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\textbf{Draw the linear part of the adaptive controller for $G_{nm}$ . Assume the filters $F$ and parameters to be known.}  The differential equation of the plant $G_{nm}(s)$ in the frequency domain  \begin{align}G_{nm}: \begin{align}G_{nm}:\;  sX_p&=(a + \underbrace{\theta_y^TF}_{=:\Upsilon(s)})X_p + (\underbrace{\theta_u^TF}_{=:\zeta(s)} + k)U\end{align} The dynamics of the closed loop are required to behave similiar to the dynamics of an arbitrary reference model:  \begin{align}G_m: \begin{align}G_m:\;  sX_m=a_mX_m+k_mR \end{align} Since all parameters are assumed to be known an adaptive controller can be designed such that the resulting dynamics of the closed loop behaves like the dynamics of the reference model  \begin{align}U&=-\frac{(a+\Upsilon)X_p}{\zeta+k}+\frac{a_mX_p+k_mR}{\zeta+k} \\ &=\frac{(-a-\Upsilon+a_m)X_p+k_mR}{\zeta+k}\end{align}