An Application of the Lurie Problem in Hopfield Neural Networks

Pinheiro, Rafael Fernandes; Colón, Diego

doi:10.1007/978-3-319-91217-2_26

Cited by 5 publications

(6 citation statements)

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“…In our bibliographic research, we found papers that relate in some way HNN to AD (Morrison et al, 2017;Sergio et al, 2009;Swietlik et al, 2019;Thuraisingham, 2015;Tabekoueng et al, 2020) and papers that connect LP with HNN (Aouiti et al, 2019;Kaskurewicz and Bhaya, 1995;Liao and Yu, 2008;Pinheiro and Colón, 2019). But, we did not find works that relate the LP, HNN and AD.…”

Section: Introductionmentioning

confidence: 66%

“…2). In Liao and Yu (2008) and Pinheiro and Colón (2019) is shown that HNN can be considered a special case of MIMO Lurie type system. Thus, considering the equation 5after some changes of variables, it can take the following form:…”

Section: Lurie's Problem (Lp)mentioning

confidence: 99%

“…This problem was studied for many researchers, as Krasovskii (1953), Popov (1961), and Kalman (1963). Research on the LP also went on to other areas, such as chaos synchronization (Liao and Yu, 2008); convex approach (Gapski and Geromel, 1994); neural networks (Pinheiro and Colón, 2019); linear parameter varying (LPV) system (Yu and Liao, 2019); and µ analysis (Abtahi and Yazdi, 2019). In works like Liao and Yu (2008) and Pinheiro and Colón (2019) is shown that HNN is a particular case of Lurie type system.…”

Section: Introductionmentioning

confidence: 99%

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Relating Lurie's problem, Hopfield's network and Alzheimer's disease

Pinheiro¹,

Colón

Reinoso

2020

Anais Do Congresso Brasileiro De Automática 2020

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Alzheimer's disease is a degenerative brain disorder that affects millions of people around the world and still without cure. A very common application of Hopfield neural networks is to simulate a human memory as well as to evaluate problems of degeneration and memory loss. On the other hand, from the control area, one has Lurie's problem, which emerged in the 1940s and which still does not have a general solution. However many works and results came in an attempt to solve it. In this paper, the Hopfield's network is shown as a particular case of Lurie's problem, then one of the consequences of Alzheimer's disease, memory failure, is modeled using Hopfield's networks and nally a recent result of Lurie's problem is applied to the computationally modeled disease to correct the problem of memory loss. The correction is made using a controller via DK-iteration. Simulations are performed to validate the computational model of the disease and to demonstrate the effectiveness of the application of the recent Lurie's problem theorem. Therefore, in addition to the results presented, this work aims at encouraging the researches in the area, so that in the future, better diagnostic and treatment conditions will be achieved.

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Section: Introductionmentioning

confidence: 66%

Section: Lurie's Problem (Lp)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Relating Lurie's problem, Hopfield's network and Alzheimer's disease

Pinheiro¹,

Colón

Reinoso

2020

Anais Do Congresso Brasileiro De Automática 2020

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“…Many researchers have worked on this problem since the years 1950s and 1960s, among whom can be cited as Aizerman (1947), with the well known Aizerman conjecture, and Krasovskii (1953), Popov (1961), and Kalman (1963). In a way, research on the Lurie’s problem took a bigger leap in the 1980s, when works began to appear that linked the problem to other areas and approaches such as neural networks (Liao and Yu, 2008; Pinheiro and Colón, 2019); complex network (Li et al, 2012); chaos and chaos synchronization (Kazemy and Farrokhi, 2017); convex approach to the Lurie problem (Gapski and Geromel, 1994); linear parameter varying (LPV) system (Yu and Liao, 2019); uncertain systems (Tan and Atherton, 2003); Integral Quadratic Constraints (IQC) and Zames-Falb multipliers (Carrasco et al, 2016); μ analysis (Abtahi and Yazdi, 2019; Lee and Juang, 2005); and more recently the application of Lurie’s problem in modern control systems such as Hopfield neural network controls (Pinheiro and Colón, 2021), modeling Alzheimer’s disease (Pinheiro and Colón, 2020), tracking differential extended state observer (Wang et al, 2020) and in control rates of the extended state observer (ESO) for speed control system for the pitching axis of a remote sensing camera (Liu, 2020). In addition to all these new lines of application and study of the Luries problem, its study remains current in the aeronautical field, as can be seen in Imani and Montazeri-Gh (2019).…”

