it should be also noted from (\[bound:elu\]) that keeping $\gamma$ constant but increasing $\kappa$ may result in a larger distance between $x(t)$ and $x_\rom{r_i}(t)$, and therefore, both should be increased simultaneously in order to keep this distance consistent during transient time.

 

Suppressing High-Frequency System Error Dynamics in Transient Time
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We now show that the high-frequency content of the system error $e_\rom{H}(t)=e(t)-e_\rom{L}(t)$ can be effectively suppressed as one increases $\kappa$ design parameter of the modified reference system (\[ref:nonideal\]). The next theorem presents the second result of this paper.

*Theorem 6.1*. Consider the system error dynamics given by (\[err:sys:mod\]) and the low-pass filtered system error dynamics given by (\[filter\]). Then, e\_(t,) &=& \^[-t]{}e\_[\_]{}+(\^[-1]{}), \[sing:per1\] holds for a sufficiently high $\kappa$, where $e_{_\rom{H_0}}\triangleq e(0)-e_\rom{L}(0)=e_0$[^6].

*Proof*. Let $\varepsilon\triangleq\kappa^{-1}$. Then, (\[err:sys:mod\]) and (\[filter\]) can be equivalently written as (t)&=&A\_e(t)-B (t)(x(t))-(e(t)-e\_(t)), \[hf:1\]      \
\_(t)&=& A\_ e\_(t)+(e(t)-e\_(t)). \[hf:2\] Since setting $\varepsilon=0$ results in 0&=&e(t)-e\_(t), \[hf:3\]\
\_(t)&=&A\_e\_(t), \[hf:4\] then the system given by (\[hf:1\]) and (\[hf:2\]) is said to be the singularly perturbed model form, where $e(t)=e_\rom{L}(t)$ captures the isolated root. To shift the quasi steady-state of $e(t)$ to the origin, consider e\_(t)=e(t)-e\_(t), as a change of variables, which yields &=&-e\_(), e\_(0)=e\_[\_]{}, \[adasd\] where $\tau$ is related 