnosis and fault tolerant tracking control for the non-Gaussian singular time-delayed stochastic distribution system with PDF approximation error

In this paper, fault diagnosis and fault tolerant tracking control algorithms are proposed for the non-Gaussian singular time-delayed stochastic distribution control system. The square root B-spline model is used to approximate the output probability density function (PDF), and the PDF approximation error is taken into consideration. A fault diagnosis approach based on the adaptive observer is constructed to diagnose the size of fault in the singular stochastic distribution control (SDC) system. When fault occurs, in order to track the expected PDF, a distribution tracking error dynamic system is established. The purpose of fault tolerant tracking control is transformed into making the distribution tracking error at each time instant satisfy a certain upper bound beyond a limited time. Then the fault tolerant controller is designed using the fault diagnosis information and other measurable information. A simulation example is included to illustrate the effectiveness of the proposed algorithms and encouraging results have been obtained.

Invariant methods for an ensemble-based sensitivity analysis of a passive containment cooling system of an AP1000 nuclear power plant

Sensitivity Analysis (SA) is performed to gain fundamental insights on a system behavior that is usually reproduced by a model and to identify the most relevant input variables whose variations affect the system model functional response. For the reliability analysis of passive safety systems of Nuclear Power Plants (NPPs), models are Best Estimate (BE) Thermal Hydraulic (TH) codes, that predict the system functional response in normal and accidental conditions and, in this paper, an ensemble of three alternative invariant SA methods is innovatively set up for a SA on the TH code input variables. The ensemble aggregates the input variables raking orders provided by Pearson correlation ratio, Delta method and Beta method. The capability of the ensemble is shown on a BE–TH code of the Passive Containment Cooling System (PCCS) of an Advanced Pressurized water reactor AP1000, during a Loss Of Coolant Accident (LOCA), whose output probability density function (pdf) is approximated by a Finite Mixture Model (FMM), on the basis of a limited number of simulations.

Short-term wind speed and power forecasting using an ensemble of mixture density neural networks

An ensemble of mixture density neural networks is used for short-term wind speed and power forecasting. Predicted wind speeds obtained from a numerical weather prediction model are used as the input data for the mixture density network, whose outputs are the mixture density parameters (used to represent the probability density function of the uncertain output or target variable). All mixture density neural networks in an ensemble are assumed to have a three-layer architecture, with each architecture having different numbers of nodes in the hidden layer. Because a mixture of Gaussian distributions is used to approximate the conditional distribution of the target random variable (either wind speed or wind turbine power), the uncertainties arising from both the model structure and model output can be completely quantified. In consequence, rigorous confidence intervals reflecting these sources of uncertainty in the prediction can be obtained and used to assess the performance for the wind speed and wind turbine power forecasting. An application of the proposed approach to a data set of the measured wind speed and power from an operational wind turbine in a wind farm in Taiwan is used to test the methodology. The results of this application demonstrate that the proposed methodology works well for the multi-step ahead wind speed and power forecasting.

A New Measure of Uncertainty and the Control Loop Performance Assessment for Output Stochastic Distribution Systems

Minimization of output uncertainty is an important control target for stochastic systems subject to bounded random inputs. Firstly, causes that prevent realization of the minimum Shannon entropy (SE) control are examined based on the analysis of the SE definition in the continuous random variable (CRV) and on this basis, a new measure of uncertainty, which is called rational entropy (RE), is proposed, and the key properties of the RE are proved. Next, results are extended to the output stochastic distribution control (SDC) systems whose output probability density functions (PDFs) are approximated by a linear B-spline basis functions model. Then, two types of minimum RE controller with the mean constraint are given and several controller performance assessment (CPA) benchmarks for output SDC systems are presented. Finally, simulations are included to discuss the feasibility and effectiveness of the proposed measure of uncertainty and performance assessment methods.

