Prove stability of the resulting closed-loop and prove convergence of the control error to zero The idea behind Closed-loop reference models is to force the trajectories of the closed loop reference model to move towards the plant model. The differential equations are given as \[\begin{aligned} \dot x_m^o &= a_m x_m^o + k_mr,& a_m <0 \\ \dot x_m^c &= a_m x_m^c +k_mr - le^c, \qquad&l, a_m < 0 \\ e^c&=x_p-x_m^c \end{aligned}\]
The closed loop dynamics are described by \(x_m^c\) while \(e^c\) denotes the feedback from control error. The additional term \(e^c\) in the dynamics of the closed loop reference model can be interpreted as the reference model moving towards the plant dynamics. Since the open loop and the closed loop dynamics differ, convergence of the closed to the open loop has to be ensured. While this is beyond the scope of the exercise, a proof for convergence of the error dynamics \(e^c\) is given.
Considering the control input \(u\) \[\begin{aligned} u &= \theta_a x_p + kr\end{aligned}\] the dynamics of the closed loop for the plant can be derived as follows \[\begin{aligned} \dot x_p &= a_p x_p + k_p u \\ &= a_p x_p + k_p ( \theta_a x_p + kr ) \\ &= ( a_p + k_p\theta_a^* ) x_p + k_p(\theta_a-\theta_a^*)x_p + k_p(k-\frac{k_m}{k_p})r +k_mr \\ &= a_mx_p + k_p\tilde \theta_a x_p + k_p\tilde k r + k_mr \\ &= a_mx_p + k_mr + k_p \tilde \theta^T \phi\end{aligned}\]
where \(\tilde \theta = \begin{bmatrix} \tilde \theta_a & \tilde k\end{bmatrix}\). In combination with the closed loop dynamics \(x_m^c\) from equation (\ref{eq:xmc}) the error dynamics can be evaluated \[\begin{aligned} \dot e_c &= \dot x_p - \dot x_m^c \\ &= a_m x_p + k_mr + k_p \tilde \theta^T\phi - a_mx_m^c - k_mr + le^c \\ &= a_m ( x_p - x_m^c ) + le^c + k_p \tilde \theta^T\phi \\ &= \underbrace{( a_m + l )}_{< 0} e^c + k_p\tilde \theta^T \phi\end{aligned}\]
which leads to \[\begin{aligned} e^{c}=\frac{1}{k^{*}}\frac{k_m}{s-(a_m+l)}\tilde{\theta}^T\phi=\frac{1}{k^{*}}M(s)[\tilde{\theta}^T\phi]\end{aligned}\] where \(M(s)\) is SPR. Thus \(e^c\) converges to zero because M(s) is SPR and therefore Hurwitz.
Using the MKY-Lemma, we know, that a Lyapunov function exits s.t. the derivative is negative definite. Hence the dynamics of the closed loop errror is asymptotically stable.