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 2f27d88..35c0793 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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which leads to   \begin{align}  e^{c}=\frac{1}{k^{*}}\underbrace{\frac{km}{s-(am+l)}\tilde{\theta^T}\phi}_{M(s)[\tilde{\theta^T\phi}])} e^{c}=\frac{1}{k^{*}}\underbrace{\frac{km}{s-(am+l)}\tilde{\theta^T}\phi}_{M(s)[\tilde{\theta^T\phi}]}  \end{align}  where $M(s)$ is SPR. Using the KY-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.. stable.  %Using Lyapunov's second method with the Lyapunov candidate function  %\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