Elsevier

Neural Networks

Volume 106, October 2018, Pages 67-78
Neural Networks

Passivity and stability analysis of neural networks with time-varying delays via extended free-weighting matrices integral inequality

https://doi.org/10.1016/j.neunet.2018.06.010Get rights and content

Abstract

This paper is concerned with the problem of passivity for uncertain neural networks with time-varying delays. First, the recently developed integral inequality called generalized free-matrix-based integral inequality is extended to estimate further tight lower bound of integral terms. By constructing a suitable augmented LKF, an enhanced passivity condition for the concerned network is derived in terms of linear matrix inequalities (LMIs). Here, the integral terms having three states in its quadratic form is estimated by the proposed Lemma. As special cases of main results, for neural networks without uncertainties, passivity and stability conditions are derived. Through three numerical examples, it will be shown that the developed conditions can promote the level of passivity and stability criteria.

Introduction

The stability of neural networks has received a great attention during the last decades owing to their excellent extensions in many scientific and engineering fields such as parallel computing, pattern recognition, associative memories, image processing, secure communication, and other scientific areas. For details, see Kwon, Park, Lee, Park, and Cha (2013), Zhang, He, Jiang, and Wu (2016) and Zhang, Wang, and Liu (2014) and reference therein.

It is well known that time-delay is a natural phenomenon but very important factor that should be considered in mathematical modeling of physical systems because of their occurrence caused by the finite switching speed of amplifiers and signal propagation. Time-delays are often a source of performance degradation such as oscillation and divergence. Since the dynamic behavior of equilibrium points has a major influence on the application of neural networks, considerable time and efforts have been concentrated on stability analysis for dynamic systems with time-delay by many researchers Fu et al. (2010), Gu (1997), Kwon, Park, Park et al. (2014), Kwon et al. (2016a), Kwon et al. (2016b), Kwon, Park, Park, Lee et al. (2014), Lee et al. (2013), Li and Cao (2017), Li and Song (2017), Liu et al. (2006), Park, Kwon et al. (2015), Park et al. (2018a), Park et al. (2018b), Park et al. (2011), Park, Lee et al. (2015), Park et al. (2016), Seuret and Gouaisbaut (2013), Velmurugan et al. (2017), Zeng, He, Wu et al. (2015), Zhang and Han (2011). Especially, with the use of Lyapunov–Krasovskii theorem and some mathematical techniques, delay-dependent stability conditions are derived within the framework of tractable LMIs of which the solutions can be easily found by various algorithms. In developing delay-dependent stability criteria for systems with time-delays, one of the main issue is how to increase maximum delay bounds ensuring the asymptotic stability since the maximum value of upper bounds of time-varying delay has been recognized as one of the key demonstration for the superiority and less conservatism of developed stability criteria.

Roughly speaking, there are two major methods to enhance the feasible region: (1) more tighter estimation of the quadratic integral terms coming out from the time-derivative of LKF and (2) the construction of Lyapunov–Krasovskiifunctional (LKF). Since Seuret & Gouaisbaut (2013) have introduced Wirtinger-based integral inequality(WBII) which breaks the conservatism of Jensen inequality, some remarkable methods for bounding integral inequality have emerged as such a rapid pace. Some representative integral inequalities for estimating tight lower bound of integral terms are Wirtinger-based double integral inequality (Park, Kwon et al., 2015), auxiliary function-based integral inequality Park, Lee et al. (2015), Park et al. (2016), free-matrix-based integral inequality (Zeng, He, Wu et al., 2015), generalized free-weighting-matrix integral inequality (Zhang et al., 2017), generalized integral inequality (Park et al., 2018b), and so on. Other well known techniques for reducing the conservatism are delay-partitioning (Gu, 1997), reciprocally convex optimization (Park et al., 2011), zero equality (Kim et al., 2010), and so on. Some recent progress in stability and stabilization of dynamic systems with time-delays can be found in Mahmoud (2017).

