he estimation of the weight matrix in the RBFN is derived from the Lyapunov theory, which guarantees stability of the closed-loop system as presenting in the following theorem.

**Theorem 2.** Given the adaptation mechanism (\[eq34\]), the proposed control scheme (\[eq35\]) can guarantee the closed-loop DAR system (\[eq15\]) to be input-to-state stable [@RN28] with the attractor $$\label{eq36}
D=\left\{ {s}\in {{R}^{4}}|\left\| {s} \right\|> \frac{{{\varepsilon }_{N}}+\varsigma \frac{{{\left\| W \right\|}_{F}}^{2}}{4}}{{{c}_{3\min }}} \right\},$$ where ${c}_{3\min }$ is the minimum value of $c_3$, and ${\varepsilon }_{N}$ is a small positive number so that the approximation error $\varepsilon=\delta-\hat{\delta}$ satisfies $\left\| {\varepsilon}  \right\|<{\varepsilon }_{N}$.

Let $$\tilde{W}=W-\hat{W}$$ define the error between the ideal weight $W$ and the estimated weight $\hat{W}$ Considering the Lyapunov function candidate $$\label{eq37}
{{V}_{2}}={{V}_{1}}+\frac{1}{2}{{{s}}^{T}}{s}+tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\tilde{W} \right)$$ and differentiating it with respect to time, one obtains $$\label{eq38}
\begin{array}{r@{}l@{\qquad}l}
{{{\dot{V}}}_{2}}=&{{{\dot{V}}}_{1}}+{{{{s}}}^{T}}\dot{{s}}+tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\tilde{W}} \right) \\ 
 =&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}\left( {\delta }-{\hat{\delta }} \right) \\
 &-tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\hat{W}} \right) \\ 
=&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}{{W}^{T}}{h}\\
&-{{{{s}}}^{T}}{{{\hat{W}}}^{T}}{h}-tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\hat{W}} \right)  +{{{{s}}}^{T}}{\varepsilon}  \\ 
=&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}{\varepsilon}\\
&+tr\left( {{{\tilde{W}}}^{T}}\left( {{{h}{{{s}}}^{T}-\Gamma }^{-1}}\dot{\h