Entropy

Research

148 KiB

Open AccessArticle

Robust H_∞ Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions

by Haiyang Chen, Meiqin Liu and Senlin Zhang

Entropy 2015, 17(1), 346-367; https://doi.org/10.3390/e17010346 - 14 Jan 2015

Cited by 15 | Viewed by 6051

This paper is concerned with the robust H_∞ finite-time control for discrete delayed nonlinear systems with Markovian jumps and external disturbances. It is usually assumed that the disturbance affects the system states and outputs with the same influence degree of 100%, which [...] Read more.

This paper is concerned with the robust H_∞ finite-time control for discrete delayed nonlinear systems with Markovian jumps and external disturbances. It is usually assumed that the disturbance affects the system states and outputs with the same influence degree of 100%, which is not evident enough to reflect the situation where the disturbance affects these two parts by different influence degrees. To tackle this problem, a probabilistic distribution denoted by binomial sequences is introduced to describe the external disturbance. Throughout the paper, the definitions of the finite-time boundedness (FTB) and the H_∞ FTB are firstly given respectively. To extend the results further, a model which combines a linear dynamic system and a static nonlinear operator is referred to describe the system under discussion. Then by virtue of state feedback control method, some new sufficient criteria are derived which guarantee the FTB and H_∞ FTB performances for the considered system. Finally, an example is provided to demonstrate the effectiveness of the developed control laws. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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306 KiB

Open AccessArticle

Chaos Control and Synchronization of a Hyperchaotic Zhou System by Integral Sliding Mode control

by Yashar Toopchi and Jidong Wang

Entropy 2014, 16(12), 6539-6552; https://doi.org/10.3390/e16126539 - 12 Dec 2014

Cited by 25 | Viewed by 6701

Abstract

In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at [...] Read more.

In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at the origin. The control is then applied to the synchronization of two identical systems, i.e., a slave and a master hyperchaotic Zhou system with unknown parameters. The adaptive control mechanism introduced synchronizes the systems by estimating the unknown parameters. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view, and it outperforms Vaidyanathan’s scheme, which is a well-recognized scheme in this area. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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209 KiB

Open AccessArticle

Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances

by Zheping Yan, Di Wu and Yibo Liu

Entropy 2014, 16(12), 6524-6538; https://doi.org/10.3390/e16126524 - 11 Dec 2014

Cited by 5 | Viewed by 5314

Abstract

In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With [...] Read more.

In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With construction of discrete dynamics of state error vector, it is proved by applying Halanay inequality that consensus of the system can be reached. Further, the definition of bounded consensus is proposed in the situation where environmental disturbances exist. In order to handle this problem, the Halanay inequality is extended into a more general one with boundedness property. Based on the new Halanay inequality obtained, the boundedness of consensus error is guaranteed. At last, simulation examples are presented to demonstrate the theoretical conclusions. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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1505 KiB

Open AccessArticle

A Memristive Diode Bridge-Based Canonical Chua’s Circuit

by Mo Chen, Jingjing Yu, Qing Yu, Changdi Li and Bocheng Bao

Entropy 2014, 16(12), 6464-6476; https://doi.org/10.3390/e16126464 - 8 Dec 2014

Cited by 49 | Viewed by 7825

Abstract

A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It [...] Read more.

A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It can be found that the circuit has three determined equilibrium points, including a zero saddle point and two nonzero saddle-foci with index 2. Specially, the circuit is non-dissipative in the neighborhood of the zero saddle point, and there exists complex nonlinear phenomena of coexisting bifurcation modes and coexisting chaotic attractors. Experimental observations are performed to verify the numerical simulation results. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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1714 KiB

Open AccessArticle

Detecting Chaos from Agricultural Product Price Time Series

by Xin Su, Yi Wang, Shengsen Duan and Junhai Ma

Entropy 2014, 16(12), 6415-6433; https://doi.org/10.3390/e16126415 - 5 Dec 2014

Cited by 26 | Viewed by 6850

Abstract

Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos [...] Read more.

Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos of the data. The approaches used include the R/S analysis, the BDS test, the power spectra, the recurrence plot, the largest Lyapunov exponent, the Kolmogorov entropy, and the correlation dimension. The results show that there is chaos in agricultural wholesale price data, which provides a good theoretical basis for selecting reasonable forecasting models as prediction techniques based on chaos theory can be applied to forecasting agricultural prices. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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1391 KiB

Open AccessArticle

Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays

by Weiyuan Ma, Changpin Li, Yujiang Wu and Yongqing Wu

Entropy 2014, 16(12), 6286-6299; https://doi.org/10.3390/e16126286 - 1 Dec 2014

Cited by 45 | Viewed by 6686

Abstract

In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain [...] Read more.

In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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2785 KiB

Open AccessArticle

Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos

by Jian Liu

Entropy 2014, 16(12), 6195-6211; https://doi.org/10.3390/e16126195 - 27 Nov 2014

Cited by 25 | Viewed by 5837

Abstract

This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based [...] Read more.

This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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503 KiB

Open AccessArticle

New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis

by Carson Ingo, Richard L. Magin and Todd B. Parrish

Entropy 2014, 16(11), 5838-5852; https://doi.org/10.3390/e16115838 - 6 Nov 2014

Cited by 32 | Viewed by 7453

Abstract

Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of [...] Read more.

Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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161 KiB

Open AccessArticle

A Note on Distance-based Graph Entropies

by Zengqiang Chen, Matthias Dehmer and Yongtang Shi

Entropy 2014, 16(10), 5416-5427; https://doi.org/10.3390/e16105416 - 20 Oct 2014

Cited by 133 | Viewed by 7676

Abstract

A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties [...] Read more.

A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties such as a meaningful structural interpretation and uniqueness. As classical work, many distance-based graph entropies, e.g., the ones due to Bonchev et al. and related quantities have been proposed and studied. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices with distance \(k\) to a given vertex. In particular, we investigate some properties thereof leading to a better understanding of this new information-theoretic quantity. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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556 KiB

Open AccessArticle

Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics

by Junwei Wang, Kairui Chen and Qinghua Ma

Entropy 2014, 16(9), 5020-5031; https://doi.org/10.3390/e16095020 - 22 Sep 2014

Cited by 18 | Viewed by 6933

Abstract

This paper deals with the leader-following consensus of multi-agent systems with matched nonlinear dynamics. Compared with previous works, the major difficulty here is caused by the simultaneous existence of nonidentical agent dynamics and unknown system parameters, which are more practical in real-world applications. [...] Read more.

This paper deals with the leader-following consensus of multi-agent systems with matched nonlinear dynamics. Compared with previous works, the major difficulty here is caused by the simultaneous existence of nonidentical agent dynamics and unknown system parameters, which are more practical in real-world applications. To tackle this difficulty, a distributed adaptive control law for each follower is proposed based on algebraic graph theory and algebraic Riccati equation. By a Lyapunov function method, we show that the designed control law guarantees that each follower asymptotically converges to the leader under connected communication graphs. A simulation example demonstrates the effectiveness of the proposed scheme. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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Graphical abstract

216 KiB

Open AccessArticle

Existence of Entropy Solutions for Nonsymmetric Fractional Systems

by Rabha W. Ibrahim and Hamid A. Jalab

Entropy 2014, 16(9), 4911-4922; https://doi.org/10.3390/e16094911 - 12 Sep 2014

Cited by 11 | Viewed by 5182

Abstract

The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L^∞ . The solutions attained are unique and exclusive. Performance is established by utilizing the [...] Read more.

The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L^∞ . The solutions attained are unique and exclusive. Performance is established by utilizing the maximum principle for certain generalized time and space-fractional diffusion equations. The fractional differential operator is inspected based on the interpretation of the Riemann–Liouville differential operator. Fractional entropy inequalities are imposed. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

1661 KiB

Open AccessArticle

Information Entropy-Based Metrics for Measuring Emergences in Artificial Societies

by Mingsheng Tang and Xinjun Mao

Entropy 2014, 16(8), 4583-4602; https://doi.org/10.3390/e16084583 - 15 Aug 2014

Cited by 18 | Viewed by 7763

Abstract

Emergence is a common phenomenon, and it is also a general and important concept in complex dynamic systems like artificial societies. Usually, artificial societies are used for assisting in resolving several complex social issues (e.g., emergency management, intelligent transportation system) with the aid [...] Read more.

