A new upper bound for the norm of interval matrices with application to robust stability analysis of delayed neural networks

Abstract

The main problem with the analysis of robust stability of neural networks is to find the upper bound norm for the intervalized interconnection matrices of neural networks. In the previous literature, the major three upper bound norms for the intervalized interconnection matrices have been reported and they have been successfully applied to derive new sufficient conditions for robust stability of delayed neural networks. One of the main contributions of this paper will be the derivation of a new upper bound for the norm of the intervalized interconnection matrices of neural networks. Then, by exploiting this new upper bound norm of interval matrices and using stability theory of Lyapunov functionals and the theory of homomorphic mapping, we will obtain new sufficient conditions for the existence, uniqueness and global asymptotic stability of the equilibrium point for the class of neural networks with discrete time delays under parameter uncertainties and with respect to continuous and slope-bounded activation functions. The results obtained in this paper will be shown to be new and they can be considered alternative results to previously published corresponding results. We also give some illustrative and comparative numerical examples to demonstrate the effectiveness and applicability of the proposed robust stability condition.

Introduction

In this paper, we consider the neural network model whose dynamical behavior is described by the following sets of nonlinear differential equations

$$\frac{d{x}_i\left(t\right)}{dt}=-c_i{x}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{f}_j\left(x_j(t-{\tau }_j)\right)+u_i\text{,}i=1,2,\dots ,n$$

where $n$ is the number of neurons, $x_i\left(t\right)$ denotes the state of the neuron $i$ at time $t,f_i(\cdot )$ denote activation functions, $a_{ij}$ and $b_{ij}$ denote the strengths of connectivity between neurons $j$ and $i$ at time $t$ and $t-{\tau }_j$, respectively; ${\tau }_j$ represents the time delay required in transmitting a signal from the neuron $j$ to the neuron $i,u_i$ is the constant input to the neuron $i$, and $c_i$ is the charging rate for the neuron $i$.

The neural network model defined by (1) can be written in the matrix–vector form as follows:

$$\dot{x}\left(t\right)=-Cx\left(t\right)+Af\left(x\left(t\right)\right)+Bf\left(x(t-\tau )\right)+u$$

where $x\left(t\right)={(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right))}^T\in R^n,A={\left(a_{ij}\right)}_{n\times n},B={\left(b_{ij}\right)}_{n\times n},C=\mathrm{diag}(c_i>0),u={(u_1,u_2,\dots ,u_n)}^T\in R^n,f\left(x\left(t\right)\right)={(f_1\left(x_1\left(t\right)\right),f_2\left(x_2\left(t\right)\right),\dots ,f_n\left(x_n\left(t\right)\right))}^T\in R^n$ and $f\left(x(t-\tau )\right)={(f_1\left(x_1(t-{\tau }_1)\right),f_2\left(x_2(t-{\tau }_2)\right),\dots ,f_n\left(x_n(t-{\tau }_n)\right))}^T\in R^n$.

The main approach to establishing the desired equilibrium and stability properties of dynamical neural networks is first to determine the character of the activation functions $f_i$, and then, impose some constraint conditions on the interconnected weight matrices $A$ and $B$.

The activation functions $f_i$ are assumed to be slope-bounded in the sense that there exist some positive constants $k_i$ such that the following conditions hold:

$$0\le \frac{f_i\left(x\right)-f_i\left(y\right)}{x-y}\le k_i\text{,}i=1,2,\dots ,n,\forall x,y\in R,x\ne y\text{.}$$

This class of functions will be denoted by $f\in \mathcal{K}$. The functions of this class do not require to be bounded, differentiable and monotonically increasing. In the recent literature, many authors have considered unbounded but slope bounded activation functions in equilibrium and stability analysis of different types of neural networks and presented various robust stability conditions for the neural network models considered (Baese et al., 2009, Balasubramaniam and Ali, 2010, Cao, 2001, Cao and Ho, 2005, Cao et al., 2005, Cao and Wang, 2005, Deng et al., 2011, Ensari and Arik, 2010, Faydasicok and Arik, 2012, Guo and Huang, 2009, Han et al., 2011, Huang et al., 2007, Huang et al., 2009, Kao et al., 2012, Kwon and Park, 2008, Liao and Wong, 2004, Liu et al., 2009, Lou et al., 2012, Mahmoud and Ismail, 2010, Ozcan and Arik, 2006, Pan et al., 2011, Qi, 2007, Shao et al., 2010, Shen and Wang, 2012, Shen et al., 2011, Singh, 2007, Wang et al., 2010, Wu et al., 2012, Zeng and Wang, 2009, Zhang et al., 2010, Zhang et al., 2011, Zhou and Wan, 2010). It is well-known from Brouwer’s fixed point theorem that there always exists an equilibrium point of a neural network if the activation functions are bounded. However, if the activation functions are not bounded, then we cannot always guarantee the existence of an equilibrium point of a neural network. Therefore, in the next sections, we first focus on investigating the existence of a unique equilibrium point for neural networks, which is necessary for global robust asymptotic stability of the neural network model  (1).

When we desire to employ a stable neural network which is robust against the deviations in values of the elements of the non-delayed and delayed interconnection matrices of the neural system, we need to know the exact parameter intervals for these matrices. A standard approach to formulating this problem is to intervalize the system matrices $A=\left(a_{ij}\right),B=\left(b_{ij}\right)$ and $C=\mathrm{diag}(c_i>0)$ as follows

$$C_I≔\{C=\mathrm{diag}\left(c_i\right):0<\underset{¯}{C}\le C\le \overline{C},\text{ i.e.,}0<{\underset{¯}{c}}_i\le c_i\le {\overline{c}}_i,\forall i\}$$$$A_I≔\{A=\left(a_{ij}\right):\underset{¯}{A}\le A\le \overline{A},\text{i.e.,}{\underset{¯}{a}}_{ij}\le a_{ij}\le {\overline{a}}_{ij},i,j=1,2,\dots ,n\}$$$$B_I≔\{B=\left(b_{ij}\right):\underset{¯}{B}\le B\le \overline{B},\text{i.e.,}{\underset{¯}{b}}_{ij}\le b_{ij}\le {\overline{b}}_{ij},i,j=1,2,\dots ,n\}\text{.}$$

In most of the robust stability analysis of neural network model (1), the norms of the matrices $A=\left(a_{ij}\right)$ and $B=\left(b_{ij}\right)$ indispensably involve the stability conditions. When the matrices $A$ and $B$ are given by the parameter ranges defined by (3), then, in order to express the exact involvement of the norms of the matrices $A$ and $B$ in the stability conditions, we need to know an upper bound for the norms of these matrices. In the previous literature, three different upper bound norms for the matrices $A$ and $B$ have been reported. A key contribution of the current paper is to define a new upper bound for the norms of $A$ and $B$.