to the original $t$ through $\tau=t/\varepsilon$. Now, as a direct consequence of \[, Theorem 11.2\] e(t,)&=&e\^[A\_t]{} e\_(0)+ \^[-t/]{}e\_[\_]{} + ()=\^[-t/]{}e\_[\_]{}+(),\
e\_(t,)&=&e\^[A\_t]{} e\_(0)+()=(), which leads to the result given by (\[sing:per1\]). This completes the proof. $\square$

*Remark 6.1*. Note that (\[hf:4\]) and (\[adasd\]) are referred as the reduced-order system and the boundary-layer system, respectively, where the former describes the asymptotic behavior and the latter describes the transient behavior. In particular, Theorem 6.1 shows that the transient high-frequency content of the system error $e_\rom{H}(t)$ is globally exponentially stable for a sufficiently high $\kappa$, and hence, it vanishes in a fast manner. If, in addition, $x_\rom{r_0}=x_0$ in (\[ref:nonideal\]), then it follows from (\[sing:per1\]) that $e_\rom{H}(t,\kappa) = \mathcal{O}(\kappa^{-1})$.

 

Extensions to Time-Varying Uncertainties
========================================

This section discusses robustness properties of the proposed adaptive control architecture with respect to time-varying uncertainties. For this purpose, we relax the uncertainty parametrization given by (\[unc:prm\]) to \_(t, x\_)&=&W\_(t) \_(x\_), x\_\^[n\_]{}, \[unc:prm:rlx\] where $W_\rom{p}(t)\in \IR^{s\times m}$ is an *unknown* time-varying weight matrix subject to $\norm{W_\rom{p}(t)}_\rom{F}\le w_\rom{p,max}$, $w_\rom{p,max} \in \IR_+$, and $\norm{\dot{W}_\rom{p}(t)}_\rom{F}\le \dot{w}_\rom{p.max}$, $\dot{w}_\rom{p,max}\in\IR_+$ [^7]. Note that, as a consequence of (\[unc:prm:rlx\]), the *unknown* (aggregated) weight matrix has the form $W(t)\mT\triangleq\bigl[ \Lambda^{-1} W_\rom{p}\mT(t), (\Lambda^{-1}-I_{m\times m})K\bigl]\in\IR^{(s+n)\times m}$ for this case. We introduce 