Exponential Generalized $H_2$ Filtering of Delayed Static Neural Networks

Abstract

This paper is concerned with the problem of generalized $H_2$ filter design for static neural networks with time-varying delay. A double-integral inequality and the reciprocally convex combination technique are employed to handle the cross terms appeared in the time-derivative of the Lyapunov functional. An improved delay-dependent design criterion is presented by means of linear matrix inequalities. It is shown that the gain matrix of the desired filter and the optimal performance index are simultaneously achieved by solving a convex optimization problem. Moreover, the upper bound of the exponential decay rate of the filtering error system can be also easily obtained. An example with simulation is exploited to illustrate the effectiveness of the developed result.

Appendix A: Proof of Lemma 1

Let $\mathcal {T}_N=\left[ \begin{array}{c@{\quad }c} I&{}0\\ -T^{-1}N&{}I \end{array} \right]$. It is known that

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c} N^TT^{-1}N&{}N^T\\ N&{}T \end{array} \right] \ge 0 \end{aligned}$$

resulting from

$$\begin{aligned} \mathcal {T}_N^T\left[ \begin{array}{c@{\quad }c} N^TT^{-1}N&{}N^T\\ N&{}T \end{array} \right] \mathcal {T}_N=\left[ \begin{array}{c@{\quad }c} 0&{}0\\ 0&{}T \end{array} \right] \ge 0. \end{aligned}$$

Then, one has

$$\begin{aligned} 0&\le \int \limits _{-\tau }^0\int \limits _{t+\theta }^t\left[ \begin{array}{c} \xi (t)\\ \omega (s) \end{array} \right] ^T\left[ \begin{array}{c@{\quad }c} N^TT^{-1}N&{}N^T\\ N&{}T \end{array} \right] \left[ \begin{array}{c} \xi (t)\\ \omega (s) \end{array} \right] dsd\theta \\&= \frac{1}{2}\tau ^2\xi ^T(t)N^TT^{-1}N\xi (t)+2\xi ^T(t)N^T \int \limits _{-\tau }^0\int \limits _{t+\theta }^t\omega (s)dsd\theta \\&+\int \limits _{-\tau }^0\int \limits _{t+\theta }^t\omega ^T(s)T\omega (s)dsd\theta . \end{aligned}$$

That is, (7) holds. This completes the proof. $\square$