Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays☆
Introduction
Quaternion, which was found by the British mathematician W. R. Hamilton in 1843, didn’t get much attention for quite a long time, let alone the actual applications. Quaternion multiplication does not meet the commutative law, so the investigation on quaternion is much harder than that on plurality, which is one reason for the slow development of quaternion. Fortunately, over the past 20 years, especially in algebra area, quaternion has been a hot topic for the effective applications in the real world. In the 1990s, the determinant of quaternion matrix was defined from the viewpoint of group theory, and many important results on quaternion have been obtained since then [3]. So far, quaternion algebra has been already applied successfully to communications problems and signal processing, such as array processing [24], color image processing [9], wind forecasting [32]. There is no deny that quaternion is an important topic needing much more attentions. Besides, neural networks have been one of the most interesting focuses due to the promising development and wide applications in diverse science and engineering fields such as associative memories [50], pattern recognition [8], control engineering [1]. So far, neural networks have been extensively investigated, see [4], [44], [45], [46], [51] and the references therein.
Quaternion-valued neural networks (QVNNs), which can be seen as a generic extension of complex-valued neural networks (CVNNs) or real-valued neural networks (RVNNs), are much more complicated than CVNNs for their quaternion-valued states, quaternion-valued connection weights and quaternion-valued activation functions. In complex domain, according to Liouville’s theorem [25], every bounded entire function must be constant, i.e., g(z) is a constant function if g(z) is bounded and analytic. That is, the activation functions of CVNNs cannot be both bounded and analytic, otherwise they are constants. However, in quaternion field the analyticity of general quaternion-valued functions has not been rigorously examined even if such functions have been implemented in [38]. Stringent Cauchy–Riemann–Fueter (CRF) and the generalized Cauchy–Riemann (GCR) conditions [10], [29] guarantee that globally analytical quaternion-valued functions are only linear functions and constants, respectively, to ensure the class of the quaternion-valued functions analytic. Therefore, choosing appropriate quaternion-valued activation functions of QVNNs remains an open issue. Fortunately, many authors have considered this analyticity problem [26] and a solution to it has also been given. That is, they adopt a local alternative to the CRF conditions, namely the local analyticity condition (LAC) [29] which can allow us to use standard activation functions, such as the function tanh .
Additionally, in the last 20 years, basically trying to combine basic heuristic methods in higher level frameworks, a new kind of approximate algorithm has been proposed aiming at exploring a search space efficiently and effectively, which are nowadays commonly called meta-heuristics. The term meta-heuristic derives from the composition of two Greek words. Heuristic derives from the verb heuriskein (euriskein) meaning “to find”, while the suffix meta means “beyond, in an upper level” [39]. Recently, meta-heuristic algorithms have gained noteworthy attention for their abilities to solve difficult optimization problems in engineering, economics, business, finance, etc. Moreover, meta-heuristics are powerful in solving the hard optimization problems, which offer higher flexibility and perform better than others for each problem. Compared with other meta-heuristics methods, such as simulated annealing, threshold accepting and tabu search, the usage frequencies of artificial neural networks from 2007 to 2011 are all higher than others, which shows that artificial neural networks have much more advantages [40], more detailed comparisons can be found in [39], [41], [42].
Generally speaking, there are two kinds of artificial neural networks according to the connectionism: feedforward neural networks and recurrent neural networks. As a kind of recurrent neural networks, despite the number of applications of QVNNs is comparatively less than that of RVNNs or CVNNs , it has been increasing recently. As is known to all, CVNNs play an important role in practical applications in various fields of engineering such as communication and quantum mechanics [12], which can result in improved performance of the geometrical affine transformation in two-dimensional space. For another case where the data is three dimensional, such as color images and body images which can be also processed by many neurons of RVNNs or CVNNs, the processing efficiency can be increased by implementing directly encoding in terms of QVNNs. Because quaternion, as a class of hypercomplex number systems, can be regarded as a four-dimensional extension of complex numbers. In recent years, the applications of QVNNs have been also widely investigated. One practical application by QVNNs is the 3D geometrical affine transformation, especially spatial rotation, which can be represented based on QVNNs efficiently and compactly [15], [27]. Other practical applications of QVNNs are image impression, color night vision [16], etc. Therefore, QVNNs deserve further investigation for the broad applications in various fields.
To the best knowledge of the readers, time delays are ubiquitous due to the processing and transmission of signals and the finite switching speed of amplifiers, which may cause oscillation, instability and other poor performance [33], [52]. Up to now, a large amount of research results for the stability analysis of neural networks with time delays have been reported, see [5], [34], [35] and the references therein. Then, we want to know if the equilibrium of a QVNN with time delays can be still stable. In some literatures, for CVNNs, time delays are assumed to be bounded [13], [47], i.e., τpq(t) ≤ τ, where τpq(t) is a time-varying delay and τ is a fixed constant. Even more, there are also many stability results on the bounded derivative of time delay τpq(t) [6], i.e., . In [43], the problem of the global μ-stability of CVNNs with unbounded time-varying delays is considered where some useful results are presented in the form of linear matrix inequalities (LMIs) using two different approaches. In [14] and [48], authors consider time delays in predator–prey models and in the input, respectively, which can help to promote the study of delayed QVNNs, especially the investigation of practical applications of QVNNs in the real world. However, traditional methods used to solve the stability problems in complex domain above cannot be directly applied to the similar problems in quaternion field. Therefore, studying dynamical behaviors of QVNNs with different methods is still open and challenging.
