%% Created by Maple 2015.2, Windows 8
%% Source Worksheet: Applying the taylor series method to the L1 model.mw
%% Generated: Mon May 07 21:49:07 BST 2018
\documentclass{article}
\usepackage{maplestd2e}
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\begin{document}
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\DefineCharStyle{Maple 2D Math}
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\DefineCharStyle{Maple 2D Output}
\DefineCharStyle{Maple 2D Input}
\section{\textbf{Apply the taylor series method to the L1 model}}
\begin{Maple Normal}{
\begin{Maple Normal}{
\mapleinline{inert}{2d}{}{\[\displaystyle \]}
}\end{Maple Normal}
}\end{Maple Normal}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{read "LieDerProg.mpl"; -1}{\[\]}
\end{mapleinput}
\end{maplegroup}
\begin{Maple Normal}{
\begin{Maple Normal}{
Define the L2 model statespace function and output function.}\end{Maple Normal}

\begin{Maple Normal}{
Define the L2 model statespace function and output function.}\end{Maple Normal}

}\end{Maple Normal}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{F := [k[a1]*C[T]*(R-x[1])-k[d1]*x[1]]; 1; H := alpha*x[1]; 1}{\[\]}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{F := [k[a1]*C[T]*(R-x[1])-k[d1]*x[1]]}{\[\displaystyle F\, := \,[k_{{{\it a1}}}C_{{T}} \left( R-x_{{1}} \right) -k_{{{\it d1}}}x_{{1}}]\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{H := alpha*x[1]}{\[\displaystyle H\, := \,\alpha\,x_{{1}}\]}
\end{maplelatex}
\end{maplegroup}
\begin{Maple Normal}{
\begin{Maple Normal}{
}\end{Maple Normal}

\begin{Maple Normal}{
Generate the first n derivatives of the output in either phase.}\end{Maple Normal}

}\end{Maple Normal}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{n := 2; -1; La := subs([x[1] = 0, x[2] = 0, alpha = 1000], listLieDer(H, F, n)); -1; Ld := subs([C[T] = 0, alpha = 1000], listLieDer(H, F, n)); -1}{\[\]}
\end{mapleinput}
\end{maplegroup}

Consider two parameter vectors giving equal the output in the dissociation.

\begin{Maple Normal}{
\begin{Maple Normal}{
}\end{Maple Normal}
}\end{Maple Normal}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{alt := C = Ch, R = Rh, k = kh, x = xh; -1; Sd := [solve([op(subs(alt, Ld)-Ld)])]; -1; i := [NonZeroSols(Sd)]; -1; Sd := Sd[i]}{\[\]}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Sd := [{k[d1] = kh[d1], kh[d1] = kh[d1], x[1] = xh[1], xh[1] = xh[1]}]}{\[\displaystyle {\it Sd}\, := \,[ \left\{ k_{{{\it d1}}}={\it kh}_{{{\it d1}}},{\it kh}_{{{\it d1}}}={\it kh}_{{{\it d1}}},x_{{1}}={\it xh}\\
\mbox{}_{{1}},{\it xh}\\
\mbox{}_{{1}}={\it xh}\\
\mbox{}_{{1}} \right\} ]\]}
\end{maplelatex}
\end{maplegroup}
\begin{Maple Normal}{
\begin{Maple Normal}{
}\end{Maple Normal}
}\end{Maple Normal}
\begin{Maple Normal}{
\begin{Maple Normal}{
Consider two parameter vectors giving equal the output in the Association.}\end{Maple Normal}
}\end{Maple Normal}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{Sa := [solve([op(subs(Sd[1], subs(alt, La))-La)])]; -1; i := [NonZeroSols(Sa)]; -1; Sa1 := Sa[i]}{\[\]}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Sa1 := [{R = R, Rh = R*(Ch[T]*kh[a1]-k[d1]+kh[d1])/(Ch[T]*kh[a1]), C[T] = (Ch[T]*kh[a1]-k[d1]+kh[d1])/k[a1], Ch[T] = Ch[T], k[a1] = k[a1], k[d1] = k[d1], kh[a1] = kh[a1], kh[d1] = kh[d1]}]}{\[\displaystyle {\it Sa1}\, := \,[ \left\{ R=R,{\it Rh}={\frac {R \left( {\it Ch}_{{T}}{\it kh}_{{{\it a1}}}-k_{{{\it d1}}}+{\it kh}_{{{\it d1}}} \right) \\
\mbox{}}{{\it Ch}_{{T}}{\it kh}_{{{\it a1}}}}},C_{{T}}={\frac {{\it Ch}_{{T}}{\it kh}_{{{\it a1}}}-k_{{{\it d1}}}+{\it kh}_{{{\it d1}}}}{k_{{{\it a1}}}}},{\it Ch}_{{T}}={\it Ch}_{{T}},k_{{{\it a1}}}=k_{{{\it a1}}},k_{{{\it d1}}}=k_{{{\it d1}}},{\it kh}_{{{\it a1}}}={\it kh}_{{{\it a1}}\\
\mbox{}},{\it kh}_{{{\it d1}}}={\it kh}_{{{\it d1}}} \right\} ]\]}
\end{maplelatex}
\end{maplegroup}
\begin{Maple Normal}{
\begin{Maple Normal}{
For a parameter vector to give equal output in both phases it must satisfy the following.}\end{Maple Normal}

\begin{Maple Normal}{
For a parameter vector to give equal output in both phases it must satisfy the following.}\end{Maple Normal}

\begin{Maple Normal}{
For a parameter vector to give equal output in both phases it must satisfy the following.}\end{Maple Normal}

}\end{Maple Normal}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{list1 := solve(convert(`union`(Sd[1], Sa1[1]), list)); -1; `~`[print](list1)[]; 1}{\[\]}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{R = Rh}{\[\displaystyle R={\it Rh}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Rh = Rh}{\[\displaystyle {\it Rh}={\it Rh}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{C[T] = Ch[T]*kh[a1]/k[a1]}{\[\displaystyle C_{{T}}={\frac {{\it Ch}_{{T}}{\it kh}_{{{\it a1}}}}{k_{{{\it a1}}}}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Ch[T] = Ch[T]}{\[\displaystyle {\it Ch}_{{T}}={\it Ch}_{{T}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{k[a1] = k[a1]}{\[\displaystyle k_{{{\it a1}}}=k_{{{\it a1}}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{k[d1] = kh[d1]}{\[\displaystyle k_{{{\it d1}}}={\it kh}_{{{\it d1}}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{kh[a1] = kh[a1]}{\[\displaystyle {\it kh}_{{{\it a1}}}={\it kh}_{{{\it a1}}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{kh[d1] = kh[d1]}{\[\displaystyle {\it kh}_{{{\it d1}}}={\it kh}_{{{\it d1}}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{x[1] = xh[1]}{\[\displaystyle x_{{1}}={\it xh}_{{1}}\]}
\end{maplelatex}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{xh[1] = xh[1]}{\[\displaystyle {\it xh}_{{1}}={\it xh}_{{1}}\]}
\end{maplelatex}
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{2d}{}{\[\]}
\end{mapleinput}
\end{maplegroup}
\begin{Maple Normal}{
\begin{Maple Normal}{
\mapleinline{inert}{2d}{}{\[\displaystyle \]}
}\end{Maple Normal}
}\end{Maple Normal}
\end{document}