Output Probability Density Function Research Articles

Propagation of hybrid uncertainty by synthesizing B-spline chaos and augmented change of probability measure

Acquiring engineering data is frequently expensive, resulting in sparse data that may lead to a lack of knowledge for design and analysis. Thus, it is not always feasible to precisely determine the probability density functions (PDFs) of uncertain model parameters. Under such circumstances that involve simultaneous aleatory and epistemic uncertainties, repeated uncertainty propagation (UP) analysis is generally required. In this paper, a novel approach for hybrid UP is proposed by integrating B-spline chaos and augmented change of probability measure (aCOM) for meeting different goals. The B-spline chaos is adopted to represent the complicated computational model as a function of an arbitrary input random variable, while the aCOM is employed to reconstruct the PDF of the model output when the input PDF is changed due to epistemic uncertainty. In the case of small epistemic uncertainty, hybrid UP can be achieved directly by changing the assigned probabilities of existing sample results. While in the case of large epistemic uncertainty, additional samples from an augmenting PDF are generated. The proposed method is compatible with both cases. The numerical algorithm of the proposed method is presented and illustrated by four benchmark problems. Further, the accuracy and efficiency of the proposed method are substantiated by four numerical examples compared with analytical solutions or Monte Carlo simulations. An attempt to enhance the proposed method with the aid of active subspace methods to handle high-dimensional problems is also discussed in this work. The limitations and potential improvements of the proposed approach are outlined as well.

On the Internationalization of English Teaching System in the Planning of Sustainable Development of Future Renewable Energy

Energy is an essential element of human production and business activities. At present, China's energy development and utilization is facing the contradiction between increasing environmental pressure and high emission energy consumption structure. Based on the optimal power flow algorithm, this paper optimizes the scheduling and international teaching of the power system with a high proportion of renewable energy. The probability density function of fan output and photovoltaic output is obtained by wind speed and light intensity model, and the probability density function of load is also proposed. Then, interruptible load, microgrid and energy storage are used as flexible resources to improve the system’s flexibility to meet the random fluctuations caused by a high proportion of wind and light, so as to maintain the stable operation of the power grid. The mathematical model of process optimization power flow satisfying process constraints is a nonlinear time-varying system. The key to process optimization power flow algorithm is to find a simple and effective linearization and discretization method that is in line with the actual power network. On the basis of ensuring the accuracy of the model, the theory, model and algorithm of the process optimization power flow are studied on the basis of the proposed linear time period, the approximate linear time variability of the node voltage, and the goal of greatly simplifying the problem. Finally, through simulation analysis, comparing the results of model 1 and model 2, we can see that from the point of view of the power abandonment cost, the power abandonment cost of model 1 is 2.29 million yuan, which is less than the 8 million yuan of model 2. From the perspective of depth peaking and start-stop times, the total depth peaking of model 1 is 331 times, which is greater than that of model 2, 258 times. The total start-stop of model 1 is 15 times, which is more than 9 times of model 2, so the total cost of optimized electricity of wind and light in model 2 is less than that in model 1. According to the optimal scheduling model of wind and light renewable energy, the effective utilization rate of renewable energy in China is low, and the teaching system should pay attention to this problem. Therefore, this paper puts forward an international English teaching strategy for the utilization rate of renewable energy.

States estimation and resilient control for non-Gaussian stochastic distribution control system under sensor attacks

This paper studies states estimation and resilient control schemes based on model predictive control (MPC) algorithm for a class of Takagi–Sugeno (T–S) fuzzy stochastic distribution control (SDC) system subjected to sparse sensor attacks. Firstly, a T–S fuzzy model is used to approximate the dynamics of a non-Gaussian SDC system, where the outputs of the system is the output probability density functions (PDFs). Secondly, in order to estimate the states and attacks in the system, a fuzzy Luenberger observer is designed. In addition, based on the estimated states and attacks, the designed MPC resilient control method achieves a satisfied tracking performance. Finally, the feasibility of the estimation algorithm and resilient controller are verified by simulation.

Fault‐tolerant control design for stochastic distribution control systems based on k‐step fault estimation

AbstractA novel scheme of fault‐tolerant control (FTC) and fault estimation (FE) for the stochastic distribution control (SDC) system with disturbance is investigated. The output probability density function (PDF) is modelled by the weight vector of the B‐spline neural network with the given basis function, and control of output probability density function can be converted to control of the weight vector of the B‐spline neural network. A k‐step fault estimation algorithm aiming at enhancing the accuracy of fault estimation is first applied to the stochastic distribution control system. Subsequently, a dynamic output feedback fault‐tolerant control algorithm is constructed by using the k‐step fault estimation information to guarantee that the target weight vector can be still approximated by the output weight vector in case of actuator fault. A simulation example is used to confirm the availability of the presented method.

