that the projection norm bound imposed on each column of $\Theta \in \IR^{n \times m}$ is $\theta_\rom{max}$.

Next, let the adaptive control law be given by (\[adapt:ctrl\]), where $\hat{W}(t)\in\IR^{(s+n)\times m}$ is the estimate of $W(t)$ satisfying the update law (t)&=&\_ , (0)=\_0, \[update:law:2\]   with $\gamma \in \IR_+$ being the learning rate, $e(t)\triangleq x(t)-x_\rom{r}(t)$ being the system error such that $x_\rom{r}(t)\in\IR^n$ satisfies (\[ref:nonideal\]), and $P \in \IR^{n \times n}_+ \cap \IS^{n \times n}$ being a solution of (\[lyap:eqn\]).

We note here that the system error, weight update error, low-pass filtered system error, and the deviation error dynamics are, respectively, given by (\[err:sys:mod\]), (\[filter\]), (t)&=&\_ -(t), (0)=\_0, \[update:law:err2\]   and (\[deviation:error\]), where $\tilde{W}_0 \triangleq \hat{W}_0-W$. Furthermore, since $W_\rom{p}(t)$ and $\dot{W}_\rom{p}(t)$ are bounded, there exists norm bounds $w_\rom{max}$ and $\dot{w}_\rom{max}$ such that $\norm{W(t)}_\rom{F}\le w_\rom{max}$ and $\norm{\dot{W}(t)}_\rom{F}\le \dot{w}_\rom{max}$ for all $t \in \overline{\IR}_+$. Similarly, since the update law for $\hat{W}(t)$ is predicated on the projection operator and $W(t)$ is bounded, there also exists a norm bound $\tilde{w}_\rom{max}$ such that $\norm{\tilde{W}(t)}_\rom{F}\le\tilde{w}_\rom{max}$ for all $t \in \overline{\IR}_+$. Finally, as it is standard for the projection operator-based update laws, we assume that $\norm{\hat{W}(0)}_\rom{F}\le \hat{w}_\rom{max}$, where $\hat{w}_\rom{max}$ is the projection norm bound imposed on $\hat{W}(t)$. The next theorem presents 