rsmc_report.tex.bak

用latex编辑适用于powerpoint格式的PDF文档。

          \vspace{-9mm}\item [\small\NibRight]\xiaosihao\hei The sliding surface(switch surface) $s(\vec{x})=s(x_1,x_2,\dots,x_n)=0$ \\
          \vspace{-8mm}\item [\small\NibRight] \xiaosihao\hei $s(\vec{x})= 0$ separates the states space into two part: $s<0$ and $s>0$\\
          \vspace{-8mm}\item [\small\NibRight] \xiaosihao\hei \textcolor[rgb]{1.00,0.00,0.00}{Slide Mode}: state comes near switch surface like \textcolor[rgb]{1.00,0.00,0.00}{\hei C},it will be arrested by the surface
          \\[-10mm]
          and terminate at the surface finally
          \vspace{-12mm}\item[\small\NibRight] \hei for all states in the sild mode area,it's required that\\[-12mm]
           \sihao ${\sihao\lim \atop {\liuhao s\rightarrow 0^+}}\dot{s}\leq 0 $\quad and \quad ${\lim \atop {\liuhao s\rightarrow
          0^-}}\dot{s}\geq 0$\\[-10mm]
          witch is equal to :\quad \quad \quad \hei\sihao$\label{eq.LA}{\lim \atop \liuhao s\rightarrow 0}~\textcolor[rgb]{1.00,0.00,0.00}{\sihao\hei s\dot{s}}\leq
          0$\\[-5mm]
          \end{itemize}
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                 define  SMC                                        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionSplitHO
\begin{slide}
\begin{itemize}
\sihao\item [\small\PlusCenterOpen ]\sihao\hei \textcolor[rgb]{1.00,0.00,0.00}{Definition of Sliding Mode Control}\\
Consider the simple dynamic system of the form: \\[-14mm]
\be\dot{x}=f(x,u,t)]\quad \hei x\in R^n,u\in R^m,t\in R\ee\\[-14mm]
it is required to find a switch functional\\[-14mm]
\be s(x)\quad\quad s\in R^n\nm\ee\\[-14mm]
then,obtain the control functional:\\[-10mm]
\be \label{eq.contol} u=\lt\{\ba{cc}
       u^+(x),&{\rm if}~ s(x)\ge 0\\
       u^-(x),&{\rm if}~ s(x)\le 0\\\ea\rt.\ee\\[-12mm]
       where $u^+(x)\neq u^-(s)$,and\\[-12mm]
       \begin{itemize}
       \item[\ding {172}] the slide mode exit ,that is ,the inequality (\ref{eq.contol}) holds  \\[-14mm]
       \item[\ding {173}] the states outsides the surface s(x)=0 terminate at the surface at limit
       interval;\\[-14mm]
       \item[\ding {174}] the slide mode is stable \\[-14mm]
       \item[\ding {174}] satisfied the needed dynamic performance of the control system\\[-14mm]
       \end{itemize}