Adaptive diagnosis and non-fragile predictive control for HFV engine with non-Gaussian uncertain output

This study presents an adaptive fault estimation and model predictive fault-tolerant control (FTC) for the hypersonic vehicle engine with non-Gaussian uncertain output jet plume. The non-Gaussian probability density function (PDF) describing the plume velocity in a stochastic system is approximated by Type II fuzzy radial basis functions. In the fault observer, the novel prey adaptive estimation method is designed for the valve faults with different amplitudes. Ultimately, this fuzzy-prey fusion adaptive observer shields perturbation and accurately estimates complex faults including incipient and intermittent faults. A predictive tolerant controller compensates for all the faults effects and non-fragile compensation factors of FTC eliminate the perturbation interference, thence the output PDF matches with the expected value. Finally, the more refined internal structure of plume output distribution replaces the local and rough valve position variables, helping this study realizes the hypersonic vehicle engine integrated refine control. Simulation result verifies the effectiveness of the proposed approaches and the superiority compared with the existing method.

Fault isolation and fault‐tolerant control for nonlinear stochastic distribution control systems with multiplicative faults

AbstractIn this paper, a fault isolation, diagnosis and fault tolerant control algorithm is proposed for nonlinear multiple multiplicative faults stochastic distribution control systems employing Takagi–Sugeno fuzzy system. To obtain the detailed fault information, a fault detection algorithm is introduced to discover the fault occurrence time. Then a fault isolation observer is built to produce the residual, and the error system is separated to subsystems affected only by disturbance and multiplicative faults. Moreover, a fault estimation scheme is presented to obtain the fault magnitude information. When faults occur, the system output probability density function will deviate from the desired distribution. So the model predictive control fault tolerant control scheme is needed to minimize the impact of faults as much as possible to make sure that the post fault output probability density function track the desired probability density function. The validity of the designed algorithm is demonstrated through a simulation example, where the fault tolerant control algorithm ensures that the system output probability density function still track the given output probability density function despite the complex case of multiple multiplicative faults occurring simultaneously.

Rational entropy‐based fuzzy fault tolerant control for descriptor stochastic distribution networked control systems with packet dropout

AbstractThe fault tolerant control (FTC) problem based on rational entropy performance criteria is researched for fuzzy descriptor stochastic distribution networked control (SDNC) systems with packet dropout. The independent Bernoulli distribution is employed to describe the packet dropout in the feedback channel. The static model and the dynamic model of descriptor SDNC systems are built up by the rational square‐root fuzzy logic model (FLM) and the T‐S fuzzy model, respectively. When the given output probability density function (PDF) is not known, the minimization of output randomness becomes an important control target. First, the unknown fault is estimated by developing the fuzzy fault estimation observer. Then, the fault tolerant controller with the fault compensation is designed so that the output of the descriptor SDNC system remains with the minimum uncertainty after the fault occurs. A simulation example is supplied to prove the proposed scheme.

Tracking Control for Output Probability Density Function of Stochastic Systems Using FPD Method

Output probability density function (PDF) tracking control of stochastic systems has always been a challenging problem in both theoretical development and engineering practice. Focused on this challenge, this work proposes a novel stochastic control framework so that the output PDF can track a given time-varying PDF. Firstly, the output PDF is characterised by the weight dynamics following the B-spline model approximation. As a result, the PDF tracking problem is transferred to a state tracking problem for weight dynamics. In addition, the model error of the weight dynamics is described by the multiplicative noises to more effectively establish its stochastic dynamics. Moreover, to better reflect the practical applications in the real world, the given tracking target is set to be time-varying rather than static. Thus, an extended fully probabilistic design (FPD) is developed based on the conventional FPD to handle multiplicative noises and to track the time-varying references in a superior way. Finally, the proposed control framework is verified by a numerical example, and a comparison simulation with the linear-quadratic regulator (LQR) method is also included to illustrate the superiority of our proposed framework.

