deletions | additions       diff --git a/CRM MRAC of a First-Order Nonlinear System1.tex b/CRM MRAC of a First-Order Nonlinear System1.tex  index 1c68364..328812c 100644  --- a/CRM MRAC of a First-Order Nonlinear System1.tex  +++ b/CRM MRAC of a First-Order Nonlinear System1.tex 
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\begin{align}V(e^c,\tilde \theta) = \frac{1}{2}(e^c)^2 + \frac{1}{2}\gamma^{-1}|k_p|\tilde \theta^T\tilde \theta\end{align}  the derivative along the system trajectories leads to  \begin{align}\dot V &= \dot e^c e^c + \gamma^{-1}|k_p| \dot{\tilde \theta}^T\tilde \theta \\  &=(a_m+l)(e^c)^2+e^c k_p \tilde\theta^T\phi + \gamma^{-1}|k_p|\dot{\tilde\theta}^T\tilde \theta \\  &=(a_m+l)(e^c)^2 \leq 0  \end{align} for   \begin{equation}e^c k_p \tilde\theta^T\phi + \gamma^{-1}|k_p|\dot{\tilde\theta}^T\tilde \theta = 0.\end{equation}