Quantized synchronization of memristive neural networks with time-varying delays via super-twisting algorithm
Introduction
Memristor is a kind of circuit element which shows the relationship between magnetic flux and charge. Its resistance varies and is decided by the charge flowing through it. Hence, by measuring the resistance of memristor, you can know the amount of charge flowing through it, thereby having the effect of memory charge. In 1971, Chua [1] pointed out from the logic and axiom point of view that there should also be a circuit component in the world, which represents the relationship between magnetic flux and charge. In 2008, Hewlett-Packard [2] studies made nano-meta-resistance devices for the first time, setting off a research boom in memristor. Additionally, memristor is an excellent method to implement artificial neural network synapses, and there are many applications with memristive neural networks (MNNs) such as [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [61], [62], [64], [65], [66], [67], [68].
Over the last few years, the dynamic analysis has been an important factor for the design of MNNs with many researches in recent years, such as passivity and passification [16], [17], [18], [19], [20], stability [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], etc. Synchronization problem, a vital part of dynamic research, has been fully studied. For instance, Yang et al. [31] considered synchronization of delayed MNNs with robust analysis approach. Wu et al. [32] considered anti-synchronization control of a type of MNNs. Cao et al. [33] discussed exponential synchronization control for MNNs with delays with interval matrix method. Wen et al. [34] researched synchronization control based on event triggering of MNNs with time-varying delay.
Recent years, with the vigorous development of the memristive neural networks, there is a need to design a new control scheme to suit for the variable MNNs. As well known, MNNs are synchronized via the design of the feedback controller. Therefore, we import a novel controller based on sliding mode control, which attracted wide concentration because of its effectiveness in robust control systems [35], [36]. Super-twisting algorithm (STA) is a significant category of the second-order sliding mode algorithm which has been proved to be an effective way to obtain chattering attenuation and accuracy improvement [37]. By using geometric method and time domain method [38] and by designing various Lyapunov functions [39], [40], the convergence time of STA is estimated.
It is generally known that network transfers the shared information in neural network system, which has certain advantages such as stability and flexibility of the system [41], [42]. At the same time, there still exist some problems due to specific limits caused by the insertion of network equipment including quantized signal, communication latency, loss of data, and other things. Therefore, the second-order sliding mode (SOSM) control scheme with signal quantization has been imported [43].
Inspired by the above discussion, we discuss synchronization via quantized super-twisting algorithm. Highlights of the main contributions in this paper are in details:
- (1)
The most important issue of synchronization control is the choice of control strategy. Most of the current synchronization control problems, as far as the reader is aware, include event triggering mechanisms, periodic control, and coupling control. Sliding mode control has not been taken into account due to the extensiveness of sliding mode control problems and the difficulty of Lyapunov functional selection. We import a feedback controller via super-twisting algorithm which aims at synchronizing the drive-response systems, which appears in memristive neural networks with time-varying delays for the first time.
- (2)
Another big problem to be overcome in this paper is the processing of time-varying delays. In this paper, the time-varying delays are continuous and derivable. The time-varying delays are common in existing neural networks. This paper cites some assumptions and lemmas to deal with the negative effects of time-varying delays, and gives the relevant constraints in the theorem. To reduce computational complexity, we provide two quantitative schemes with uniform quantizer and logarithmic quantization, which is shown its effectiveness in the example. Although there will be error before and after quantization, it has been proved by us that this error has no great influence on the conclusion.
- (3)
The processing of the memristive part of the MNNs also has difficulties, such as parameter mismatch caused by nonlinearity and switching. This paper has done some special treatment for this problem. The combination of memristive and sliding mode control is also a highlight of this paper. By using the method of Lyapunov functional, sufficient conditions are obtained to ensure global synchronization for a drive-response system.
The distribution in the paper is listed below. In Section 2, we will describe the system model, some essential preparations and the required assumptions. We give the main theoretical results in Section 3. Then, some illustrative examples and simulations will be presented in support of the theoretical results in Section 4. To the end, we present the conclusions in Section 5.
Notations: In this paper, R denotes the space of real number; N denotes the positive integer; (aij)n*n denotes the n × n matrix; || · ||1 denotes the 1-norm for a matrix or for a vector and || · || denotes the 2-norm for a matrix or for a vector. |a| is used to denote the absolute value of a. is used to represent a diagonal matrix. sign( · ) is used to denote the sign function. AT is used to denote the transpose of matrix A. λmax( · ) is used to represent the maximum eigenvalue of a matrix, λmin( · ) is used to represent the minimum eigenvalue of a matrix. q(z) denotes the quantized value of a vector z.
Section snippets
Preliminaries
In this part, we present a mathematic model, which is based on the quantized super-twisting algorithm model for MNNs.
