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Wang J, Zhu S, Mu C, Liu X, Wen S. Unified analysis on multistablity of fraction-order multidimensional-valued memristive neural networks. Neural Netw 2024; 179:106498. [PMID: 38986183 DOI: 10.1016/j.neunet.2024.106498] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2024] [Revised: 04/29/2024] [Accepted: 06/26/2024] [Indexed: 07/12/2024]
Abstract
This article provides a unified analysis of the multistability of fraction-order multidimensional-valued memristive neural networks (FOMVMNNs) with unbounded time-varying delays. Firstly, based on the knowledge of fractional differentiation and memristors, a unified model is established. This model is a unified form of real-valued, complex-valued, and quaternion-valued systems. Then, based on a unified method, the number of equilibrium points for FOMVMNNs is discussed. The sufficient conditions for determining the number of equilibrium points have been obtained. By using 1-norm to construct Lyapunov functions, the unified criteria for multistability of FOMVMNNs are obtained, these criteria are less conservative and easier to verify. Moreover, the attraction basins of the stable equilibrium points are estimated. Finally, two numerical simulation examples are provided to verify the correctness of the results.
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Affiliation(s)
- Jiarui Wang
- School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China.
| | - Song Zhu
- School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China.
| | - Chaoxu Mu
- School of Electrical and Automation Engineering, Tianjin University, Tianjin, 300072, China.
| | - Xiaoyang Liu
- School of Computer Science and Technology, Jiangsu Normal University, Xuzhou, 221116, China.
| | - Shiping Wen
- Centre for Artificial Intelligence, University of Technology Sydney, Ultimo, NSW 2007, Australia.
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Liu P, Wang J, Zeng Z. An Overview of the Stability Analysis of Recurrent Neural Networks With Multiple Equilibria. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:1098-1111. [PMID: 34449396 DOI: 10.1109/tnnls.2021.3105519] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The stability analysis of recurrent neural networks (RNNs) with multiple equilibria has received extensive interest since it is a prerequisite for successful applications of RNNs. With the increasing theoretical results on this topic, it is desirable to review the results for a systematical understanding of the state of the art. This article provides an overview of the stability results of RNNs with multiple equilibria including complete stability and multistability. First, preliminaries on the complete stability and multistability analysis of RNNs are introduced. Second, the complete stability results of RNNs are summarized. Third, the multistability results of various RNNs are reviewed in detail. Finally, future directions in these interesting topics are suggested.
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Wan L, Liu Z. Multiple exponential stability and instability for state-dependent switched neural networks with time-varying delays and piecewise-linear radial basis activation functions. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2022.12.040] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
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Guo Z, Ou S, Wang J. Multistability of Switched Neural Networks With Gaussian Activation Functions Under State-Dependent Switching. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2022; 33:6569-6583. [PMID: 34077372 DOI: 10.1109/tnnls.2021.3082560] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
This article presents theoretical results on the multistability of switched neural networks with Gaussian activation functions under state-dependent switching. It is shown herein that the number and location of the equilibrium points of the switched neural networks can be characterized by making use of the geometrical properties of Gaussian functions and local linearization based on the Brouwer fixed-point theorem. Four sets of sufficient conditions are derived to ascertain the existence of 7p15p23p3 equilibrium points, and 4p13p22p3 of them are locally stable, wherein p1 , p2 , and p3 are nonnegative integers satisfying 0 ≤ p1+p2+p3 ≤ n and n is the number of neurons. It implies that there exist up to 7n equilibria, and up to 4n of them are locally stable when p1=n . It also implies that properly selecting p1 , p2 , and p3 can engender a desirable number of stable equilibria. Two numerical examples are elaborated to substantiate the theoretical results.
