Neural networks letterDissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches
Highlights
► We investigate the quasi-synchronization of discontinuous neural networks. ► The global dissipativity for the discontinuous system can be guaranteed. ► The quasi-synchronization of two non-identical systems can further be ensured. ► We give some numerical examples to verify the theoretical results.
Introduction
Neural networks with discontinuous (or non-Lipschitz, or nonsmooth) neuron activations, have been found useful to address a number of interesting engineering tasks, such as dry friction, impacting machines, systems oscillating under the effect of an earthquake, power circuits, switching in electronic circuits and many others, and therefore have received a great deal of attention in the literature (Cortés, 2008, Danca, 2002, Forti and Nistri, 2003, Forti et al., 2005, Liu and Cao, 2009, Lu and Chen, 2006, Lu and Chen, 2008). In linear and nonlinear programming, the discontinuous neural networks (DNNs) are able to execute the circuit equilibrium points coinciding with the constrained critical points of the objective function (Chong et al., 1999, Ferreira et al., 2005, Forti and Tesi, 1995). The best property of such networks that should be stressed is the global convergence in finite time, in comparison to smooth dynamical systems which can only converge as time goes to infinity. Such a property seems especially important in a global optimization problem since the minimum can be computed in real time (Forti and Nistri, 2003, Wang and Xiao, 2010).
In the literature of analyzing DNNs, fundamental results have been established on (robust) stability or convergence of the equilibrium point or periodic solutions for delayed Hopfield DNNs (Forti and Nistri, 2003, Forti et al., 2005, Liu and Cao, 2009) and Cohen–Grossberg DNNs (Lu and Chen, 2006, Lu and Chen, 2008). The stability problem of an equilibrium point is indeed central to the analysis of a dynamic system. Nevertheless, from a practical point of view, it is not always the case that the orbits of the neural network approach a single equilibrium point. It is possible that there is no equilibrium point in some situations. Therefore, the concept on dissipativity was introduced (Cao et al., 2006, Hale, 1989, Liao and Wang, 2003, Song and Cao, 2008) and has applications in the areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control (Liao & Wang, 2003). In this paper, we continue to consider the global dissipativity problem of neural networks, but the activations are not assumed to be continuous.
Synchronization, that means two or more systems share a common dynamical behavior, which can be induced by coupling or by external forcing, is a basis to understand an unknown dynamical system from one or more well-known dynamical systems. From Pecora and Carroll (1990), chaotic synchronization has become a hot topic in nonlinear dynamics due to theoretical significance and potential applications. So far, many types of synchronization have been presented, such as identical or complete synchronization, generalized synchronization, phase synchronization, anticipated and lag synchronization (Liang et al., 2008, Luo, 2009). Recently, the quasi-synchronization issue has received a great deal of attention in the literature mainly due to the unavoidability of parameter mismatches between two systems in practical synchronization implementations (Astakhov et al., 1998, Huang et al., 2009, Jalnine and Kim, 2002, Masoller, 2001, Shahverdiev et al., 2002). Generally, mismatched parameters always implies that the synchronization error could not approach zero with time, but fluctuates. However, it is important to know the region of the synchronization error and control it within a small region around zero, i.e., quasi-synchronization. So, in this paper, the quasi-synchronization of master–response systems with mismatched parameters is investigated.
For these purposed, matrix measure is introduced to deal with matrix inequalities, which can be positive or negative, in comparison to the matrix norm which should always be nonnegative. Owing to these special properties of matrix measure, the results obtained via this approach are usually less restrictive than those via matrix norm (He and Cao, 2009, Vidyasagar, 1993). Another advantage of this approach is avoiding constructing Lyapunov function in the proof. The main contribution of this paper includes three aspects. First, for the differential equations with discontinuous right-hand sides, the concept of Filippov solution (Filippov, 1988) is introduced and the existence of solution is proved for the DNNs. Second, the global dissipativity of Filippov solution is considered by using the matrix measure approach and the generalized Halanay inequalities (Halanay, 1996, Wen et al., 2008). Third, for the two DNNs with parameter mismatches, the quasi-synchronization issue is discussed also by the matrix measure approach. Furthermore, the complete synchronization between two coupled identical systems can be studied as a special case of the quasi-synchronization.
The rest of the paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, the existence of Filippov solutions of the DNNs is considered and the global dissipativity conditions is obtained by matrix measure approach. In Section 4, the quasi-synchronization of master–response systems with discontinuous activations and parameter mismatches is discussed by the matrix measure method. In Section 5, simulation results aiming at substantiating the theoretical analysis are presented. This paper is concluded in Section 6.
Section snippets
Model formulation and preliminaries
In this paper, we consider the following neural networks described by the following differential equations where is the state vector associated with the neurons; is an diagonal matrix with ; and are the time-varying connection weight matrix and the delayed connection weight matrix, respectively;
Existence and global dissipativity of Filippov solutions
In this section, we prove that under some conditions, system (1) exists solutions globally in the sense of Filippov, and further show the dissipativity of the existed solutions under the same conditions.
Theorem 1 Suppose that satisfies a growth condition ( ): there exist constants , with such thatThen, there exists at least one solution of system (1) in the sense of Eqs. (7).
Proof Based on the detailed discussions in Section 2, the set-valued map
Quasi-synchronization of discontinuous neural networks
In this section, we consider the quasi-synchronization issue of DNNs with parameter mismatches.
Consider the model (1) as the master system, the response system is where , , are coefficient matrices with or or , i.e., there exist parameter mismatches between the master system (1) and response system (23). is the feedback controller. Let , and
Numerical examples
Example 1 Consider the second-order master system of a discontinuous delayed neural network (1) with , , , , , the activation function is defined as are the state vectors. Let , from Theorem 2, the neural network (1) is global dissipativity with , as shown by Fig. 1. In Example 1, let the
Conclusions
In this paper, based on the concept of Filippov solution, dissipativity and quasi-synchronization of DNNs are studied. By the matrix measure approach and generalized Halanay inequalities, several sufficient conditions are derived to guarantee the global dissipativity for the discontinuous single system and the quasi-synchronization of two non-identical systems with parameter mismatches respectively. These results are novel since there are few works on the dissipativity or quasi-synchronization
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China under Grants No. 60874088, 11072059, 60974015 and 60804044, the Domestic Ph.D. student visiting program, and the First Doctoral Academic New People Awards.
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