To address those issues aforementioned, in this paper, we study the global μ-stability of QVNNs with unbounded time-varying delays by using a plural decomposition method of quaternion, i.e., a quaternion-valued matrix can be uniquely expressed as the sum of two complex-valued matrices. Compared with existing methods in complex domain [43], the plural decomposition method of quaternion can help to avoid many redundant computations. The global μ-stability problem of QVNNs is firstly studied by decomposing a quaternion-valued matrix into two complex-valued matrices uniquely and they are mapped into two systems with the same dimensions.
However, there are two biggest difficulties in dealing with the global μ-stability of QVNNs with unbounded time-varying delays: the non-commutativity of quaternion multiplication and the choice of an appropriate Lyapunov–Krasovskii functional. For the first one, fortunately, we can transform into complex-valued systems with the plural decomposition method of quaternion, i.e., where . Then the QVNN is decomposed into two complex-valued systems which are equivalent to the original QVNN. Consequently, the complexity of calculations is greatly reduced for avoiding the non-commutativity of quaternion multiplication. Whereas the second difficulty can be solved by the form of the quaternion-valued model of the considered QVNN. Considering from Hermitian matrices in complex field, the Lyapunov–Krasovskii functional can be chosen by certain Hermitian matrix and the activation function, which gives the complex-valued LMIs, i.e., negative definite Hermitian matrices. That is to say, the two main difficulties and challenges above mentioned can be well solved with proper methods adopted in this paper.
The remaining part of this paper is organized as follows. In Section 2, the system model description and preliminaries are provided. In Section 3, some sufficient conditions for the global μ-stability of QVNNs are derived by decomposing a QVNN into two equivalent complex-valued systems, and a corollary on power stability is obtained as well. In Section 4, two numerical examples are given to show the effectiveness of our theoretical analysis. Some necessary discussions are presented in Section 5. Finally, Section 6 concludes this paper.
Notations: Throughout this paper, denote, respectively, the set of all m × n real-valued, complex-valued and quaternion-valued matrices. is the set of all quaternion self-conjugate matrices and quaternion positive definite matrices, respectively. Function denotes the quaternion-valued function, where z11(t), z12(t), z21(t), z22(t) are all real-valued functions and i, j, k obey Hamilton rules: . For any where denotes conjugate transpose of h. If g is a quaternion-valued function, denotes conjugate of g. If vector x* denotes conjugate transpose of x. If matrix A* and λmax (A)(λmin (A)) denote the conjugate, the conjugate transpose and the maximum (minimum) eigenvalue of A, respectively.
Section snippets
Preliminaries
Consider the following QVNN with unbounded time-varying delays where is the state vector of neurons at time t. diag with dp > 0, is the constant matrix, whose diagonal element dp denotes the self-feedback connection weight of the pth neuron, for . and denote the vector-valued
Main results
The stability analysis for CVNNs has been obtained, but the theoretical result of the global μ-stability of QVNNs has not been given so far. In this section, we will study the global μ-stability of QVNNs with the plural decomposition method of quaternion.
Theorem 1 Assume that Assumptions 1, 2 and 3 hold. The equilibrium point of QVNN (1) is μ-stable, if there exist constants α1, α2 ≥ 0, three Hermitian matrices P, Q, R > 0, a diagonal matrix G > 0, and a non-negative continuous differential function μ(t
Examples
In this section, two examples are provided to illustrate the effectiveness of the proposed method.
Example 1 Consider a two-dimensional QVNN with unbounded time-varying delays:
where and
Discussion
As shown above, time delays often occur in the interaction among neurons for neural networks in the application to electronic implementation. On one hand, the upper bound of delays is not sure in this paper which is an uncertain factor, so the addressed QVNN has robustness, and the detailed analysis can be seen in [18] where the robust bifurcation of the considered system is investigated and an innovative approach is used to analyze the system parameters which will be very helpful to further
Conclusion
In this paper, the global μ-stability of QVNNs with unbounded time-varying delays is investigated. Due to Hamilton rules of quaternion: the multiplication of quaternion is not commutative which is the biggest difficulty to study the stability of delayed QVNNs. Despite they can be dealt by decomposing a QVNN into real-valued systems, it is still a fussy work needing a lot of calculations. Fortunately, the plural decomposition method gives a good
Acknowledgments
The authors would like to thank the Editor in Chief, the Associate Editor and the anonymous reviewers for a number of valuable comments and constructive suggestions that have improved the quality of this paper.
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This work was partially supported by the Zhejiang Provincial Natural Science Foundation of China under grant LY14A010008, the China Postdoctoral Science Foundation under Grant no. 2015M580378, and the National Natural Science Foundation of China under Grant nos. 61573102 and 61573096.