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\begin{document}
\rightskip\rightmargin
\title{Control and stabilization of delayed systems }
\author{ \Large \textbf{Xxxx YYYYYY} }
\institute{\large\textbf{Master Automatic control and systems }\\
---------\\
Faculty Sciences and Technology\\
---------\\
University of Djelfa}
\footnotesize{\date{\today }
\begin{frame}
\maketitle
\end{frame}
\begin{frame}{ Summery}
\footnotesize \tableofcontents
\end{frame}
\section{Introduction}
\begin{frame}
\large Motivations:
\begin{itemize}
\item Most systems are nonlinear
\item Delay complicates the system analysis
\item Delay can lead to the system instability
\item ...
\end{itemize}
\end{frame}
\section{Modeling}
\begin{frame}
\footnotesize
The TS fuzzy model can be justified by:
\begin{itemize}
\item Its simplicity
\item Uncertainties
\item Its acceptable accuracy
\item ....
\end{itemize}
\pause
\quad \\
The system can be modeled as:
\begin{equation*}
\left\{\begin{array}{l}
{ }^C D^\alpha x(t)=f(x(t),x(t-\tau(t)),u(t)),\, t \geq 0, \\
x(s)=\varphi(s),\, s \in[-\tau, 0]
\end{array}\right.
\end{equation*}
$x(t) \in \Re ^{n}$ the system state\\
$u(t) \in \Re ^{m}$ the control vector \\
$\tau$: the delay
\end{frame}
\section{Lemmas}
\begin{frame}
\footnotesize
\begin{block}{Lemma 1}
\begin{equation*}
I(x,t)=\int_{a(t)}^{b(t)} f(x,t) \,dx \ \ , \ a(t) \And b(t)<\infty
\end{equation*}
with
$a(t)$ و $b(t)$
and $f(x,t)$
\begin{equation*}
\frac{d(I(x,t))}{dt}=\frac{db(t)}{dt}f(b(t),t)-\frac{da(t)}{dt}f(a(t),t)+\int_{a(t)}^{b(t)} \frac{\partial f(x,t)}{\partial t} \,dx
\end{equation*}
\end{block}
\begin{block}{Lemma 2}
\begin{equation*}
I(x,t)=\int_{a(t)}^{b(t)} f(x,t) \,dx \ \ , \ a(t) \And b(t)<\infty
\end{equation*}
with
$a(t)$ و $b(t)$
and $f(x,t)$
\begin{equation*}
\frac{d(I(x,t))}{dt}=\frac{db(t)}{dt}f(b(t),t)-\frac{da(t)}{dt}f(a(t),t)+\int_{a(t)}^{b(t)} \frac{\partial f(x,t)}{\partial t} \,dx
\end{equation*}
\end{block}
\end{frame} %.............................................................
\footnotesize
\section{Control without delay}
\begin{frame}
\footnotesize
\begin{block}{Theorem}
Let's consider:
\begin{equation*}
\left\{\begin{array}{l}
{\Omega_{ii}}< 0,\ (i=1,2,...r) , \\
{\Omega_{ij}}+{\Omega_{ji}}<0,\ i<j ,\ i,j=1,2,...r
\end{array}\right.
\end{equation*}
\end{block}
with:
\tiny
\begin{equation*}
\Omega_{ij}= \left[\begin{array}{cccc}
PA_i+A_i^TP^T+PB_iK_j+K_j^TB_i^TP^T+Q& PA_{di}& A_i^TN^T+K_j^TB_i^TN^T & 0 \\ *& -(1-\mu)Q& A_{di}^TN^T & 0\\ *&*&\tau^2R-N-N^T & 0\\ * &* &* &-R
\end{array}\right]
\end{equation*}
\end{frame}
\section{Proposed controller}
\footnotesize
\begin{frame}
We consider:
$u(t)=\sum_{i=1}^{r} h_{i}(\theta(t))[K_{i}x(t)+K_{di}x(t-\tau(t)]$
\begin{block}{Theorem }
\begin{equation*}
\left\{\begin{array}{l}
\Bar{\Omega_{ii}}< 0,\ i=1,2,...r \\
\Bar{\Omega_{ij}}+\Bar{\Omega_{ji}}<0,\ i<j ,\ i,j=1,2,...r
\end{array}\right.
