# Sets of generalized ${\mathcal{H}}_{2}$ exponential stability criteria for switched multilayer dynamic neural networks

- Choon Ki Ahn
^{1}Email author and - Young Sam Lee
^{2}

**2012**:150

https://doi.org/10.1186/1687-1847-2012-150

© Ahn and Lee; licensee Springer 2012

**Received: **26 March 2012

**Accepted: **20 July 2012

**Published: **3 September 2012

## Abstract

This paper investigates new sets of generalized ${\mathcal{H}}_{2}$ exponential stability criteria for switched multilayer dynamic neural networks. These sets of sufficient stability criteria in forms of linear matrix inequality (LMI) and matrix norm are presented, under which switched multilayer dynamic neural networks reduce the effect of external input to a predefined level. The proposed sets of criteria also guarantee exponential stability for switched multilayer dynamic neural networks without external input.

## 1 Introduction

Switched systems are a class of hybrid systems consisting of a family of continuous (or discrete) time subsystems and a logical rule that orchestrates the switching between these subsystems. Switched systems have been extensively researched, and several efforts have been focused on the analysis of switched systems [1, 2]. Recently, switched recurrent neural networks were introduced to represent some complex nonlinear systems efficiently by integrating the theory of switched systems with recurrent neural networks [3–6]. Switched dynamic neural networks have found applications in the field of artificial intelligence and high speed signal processing [7, 8]. In [3–6], some stability criteria for switched dynamic neural networks were studied.

There always exist noise disturbances and model uncertainties in real physical systems. Recently, this has led to an interest in a generalized ${\mathcal{H}}_{2}$ approach [9–19]. The generalized ${\mathcal{H}}_{2}$ approach has been known as a significant concept to examine the stability of various nonlinear dynamical systems. Here, we have the following natural question: Can we obtain a generalized ${\mathcal{H}}_{2}$ stability criterion for switched dynamic neural networks. This paper provides an answer to this question. To the best of the authors’ knowledge, the generalized ${\mathcal{H}}_{2}$ analysis of switched dynamic neural networks has not yet been studied in the literature.

In this paper, we propose new sets of generalized ${\mathcal{H}}_{2}$ exponential stability criteria for switched multilayer dynamic neural networks. The sets of conditions proposed in this paper are a new contribution to the stability evaluation of switched neural networks. The proposed sets of sufficient stability criteria in forms of linear matrix inequality (LMI) and matrix norm guarantee that switched multilayer dynamic neural networks reduce the effect of external input to a predefined level. This paper is organized as follows. In Section 2, new sets of generalized ${\mathcal{H}}_{2}$ exponential stability criteria are derived. Finally, conclusions are presented in Section 3.

## 2 New sets of generalized ${\mathcal{H}}_{2}$ exponential stability criteria

*α*is a switching signal which takes its values in the finite set $\mathcal{I}=\{1,2,\dots ,N\}$. The matrices $({A}_{\alpha},{W}_{\alpha},{V}_{\alpha},{H}_{\alpha})$ are allowed to take values, at an arbitrary time, in the finite set $\{({A}_{1},{W}_{1},{V}_{1},{H}_{1}),\dots ,({A}_{N},{W}_{N},{V}_{N},{H}_{N})\}$. Throughout this paper, we assume that the switching rule

*α*is not known a priori and its instantaneous value is available in real time. Define the indicator function $\xi (t)={({\xi}_{1}(t),{\xi}_{2}(t),\dots ,{\xi}_{N}(t))}^{T}$, where

under zero-initial conditions for all nonzero $J(t)\in {L}_{2}[0,\mathrm{\infty})$, where ${L}_{2}[0,\mathrm{\infty})$ is the space of square integrable vector functions over $[0,\mathrm{\infty})$.

A set of generalized ${\mathcal{H}}_{2}$ exponential stability criterion of the switched multilayer neural network (3)-(4) is derived in the following theorem.

**Theorem 1**

*For given*$\gamma >0$

*and*$\kappa >0$,

*the switched multilayer neural network*(3)-(4)

*is generalized*${\mathcal{H}}_{2}$

*exponentially stable if*

*for*$i=1,\dots ,N$, *where*${\lambda}_{min}(\cdot )$*is the minimum eigenvalue of the matrix and* *P* *satisfies the Lyapunov inequality*${A}_{i}^{T}P+P{A}_{i}<-{k}_{i}I$.

*Proof*We consider the Lyapunov function $V(t)=exp(\kappa t){x}^{T}(t)Px(t)$. The time derivative of the function along the trajectory of (3) satisfies

Taking the supremum over $t>0$ leads to (5). This completes the proof. □

**Corollary 1** *When*$J(t)=0$, *the conditions* (6)-(9) *ensure that the switched multilayer neural network* (3)-(4) *is exponentially stable*.

*Proof*When $J(t)=0$, from (13), $\dot{V}(t)<0$ for all $x(t)\ne 0$. Thus, for any $t\ge 0$, it implies that

This completes the proof. □

In the next theorem, we find a new set of LMI criteria for the generalized ${\mathcal{H}}_{2}$ exponential stability of the switched multilayer neural network (3)-(4). This set of LMI criteria can be facilitated readily via standard numerical algorithms [21, 22].

**Theorem 2**

*For given level*$\gamma >0$

*and*$\kappa >0$,

*the switched multilayer neural network*(3)-(4)

*is generalized*${\mathcal{H}}_{2}$

*exponentially stable if there exist a positive symmetric matrix*

*P*

*and a positive scalar*

*ϵ*

*such that*

*for*$i=1,\dots ,N$.

*Proof*Consider the Lyapunov function $V(t)=exp(\kappa t){x}^{T}(t)Px(t)$. Applying Young’s inequality [20], we have $\u03f5[{L}_{\varphi}^{2}{x}^{T}(t){V}_{i}^{T}{V}_{i}x(t)-{\varphi}^{T}({V}_{i}x(t))\varphi ({V}_{i}x(t))]\ge 0$. By using this inequality, the time derivative of $V(t)$ along the trajectory of (3) is

Taking the supremum over $t>0$ leads to (5). This completes the proof. □

**Corollary 2** *When*$J(t)=0$, *a set of LMI conditions* (19)-(20) *ensure that the switched multilayer neural network* (3)-(4) *is exponentially stable*.

*Proof*When $J(t)=0$, from (22), we have

This completes the proof. □

*Z*is the whole set of nonnegative integers. Figure 1 shows state trajectories when $x(0)={[-1.5\phantom{\rule{0.5em}{0ex}}0.5]}^{T}$ and ${J}_{i}(t)$ ($i=1,2$) is a white noise.

## 3 Conclusion

In this paper, we have proposed new sets of generalized ${\mathcal{H}}_{2}$ exponential stability criteria for switched multilayer dynamic neural networks. These sets of sufficient stability criteria are represented by matrix norm and LMI. The proposed sets of criteria ensured that switched multilayer dynamic neural networks attenuate the effect of external input on the state vector. These sets of criteria also guaranteed exponential stability for switched multilayer dynamic neural networks when there is no external input.

## Declarations

### Acknowledgements

This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the CITRC (Convergence Information Technology Research Center) support program (NIPA-2012-H0401-12-1007) supervised by the NIPA (National IT Industry Promotion Agency).

## Authors’ Affiliations

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