Section: Introductionmentioning

confidence: 99%

Analysis and synthesis of single-input-single-output Lurie type systems via H∞ mixed-sensitivity

Pinheiro

Colón

2021

Transactions of the Institute of Measurement and Control

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The goal of this paper is to present a different approach to the analysis of the absolute stability of Lurie type systems in the single-input-single-output (SISO) case using robust control theory. The proposed technique enables the design of controllers via [Formula: see text] mixed-sensitivity (S/KS/T), where, besides making the system absolutely stable, the performance problem can also be solved. In addition, it is also demonstrated that it is possible to make use of this new approach in time-delay Lurie type systems. Thus, through a new methodology, this work paves the way to the study of the absolute stability of multiple-inputs-multiple-outputs (MIMO) systems, aiming at a better generalization of the theory and enabling applications in other areas, such as neural networks. Examples, numerical simulations and application in Chua’s circuit are given to illustrate the results.

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“…and Kalman (KALMAN, 1963). The research in the LP has grown since 1980s, when different papers were published linking this problem to other areas and approaches, such as neural networks (FORTI et al, 1994;KASZKUREWICZ;BHAYA, 1995;PINHEIRO;COL ÓN, 2019;TOWNLEY et al, 2000;YI;HENG;VADAKKEPAT, 2002) LIAO, 2019). In addition to all these new lines of applications, its study remains current in the aeronautical field, as can be seen in (IMANI, 2019).…”

Section: Related Workmentioning

confidence: 99%

The Lurie problem and its relationships with artificial neural networks and alzheimer-like disease.

Pinheiro¹

Self Cite

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I am a Christian, that means: I believe in the divinity of Jesus Christ together with Tycho Brahe, Copernicus, Descartes, Newton, Fermat, Leibniz, Pascal, Grimaldi, Euler, Guldin, Boscowitsch, Gerdil, with all the great astronomers, all the great researchers of the natural sciences, all the great mathematicians of past centuries. And if you ever ask me by the reason, I will be happy to explain it. You will see that my conviction is the result of careful study and not of prejudice."-Augustin Louis Cauchy (1789-1857), mathematician and physicist-(GUTZWILLER, 2000) RESUMONesta tese, são apresentados estudos que fornecem novas condições suficientes para a análise da estabilidade absoluta de sistemas do tipo Lurie para os casos Single-Input-Single-Output (SISO) e Multiple-Input-Multiple-Output (MIMO). Técnicas de projetos de controladores para sistemas tipo Lurie são deselvolvidas e aplicadas em redes neurais artificiais. Os resultados apresentados são baseados na teoria de controle H ∞ , sendo, para o caso SISO, usada a técnica de sensibilidade mista (S/KS/T) e para o caso MIMO utilizada a técnica de análise e síntese µ. Além disso, obtem-se condições para sistemas com atraso no tempo, mostrando que os resultados podem também ser utilizados quando existem atrasos. Na aplicação, primeiramente se faz a modelagem de uma neuropatologia que simula perda de memória usando redes de Hopfield em tempo contínuo, que é denominada neste trabalho como doença tipo-Alzheimer. Em seguida, o controlador desenvolvido, baseado na teoria deste trabalho, é aplicado para corrigir o problema de falha de memória. Exemplos são apresentados para ilustrar a teoria e simulações são realizadas para validar e mostrar a eficácia dos resultados.

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An Application of the Lurie Problem in Hopfield Neural Networks

Cited by 5 publications

References 6 publications

Relating Lurie's problem, Hopfield's network and Alzheimer's disease

Relating Lurie's problem, Hopfield's network and Alzheimer's disease

Analysis and synthesis of single-input-single-output Lurie type systems via H∞ mixed-sensitivity

The Lurie problem and its relationships with artificial neural networks and alzheimer-like disease.

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