Another key point to increase the feasible region of stability criteria is how to construct LKF and choose augmented vectors to derive negative definite conditions of the time-derivative of LKF in the form of quadratic forms. In Kwon, Park, Park, Lee et al. (2014), the authors showed the importance and superiority of augmented LKF by comparing the results with and without augmented LKFs. Since then, by constructing the following form of LKF, VAug(t)=thMtstẋ(u)x(u)utẋ(v)dvTRẋ(u)x(u)utẋ(v)dvdudswhere t denotes time, hM means a maximum admissible delay bound, x() is a state vector, and R is a positive definite matrix with an appropriate dimension, stability criteria which provide significant improved results have been derived in various problems such as H performance analysis and stability of linear systems with interval time-varying delay (Kwon et al., 2016a), stability and stabilization of Takagi–Sugeno fuzzy systems with time-varying delays (Kwon et al., 2016b), and linear systems with various kinds of constraints on the derivative of time-delays (Park et al., 2018a). Thus, it can be seen that the utilization of LKF (1) plays a leading role in increasing delay bounds.

The basic concept of dissipativity is that the increase in storage over a time interval cannot exceed the supply delivered to the system during this time-interval (Willems, 2007). A dissipative system with a particular form of supply rate function such as an inner produce of system and output vectors is a passive system. Passivity theory which is known as part of a general theory of dissipativity has been received a great attention since 1970s due to its advantage in stability analysis for nonlinear systems, high-order systems, and so on. The application of passivity theory can be found in various problems such as synchronization, chaos control, passive control, and so on. Also, owing to the rapid development of integral inequality and mathematical techniques mentioned before, many noticeable results in passivity and dissipativity analysis for neural networks with time-varying delays can be found in Chen, Li, & Bi (2009) Chen, Li, Lin, & Zhou (2009) Chen, Wang, Zhong, & Yang (2017) Chen, Xia, & Zhuang (2016) Fu, Zhang, Ma, & Zhang (2010) Kwon, Lee, & Park (2012) Kwon, Park, Lee et al. (2013), Kwon, Park, Park et al. (2013) Manivannan, Mahendrakumar, Samidurai, Cao, & Alsaedi (2017) Manivannan, Mahendrakumar et al. (2017), Manivannan, Samidurai et al. (2017) Manivannan, Samidurai, & Zhu (2017) Shu, Liu, Qiu, & Wang (2017) Thuan, Trinh, & Hien (2016) Wu, Park, Su, & Chu (2012) Xiao, Lian, Zeng, Chen, & Zheng (2017) Zeng, He, Shi, Wu, & Xiao (2015) Zeng et al. (2011), Zeng et al. (2014) Zeng, Park, & Shen (2015) Zeng, Park, & Xia (2015). For neural networks with randomly uncertain neural networks and time-varying delays, passivity analysis was conducted in Zeng, Park, and Xia (2015) as a special case of dissipativity by utilizing free-matrix-based integral inequality and reciprocally convex optimization. Thuan et al. (2016) investigated passivity for neural networks with interval time-varying delays by proposing refined two integral inequalities for single and double integral terms and applying these inequalities to estimate the integral terms calculated by LKFs. In Xiao et al. (2017), Wirtinger-based integral inequality (Seuret & Gouaisbaut, 2013), auxiliary function-based integral inequality Park, Lee et al. (2015), Park et al. (2016) and free-matrix-based integral inequality (Zhang et al., 2017) were simultaneously utilized in studying passivity for uncertain neural networks with time-varying delays. However, most of the previous results concerned to passivity for neural networks have utilized only one state in double integral form of LKF. Therefore, there are still a spare space to lift up passivity criteria.

In this paper, passivity analysis for neural networks with time-varying delays is revisited. Here, norm-bounded parameter uncertainties are considered to reflect the real situation caused by unidentified dynamics, external errors, and some noises. In order to achieve that, the followings are performed:

  • Based on generalized free-weighting-matrix integral inequality (Zhang et al., 2017), a further extended generalized free-weighting-matrix integral inequality is introduced in Lemma 2 by utilizing orthogonal polynomials.

  • Unlike the previous results such as Thuan et al. (2016) and Xiao et al. (2017), a triple integral form of LKF is not utilized in this paper. Instead, by constructing LKF including the double integral form introduced in Kwon, Park, Lee et al. (2013) and utilizing the proposed Lemma 2, a significantly improved passivity criterion is introduced in Theorem 1 with the framework of tractable LMIs.