\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                            Problem Formulation                                        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionSplitHO
\begin{slide}
\yihao\color{blue}\centering \li Problem Formulation \\
\color{black}
\begin{itemize}
\item[\small\HandRight ]\xiaoerhao \hei{ The uncertain
time-delay system of the form}
 \sihao \hei \color{black}
 \be
\label{eq1.1}
 \ba{rl}
 \dot{x}(t)\hs{-3mm} & =(A+\Delta A)x(t)+(A_d+\Delta A_d)x(t-\tau(t))\\
 \hspace{-3mm}&~~+B(u(t)+w(x, x(t-\tau(t)), t))\\
 \hspace{-10mm} x(t)&=\varphi (t),  t\in \left[ \ba{cc} -\tau,0
 \ea \right]
 \ea
 \ee
 \end{itemize}
\sihao\hei{where $x(t)\in R^n$ is the state, $w\in R^l$ are the
matched uncertainties and disturbance. $\Delta A$ and $\Delta A_d$
are unmatched uncertainties, $\tau$ is a constant which is the upper
bound of $\tau(t)$. $u(t)\in R^m$ is the control input, $A, A_d$ and
$B$ are real constant matrices with appropriate dimensions and
$rank(B)=m$.}\hspace*{\fill}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                          Assumption  1                                                 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
\sihao\hei \color{black}
\begin{itemize}
\item[\HandRight]\raisebox{-1ex}{\sanhao\color{blue}\centering \li Assumption: 1\hspace*{\fill}}\\
The uncertainties $\Delta A$ and $\Delta A_d$ in (1) are assumed
to have the following form\\[3mm]
\vspace{-8mm}
 \be\label{eq.uncertainform} \ba{rl}
 \Delta A&=\sum_{i=1}^p\theta _iA_i\\
 \theta&=[\theta_1~\theta_2~\cdots~\theta_p]^T\in \Omega=\{\theta|~|\theta_i|\le
 1\}\\[3mm]
\Delta A_d&=\sum_{i=1}^q\beta _iA_{di}\\
 \beta&=[\beta_1~\beta_2~\cdots~\beta_q]^T\in
 \Omega_d=\{\beta|~|\beta_i|\le 1\}\\ \ea \ee\\[3mm]
 where the matrices ${A_i}$ and ${A_{di}}, i=1,\cdots, p$ are known with $rank(A_i)=a_i$ and $rank(A_{di})=a_{di}$, $\theta _i$ and $\beta _i$ are unknown
 constants.\hspace*{\fill}
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                          Assumption  1 ~                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
\sihao\hei \color{black} Suppose that $A_i$ and $A_{di}$ have full
rank factorization of $A_i =G_iH_i$ and $A_{d_i}=G_{di}H_{di}$,
respectively, where $G_i\in R^{n\times a_i}$, $H_i \in R^{a_i \times
n}$, $G_{di}\in R^{n\times a_{di}}$ and $H_{di} \in R^{a_{di} \times
n}$. Then\\[3mm]

\vspace{-10mm}
\be\label{uncertainform}
\ba{rl}
\hs{-4mm}\Delta A\hs{-2mm}&=\sum_{i=1}^p\theta _iA_i=\left[ \ba{cccc}G_1 &G_2&\cdots\hs{-2mm}&G_p \ea \right]D\times\\
                                               &~~~\left[ \ba{cccc} H_1&H_2&\cdots& H_p\ea \right]^T=GDH\\\\[3mm]
 \hs{-4mm}\Delta A_d\hs{-2mm}&=\sum_{i=1}^q\beta_iA_{di}=\left[\ba{cccc}G_{d1}&G_{d2}&\cdots\hs{-2mm}&G_{dq}
\ea\right]D_d \times\\
&~~~\left[ \ba{cccc} H_{d1}&H_{d2}&\cdots & H_{dq}\ea
\right]^T=G_dD_dH_d \ea \ee\\[3mm]
 where
 \be\label{eq.ddd}\small\ba{rl}
 D\hs{-3mm}&={\rm blockdiag}\left[ \ba{cccc} \theta _1I_{a_1\times a_1}
& \theta _2I_{a_2\times a_2}&\cdots &\theta _pI_{a_p\times a_p} \ea \right]\\
D_d\hs{-3mm}&={\rm blockdiag}\left[ \ba{cccccc} \beta_1I_{a_{d1}\times a_{d1}} & \beta _2I_{a_{d2}\times
a_{d2}}\ea\rt.\\
&~~~~~~~~~~~~~~~~~~~~\lt.\ba{ccccc}\cdots &\beta _qI_{a_{dq}\times
a_{dq}} \ea \right]\ea\ee

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                          Assumption  2                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
\begin{itemize}
\item [\HandRight]\raisebox{-1ex}{\textcolor[rgb]{0.00,0.00,1.00}{~~~Assumption 2}}\\[6mm]
\sihao\hei \color{black} The matched uncertainties $w(x,x(t-\tau),
t)$ are assumed to satisfy the following condition
 \be ||w(x,x(t-\tau), t)||\le \textcolor[rgb]{0.00,0.00,1.00}{c}+ \textcolor[rgb]{0.00,0.00,1.00}{k}||x(t)||=\textcolor[rgb]{0.00,0.00,1.00}{\rho}\ee
where $c$ and $k$ are constants, but it may not be easily obtained
due to the complexity of the structure of the uncertainty.

