Commit 873271d1 authored by Kristian Soltesz's avatar Kristian Soltesz
Browse files

Added placeholder for final plot

parent 67eb61f7
% This is the LaTeX source used to generate the doumentation file PIDopt.pdf
% Downloaded from git@gitlab.control.lth.se:kristian/PIDopt.git
\documentclass[a4paper]{article}
\usepackage[left=3cm,right=3cm,top=3cm,bottom=3cm]{geometry}
\usepackage{amsmath}
\usepackage{filecontents}
\usepackage{xspace}
\usepackage{tikz}
\usepackage{tikz,pgfplots}
\usetikzlibrary{shapes,positioning}
\usetikzlibrary{narrow}
......@@ -203,8 +205,30 @@ The filter is implemented with relative damping $\zeta=1/\sqrt{2}$. The role of
The robustness constraints are generally not convex in the controller parameters. The optimum (when either IE or IAE is minimized) is still unique for many processes \cite{garpinger15}, although it is possible to construct pathological examples, for which this is not true.
Minimizing IE \eqref{eq:ie}, as is done by \pidIE, has the advantage of being equivalent to maximizing $k_i$ of the controller \eqref{eq:pid}. This makes the objective convex (actually linear) in the controller parameters, while also being independent of the process dynamics. The resulting constrained optimization problem makes it possible to apply a very fast convex-concave method \cite{pidIE}. Furthermore, for closed-loop systems with a non-oscillating load step response it holds that IE and IAE are equal. Combinations of poor performance and low IE-values are, however, possible for oscillating responses, as illustrated in \figref{ievsiae}. In practice, this can be avoided by reasonable constraints on sensitivity and complementary sensitivity. For instance, the controllers optimized by \pidIE and \pidIAE, respectively, yield almost equivalent IAE-values for all processes of the extensive test batch reported in \cite{}, when $M_s=M_t=1.5$ is used.
\begin{figure}[b]
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.14]
\begin{axis}[
xlabel=Cost,
ylabel=Error]
\addplot[color=red,mark=x] coordinates {
(2,-2.8559703)
(3,-3.5301677)
(4,-4.3050655)
(5,-5.1413136)
(6,-6.0322865)
(7,-6.9675052)
(8,-7.9377747)
};
\end{axis}
\end{tikzpicture}
%\resizebox{.8\columnwidth}{!}{
%\begin{tikzpicture}[scale=.5]
%\begin{axis}[xlabel=$t$,ylabel=$e(t)$]
%\end{axis}
%\end{tikzpicture}
%}
\caption{The figure shows the load step response of two closed-loop control systems. They both have equal IE values, while the IAE of the blue response is only half of that of the red response. IAE is consequently a more reliable performance measure for systems where oscillating load step responses cannot be ruled out.}
\label{fig:ievsiae}
\end{figure}
......@@ -240,8 +264,6 @@ It is straight forward to impose constraints on other closed-loop transfer funct
\item\emph{Active constraints} For most practical design scenarios, at least one of the robustness constraints will be active. There are many situations, where the degrees of freedom result in only one active constraints.
\item\emph{Hessian matrix} The current implementations of \pidIAE and \pidfIAE relies on a Jacobian obtained through exact gradient computations. It is likely that computational efficiency could be improved by also providing a Hessian computed by similar means (rather than by finite differences).
\end{itemize}
%\nobibliography{\jobname} % FIXME: add references here
\end{itemize}\clearpage
\bibliography{\jobname}
\end{document}
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