\^\*(e(t),(t),e\_(t),(t))&& \_[(e,,e\_,)\_]{}(e,,e\_,)=\_. Now, by following the final steps in the proof of Theorem 5.1, (\[ultim:bound\]) is immediate. This completes the proof. $\square$

*Remark 7.2*. By following an identical analysis to \[, Theorem 9\], the results of Theorem 7.1 can be further extended to show that the Lyapunov-like function candidate given by (\[proof:1\]) exponentially converges to the set defined by the ultimate bound (\[ultim:bound\]).

Illustrative Numerical Example
==============================

Consider the nonlinear dynamical system representing a controlled wing rock aircraft dynamics model given by

\_(t)\
\_(t)

&=&

0 & 1\
0 & 0

[x]{}\_(t)\
[x]{}\_(t)

+

0\
1

,\[083\] where $x_\rom{p_1}(0)=0$, $x_\rom{p_2}(0)=0$, $x_\rom{p_1}$ represents the roll angle in radians, and $x_\rom{p_2}$ represents the roll rate in radians per second. In (\[083\]), $\delta_\rom{p}(t,x_\rom{p})$ and $\Lambda$ represent uncertainties of the form \_(t,x\_)=\_1(t)+\_2x\_+\_3x\_+\_4|[x\_]{}|x\_+\_5|[x\_]{}| x\_+\_6x\_\^3, and =0.75, where $\alpha_i$, $i=1,\ldots,6$, are unknown parameters. For our numerical example, we set $\alpha_1=0.25$, $\alpha_2=0.5$, $\alpha_3=1.0$, $\alpha_4=-5.0$, $\alpha_5=5.0$, and $\alpha_6=10.0$. We chose $K=[2.0, \ 2.0, \ 1.0]$ for the nominal controller design. For the proposed adaptive control architecture (Theorem 7.1), $
\sigma(x)=\bigl[1, \ x_\rom{p_1}, \ x_\rom{p_2}, \ |{x_\rom{p_1}}|x_\rom{p_2}, \ |{x_\rom{p_2}}| x_\rom{p_2}, \ x_\rom{p_1}^3, \ x\mT \bigl]\mT,
$ is chosen as the basis function and we set $R=I_3$. Figures \[figure:sim:1\]–\[figure:sim:4\] present 