Emergence is a common phenomenon, and it is also a general and important concept in complex dynamic systems like artificial societies. Usually, artificial societies are used for assisting in resolving several complex social issues (e.g., emergency management, intelligent transportation system) with the aid of computer science. The levels of an emergence may have an effect on decisions making, and the occurrence and degree of an emergence are generally perceived by human observers. However, due to the ambiguity and inaccuracy of human observers, to propose a quantitative method to measure emergences in artificial societies is a meaningful and challenging task. This article mainly concentrates upon three kinds of emergences in artificial societies, including emergence of attribution, emergence of behavior, and emergence of structure. Based on information entropy, three metrics have been proposed to measure emergences in a quantitative way. Meanwhile, the correctness of these metrics has been verified through three case studies (the spread of an infectious influenza, a dynamic microblog network, and a flock of birds) with several experimental simulations on the Netlogo platform. These experimental results confirm that these metrics increase with the rising degree of emergences. In addition, this article also has discussed the limitations and extended applications of these metrics. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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692 KiB

Open AccessArticle

Chaos Synchronization Error Technique-Based Defect Pattern Recognition for GIS through Partial Discharge Signal Analysis

by Hung-Cheng Chen, Her-Terng Yau and Po-Yan Chen

Entropy 2014, 16(8), 4566-4582; https://doi.org/10.3390/e16084566 - 13 Aug 2014

Cited by 12 | Viewed by 6657

Abstract

The work is aimed at using the chaos synchronization error dynamics (CSED) technique for defect pattern recognition in gas insulated switchgear (GIS). The radiated electromagnetic waves generated due to internal defects were measured by the self-made ultrahigh frequency (UHF) micro-strip antenna, so as [...] Read more.

The work is aimed at using the chaos synchronization error dynamics (CSED) technique for defect pattern recognition in gas insulated switchgear (GIS). The radiated electromagnetic waves generated due to internal defects were measured by the self-made ultrahigh frequency (UHF) micro-strip antenna, so as to determine whether partial discharge will occur. Firstly, a data pretreatment is performed on the measured raw data for the purpose of computational burden reduction. A characteristic matrix is then constructed according to dynamic error trajectories in a chaos synchronization system, subsequent to which characteristics are extracted. A comparison with the existing Hilbert-Huang Transform (HHT) method reveals that the two characteristics extracted from the CSED results presented herein using the fractal theory were recognized at a higher rate pattern. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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4266 KiB

Open AccessArticle

Fractional Order Generalized Information

by José Tenreiro Machado

Entropy 2014, 16(4), 2350-2361; https://doi.org/10.3390/e16042350 - 24 Apr 2014

Cited by 120 | Viewed by 8244

Abstract

This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow [...] Read more.

This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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542 KiB

Open AccessArticle

A Fuzzy Parallel Processing Scheme for Enhancing the Effectiveness of a Dynamic Just-in-time Location-aware Service System

by Toly Chen

Entropy 2014, 16(4), 2001-2022; https://doi.org/10.3390/e16042001 - 8 Apr 2014

Cited by 12 | Viewed by 5811

Abstract

Location-aware service systems are a hot topic in diverse research fields including mobile commerce, ambient intelligence, remote sensing and ubiquitous computing. However, the timeliness and efficiency of such systems are two issues that have rarely been emphasized. For this reason, this study tries [...] Read more.

Location-aware service systems are a hot topic in diverse research fields including mobile commerce, ambient intelligence, remote sensing and ubiquitous computing. However, the timeliness and efficiency of such systems are two issues that have rarely been emphasized. For this reason, this study tries to establish a location-aware service system in which both the timeliness and efficiency of service provision are addressed. To this end, some innovative treatments have been used in the proposed methodology. First, the uncertainty of detecting a user’s location using the global positioning system is considered by modeling the location and speed of the user with fuzzy numbers. Subsequently, a fuzzy integer-nonlinear programming model is formulated to address the problem of finding the dynamic just-in-time service location and path for the user. To help solve the problem, the maximum entropy weighting function and the basic defuzzification distribution (BADD) method are applied to defuzzify the fuzzy variables. In addition, to enhance the efficiency of solving the problem, a fuzzy parallel processing scheme is also proposed for decomposing the problem into smaller pieces that can be handled by separate processing modules. An illustrative example is used to illustrate the proposed methodology. Finally, the effectiveness of the proposed methodology has been confirmed with an experiment. According to the results, using the proposed methodology the waiting time could be reduced by 60%. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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