given by using (\[unc:sys3\]), (\[adapt:ctrl\]), and (\[ref:nonideal\]) as (t)&=&A\_e(t)-B(t)(x(t))-(e(t)-e\_(t)), e(0)=e\_0.     \[err:sys:mod\] Finally, note for the rest of this paper that the update law (\[update:law\]) is driven by the system error $e(t)=x(t)-x_\rom{r}(t)$, where $x_\rom{r}(t)$ is obtained from (\[ref:nonideal\]) (not (\[ref:ideal\])).

*Remark 4.1*. The reference system (\[ref:nonideal\]) captures a desired closed-loop dynamical system behavior modified by a mismatch term $\kappa\bigl(e(t)-e_\rom{L}(t)\bigl)$ representing the high-frequency content between the uncertain dynamical system and this reference system. Although this implies a modification of the ideal (unmodified) reference system (\[ref:ideal\]) during transient time, as we see in the following sections, this mismatch term allows to limit the frequency content of the system error dynamics (\[err:sys:mod\]), which is used to drive the adaptive controller. In other words, the purpose of our methodology is to prevent the update law from attempting to learn through the high-frequency content of the system error.

*Remark 4.2*. As it is noted, the filter gain $\eta$ needs to be chosen such that $\eta\le\eta^*$, where $\eta^*$ needs to be small enough to cut off the high-frequency content of $e(t)$. To see the negative effect of high filter gain, let $\eta$ be sufficiently large. Then, $e(t)-e_\rom{L}(t) \approx 0$ as a consequence of (\[filter\]), and hence, we 