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The dynamics of the closed loop are required to behave similiar to the dynamics of an arbitrary reference model:  \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} The block diagram of the adaptive controller is displayed in the following figure.