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Xu F, Liu L, Zurada JM, Yi Z. Analysis of Equilibria for a Class of Recurrent Neural Networks With Two Subnetworks. IEEE TRANSACTIONS ON CYBERNETICS 2022; 52:4701-4716. [PMID: 33296319 DOI: 10.1109/tcyb.2020.3034249] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
This article is concerned with the problem of the number and dynamical properties of equilibria for a class of connected recurrent networks with two switching subnetworks. In this network model, parameters serve as switches that allow two subnetworks to be turned ON or OFF among different dynamic states. The two subnetworks are described by a nonlinear coupled equation with a complicated relation among network parameters. Thus, the number and dynamical properties of equilibria have been very hard to investigate. By using Sturm's theorem, together with the geometrical properties of the network equation, we give a complete analysis of equilibria, including the existence, number, and dynamical properties. Necessary and sufficient conditions for the existence and exact number of equilibria are established. Moreover, the dynamical property of each equilibrium point is discussed without prior assumption of their locations. Finally, simulation examples are given to illustrate the theoretical results in this article.
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Wan L, Liu Z. Multiple O(t-q) stability and instability of time-varying delayed fractional-order Cohen-Grossberg neural networks with Gaussian activation functions. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.05.018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Nie X, Liu P, Liang J, Cao J. Exact coexistence and locally asymptotic stability of multiple equilibria for fractional-order delayed Hopfield neural networks with Gaussian activation function. Neural Netw 2021; 142:690-700. [PMID: 34403909 DOI: 10.1016/j.neunet.2021.07.029] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2021] [Revised: 06/08/2021] [Accepted: 07/26/2021] [Indexed: 11/30/2022]
Abstract
This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of 3n equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that 2n of 3n total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.
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Affiliation(s)
- Xiaobing Nie
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China.
| | - Pingping Liu
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China
| | - Jinling Liang
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China
| | - Jinde Cao
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China; Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea
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Multi-periodicity of switched neural networks with time delays and periodic external inputs under stochastic disturbances. Neural Netw 2021; 141:107-119. [PMID: 33887601 DOI: 10.1016/j.neunet.2021.03.039] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2020] [Revised: 03/11/2021] [Accepted: 03/29/2021] [Indexed: 11/21/2022]
Abstract
This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5n regions in which 3n ones are shown to be positively invariant with probability one. Furthermore, by using Itô's formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3n from 2n in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.
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Comparison of Diagnosis Accuracy between a Backpropagation Artificial Neural Network Model and Linear Regression in Digestive Disease Patients: an Empirical Research. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2021; 2021:6662779. [PMID: 33727951 PMCID: PMC7937476 DOI: 10.1155/2021/6662779] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/18/2020] [Revised: 12/10/2020] [Accepted: 02/18/2021] [Indexed: 02/08/2023]
Abstract
Introduction A Noninvasive diagnosis model for digestive diseases is the vital issue for the current clinical research. Our systematic review is aimed at demonstrating diagnosis accuracy between the BP-ANN algorithm and linear regression in digestive disease patients, including their activation function and data structure. Methods We reported the systematic review according to the PRISMA guidelines. We searched related articles from seven electronic scholarly databases for comparison of the diagnosis accuracy focusing on BP-ANN and linear regression. The characteristics, patient number, input/output marker, diagnosis accuracy, and results/conclusions related to comparison were extracted independently based on inclusion criteria. Results Nine articles met all the criteria and were enrolled in our review. Of those enrolled articles, the publishing year ranged from 1991 to 2017. The sample size ranged from 42 to 3222 digestive disease patients, and all of the patients showed comparable biomarkers between the BP-ANN algorithm and linear regression. According to our study, 8 literature demonstrated that the BP-ANN model is superior to linear regression in predicting the disease outcome based on AUROC results. One literature reported linear regression to be superior to BP-ANN for the early diagnosis of colorectal cancer. Conclusion The BP-ANN algorithm and linear regression both had high capacity in fitting the diagnostic model and BP-ANN displayed more prediction accuracy for the noninvasive diagnosis model of digestive diseases. We compared the activation functions and data structure between BP-ANN and linear regression for fitting the diagnosis model, and the data suggested that BP-ANN was a comprehensive recommendation algorithm.