Recent Advances in Non-Gaussian Stochastic Systems Control Theory and Its Applications

Survey/review study Recent Advances in Non-Gaussian Stochastic Systems Control Theory and Its Applications Qichun Zhang 1, and Yuyang Zhou 2,* 1 Department of Computer Science, University of Bradford, Bradford BD7 1DP, United Kingdom 2 School of Computing, Engineering and the Built Environment, Edinburgh Napier University, Edinburgh EH10 5DT, United Kingdom * Correspondence: y.zhou@napier.ac.uk Received: 24 October 2022 Accepted: 18 November 2022 Published: 22 December 2022 Abstract: Non-Gaussian randomness widely exists in complex dynamical systems, in which the traditional mean-variance index cannot fully reflect the systematic characteristics. To improve the performance of control design subjected to non-Gaussian noises, stochastic distribution control (SDC) theory was proposed in the 1990s, where the output probability density function (PDF) has been investigated as an additional system variable. Following this framework, SDC has been extended to other research subjects in control systems such as filter design, fault diagnosis, and so on. It shows that SDC supplies an important solution to enhance the accuracy of system design, which is further beneficial to almost all the topics subject to non-Gaussian randomness. Meanwhile, the theoretical results of the SDC have been applied to several practical industrial applications. As data science raises based on the development of industrial artificial intelligence, SDC has been further developed recently focusing on data-driven design and multi-agent systems. To explore the new challenges with the evolution of SDC, e.g. unknown system models, unknown noise distributions, strong non-stationary transient dynamics, stability analysis and industrial applications, this survey summarises the most recent published results in the last 5 years of SDC work in terms of modelling, control, filtering, fault diagnosis, and industrial applications. Based on the technical analysis, potential future work is discussed in the end.

Bionic adaptive fault-tolerant control of non-Gaussian stochastic attitude hypersonic vehicle

This study investigates an adaptive fault-tolerant control (FTC) for hypersonic flight vehicles (HFVs) with incipient faults and non-Gaussian stochastic output attitudes. In the nonlinear HFV dynamics, a hybrid fuzzy approximation method achieves the linearization, then the stochastic outputs are transformed into probability density functions (PDFs) via rational square root B-spline. The disturbance and faults are estimated simultaneously by an adaptive augmented observer. Then actuator faults are compensated by an bionic adaptive fault-tolerant controller to ensure that the output PDFs accurately track the expected PDFs, thereby matching actual attitude angles with the desired ones, the bionic prey adaptive law can make FTC accurately repair the incipient fault deviations. Lyapunov theory proves the robust stability of the scheme, and simulation illustrates the effectiveness.

Two-Step Neural-Network-Based Fault Isolation for Stochastic Systems

This paper studies a fault isolation method for an optical fiber vibration source detection and early warning system. We regard the vibration sources in the system as faults and then detect and isolate the faults of the system based on a two-step neural network. Firstly, the square root B-spline expansion method is used to approximate the output probability density functions. Secondly, the nonlinear weight dynamic model is established through a dynamic neural network. Thirdly, the nonlinear filter and residual generator are constructed to estimate the weight, analyze the residual, and estimate the threshold, so as to detect, diagnose, and isolate the faults. The feasibility criterion of fault detection and isolation is given by using some linear matrix inequalities, and the stability of the estimation error system is proven according to the Lyapunov theorem. Finally, simulation experiments based on a optical fiber vibration source system are given to verify the effectiveness of this method.

Optimal criteria and their asymptotic form for data selection in data-driven reduced-order modelling with Gaussian process regression

We derive criteria for the selection of datapoints used for data-driven reduced-order modelling and other areas of supervised learning based on Gaussian process regression (GPR). While this is a well-studied area in the fields of active learning and optimal experimental design, most criteria in the literature are empirical. Here we introduce an optimality condition for the selection of a new input defined as the minimizer of the distance between the approximated output probability density function (pdf) of the reduced-order model and the exact one. Given that the exact pdf is unknown, we define the selection criterion as the supremum over the unit sphere of the native Hilbert space for the GPR. The resulting selection criterion, however, has a form that is difficult to compute. We combine results from GPR theory and asymptotic analysis to derive a computable form of the defined optimality criterion that is valid in the limit of small predictive variance. The derived asymptotic form of the selection criterion leads to convergence of the GPR model that guarantees a balanced distribution of data resources between probable and large-deviation outputs, resulting in an effective way of sampling towards data-driven reduced-order modelling. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Distributed Active Power Optimal Dispatching of Wind Farm Cluster Considering Wind Power Uncertainty

