all initial conditions be zero. For this special case, the system loop transfer function $\mathcal{G}(s)$ (broken at the control input) can be equivalently written as a linear time-invariant dynamical system, and hence, we can resort to classical control theory tools, such as Bode plots, to analyze the closed-loop system with respect to different choices of $\gamma$, $\kappa$, and $\eta$. Specifically, the system loop transfer function is given by (s)&=&\_[(s)]{}\_[(s)]{}, where $\mathcal{C}(s)$ and $\mathcal{P}(s)$ denote the controller and the plant, respectively. Furthermore, for the standard model reference adaptive controller ($\eta$ is sufficiently large, or simply, $\kappa=0$), note that $\mathcal{C}(s)=\frac{\gamma}{s}$. For the controller $\mathcal{C}(s)$ of proposed approach, since $\frac{\gamma}{s}$ is multiplied by a lead compensator $\frac{s+\alpha+\eta}{s+\alpha+\kappa+\eta}$, it can improve stability margins of the closed-loop system positively. To further see the effects of $\gamma$, $\kappa$, and $\eta$, consider the Bode plot of $\mathcal{G}(s)$ in Figure \[figure:bode\].

Here, we tune these design parameters in order to obtain a large loop gain at low frequencies (from $0$ rad/s to $5/2\pi$ rad/s) for good rejection of low-frequency disturbances and a small loop gain at high frequencies to avoid injecting too much measurement noise into the plant \[, Section 11.4\]. Furthermore, we also would like to have 