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Liu P, Wang J, Guo Z. Multiple and Complete Stability of Recurrent Neural Networks With Sinusoidal Activation Function. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2021; 32:229-240. [PMID: 32203032 DOI: 10.1109/tnnls.2020.2978267] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
This article presents new theoretical results on multistability and complete stability of recurrent neural networks with a sinusoidal activation function. Sufficient criteria are provided for ascertaining the stability of recurrent neural networks with various numbers of equilibria, such as a unique equilibrium, finite, and countably infinite numbers of equilibria. Multiple exponential stability criteria of equilibria are derived, and the attraction basins of equilibria are estimated. Furthermore, criteria for complete stability and instability of equilibria are derived for recurrent neural networks without time delay. In contrast to the existing stability results with a finite number of equilibria, the new criteria, herein, are applicable for both finite and countably infinite numbers of equilibria. Two illustrative examples with finite and countably infinite numbers of equilibria are elaborated to substantiate the results.
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Zhang XM, Han QL, Ge X, Zhang BL. Passivity Analysis of Delayed Neural Networks Based on Lyapunov-Krasovskii Functionals With Delay-Dependent Matrices. IEEE TRANSACTIONS ON CYBERNETICS 2020; 50:946-956. [PMID: 30346302 DOI: 10.1109/tcyb.2018.2874273] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper is concerned with passivity of a class of delayed neural networks. In order to derive less conservative passivity criteria, two Lyapunov-Krasovskii functionals (LKFs) with delay-dependent matrices are introduced by taking into consideration a second-order Bessel-Legendre inequality. In one LKF, the system state vector is coupled with those vectors inherited from the second-order Bessel-Legendre inequality through delay-dependent matrices, while no such coupling of them exists in the other LKF. These two LKFs are referred to as the coupled LKF and the noncoupled LKF, respectively. A number of delay-dependent passivity criteria are derived by employing a convex approach and a nonconvex approach to deal with the square of the time-varying delay appearing in the derivative of the LKF. Through numerical simulation, it is found that: 1) the coupled LKF is more beneficial than the noncoupled LKF for reducing the conservatism of the obtained passivity criteria and 2) the passivity criteria using the convex approach can deliver larger delay upper bounds than those using the nonconvex approach.
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Guo Z, Ou S, Wang J. Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching. Neural Netw 2020; 122:239-252. [DOI: 10.1016/j.neunet.2019.10.012] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Revised: 09/04/2019] [Accepted: 10/17/2019] [Indexed: 11/12/2022]
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Zarra T, Galang MG, Ballesteros F, Belgiorno V, Naddeo V. Environmental odour management by artificial neural network - A review. ENVIRONMENT INTERNATIONAL 2019; 133:105189. [PMID: 31675561 DOI: 10.1016/j.envint.2019.105189] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2019] [Revised: 09/11/2019] [Accepted: 09/13/2019] [Indexed: 06/10/2023]
Abstract
Unwanted odour emissions are considered air pollutants that may cause detrimental impacts to the environment as well as an indicator of unhealthy air to the affected individuals resulting in annoyance and health related issues. These pollutants are challenging to handle due to their invisibility to the naked eye and can only be felt by the human olfactory stimuli. A strategy to address this issue is by introducing an intelligent processing system to odour monitoring instrument such as artificial neural network to achieve a robust result. In this paper, a review on the application of artificial neural network for the management of environmental odours is presented. The principal factors in developing an optimum artificial neural network were identified as elements, structure and learning algorithms. The management of environmental odour has been distinguished into four aspects such as measurement, characterization, control and treatment and continuous monitoring. For each aspect, the performance of the neural network is critically evaluated emphasizing the strengths and weaknesses. This work aims to address the scarcity of information by addressing the gaps from existing studies in terms of the selection of the most suitable configuration, the benefits and consequences. Adopting this technique could provide a new avenue in the management of environmental odours through the use of a powerful mathematical computing tool for a more efficient and reliable outcome.