  • Based on Theorem 1, a passivity condition for neural networks without uncertainties is presented in Theorem 2.

  • To show the superiority of the proposed conditions introduced in Theorem 1, Theorem 2 more clearly, a stability criterion for the concerned network is introduced without modifying LKFs utilized in Theorem 1, Theorem 2.

By comparing maximum delay bounds with the previous results, the superiority and less conservatism of the developed passivity and stability criteria presented in this paper will be shown.

Notation: Throughout this paper, the used notations are standard. Rn and Rm×n denote the n-dimensional Euclidean space with the Euclidean vector norm and the set of all m×n real matrices, respectively. Sn, S+n and Dn+ are the sets of symmetric, positive definite and positive diagonal n×n matrices, respectively. In means the identity matrix with n×n dimension. 0n(0mn) denotes the zero matrix with n×n(m×n) dimension. X>0 (<0) means symmetric positive (negative) definite matrix. X denotes a basis for the null-space of X. diag{}, Sym{X}, col{x1,,xn} and {yi}i=1n stand for, respectively, the (block) diagonal matrix, the sum of X and XT, the column vector with the vectors x1,,x2, and the set of the elements y1,,yn. The symmetric terms in symmetric matrices and in quadratic forms will be denoted by (This is used if necessary.). X[f(t)] means the sum of a constant matrix X1 and a linear matrix f(t)X2 for all real scalars f(t); i.e., X[f(t)]=X1+f(t)X2.

Section snippets

Problem statements

Consider the following neural networks : ẋ(t)=(A+ΔA(t))x(t)+(W0+ΔW0(t))f(x(t))+(W1+ΔW1(t))f(x(th(t)))+u(t),y(t)=C0f(x(t))+C1f(x(th(t))),where x(t)Rn is the neuron state vector, u(t)Rn is the input vector, y(t)Rn is the output vector, f()Rn=f1(),,fn()T is the activation function, ARn×n is the positive diagonal matrix, C0Rn×n and C1Rn×n are given real matrices, Wi(i=0,1)Rn×n are the interconnection weight matrices. h(t) means time-varying delays which satisfy ḣ(t)hD and 0h(t)hM

Main results

To simplify matrix and vector representations of proposed results, block entry matrices, ei(i=1,2,,20)R20n×n and e0=020nn which will be used in Theorem 1 are defined. For an example, e12=[0n11n,In,0n8n]T. The other notations are defined as

ζ(t)=col{ξ(t),u(t),p(t)},Q1=Q+Sym0n0nIn0n0nInP1In0n0nIn0n0n,Q2=Q+Sym0n0nIn0n0nInP2In0n0nIn0n0n,Γ1[h(t)]=Sym{[e1,e3,e6+e7,h(t)e8+(hMh(t))(e9+e6)]R[e4,e5,e1e3,hMe1e6e7]T},Γ2=[e1,e4,e16]N[e1,e4,e16]T[e3,e5,e18]N[e3,e5e18]T+Sym{(e16e1K

Numerical examples

In this section, three numerical examples will be provided to illustrate the effectiveness of the proposed stability criteria. The system parameters are as follows.

Example 1

Consider the uncertain neural networks (2) with the parameters A=2.2001.5,W0=10.60.10.3,W1=10.10.10.2,D=0.1000.1,E0=0.1000.1,E1=0.2000.2,E2=0.3000.3,C0=1001,C1=0000,Kp=1001,Km=0000.

In Table 1, the results of maximum delay bounds obtained by Theorem 1 are shown and some recent results

Conclusions

The passivity and stability of neural networks have been investigated in this paper. In Lemma 2, inspired by the work of Zhang et al. (2017), the further extended generalized free-matrix-based integral inequality has been proposed. By constructing the suitable LKFs which partially include new terms, the passivity condition for uncertain neural networks has been proposed in Theorem 1. Based on the result of Theorem 1, the passivity condition for nominal neural network has been presented in

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1D1A1A09917886) and by the Brain Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2017M3C7A1044815). This work was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government

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