Uniform quantization scheme
Consider the uniform quantization scheme as follows:where z1 ∈ R is the state, which is ready to be quantized and ρ is positive and denotes the quantitative parameter. r( · ) represents the nearest integer operation which means that the positive element of decimal part 0.5 is rounded up to the nearest positive integer whereas negative element of decimal part 0.5 is rounded down to the nearest negative integer. We define that which follows that |e(t)| ≤ 0.5ρ.
Examples
We consider two-neuron memristive neural networks with time-varying delays as follows:where is the state vector, A⋆(x(t)) and B⋆(x(t))is defined as:where and
Conclusions
In this paper, we discussed the quantized synchronization of MNNs with time-varying delays via super-twisting algorithm. A feedback controller via super-twisting algorithm was introduced with two quantized schemes proposed with uniform quantizer and logarithmic quantization, which has been presented to MNNs for the first time. By using the method of Lyapunov functional, sufficient conditions were obtained to guarantee global synchronization of drive-response systems. Finally, the effectiveness
Declaration of Competing Interest
There is no interest for anything in this paper.
Bo Sun was admitted to School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China in 2017. His research interests include neural network, memristor, computer version and deep learning.
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Bo Sun was admitted to School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China in 2017. His research interests include neural network, memristor, computer version and deep learning.
Shiping Wen received the M.S degree in Control Science and Engineering, from School of Automation, Wuhan University of Technology, Wuhan, China, in 2010, and received the Ph.D degree in Control Science and Engineering, from School of Automation, Huazhong University of Science and Technology, Wuhan, China, in 2013. He is currently a Professor at School of Computer Science and Engineering, University of Electronic Science and Technology, Chengdu, China. His current research interests include memristor-based circuits and systems, computer version and deep learning. Currently, he serves as an associate editor for IEEE Access.
Shengbo Wang was admitted to School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China in 2016. He will work towards the M. Eng. degree from University of Electronic Science and Technology of China, Chengdu, China. His research interests include neural network, memristor, computer version and deep learning.
Tingwen Huang is a professor at Texas A & M University-Qatar. He received his B.S. degree from Southwest Normal University (now Southwest University), China, 1990, his M.S. degree from Sichuan University, China, 1993, and his Ph.D. degree from Texas A & M University, College Station, Texas, 2002. After graduated from Texas A & M University, he worked as a Visiting Assistant Professor there. Then he joined Texas A & M University at Qatar (TAMUQ) as an Assistant Professor in August 2003, then he was promoted to Professor in 2013. His research interests include neural networks based computational intelligence, distributed control and optimization, nonlinear dynamics and applications in smart grids. He has published more than three hundred peer-review reputable journal papers, including more than one hundred papers in IEEE Transactions. Currently, he serves as an associate editor for four journals including IEEE Transactions on Neural Networks and Learning Systems, IEEE Transactions on Cybernetics, and Cognitive Computation.
Yiran Chen received B.S and M.S. from Tsinghua University and Ph.D. from Purdue University in 2005. After five years in industry, he joined University of Pittsburgh in 2010 as Assistant Professor and then promoted to Associate Professor with tenure in 2014, held Bicentennial Alumni Faculty Fellow. He now is a tenured Professor of the Department of Electrical and Computer Engineering at Duke University and serving as the co-director of Duke Center for Evolutionary Intelligence (CEI), focusing on the research of new memory and storage systems, machine learning and neuromorphic computing, and mobile computing systems. Dr. Chen has published one book and more than 300 technical publications and has been granted 93 US patents. He is the associate editor of IEEE TNNLS, IEEE TCAD, IEEE D&T, IEEE ESL, ACM JETC, ACM TCPS, and served on the technical and organization committees of more than 40 international conferences. He received 6 best paper awards and 14 best paper nominations from international conferences. He is the recipient of NSF CAREER award and ACM SIGDA outstanding new faculty award. He is the Fellow of IEEE.
Peng Li received the Ph.D. degree in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in 2003. He is currently a Professor with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA. His current research interests include integrated circuits and systems, computer-aided design, brain-inspired computing, computational brain modeling, and machine learning and its hardware realization in very large scale integration. Dr. Li was a recipient of various distinctions including four IEEE/ACM Design Automation Conference Best Paper Awards for his research, the ISCAS Honorary Mention Best Paper Award from the Neural Systems and Applications Technical Committee of IEEE Circuits and Systems Society, the IEEE/ACM William J. McCalla ICCAD Best Paper Award, the U.S. National Science Foundation CAREER Award, two Inventor Recognition Awards from Microelectronics Advanced Research Corporation, two Semiconductor Research Corporation Inventor Recognition Awards, and the William and Montine P. Head Fellow Award and TEES Fellow Award from the College of Engineering, Texas A&M University. He was an Associate Editor of the IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems from 2008 to 2013 and the IEEE Transactions on Circuits and Systems-Part II: Express Briefs from 2008 to 2016.