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This paper examines a passivity analysis for a class of discrete-time recurrent neural networks (DRNNs) with norm-bounded time-varying parameter uncertainties and interval time-varying delay. The activation functions are assumed to be globally Lipschitz continuous. Based on an appropriate type of Lyapunov functional, sufficient passivity conditions for the DRNNs are derived in terms of a family of linear matrix inequalities (LMIs). Two numerical examples are given to illustrate the effectiveness and applicability.

Recurrent neural networks have been extensively studied in the past decades. Two popular examples are Hopfield neural networks and cellular neural networks. Increasing attention has been draw to the potential applications of recurrent neural networks in information processing systems such as signal processing, model identification, optimization, pattern recognition, and associative memory. However, these successful applications are greatly dependent on the dynamic behavior of recurrent neural networks (RNNs). On the other hand, time delay is inevitably encountered in RNNs, since the interactions between different neurons are asynchronous. Generally, time delays, both constant and time varying, are often encountered in various engineering, biological, and economic systems due to the finite switching speed of amplifiers in electronic networks, or to the finite signal propagation time in biological networks [

The theory of passivity plays an important role for analyzing the stability of nonlinear system [

Recently, the stability analysis problems for discrete-time neural networks with time delay have received considerable research interests. For instance in [

The purpose of this paper is to deal with the problem of passivity conditions for discrete-time uncertain recurrent neural networks with interval time-varying delay. The interval time-varying delay includes both lower and upper bounds of delay, and the parameter uncertainties are assumed to be time varying but norm bounded which appear in all the matrices in the state equation. It is then established that the resulting passivity condition can be cast in a linear matrix inequality format which can be conveniently solved by using the numerically effective Matlab LMI Toolbox. In particular, when the interval time-delay factor is known, it is emphasized that delay-range-dependent passivity condition yields more general and practical results. Finally, two numerical examples are given to demonstrate the effectiveness.

Throughout this paper, the notation

Consider a discrete-time recurrent neural network with interval time-varying delay described by

In order to obtain our main results, the activation functions in (

The activation functions

System (

This section explores the globally robust delay-range-dependent passivity conditions of the discrete-time recurrent uncertain neural network with interval time-varying delay given in (

Let

For any

For vectors

To study the globally robust delay-range-dependent passivity conditions of the discrete-time uncertain recurrent neural network with interval time-varying delay, the following theorem reveals that such conditions can be expressed in terms of LMIs.

Under Assumption

Choose the Lyapunov-Krasovskii functional candidate for the system in (

Following from Lemma

Theorem

Under Assumption

In the stochastic context, robust delay-dependent passivity conditions are studied in [

Two numerical examples are now presented to demonstrate the usefulness of the proposed approach.

Consider the following discrete-time uncertain recurrent neural network:

The various lower bounds

0 | 5.9189 |

1 | 5,9228 |

2 | 5.8145 |

3 | 5.9832 |

4 | 5.8014 |

5 | 5.9522 |

6 | 5.9744 |

7 | 6.0126 |

8 | 6.0186 |

9 | 6.1851 |

Consider the discrete-time uncertain recurrent neural network with the following parameters:

This study has investigated the problem of globally robust passivity conditions for a discrete-time recurrent uncertain neural network with interval time-varying delay. A sufficient condition for the solvability of this problem, which takes into account the range for the time delay, has been established that the passivity conditions can be cast in linear matrix inequalities format. It has been shown that the bound for the time-varying delay in a range which ensures that the discrete-time recurrent uncertain neural network with interval time-varying delay attains globally robust passivity conditions can be obtained by solving a convex optimization problem. Two numerical examples have been presented to demonstrate the effectiveness of the proposed approach.