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Shan edited CRM MRAC of a First-Order Nonlinear System1.tex
over 9 years ago
Commit id: 13d804fa5520ca224cae91537bb0b4979cf64a1c
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\begin{align}
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, 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.
%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