Finite Time Anti-synchronization of Quaternion-Valued Neural Networks with Asynchronous Time-Varying Delays
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Finite Time Anti-synchronization of Quaternion-Valued Neural Networks with Asynchronous Time-Varying Delays Zihan Li1 · Xiwei Liu1,2,3 Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider the finite time anti-synchronization (A-SYN) of master-slave coupled quaternion-valued neural networks, where the time-varying delays can be asynchronous and unbounded. Without adopting the general decomposition method, the quaternion-valued state is considered as a whole, which greatly reduces the hassle of further analysis and calculations. The designed controller is delay-free, and the terms with time delay do not need to be bounded globally. Several sufficient conditions for ensuring the finite time A-SYN are obtained under 1-norm and 2-norm respectively. The A-SYN error will be proved to evolve from the initial value to 1 in finite time, and evolve from 1 to 0 also in finite time, hence the finite time A-SYN is proved, which is called two-phases-method. Moreover, adaptive rules for control strengths are also designed to realize the finite time A-SYN. Lastly, a numerical example is presented to demonstrate the correctness and effectiveness of our obtained results. Keywords Anti-synchronization · Asynchronous · Finite time · Quaternion-valued neural network · Time delay · Unbounded
1 Introduction In 1843, British mathematician Hamilton introduced quaternion, which was an extension of complex numbers. However, quaternion did not get too much attention or development for quite a long time, where one significant reason was that, unlike complex numbers, quaternion multiplication did not satisfy the commutative law. By the late twentieth century, quaternion ushered in recovery due to its effectiveness in describing spatial rotations. Specifically,
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Xiwei Liu [email protected] Zihan Li [email protected]
1
Department of Computer Science and Technology, Tongji University, Shanghai 201804, China
2
Key Laboratory of Embedded System and Service Computing (Tongji University), Ministry of Education, Shanghai 201804, China
3
Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China
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Z. Li, X. Liu
researchers found that quaternion gave a simple way to encode the rotation information into four numbers, which was more compact and simpler than matrices and Euler angles. Hence, in recent years, quaternion has been widely applied in computer graphics, computer vision, robotics, navigation, and so on. Neural network [1,2] has become one of the most popular research fields in the past 30 years due to the promising development and wide applications in signal processing, pattern recognition, optimization problems, deep learning, etc. Just as real-valued neural networks (RVNNs) are extended by complex-valued neural networks (CVNNs), QVNNs can also be regarded as an extension of CVNNs, where the neurons’ state, activation functions, self-feedback weights, connections weights, and external inputs are all quate
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