the case $(\gamma,\kappa,\eta)=(1000, 50, 0)$ for achieving the same level of low-frequency disturbance rejection characteristics, however, as noted, this amplifies the measurement noise excessively and has less time-delay margin as compared to the case $(\gamma,\kappa,\eta)=(100, 50, 10)$. Finally, the effect of increasing $\eta$ to a moderate value can be seen from the cases $(\gamma,\kappa,\eta)=(100, 50, 0)$, $(\gamma,\kappa,\eta)=(100, 50, 1)$, and $(\gamma,\kappa,\eta)=(100, 50, 10)$. That is, we can recover the desired low-frequency disturbance rejection characteristics without increasing $\gamma$, and hence, without amplifying the measurement noise.

 

Transient and Steady-State Performance Guarantees
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This section establishes transient and steady-state performance properties of the proposed adaptive control architecture. For this purpose, consider $e(t)=x(t)-x_\rom{r}(t)$ with $x_\rom{r}(t)$ satisfying (\[ref:nonideal\]) and $\tilde{W}(t)=\hat{W}(t)-W$. Furthermore, let the ideal (unmodified) reference system[^4] be \_(t)&=&A\_x\_(t)+B\_c(t), x\_(0)=x\_, \[ref:ideal:2\] where $x_\rom{r_i}(t)\in\IR^n$ being the ideal reference state vector. Finally, let $\tilde{x}(t)\triangleq x_\rom{r}(t)-x_\rom{r_i}(t)$ be the deviation error from the ideal reference system with $x_\rom{r}(t)$, once again, satisfying (\[ref:nonideal\]). Then, 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)&=&(x(t)) e(t) PB, (0)=\_0, \[update:law:error\]\
(t)&=&A\_ (t)+(e(t)-e\_(t)), (0)=0, \[deviation:error\]    where $\tilde{W}_0 \triangleq \hat{W}_0-W$. The next theorem presents the first result of this paper.

*Theorem 5.1*. 