Fuzzy Neural Networks for Real Time Control Applications

$\begin{array}{ll}\hfill \dot{V}& =-\mid e\mid 2\alpha +e\left[\sum _{r=1}^N\left[\sum _{i=1}^I\left(-\alpha \text{sgn}\left(e\right)x_i^2\right.\right.\right.\hfill \\ \left.\left.\left.+a_{ri}{\dot{x}}_i\left(q{\widetilde{\underset{¯}{w}}}_r+(1-q){\tilde{\overline{w}}}_r\right)\right)\right]-\dot{y}+f_r\mathrm{\Delta }{\widetilde{\underset{¯}{w}}}_r+f_r\mathrm{\Delta }{\tilde{\overline{w}}}_r\right]+\frac{1}{\gamma }\dot{\alpha }(\alpha -{\alpha }^{*})\hfill \\ \hfill \dot{V}& <-\mid e\mid 2\alpha +\mid e\mid \left[\sum _{r=1}^N\left[\sum _{i=1}^I\left(-\alpha B_{x^2}\right.\right.\right.\hfill \\ \left.\left.\left.+B_a{B}_{\dot{x}}\left(q{\widetilde{\underset{¯}{w}}}_r+(1-q){\tilde{\overline{w}}}_r\right)\right)\right]+B_{\dot{y}}{B}_f{B}_{{\mathrm{\Delta }}_w}\right]+\frac{1}{\gamma }\dot{\alpha }(\alpha -{\alpha }^{*})\hfill \end{array}$

Considering the upper bounds of x_i’s, ${\dot{x}}_i$, and a_ri’s as in the assumptions of Theorem 7.2, we have:

$\begin{array}{ll}\hfill \dot{V}& <-\mid e\mid 2\alpha +\mid e\mid \left[\sum _{r=1}^N\left[-I\alpha B_{x^2}+I{B}_a{B}_{\dot{x}}\left(q{\widetilde{\underset{¯}{w}}}_r\right.\right.\right.\hfill \\ \left.\left.\left.+(1-q){\tilde{\overline{w}}}_r\right)\right]+B_{\dot{y}}+B_f{B}_{{\mathrm{\Delta }}_w}\right]+\frac{1}{\gamma }\dot{\alpha }(\alpha -{\alpha }^{*})\hfill \\ <-\mid e\mid 2\alpha +\mid e\mid \left[-I\alpha B_{x^2}+I{B}_a{B}_{\dot{x}}\left(q\sum _{r=1}^N{\widetilde{\underset{¯}{w}}}_r\right.\right.\hfill \\ \left.\left.+(1-q)\sum _{r=1}^N{\tilde{\overline{w}}}_r\right)+B_{\dot{y}}\right]+\frac{1}{\gamma }\dot{\alpha }(\alpha -{\alpha }^{*})\hfill \end{array}$

So that:

$\begin{array}{ll}\hfill \dot{V}& <-\mid e\mid 2\alpha +\mid e\mid \left[-I\alpha B_{x^2}+I{B}_a{B}_{\dot{x}}+B_{\dot{y}}+B_f{B}_{{\mathrm{\Delta }}_w}\right]\hfill \\ +\frac{1}{\gamma }\dot{\alpha }(\alpha -{\alpha }^{*})\hfill \end{array}$