With the large-scale volatility and uncertainty of the centralized grid connection of wind power, the dimensionality disaster problem of wind farm cluster (WFC) scheduling optimization calculation becomes more and more significant. In view of these challenges, a distributed active power optimal dispatch model for WFC based on the alternating direction multiplier method (ADMM) is proposed, and the analytical description of the distribution characteristics of the active power output of wind farms is introduced into the proposed model. Firstly, based on the wake effect, the Weibull distribution of wind speed is transformed by the impulse function to establish an analytical model of the active output distribution of the wind farm. Secondly, the optimization goal is to minimize the expected sum of the deviations of the dispatch instructions and the output probability density function of each wind farm, constructing a WFC active power optimal dispatch model considering uncertainty. Finally, the proposed model is decoupled in space and time into sub-optimization problems, and the ADMM is improved to construct an efficient solution algorithm that can iterate in parallel and decouple a large number of decision variables at the same time. The model is compared with other current classical models through the evaluation of multiple recommendation evaluation metrics, and the experimental results show that the model has a 3–7% reduction in dispatched power shortfalls and a 4–21% improvement in wind power utilization. The optimization algorithm for model construction has extremely high computational efficiency and good convergence. The results show that when the update step size is three, the convergence is the fastest, and when the update step size is six, the convergence is the slowest; in addition, when the number of wind farms is greater than eight, the distributed computing efficiency has an incomparable advantage over the centralized one. It plays a helpful role in wind power consumption and the efficient calculation of the power grid and effectively improves the reliability of the power grid.

Fault Detection for Non-Gaussian Stochastic Distribution Systems Based on Randomized Algorithms

This paper presents a new fault detection scheme for stochastic distribution systems with non-Gaussian variables based on a probabilistic framework. The available information is the output probability density function (PDF) rather than the measured outputs themselves. The square-root B-spline function is utilized to formulate the output PDFs. Probabilistic parameter models are developed to characterize system uncertainties and faults. An observer is designed to detect the multiplicative and additive faults. The randomized algorithm is adopted to design the threshold to achieve an optimal balance between the false alarm and fault detection rate. The effectiveness of the proposed method is demonstrated and compared against existing work by utilizing a continuous stirred tank reactor system.

Survey on stochastic distribution systems: A full probability density function control theory with potential applications

AbstractComplex systems seen either in general engineering practice or economics are subjected to ever increased uncertainties that are mostly represented as random variables or parameters, and the characteristics of random variables are represented by their probability density functions (PDFs). Controlling their PDFs means to shape their stochastic distributions and in general it would provide a full treatment for system analysis and operational control and optimization. This leads to the development of stochastic distribution control (SDC) systems theory in the past decades, where the original aim of the controller design is to realize a shape control of the distributions of certain random variables in their PDFs sense for some engineering processes. Indeed, once the PDFs of these random variables or parameters are used to describe their distribution characters, the control task is to obtain control signals so that the output PDFs of stochastic systems are made to follow their target PDFs. The subject of SDC was initially originated for non‐Gaussian stochastic control systems design but has found a wide spectrum of applications in general systems in terms of data‐driven modeling, analysis, signal processing (filtering), data mining via multivariable statistics, decision‐making (optimization) for systems subjected to uncertainties and even in economics. In this context, SDC constitutes an effective primer tool for complex system analysis, control and operational optimizations. In this review paper, a detailed survey of the developments on the research of SDC systems will be made together with their wide spectrum applications and future perspectives.

Uncertainty and multi-criteria global sensitivity analysis of structural systems using acceleration algorithm and sparse polynomial chaos expansion

Sparse polynomial chaos expansion (PCE) can be used to emulate the stochastic model output where the original model is computationally expensive. It is a powerful tool in efficient uncertainty quantification and sensitivity analysis. Structural systems are usually associated with high dimensional and probabilistic input. The number of candidate basis functions increases significantly with input dimension, resulting in high computational burden for establishing sparse PCE. In this study, acceleration techniques are integrated to formulate an algorithm for efficient computation of sparse PCE (ASPCE). The integrated algorithm can improve efficiency of computational process compared with conventional greedy algorithm while ensuring the satisfying predictive performance. Once the sparse PCE model is obtained, the statistic moments, probability density function of stochastic output, and global sensitivity index could be computed efficiently. Traditional PCE based global sensitivity analysis only assesses the sensitivity on individual structural performance criterion. Assessing the global sensitivity considering multiple criteria is challenging as the sensitive parameters may not be consistent for different performance criteria. To address this issue, a two-stage multi-criteria global sensitivity analysis algorithm is proposed by coupling ASPCE and the technique for order preference by similarity to ideal solution (TOPSIS). A holistic global sensitivity index is proposed to identify the sensitive parameters incorporating multiple performance criteria. In order to illustrate the efficiency, accuracy, and applicability of the proposed approach, two illustrative cases are presented.