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Affiliation(s)
- Tiziano Zarra
- SEED - Sanitary Environmental Engineering Division, Department of Civil Engineering, University of Salerno, via Giovanni Paolo II 132, 84084 Fisciano, SA, Italy.
| | - Mark Gino Galang
- Environmental Engineering Program, University of the Philippines, Diliman, Quezon City 1101, Philippines
| | - Florencio Ballesteros
- Environmental Engineering Program, University of the Philippines, Diliman, Quezon City 1101, Philippines
| | - Vincenzo Belgiorno
- SEED - Sanitary Environmental Engineering Division, Department of Civil Engineering, University of Salerno, via Giovanni Paolo II 132, 84084 Fisciano, SA, Italy
| | - Vincenzo Naddeo
- SEED - Sanitary Environmental Engineering Division, Department of Civil Engineering, University of Salerno, via Giovanni Paolo II 132, 84084 Fisciano, SA, Italy
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Guo Z, Liu L, Wang J. Multistability of Switched Neural Networks With Piecewise Linear Activation Functions Under State-Dependent Switching. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:2052-2066. [PMID: 30418927 DOI: 10.1109/tnnls.2018.2876711] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
This paper is concerned with the multistability of switched neural networks with piecewise linear activation functions under state-dependent switching. Under some reasonable assumptions on the switching threshold and activation functions, by using the state-space decomposition method, contraction mapping theorem, and strictly diagonally dominant matrix theory, we can characterize the number of equilibria as well as analyze the stability/instability of the equilibria. More interesting, we can find that the switching threshold plays an important role for stable equilibria in the unsaturation regions of activation functions, and the number of stable equilibria of an n -neuron switched neural network with state-dependent parameters increases to 3n from 2n in the conventional one. Furthermore, for two-neuron switched neural networks, the precise attraction basin of each stable equilibrium point can be figured out, and its boundary is composed of the stable manifolds of unstable equilibrium points and the switching lines. Two simulation examples are discussed in detail to substantiate the effectiveness of the theoretical analysis.
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Zhang F, Zeng Z. Multiple ψ -Type Stability and Its Robustness for Recurrent Neural Networks With Time-Varying Delays. IEEE TRANSACTIONS ON CYBERNETICS 2019; 49:1803-1815. [PMID: 29993797 DOI: 10.1109/tcyb.2018.2813979] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
In this paper, the ψ -type stability and robustness of recurrent neural networks are investigated by using the differential inequality. By utilizing ψ -type functions combined with the inequality techniques, some sufficient conditions ensuring ψ -type stability and robustness are derived for linear neural networks with time-varying delays. Then, by choosing appropriate Lipschitz coefficient in subregion, some algebraic criteria of the multiple ψ -type stability and robust boundedness are established for the delayed neural networks with time-varying delays. For special cases, several criteria are also presented by selecting parameters with easy implementation. The derived results cover both ψ -type mono-stability and multiple ψ -type stability. In addition, these theoretical results contain exponential stability, polynomial stability, and μ -stability, and they also complement and extend some previous results. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed criteria.
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Guo T, Wang L, Zhou M, Duan S. A multi-layer memristive recurrent neural network for solving static and dynamic image associative memory. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.12.056] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Liu P, Nie X, Liang J, Cao J. Multiple Mittag-Leffler stability of fractional-order competitive neural networks with Gaussian activation functions. Neural Netw 2018; 108:452-465. [DOI: 10.1016/j.neunet.2018.09.005] [Citation(s) in RCA: 43] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2018] [Revised: 09/05/2018] [Accepted: 09/07/2018] [Indexed: 11/28/2022]
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Liu P, Zeng Z, Wang J. Multistability of Recurrent Neural Networks With Nonmonotonic Activation Functions and Unbounded Time-Varying Delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2018; 29:3000-3010. [PMID: 28678718 DOI: 10.1109/tnnls.2017.2710299] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
This paper is concerned with the coexistence of multiple equilibrium points and dynamical behaviors of recurrent neural networks with nonmonotonic activation functions and unbounded time-varying delays. Based on a state space partition by using the geometrical properties of the activation functions, it is revealed that an -neuron neural network can exhibit equilibrium points with . In particular, several sufficient criteria are proposed to ascertain the asymptotical stability of equilibrium points for recurrent neural networks. These theoretical results cover both monostability and multistability. Furthermore, the attraction basins of asymptotically stable equilibrium points are estimated. It is shown that the attraction basins of the stable equilibrium points can be larger than their originally partitioned subsets. Finally, the results are illustrated by using the simulation results of four examples.
