Entropy Optimization Principle Research Articles

Disturbance Observer-Based Minimum Entropy Control for a Class of Disturbed Non-Gaussian Stochastic Systems

In this article, a novel control algorithm is developed for a class of nonlinear stochastic systems subject to multiple disturbances, including exogenous dynamic disturbance and general non-Gaussian noise. An observer is designed to estimate the exogenous disturbance, and then the disturbance compensation is incorporated into a feedback control strategy for the non-Gaussian system. Considering the ability of entropy in randomness quantification, a performance index is established based on the generalized entropy optimization principle. Furthermore, it is adjusted to be available for the controller solution, which also solves the coupling between two kinds of disturbances. On this basis, the optimal controller is provided in a recursive way, with which the closed-loop stability and good antidisturbance ability can be guaranteed simultaneously. Compared with the existing studies on the non-Gaussian stochastic systems, the proposed control algorithm has merits in multiple disturbances decoupling and enhanced antidisturbance performance. Finally, a simulation example is given to demonstrate the effectiveness of theoretical results.

Multi-period Multi-criteria (MPMC) Valuation of American Options Based on Entropy Optimization Principles

The multi-period multi-nomial models have gained wider popularity in option pricing researches; however, there is still a need to focus researches on the efficacy of the effective criteria in determining the future values of the underlying assets. We, in this paper, employ probability distributions of the effective criteria in-price fluctuations. The Black–Scholes differential equation is used, and the EM algorithm is applied to estimate parameters of probability distributions. We employ entropy optimization measures to determine weights of each criterion in each period. Then, we carry out Monte Carlo simulations to access the initial option price. Finally, an illustrative example is given to verify the proposed approach.

Contribution of maximum entropy principle in the field of queueing theory

ABSTRACTIn the literature of information theory, there exist many well known measures of entropy suitable for entropy optimization principles towards applications in different disciplines of science and technology. The object of this article is to develop a new generalized measure of entropy and to establish the relation between entropy and queueing theory. To fulfill our aim, we have made use of maximum entropy principle which provides the most uncertain probability distribution subject to some constraints expressed by mean values.

An Algorithm to Generate Probabilities with Specified Entropy

The present communication offers a method to determine an unknown discrete probability distribution with specified Tsallis entropy close to uniform distribution. The relative error of the distribution obtained has been compared with the distribution obtained with the help of mathematica software. The applications of the proposed algorithm with respect to Tsallis source coding, Huffman coding and cross entropy optimization principles have been provided.

An unconstrained approach to blank localization with allowance assurance for machining complex parts

In machining large complex parts with critical stock allowance, even small deviations in the blank parts or slight inadequacy in the fixturing may result in local shortage of material (i.e., insufficient stock allowance). This paper presents an optimal localization algorithm that aligns the measured points from a blank part with the nominal model to assure the weakest allowance area with as much material to be cut as possible. Instead of exploiting extra constraints to force the allowance at each point larger than a specific value, which is a popular strategy for allowance assurance in the previous localization algorithms, we formulate the blank localization problem as an unconstrained max-min problem. To deal with the unsmoothness exhibiting in the max-min objective function, a method based on the entropy optimization principle is adopted to convert the non-differential objective function to an unconstrained differential one, which can be efficiently solved using the conventional Quasi-Newton algorithms. The unconstrained optimization result finally gives rise to localization with the maximum allowance margin. For the blank parts that the material shortage is inevitable, the method can still efficiently achieve reasonable localization results, which confine the material shortage to a least extent. The proposed method is easy to be implemented and works well for both sparse sample points and dense-scanned points. Case studies included justify the superiority of the proposed scheme over the existing non-linear constrained optimization solutions.

Function Based Fault Detection for Uncertain Multivariate Nonlinear Non-Gaussian Stochastic Systems Using Entropy Optimization Principle

In this paper, the fault detection in uncertain multivariate nonlinear non-Gaussian stochastic systems is further investigated. Entropy is introduced to characterize the stochastic behavior of the detection errors, and the entropy optimization principle is established for the fault detection problem. The principle is to maximize the entropies of the stochastic detection errors in the presence of faults and to minimize the entropies of the detection errors in the presence of disturbances. In order to calculate the entropies, the formulations of the joint probability density functions (JPDFs) of the stochastic errors are presented in terms of the known JPDFs of both the disturbances and the faults. By using the novel performance indexes and the formulations for the entropies of the detection errors, new fault detection design methods are provided for the considered multivariate nonlinear non-Gaussian plants. Finally, a simulation example is given to illustrate the efficiency of the proposed fault detection algorithm.

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.

A Procedure for Identification of Principal Variables by Least Generalized Dependence

Principal components are often used for reducing dimensions in multivariate data, but they frequently fail to provide useful results and their interpretation is rather difficult. In this article, the use of entropy optimization principles for dimensional reduction in multivariate data is proposed. Under the assumptions of multivariate normality, a four-step procedure is developed for selecting principal variables and hence discarding redundant variables. For comparative performance of the information theoretic procedure, we use simulated data with known dimensionality. Principal variables of cluster bean (Guar) are identified by applying this procedure to a real data set generated in a plant breeding experiment.