\end{equation*}
\end{block}
where:\\
$ \Bar{\Omega_{ij}}=\left[\begin{array}{cccc}
A_iX+X^TA_i^T+B_iY_j+Y_j^TB_i^T+\Bar{Q}& A_{di}X+B_iY_{dj}& \epsilon X^T A_i^T+\epsilon Y_j^TB_i^T & 0 \\ *& _(1-\mu)\Bar{Q}& \epsilon X^T A_{di}^T+\epsilon Y_{di}^TB_i^T & 0\\ *&*&\tau^2\Bar{R}-\epsilon X-\epsilon X^T & 0\\ *&*&*&-\Bar{R}
\end{array}\right]$
$K_j=Y_jX^{-1}$ و $K_{dj}=Y_{dj}X^{-1}(j=1,2,...r)$
\end{frame}
\section{Simulation results}
\begin{frame}
\footnotesize
System behavior without controller
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{x_1.eps}
\caption{System state}
\end{figure}
System unstable
\end{frame}
\begin{frame}
System behavior with controller
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{kuc.eps}
\caption{Control signal }
\label{fig3.3}
\end{figure}
\end{frame}
\begin{frame}
System behavior with controller
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{kstates.eps}
\caption{System state}
\label{fig3.4}
\end{figure}
The system is stable but it needs more enhancement
\end{frame}
\begin{frame}
System behavior with the proposed controller
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{2uc.eps}
\caption{Proposed controller signal }
\label{fig3.5}
\end{figure}
\end{frame}
\begin{frame}
System behavior with the proposed controller
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{2states.eps}
\caption{System state with the proposed controller }
\label{fig3.6}
\end{figure}
\textcolor{blue}{ Stability + better performance}
\end{frame}
\begin{frame}
\footnotesize
Quantification of the comparative study
\begin{table}[h]
\centering
\begin{tabular}{|*{4}{c|}}
\hline
& Classical controller & Proposed controller & Enhancement rate \\
\hline
Settling time & $26$ & $20$ & 23 \% \\
\hline
Pic to pic $x_3$ &$1.36$ & $1.16 $ &15 \% \\
\hline
$\int\limits_0^{ts}(x_3^2)dt $
& $2.5540$ & $0.7771$& 70 \% \\
\hline
$ \int\limits_0^{ts}(u^2)dt$ & $12.8476$ & $7.1868$& 40 \% \\
\hline
\end{tabular}
\caption{Quantification of the comparative study}
\end{table}
We remark that
\begin{itemize}
\item \textcolor{blue}{\textbf{Enhancement of the settling time of $23\%$}}
\item \textcolor{blue}{\textbf{ Reduction of the control energy by $40\%$}}
\item \textcolor{blue}{\textbf{ Overall enhancement by $70\%$ }}
\item \textcolor{blue}{\textbf{ Pic to pic reduction by $15\%$}}
\end{itemize}
\end{frame}
\section{Conclusion and Perspectives}
\begin{frame}
\footnotesize
We conclude that:
\begin{block}{Conclusion}
\begin{itemize}
\item \textcolor{blue}{\textbf{Lyapunov method efficiency.}}
\item \textcolor{blue}{\textbf{Proposed controller leads to better performance.}}
\item \textcolor{blue}{\textbf{Delayed controller enhances the performance.}}
\item \textcolor{blue}{\textbf{Proposed approach allows reduction of the control energy.}}
\end{itemize}
\end{block}
As perspectives we propose:
\begin{block}{perspectives}
\begin{itemize}
\item Perspective 1.
\item Perspective 2.
\item Perspective 3.
\end{itemize}
\end{block}
\end{frame}
\end{document}