\item [\HandRight]\raisebox{-1ex}{\textcolor[rgb]{0.00,0.00,1.00}{~~~Assumption 3}}\\[6mm]
It is assumed that $\tau(t)$ is differentiable, and $\dot \tau
(t)\leq d<1$.
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                           Factorizations                                              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
 To obtain a regular form of the systems (\ref{eq1.1}), a
nonsingular matrix $T$ can be chosen such that \bea TB=\left[\ba{c}
0_{(n-m)\times m}\\B_2\ea\right] \nm\eea where $B_2\in R^{m\times
m}$ is nonsingular. For convenience, choose \bea T=\left[\ba{c}
{U_2}^T\\{U_1}^T\ea \right] \nm\eea \noindent {where} $U_1\in
R^{n\times m}$ and $U_2\in R^{n \times (n-m)}$ are two sub-blocks of
a unitary matrix resulting from the singular value decomposition of
$B$, i.e.,
 \bea B=\left[\ba{cc} U_1&U_2 \ea \right] \left[\ba{c} \Sigma \\0_{(n-m)\times m}
\ea \right]V^T \nm\eea where $\Sigma\in R^{m\times m}$ is a diagonal
positive-definite matrix and $V\in R^{m\times m}$ is a unitary
matrix.
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                           Matlab M-File Code                                          %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
\begin{itemize}
\vspace{10mm}\item
[\HandRight]\raisebox{-1ex}{\hei\textcolor[rgb]{0.00,0.00,1.00}{~~MATLAB
Language Code for above decomposition}}\hspace*{\fill}\\%%MATLAB 语言实现上述的分解