Fault detection for non-linear non-Gaussian stochastic systems using entropy optimization principle

In this paper, a fault detection (FD) problem is studied for non-linear dynamic stochastic systems with non-Gaussian disturbances and faults (or abrupt changes of system parameters). After a filter is constructed to generate the detected error, the FD problem is reduced to an optimization problem for the error system, which is represented by a non-linear non-Gaussian stochastic system. Since generally (extended) Kalmen filtering approaches are insufficient to characterize the non-Gaussian variables, we propose the entropy optimization principle for the stochastic error system. The design objective is to maximize the entropies of the stochastic detection errors when the faults occur, and to minimize the entropies of the stochastic estimator errors resulting from the other stochastic noises. Following the formulation of the probability density functions of the stochastic error in terms of those of both the disturbances and the faults, new recursive approaches are established to calculate the entropies of the detection errors. By using the novel performance index and the formulations for the entropies, the real-time optimal FD filter design method is provided. Finally, simulations are given to demonstrate the effectiveness of the proposed FD filtering algorithms.

Dimension reduction in multivariate analysis using maximum entropy criterion

In the present communication dimension reduction criteria in Multivariate data with no external variables are studied by using Entropy Optimization Principles. Maximum entropy criterion is provided and its relation with other criteria for selection of principal variables in multivariate analysis is established. A comparative study of performance of principal variables with the corresponding number of principal components is made by considering empirical data set.

Nonparametric Modeling of Random Uncertainties for Dynamic Response of Mistuned Bladed Disks

The random character of blade mistuning is a motivation to construct probability models of random uncertainties. Recently, a new approach known as a nonparametric model of random uncertainties, based on the entropy optimization principle, was introduced for modeling random uncertainties in linear and nonlinear elastodynamics. This paper presents an extension of this nonparametric model for vibration analysis of structures with cyclic geometry. In particular this probability model allows the blade eigenfrequencies uncertainties and the blade-modal-shape uncertainties to be modeled.

Random Uncertainties Model in Dynamic Substructuring Using a Nonparametric Probabilistic Model

This paper presents a new approach, called a nonparametric approach, for constructing a model of random uncertainties in dynamic substructuring in order to predict the matrix-valued frequency response functions of complex structures. Such an approach allows nonhomogeneous uncertainties to be modeled with the nonparametric approach. The Craig-Bampton dynamic substructuring method is used. For each substructure, a nonparametric model of random uncertainties is introduced. This nonparametric model does not require identifying uncertain parameters in the reduced matrix model of each substructure as is usually done for the parametric approach. This nonparametric model of random uncertainties is based on the use of a probability model for symmetric positive-definite real random matrices using the entropy optimization principle. The theory and a numerical example are presented in the context of the finite-element method. The numerical results obtained show the efficiency of the model proposed.

Otimização de entropia: implementação computacional dos princípios MaxEnt e MinxEnt

Os princípios de otimização de entropia MaxEnt de Jaynes (1957a,b) e MinxEnt de Kullback (1959) encontram aplicações em várias áreas de investigação científica. Ambos envolvem a otimização condicionada de medidas de entropia que são funções intrinsecamente não-lineares de probabilidades. Como constituem problemas de programação não-linear, suas soluções demandam algoritmos de busca iterativa e, além disso, as condições de não-negatividade e de soma um para as probabilidades restringem de modo particular o espaço de soluções. O artigo apresenta em detalhe (com a ajuda de dois fluxogramas) uma implementação computacional eficiente desses dois princípios no caso de restrições lineares com verificação prévia de existência de solução dos problemas de otimização. Os autores também disponibilizam rotinas de fácil uso desenvolvidas em linguagem MatLab<FONT FACE=Symbol>â</FONT> .

Transient responses of dynamical systems with random uncertainties

A new approach is presented for modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The probability model is deduced from the use of the entropy optimization principle whose available information involves the algebraic properties related to the generalized mass, damping and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. Finally, a simple example is presented.

On the nature of random system matrices in structural dynamics

This work is concerned with the nature of random mass and stiffness matrices arising in linear dynamic systems due to inherent uncertainties in the system parameters. In an important recent paper, Soize [Prob. Eng. Mech. 15, 277–294 (2000)] has developed an expression for the probability density function of such matrices by using the mean value of the matrices in conjunction with the entropy optimization principle. Although mathematically optimal given knowledge of only the mean values of the matrices, it is not entirely clear how well the results obtained will match the statistical properties of a physical system. This issue is investigated here by considering the structure of random matrices arising from Guyan-reduced dynamic models. It is shown that, under rather general conditions, the random parts of the mass and the stiffness matrices of the reduced system in the modal coordinates resemble the so-called ‘‘Gaussian Orthogonal Ensemble (GOE).’’ A single parameter σ (the standard deviation of the matrix diagonals) characterizes a GOE, and this offers the possibility of a very straightforward Monte Carlo simulation technique for the system matrices and response. A limited number of numerical results have been shown in favor of the proposed result.

Maximum entropy approach for modeling random uncertainties in transient elastodynamics

A new approach is presented for analyzing random uncertainties in dynamical systems. This approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The information used does not require the description of the local parameters of the mechanical model. The probability model is deduced from the use of the entropy optimization principle, whose available information is constituted of the algebraic properties related to the generalized mass, damping, and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. The fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed. Finally, an example is presented.

Measures of Information and Their Applications.

Delves into entropy optimization principles, offering new insights into their applications. Novel concepts are illustrated with figures and numerical examples. Also presents applications to ecological sciences. The standards and information discussed fill numerous gaps in existing literature.

A review of: “ENTROPY OPTIMIZATION PRINCIPLES WITH APPLICATIONS”, by J. N. Ka-pur and H. K. Kesavan. Academic Press, San Diego, 1992. xix + 408 pages.

(1993). A review of: “ENTROPY OPTIMIZATION PRINCIPLES WITH APPLICATIONS”, by J. N. Ka-pur and H. K. Kesavan. Academic Press, San Diego, 1992. xix + 408 pages. International Journal of General Systems: Vol. 21, No. 4, pp. 403-404.