Stochastic distribution tracking control for stochastic non-linear systems via probability density function vectorisation

This paper presents a new control strategy for stochastic distribution shape tracking regarding non-Gaussian stochastic non-linear systems. The objective can be summarised as adjusting the probability density function (PDF) of the system output to any given desired distribution. In order to achieve this objective, the system output PDF has first been formulated analytically, which is time-variant. Then, the PDF vectorisation has been implemented to simplify the model description. Using the vector-based representation, the system identification and control design have been performed to achieve the PDF tracking. In practice, the PDF evolution is difficult to implement in real-time, thus a data-driven extension has also been discussed in this paper, where the vector-based model can be obtained using kernel density estimation (KDE) with the real-time data. Furthermore, the stability of the presented control design has been analysed, which is validated by a numerical example. As an extension, the multi-output stochastic systems have also been discussed for joint PDF tracking using the proposed algorithm, and the perspectives of advanced controller have been discussed. The main contribution of this paper is to propose: (1) a new sampling-based PDF transformation to reduce the modelling complexity, (2) a data-driven approach for online implementation without model pre-training, and (3) a feasible framework to integrate the existing control methods.

An extended polynomial chaos expansion for PDF characterization and variation with aleatory and epistemic uncertainties

This paper presents an extended polynomial chaos formalism for epistemic uncertainties and a new framework for evaluating sensitivities and variations of output probability density functions (PDF) to uncertainty in probabilistic models of input variables. An ”extended” polynomial chaos expansion (PCE) approach is developed that accounts for both aleatory and epistemic uncertainties, modeled as random variables, thus allowing a unified treatment of both types of uncertainty. We explore in particular epistemic uncertainty associated with the choice of prior probabilistic models for input parameters. A PCE-based Kernel Density (KDE) construction provides a composite map from the PCE coefficients and germ to the PDF of quantities of interest (QoI). The sensitivities of these PDF with respect to the input parameters are then evaluated. Input parameters of the probabilistic models are considered. By sampling over the epistemic random variable, a family of PDFs is generated and the failure probability is itself estimated as a random variable with its own PCE. Integrating epistemic uncertainties within the PCE framework results in a computationally efficient paradigm for propagation and sensitivity evaluation. Two typical illustrative examples are used to demonstrate the proposed approach.

Fault diagnosis and fault-tolerant control for T-S fuzzy time-varying delay stochastic distribution systems with actuator and sensor faults

ABSTRACT The problem of fault diagnosis (FD) and fault-tolerant control (FTC) for a class of nonlinear time-varying delay stochastic distribution control (SDC) systems with actuator and sensor faults is investigated in this paper. The Takagi–Sugeno (T-S) model is used to approximate the nonlinear dynamics. The B-spline function is used to approximate the output probability density function (PDF) of the system. According to the output equivalence principle and Laplace transformation, an augmented state vector is designed to solve the time-varying delay problem. An augmented state adaptive diagnosis observer is proposed to estimate the system state, actuator and sensor faults simultaneously. A new fault-tolerant control algorithm is designed based on the PI control strategy to compensate sensor fault and actuator faults simultaneously. The sensor fault is compensated through the information obtained by the observer. The PI controller can compensate for the influence of the actuator fault, and the output PDF of system can still track the desired PDF when fault occurs. Simulation results are provided to show the effectiveness of the proposed method.

Fault isolation and fault-tolerant control for Takagi-Sugeno fuzzy time-varying delay stochastic distribution systems

A fault isolation, estimation, and fault-tolerant control (FTC) scheme for nonlinear time-varying delay stochastic distribution control systems was presented in this paper. The Takagi-Sugeno fuzzy model was adopted to approach the nonlinear dynamics of time-varying delay systems. According to the output equivalence principle and Laplace transformation, an augmented state vector was given to solve the time-varying delay problem. When multiple actuator faults and interference occur simultaneously, fault detection, isolation and fault estimation was designed to obtained the fault information. To decouple faults and obtain the value and location information of the fault, the system was separated into two parts through the designed multiple conversion matrices, in which one subsystem was only affected by one actuator fault. This has simplified the design of fault isolation and estimation. A adaptive observer for fault estimation was given. Then, fault information such as the time, location, and size was determined. The observer gain matrices were calculated using linear matrix inequality (LMI). When a fault was detected and diagnosed, a FTC algorithm was devised using the proportional-integral control scheme to compensate the fault as much as possible. It has been shown that even if multiple faults actuator occurred simultaneously, the FTC controller still ensured the output probability density function of the system traced the desired probability density function when a fault occurred. Finally, the expected results were obtained through the simulation example, which confirmed the effectiveness of the method.