(augmented) state vector, $x_0\triangleq\bigl[x_\rom{p_0}\mT, x_\rom{c_0}\mT\bigl]\mT\in\IR^n$, A &&

A\_ & 0\_[n\_n\_]{}\
E\_ & 0\_[n\_n\_]{}

\^[n n]{},\
B&&\^[n m]{},\
B\_&&\^[n n\_]{}.

Next, consider the feedback control law given by u(t) &=& u\_(t)+u\_(t), \[total:ctrl\] where $u_\rom{n} (t) \in \IR^m$ and $u_\rom{a}(t)\in\IR^m$ are the nominal and adaptive control laws, respectively. Furthermore, let the nominal control law be u\_(t) &=& -Kx(t), K\^[m n]{}, \[nom:ctrl\] such that $A_\rom{r} \triangleq A-BK$ is Hurwitz. Using (\[total:ctrl\]) and (\[nom:ctrl\]) in (\[unc:sys2\]) yields (t)&=&A\_x(t)+B\_c(t)+B,     \[unc:sys3\] where $W\mT\triangleq\bigl[ \Lambda^{-1} W_\rom{p}\mT, (\Lambda^{-1}-I_{m\times m})K\bigl]\in\IR^{(s+n)\times m}$ is an *unknown* (aggregated) weight matrix and $\sigma\mT\bigl(x(t)\bigl)\triangleq\bigl[\sigma_\rom{p}\mT\bigl(x_\rom{p}(t)\bigl), x\mT(t)\bigl]\in\IR^{s+n}$ is a known (aggregated) basis function. Considering (\[unc:sys3\]), let the adaptive control law be u\_(t)=-(t)(x(t)), \[adapt:ctrl\] where $\hat{W}(t)\in\IR^{(s+n)\times m}$ be the estimate of $W$ satisfying the update law (t)&=&(x(t)) e(t) PB, (0)=\_0, \[update:law\]   where $\gamma \in \IR_+$ is the learning rate, $e(t)\triangleq x(t)-x_\rom{r}(t)$ is the system error with $x_\rom{r}(t)\in\IR^n$ being the reference state vector satisfying the reference system \_(t)&=&A\_x\_(t)+B\_c(t), x\_(0)=x\_, \[ref:ideal\] and $P \in \IR^{n \times n}_+ \cap \IS^{n \times n}$ is a solution of the Lyapunov equation[^3] 0 &=& A\_P + P A\_ + R, \[lyap:eqn\] with $R \in \IR^{n \times n}_+ \cap \IS^{n \times n}$.

Now, the system error dynamics is given by using (\[unc:sys3\]), (\[adapt:ctrl\]), and 