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Lu C, Zhang XM, Wu M, Han QL, He Y. Energy-to-Peak State Estimation for Static Neural Networks With Interval Time-Varying Delays. IEEE TRANSACTIONS ON CYBERNETICS 2018; 48:2823-2835. [PMID: 29994237 DOI: 10.1109/tcyb.2018.2836977] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper is concerned with energy-to-peak state estimation on static neural networks (SNNs) with interval time-varying delays. The objective is to design suitable delay-dependent state estimators such that the peak value of the estimation error state can be minimized for all disturbances with bounded energy. Note that the Lyapunov-Krasovskii functional (LKF) method plus proper integral inequalities provides a powerful tool in stability analysis and state estimation of delayed NNs. The main contribution of this paper lies in three points: 1) the relationship between two integral inequalities based on orthogonal and nonorthogonal polynomial sequences is disclosed. It is proven that the second-order Bessel-Legendre inequality (BLI), which is based on an orthogonal polynomial sequence, outperforms the second-order integral inequality recently established based on a nonorthogonal polynomial sequence; 2) the LKF method together with the second-order BLI is employed to derive some novel sufficient conditions such that the resulting estimation error system is globally asymptotically stable with desirable energy-to-peak performance, in which two types of time-varying delays are considered, allowing its derivative information is partly known or totally unknown; and 3) a linear-matrix-inequality-based approach is presented to design energy-to-peak state estimators for SNNs with two types of time-varying delays, whose efficiency is demonstrated via two widely studied numerical examples.
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Zhang XM, Han QL, Zeng Z. Hierarchical Type Stability Criteria for Delayed Neural Networks via Canonical Bessel-Legendre Inequalities. IEEE TRANSACTIONS ON CYBERNETICS 2018; 48:1660-1671. [PMID: 29621005 DOI: 10.1109/tcyb.2017.2776283] [Citation(s) in RCA: 42] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper is concerned with global asymptotic stability of delayed neural networks. Notice that a Bessel-Legendre inequality plays a key role in deriving less conservative stability criteria for delayed neural networks. However, this inequality is in the form of Legendre polynomials and the integral interval is fixed on . As a result, the application scope of the Bessel-Legendre inequality is limited. This paper aims to develop the Bessel-Legendre inequality method so that less conservative stability criteria are expected. First, by introducing a canonical orthogonal polynomial sequel, a canonical Bessel-Legendre inequality and its affine version are established, which are not explicitly in the form of Legendre polynomials. Moreover, the integral interval is shifted to a general one . Second, by introducing a proper augmented Lyapunov-Krasovskii functional, which is tailored for the canonical Bessel-Legendre inequality, some sufficient conditions on global asymptotic stability are formulated for neural networks with constant delays and neural networks with time-varying delays, respectively. These conditions are proven to have a hierarchical feature: the higher level of hierarchy, the less conservatism of the stability criterion. Finally, three numerical examples are given to illustrate the efficiency of the proposed stability criteria.
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Sliding Surface in Consensus Problem of Multi-Agent Rigid Manipulators with Neural Network Controller. ENERGIES 2017. [DOI: 10.3390/en10122127] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
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22
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Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2017.06.047] [Citation(s) in RCA: 47] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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