Fuzzy Neural Networks for Real Time Control Applications

8.4.2.3 Stability Analysis

Since the stability of each particle is independent of the others, the stability of one particle is considered. Furthermore, for brevity the s in the indices does not appear. Since ǁxǁ≤ B_x and $\parallel \stackrel{.}{x}\parallel \le B_{\stackrel{.}{x}}$, the time derivative of (8.43) has the following properties:

$\begin{array}{l}\hfill \sum _{r=1}^R{\stackrel{.}{\underset{\_}{\stackrel{~}{w}}}}_r\le B_3+B_4\parallel ċ\parallel +B_5\parallel \stackrel{.}{\sigma }\parallel \end{array}$

  (8.53)

$\begin{array}{l}\hfill \sum _{r=1}^R{\stackrel{.}{\underset{\_}{\stackrel{~}{w}}}}_r\le B_6+B_7\parallel ċ\parallel +B_8\parallel \stackrel{.}{\sigma }\parallel \end{array}$

  (8.54)

in which B₃,B₄,B₅,B₆,B₇, and B₈ are positive real values. By using the following Lyapunov ...