Exponential convergence rate estimation for uncertain delayed neural networks of neutral type
Introduction
In recent years, neural networks (NNs) were used in many applications; such as automatic control engineering, connected component detection, hole filling, image shadowing, optimization and associative memories, pattern recognition, and signal processing. Some well-known neural networks contain cellular neural networks (CNNs) introduced by Chua and Yang [1], Chua and Roska [2] bidirectional associative memory neural networks (BAMNNs) proposed by Kosko [3], Hopfield neural networks (HNNs) put forward by Hopfield [4], and Cohen–Grossberg neural networks (CGNNs) developed by Cohen and Grossberg [5]. The delayed neural networks (DNNs) are applied in many areas including processing of moving images and pattern classification. On the other hand, artificial neural networks are usually implemented by integrated circuits. In the implementation of artificial neural networks, time delay is produced from finite switching and finite propagation speed of electronic signals. During the implementation on very large-scale integrated chips, parameter perturbations and delays in transmitting time will affect the stability of neural networks. Hence the robust or global stability of DNNs has been studied by many researchers in recent years [5], [6], [7], [8], [9], [10], [11], [12]. Stability for some classes of DNNs of neutral type has recently been investigated [6], [7], [10]. DNNs of neutral type are new and more general than DNNs. Hence the robust global exponential stability for a class of DNNs of neutral type with time delay is considered in this paper. Stability and uniqueness of equilibrium point of DNNs of neutral type is also guaranteed via LMI condition.
Depending on whether the stability criterion itself contains the size of delay, criteria for DNNs can be classified into two categories, namely delay-dependent criteria [6], [7], [8], [10], [11], [13] and delay-independent criteria [9], [12]. Usually the former is less conservative when the delay is small. In recent years, some algebraic stability criteria have been derived from Razumikhin-like approach [6], Lyapunov approach [7], and Halanay inequality [12]. In [11], matrix inequality conditions were developed according to Lyapunov theory. In [8], delay-dependent result was applied by checking the Hamiltonian matrix with no eigenvalues on the imaginary axis. It is usually difficult to obtain a feasible solution using algebraic criteria and matrix inequality conditions. Hamiltonian matrix condition is simple, but it is usually conservative in many situations. LMI approach is an efficient tool for dealing with many control problems and can be solved using the toolbox of Matlab [14]. In [9], [10], [13], LMI-based stability criteria for DNNs have been proposed. In this paper, LMI-based delay-dependent and delay-independent results are proposed using Razumikhin-like approach and Leibniz–Newton formula. Some numerical examples are provided to show the improvement achieved by our results.
Section snippets
Formulations and preliminaries
The notation that will be used throughout the paper is listed as follows:
- C1
set of differentiable functions from [−h, 0] to
- AT
transpose of matrix A
- ∥x∥
Euclidean norm of vector x
- ∥A∥
spectral norm of matrix A
- ∥xt∥s
- λmax(P)
maximal eigenvalue of matrix P
- λmin(P)
minimal eigenvalue of matrix P
- diag[ai]
diagonal matrix with the diagonal elements ai, i = 1, 2, … , n
- P > 0 (resp. P < 0)
P is a positive (resp., negative) definite symmetric matrix
- I
unit matrix
∗ represents the symmetric form of
Global exponential stability analysis
In this section, we present a delay-dependent criterion for the global exponential stability and uniqueness of equilibrium point for system (1a), (1b), (5a), (5b). Theorem 1 The equilibrium point of system (1a), (1b), (5a), (5b) is unique and globally exponentially stable with convergence rate 0 < ξ = α < − (ln∥D∥)/h, if there exist some n × n positive definite symmetric matrices P, Qi, i = 1, 2, 3, 4, two positive diagonal matrices R1, R2, some matrices PE, Uj, , j = 1, 2, … , 7, and a positive constant ε, such
Illustrative examples
Example 1 Consider the following CGNNs of neutral-type: (Example 1 of [7])where di(xi(t)) = 5 + sin(xi(t)), fi(xi(t)) = tanh(xi (t)), i = 1,2. The following matrices can be obtained from systems (1a), (1b), (16)
Conclusions
In this paper, the global exponential stability for uncertain neural networks of neutral type with time delay has been investigated. Some numerical examples have been given to illustrate the improvement over other recent results. This paper provides the following main contribution:
- 1.
A new style DNN of neutral type is investigated in this paper [7], [10]. The considered DNNs are generalization of [8], [9], [11], [12], [13], [15].
- 2.
Based on LMI and Razumikhin-like approaches, global exponential
Acknowledgement
The research reported here was supported by the National Science Council of Taiwan, ROC under grant no. NSC 95-2221-E-022-019.
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