Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches

• Published in: IEEE transaction on neural networks and learning systems Vol. 29; no. 9; pp. 4201 - 4211
• Main Authors: Liu, Yang, Zhang, Dandan, Lou, Jungang, Lu, Jianquan, Cao, Jinde
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.09.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Decomposition; Delay effects; Delays; Feasibility studies; Global μ-stability; Image color analysis; Liapunov functions; Mathematical analysis; Matrix methods; Multiplication; Neural networks; quaternion-valued linear matrix inequality (LMI); quaternion-valued neural network (QVNN); Quaternions; Stability analysis; Stability criteria; time delay
• ISSN: 2162-237X; 2162-2388; 2162-2388
• DOI: 10.1109/TNNLS.2017.2755697

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: <inline-formula> <tex-math notation="LaTeX">ij=-ji=k,~jk=-kj=i </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">ki=-ik=j </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">i^{2}=j^{2}=k^{2}=ijk=-1 </tex-math></inline-formula>. With the Lyapunov function method, some criteria are, respectively, presented to ensure the global <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>-stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>-stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.