\centering \sihao [U \textcolor[rgb]{1.00,0.00,0.00}{S0} V] = \hei \textcolor[rgb]{1.00,0.00,0.00}{svd}(B);\\
\centering \quad\textcolor[rgb]{1.00,0.00,0.00}{S} = \textcolor[rgb]{1.00,0.00,0.00}{S0}([1:m],:);\\
U1 = U(:,[1:m]);\\
\quad U2 = U(:,(m+1:n));\\
\quad T = [U2';U1'];
\item[$\star$]\centering \textcolor[rgb]{0.00,0.00,0.00}{The MATLAB function \hei\textcolor[rgb]{1.00,0.00,0.00}{svd} computes the matrix singular
value decomposition with:}\hspace{\fill} \bea
\textcolor[rgb]{1.00,0.00,0.00}{S0}= \left[\ba{c}
\Sigma\\0_{(n-m)\times m} \ea \right] \nm\eea
\centering$\textcolor[rgb]{1.00,0.00,0.00}{S}$=$\Sigma$
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                           rewrite system                                          %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
\begin{itemize}
\vspace{10mm}\item
[\HandRight]\raisebox{-1ex}{\li\xiaoerhao\hei\textcolor[rgb]{0.00,0.00,1.00}{~~Rewrite the system :1 }}\hspace*{\fill}\\
 The we can obtain a system in the
form\be\label{system1.3}\ba{rl}
      \dot{z}_1(t)  =\hs{-2mm}&(\overline A_{11}+\Delta \overline A_{11})z_1(t)+(\overline A_{d11}+\Delta \overline A_{d11})z_1(t-\tau )\\
                              &+(\overline A_{12}+\Delta \overline A_{12})z_2(t)+(\overline A_{d12}+\Delta \overline A_{d12})z_2(t-\tau )\\
    \dot{z}_2(t)  =\hs{-2mm}&(\overline A_{21}+\Delta \overline A_{21})z_1(t)+(\overline A_{d21}+\Delta \overline A_{d21})z_1(t-\tau )\\
                         \hs{-2mm} &+(\overline A_{22}+\Delta \overline A_{22})z_2(t)+(\overline A_{d22}+\Delta \overline A_{d22})z_2 (t-\tau )\\
                        \hs{-2mm} & +B_2(u+w(z(t), z(t-\tau), t))\\
  z_1(t)=\hs{-2mm}&\overline \varphi _1(t), t\in \left[ \ba{cc} -\tau, & 0 \ea \right]\\
     z_2(t)=\hs{-2mm}&\overline \varphi _2(t), t\in \left[ \ba{cc} -\tau, & 0 \ea \right]\\
\ea\ee where \textcolor[rgb]{0.00,0.00,1.00}{$z_1\in R^{n-m}$,
$z_2\in R^m$, $B_2=\Sigma V^T$, $\overline A_{11}=U^T_2AU_2$,
$\overline A_{12}=U^T_2AU_1$, $\overline A_{d11}=U^T_2A_dU_2$,
$\overline A_{d12}=U^T_2A_dU_1$, $\Delta \overline
A_{11}={U_2}^TGDHU_2$,
 $\Delta \overline A_{d11}={U_2}^TG_dD_dH_dU_2$, $\Delta \overline A_{12}={U_2}^TGDHU_1$,
$\Delta \overline A_{d12}={U_2}^TG_d D_dH_dU_1$, $\ol
\varphi_1(t)\in R^{(n-m)}$ }and $\ol \varphi_2(t) \in R^m$ are the
sub-blocks of $\ol\varphi(t)$.
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                           rewrite system                                          %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pageTransitionGlitter{270}
\begin{slide}
\begin{itemize}
\vspace{10mm}\item
[\HandRight]\raisebox{-1ex}{\xiaoerhao\li\hei\textcolor[rgb]{0.00,0.00,1.00}{~~Rewrite the system :2 }}\hspace*{\fill}\\
It is obvious that the first equation of system (\ref{system1.3})
represents the sliding motion dynamics of   system ,   hence the
corresponding sliding surface can be chosen as follows:
     \be \delta(t)=\left[\ba{cc} C&I \ea \right]z(t)=Cz_1(t)+z_2(t)=0 \nm\ee
\noindent {where} $C\in R^{m \times (n-m)}$. Substituting
$z_2=-Cz_1$ into the first equation of system (\ref{system1.3})
gives the sliding motion \be\label{sliding-motion}\ba{rl}
\dot{z_1}(t) =\hs{-2mm}&(\overline A_{11}+\Delta \overline A_{11}-\overline A_{12}C-\Delta \overline A_{12}C)z_1(t)\\
\hs{-2mm} &+(\overline A_{d11}+\Delta \overline A_{d11}-\overline A_{d12}C-\Delta \overline A_{d12}C)z_1(t-\tau )  \\
z_1(t)=\hs{-2mm}&\overline \varphi _1(t), t\in \left[ \ba{cc} -\tau, & 0 \ea \right].\\
\ea\ee For simplicity, the sliding motion can be written as
\be\label{sliding-motion}\ba{rl}
\dot{z_1}(t) =\hs{-2mm}&\tilde A z_1(t)+\tilde A_d z_1(t-\tau )  \\
z_1(t)=\hs{-2mm}&\overline \varphi _1(t), t\in \left[ \ba{cc} -\tau, & 0 \ea \right].\\
\ea\nm\ee where\textcolor[rgb]{0.00,0.00,1.00}{
\be\label{eq.tildeaad}\ba{rl}\tilde A&=\overline A_{11}+\Delta
\overline A_{11}-\overline A_{12}C-\Delta \overline
A_{12}C\\
 \tilde A_d&=\overline A_{d11}+\Delta \overline
A_{d11}-\overline A_{d12}C-\Delta \overline A_{d12}C \ea\ee}
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                          Main Result             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%