Moment‐independent regional sensitivity analysis of complicated models with great efficiency

SummaryMoment‐independent regional sensitivity analysis (RSA) is a very useful guide tool for assessing the effect of a specific range of an individual input on the uncertainty of model output, while large computational burden is involved to perform RSA, which would certainty lead to the limitation of engineering application. Main tasks for performing RSA are to estimate the probability density function (PDF) of model output and the joint PDF of model output and the input variable by some certain smart techniques. Firstly, a method based on the concepts of maximum entropy, fractional moment and sparse grid integration is utilized to estimate the PDF of the model output. Secondly, Nataf transformation is applied to obtain the joint PDF of model output and the input variable. Finally, according to an integral transformation, those regional sensitivity indices can be easily computed by a Monte Carlo procedure without extra function evaluations. Because all the PDFs can be estimated with great efficiency, and only a small amount of function evaluations are involved in the whole process, the proposed method can greatly decrease the computational burden. Several examples with explicit or implicit input–output relations are introduced to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Performance analysis of SSC diversity reception over η–μ fading channel in the presence of CCI

The aim of this paper is to study the performance of a switch-and-stay combining (SSC) receiver, operating in the presence of η–μ fading and co-channel interference (CCI). The analysis will incorporate both cases when the receiver branches are independent and when a correlation arises between them. Infinite-series expressions will be presented for the probability density function (PDF) and cumulative distribution function (CDF) of the SSC output signal-to-interference ratio (SIR). Further, capitalising on them, standard wireless performance measures, outage probability (OP) and average bit error probability (ABER) will be determined and discussed in the function of various system parameters, including fading severity, observed threshold and input SIR unbalance.

Fault Diagnosis and Tolerant Control for Discrete Stochastic Distribution Collaborative Control Systems

This paper presents a novel fault-tolerant control method for a class of discrete-time and nonGaussian stochastic systems, where two subsystems are connected in series so as to operate in a collaborative way. For such systems, the output probability density function of the second subsystem is taken as the output of the whole system. The proposed method includes the design of a fault diagnosis (FD) algorithm for the first subsystem and the establishment of a fault-tolerant control algorithm for the second subsystem. At first, linear matrix inequality techniques are used to construct the FD algorithm for the first subsystem. Once the fault is diagnosed, a fault-tolerant control algorithm is designed using the well-known optimal norm-based iterative learning control approach. Different from the existing fault tolerant controller methods, the proposed fault-tolerant control is designed not for the faulty subsystem but for the healthy subsystem. As a result, when a fault occurs in the first subsystem, the reconfigured controller for the healthy second subsystem can accommodate the fault and guarantee that the whole system will still exhibit good operational performance. A simulated example is used to demonstrate the collaborative fault-tolerant control effect and desired results have been obtained.

Predictive PDF control in shaping of molecular weight distribution based on a new modeling algorithm

The aims of this work are to develop an efficient modeling method for establishing dynamic output probability density function (PDF) models using measurement data and to investigate predictive control strategies for controlling the full shape of output PDF rather than the key moments. Using the rational square-root (RSR) B-spline approximation, a new modeling algorithm is proposed in which the actual weights are used instead of the pseudo weights in the weights dynamic model. This replacement can reduce computational load effectively in data-based modeling of a high-dimensional output PDF model. The use of the actual weights in modeling and control has been verified by stability analysis. A predictive PDF model is then constructed, based on which predictive control algorithms are established with the purpose to drive the output PDF towards the desired target PDF over the control process. An analytical solution is obtained for the non-constrained predictive PDF control. For the constrained predictive control, the optimal solution is achieved via solving a constrained nonlinear optimization problem. The integrated method of data-based modeling and predictive PDF control is applied to closed-loop control of molecular weight distribution (MWD) in an exemplar styrene polymerization process, through which the modeling efficiency and the merits of predictive control over standard PDF control are demonstrated and discussed.

A simple and efficient method for global sensitivity analysis based on cumulative distribution functions

Variance-based approaches are widely used for Global Sensitivity Analysis (GSA) of environmental models. However, methods that consider the entire Probability Density Function (PDF) of the model output, rather than its variance only, are preferable in cases where variance is not an adequate proxy of uncertainty, e.g. when the output distribution is highly-skewed or when it is multi-modal. Still, the adoption of density-based methods has been limited so far, possibly because they are relatively more difficult to implement. Here we present a novel GSA method, called PAWN, to efficiently compute density-based sensitivity indices. The key idea is to characterise output distributions by their Cumulative Distribution Functions (CDF), which are easier to derive than PDFs. We discuss and demonstrate the advantages of PAWN through applications to numerical and environmental modelling examples. We expect PAWN to increase the application of density-based approaches and to be a complementary approach to variance-based GSA.