theorems.tex

\boldentry{Named Theorems and Conditions}



\entry{
    Poincaré-Bendixson theorem
}{
\begin{itemize}
    \item TODO
\end{itemize} 
}


\entry{
    Small-gain Theorem
}{
Given
\begin{itemize}
    \item $H_1$: an Input-Output System with input $e_1$ and output $y_1$ that is finite-gain $\mathcal{L}_p$-stable
    \item $H_2$: an Input-Output System with input $e_2$ and output $y_2$ that is finite-gain $\mathcal{L}_p$-stable
    \item $y_1=H_1 e_1$
    \item $y_2=H_2 e_2$
    \item $e_1=u_1-y_2$
    \item $e_2=u_2-y_1$
\end{itemize} 

By the definition of finite-gain $\mathcal{L}_p$-stable,
\[
    \norm{{y_1}_\tau}_{\mathcal{L}_p} \leq \gamma_1 \norm{{e_1}_\tau}_{\mathcal{L}_p} + \beta_1
\]
\begin{center}
    \textit{(The $\mathcal{L}_p$ norm of the $y_1$ is truncated by $\tau$, i.e. the system response is zero when $t>\tau$. This is less than or equal to The $\mathcal{L}_p$ norm of the $e_1$ truncated by $t<\tau$, multiplied by some gain value $\gamma_1$, plus some bias $\beta_1$)}
\end{center}

As long as a system does not have a finite escape time, we can compute the $\mathcal{L}_p$ norm of the system. \\

Likewise,
\[
    \norm{{y_2}_\tau}_{\mathcal{L}_p} \leq \gamma_2 \norm{{e_2}_\tau}_{\mathcal{L}_p} + \beta_2
\]

The Small-gain Theorem tells us,
\begin{equation}
    \norm{ 
        \begin{matrix} 
            {{y_1}_\tau} \\ {{y_2}_\tau} 
        \end{matrix} 
    }_{\mathcal{L}_p} 
    \le 
    \frac{1}{1-\gamma_1 \gamma_2} \left( 
        \norm{{u_1}_\tau}_{\mathcal{L}_p} + \gamma_2 \norm{{u_2}_\tau}_{\mathcal{L}_p} + \gamma_2 \beta_1 + \beta_2 
    \right)
    = \gamma_3 \left( \text{some $\mathcal{L}_p$-stable system} \right)
\end{equation}


Therefore, if $\gamma_1$ and $\gamma_2$ are less than one, the feedback connection is input/output stable (finite-gain $\mathcal{L}_p$-stable)

}