I have a system and a reference model represented in state space in the following form:
\begin{gather} \dot{x} = Ax+Bu \\\ u = -Kx+k_rr \\\ K,k_r : constants - controller \ gains \\ A_m = A-BK \\\ \dot{x_m} = A_mx_m+B_rr \end{gather}
where my goal is to design a direct model reference adaptive controller for the system $x$ to track the reference model $x_m$ when there is bounded $r(t)$ signal as reference input. By defining the error terms to be: \begin{align} e &= x - x_m \\ \tilde{K(t)}&=\hat{K(t)}-K \\ \tilde{K_r(t)}&=\hat{K_r(t)}-K_r \end{align} I have derived the system tracking error:
$$\dot{e} = A_me-B\tilde{K(t)}x-B\tilde{k_r(t)r}$$ What I want to do is derive update laws for the time changing controller parameters $\hat{K}$, $\hat{k_r}$. In order to do so I follow the Lyapunov procedure by defining a Lyapunov candidate function,computing its derivative and then trying to prove that the derivative is at least negative semi-definite. I chose the following Lyapunov candidate function:
$$V = e^TPe+|b|(\tilde{K}Γ^{-1}\tilde{K}^T+γ_r^{-1}\tilde{k_r}^2) $$
where $P$ is a positive definite matrix computed from the Lyapunov equation $A_m^TP+PA_m=-Q$ since $A_m$ is defined to be Hurwitz, $B=[0 \ \ b]^T$(sign of $b$ is known) and $Q=Q^T>0$. $V$ is the Lyapunov candidate function and the constants $Γ$ and $γ_r$ are positive adaptation gains.
$$ \dot{V} = \dot{e}^TPe+e^TP\dot{e}+|b|(2\tilde{K}Γ^{-1}\dot{\tilde{K}^T}+2γ_r^{-1}\tilde{k_r}\dot{\tilde{k_r}}) $$
Now by substituting the equation for $\dot{e}$ I have derived the following result:
$$ \dot{V} = -e^TQe+2e^TPB(-\tilde{k_r}r-\tilde{K}x)+|b|(2\tilde{K}Γ^{-1}\dot{\tilde{K}^T}+2γ_r^{-1}\tilde{k_r}\dot{\tilde{k_r}}) $$
The term $2e^TPB$ can be written $2e^T\bar{P}b$ where $\bar{P}=[p_{12} \ \ p_{22}]^T$ and since the sign of $b$ is known, $b = |b|sgn(b)$. Taking these into consideration I have reached the final form of $\dot{V}$ which is :
$$ \dot{V} = -e^TQe+2|b|\tilde{K}(-xe^T\bar{P}sgn(b)+Γ^{-1}\dot{\hat{K}^T})+2|b|\tilde{k_r}(-re^T\bar{P}sgn(b)+γ_r^{-1}\dot{\hat{k_r}}) $$
Defining the update laws to be:
$$ \dot{\hat{K}^T}=Γxe^T\bar{P}sgn(b) $$
$$ \dot{\hat{k_r}} = γ_rre^T\bar{P}sgn(b) $$
we are left with $\dot{V} = -e^TQe \le 0$. I am writing this more for validating purposes since I am not so sure if I computed the derivatives of the transposed matrices together with the normal matrices correctly (I know it is time-consuming but anyone who is willing would really help) and also is there any other good option regarding the Lyapunov candidate function which I could consider ?