in (\[unc:sys\]) is accessible \[, Section II.C\]. Since $\dot{\mathcal{V}}\bigl(e(t),\tilde{W}(t)\bigl)\le 0$, and hence, (e(t),(t))&& (e\_0,\_0)=\^[-1]{}\^2\_, \[temp:01\] then using $\mathcal{V}\bigl(e(t),\tilde{W}(t)\bigl)\ge\lambda_\rom{min}(P)\norm{e(t)}^2_2$ in (\[temp:01\]) yields $\norm{e(t)}_2\le \norm{\tilde{W}_0 \Lambda^\frac{1}{2}}_\rom{F}/\sqrt{\gamma \lambda_\rom{min}(P)}$. Since $\norm{\cdot}_\infty \le \norm{\cdot}_2$ holds, and this bound is uniform, then $\norm{e_\tau(t)}_{\mathcal{L}_\infty}\le \norm{\tilde{W}_0 \Lambda^\frac{1}{2}}_\rom{F}/\sqrt{\gamma \lambda_\rom{min}(P)}$. Finally, since this holds uniformly in $\tau$, we have \_[\_]{}\_/, which shows that the distance between $x(t)$ and $x_\rom{r}(t)$ can be made arbitrarily small in transient time by resorting to a high learning rate $\gamma$. As discussed, however, update laws with high learning rates in the face of large system uncertainties and abrupt changes may yield to signals with high-frequency oscillations, which can violate actuator rate saturation constraints and/or excite unmodeled system dynamics \[, \] resulting in system instability for practical applications.

 

Frequency-Limited System Error Dynamics
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One of the fundamental components of a model reference adaptive control scheme is the system error $e(t)$. In particular, if the system error $e(t)$ contains any high-frequency oscillations, then the adaptive control law (\[adapt:ctrl\]) can also have such oscillations, since the update law (\[update:law\]) is driven by this system error $e(t)$. Motivating from this standpoint, our aim is to limit the frequency-content of the system error dynamics (\[err:sys\]) during transient-time (learning phase), and 