Presentation template for Master and PhD by Prof. K. GUESMI Univ. Djelfa
Autor
Prof. Kamel GUESMI
Letzte Aktualisierung
vor 10 Monaten
Lizenz
Creative Commons CC BY 4.0
Abstrakt
Presentation template for Master and PhD
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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}