Commit 3b673be9 authored by Kristian Soltesz's avatar Kristian Soltesz
Browse files

epsilon update

parent 873271d1
......@@ -12,67 +12,7 @@
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\tikzset{every picture/.style={auto, line width=1pt, >=narrow,x=1mm, y=1mm, font=\small}}
\begin{filecontents}{\jobname.bib}
@article{pidfIAE,
author={Soltesz, Kristian and Grimholt, Chriss and Skogestad, Sigurd},
title={Simultaneous Design of {PID} Controller and Measurement Filter by Optimization},
journal={{IET Control Theory \& Applications}},
volume=11,
number=3,
year=2017,
pages={348--348},
doi={10.1049/iet-cta.2016.0297}
}
@inproceedings{pidIAE,
author={Grimholt, Chriss and Skogestad, Sigurd},
title={Improved Optimization-based Design of {PID} Controllers Using Exact Gradients},
booktitle={12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering},
address={Copenhagen, Denmark},
year=2015,
pages={1753--1756},
isbn={978-0-444-63445-0 (eBook)}
}
@inproceedings{pidIE,
author={Hast, Martin and {\AA}str{\"o}m, Karl Johan and Bernhardsson, Bo and Boyd, Stephen},
title={{PID} Design by Convex-Concave Optimization},
booktitle={{IEEE} European Control Conference ({ECC})},
address={Z\"{u}rish, Switzerland},
year=2013,
pages={4460--4465},
isbn={978-3-033-03962-9 (eBook)}
}
@book{astrom06,
author={{\AA}str{\"o}m, Karl Johan and H{\"a}gglund, Tore},
title={Advanced {PID} Control},
year=2006,
publisher={{ISA} - The Instrumentation, Systems and Automation Society},
isbn={978-1-55617-942-6}
}
@article{hast15,
author={Hast, Martin and H{\"a}gglund, Tore},
title={Optimal proportional-integral-derivative set-point weighting and tuning rules for proportional set-point weights},
journal={{IET Control Theory \& Applications}},
volume=9,
number=15,
year=2015,
pages={2266--2272},
doi={10.1049/iet-cta.2015.0171}
}
@phdthesis{garpinger15,
author={Garpinger, Olof},
title={Analysis and Design of Software-Based Optimal {PID} Controllers},
year=2015,
school={Department of Automatic Control, Lund University, Sweden},
number={{TFRT-1105}}
}
\end{filecontents}
\tikzset{every picture/.style={>=narrow}}%
\newcommand{\figref}[1]{Figure~\ref{fig:#1}}
\newcommand{\secref}[1]{Section~\ref{sec:#1}}
......@@ -135,17 +75,18 @@ The remainder of this document is dedicated to briefly describe the design metho
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\label{fig:cc}
......@@ -207,10 +148,10 @@ The robustness constraints are generally not convex in the controller parameters
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}[t]
\centering
\begin{tikzpicture}[scale=0.14]
\begin{tikzpicture}[scale=1]
\begin{axis}[
xlabel=Cost,
ylabel=Error]
xlabel=$t$,
ylabel=$e(t)$]
\addplot[color=red,mark=x] coordinates {
(2,-2.8559703)
(3,-3.5301677)
......@@ -229,7 +170,7 @@ ylabel=Error]
%\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.}
\caption{The figure shows the load step response of two closed-loop control systems. Most users would prefer the one drawn in solid, which has a significantly smaller IAE than the dashed. However, due to the repetitive zero crossings, the IE of the dashed response is smaller than that of the solid line.}
\label{fig:ievsiae}
\end{figure}
......@@ -259,7 +200,6 @@ The time domain evaluation of IAE in \pidIAE and \pidfIAE ensures stability. The
\item\emph{Alternative objectives and constraints}
It is straight forward to impose constraints on other closed-loop transfer functions. In \pidIAE and \pidfIAE it would also be possible to change the objective to minimization of for instance the integrated square error (ISE), being the $\mathcal{L}_2$-norm of the load response $e$ (of which IAE is the $\mathcal{L}_1$-norm). The choices in the provided code were motivated by what is most commonly used in industrial applications. It is also possible to impose frequency-dependent robustness constraints, in which each frequency grid point is independently constrained.
\item\emph{Reference handling} The transfer functions from reference $r$ to control signal $u$ and measurement $y$ has not been considered. The reason for this is that they can be shaped by adding a reference pre-filter in combination with a feed-forward path from $r$ to $u$, to shape these transfer functions once the feedback controller $K$ has been designed. See for instance \cite{hast15} for a discussion on the topic.
\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.
......@@ -267,3 +207,67 @@ It is straight forward to impose constraints on other closed-loop transfer funct
\end{itemize}\clearpage
\bibliography{\jobname}
\end{document}
\begin{filecontents}{\jobname.bib}
@article{pidfIAE,
author={Soltesz, Kristian and Grimholt, Chriss and Skogestad, Sigurd},
title={Simultaneous Design of {PID} Controller and Measurement Filter by Optimization},
journal={{IET Control Theory \& Applications}},
volume=11,
number=3,
year=2017,
pages={348--348},
doi={10.1049/iet-cta.2016.0297}
}
@inproceedings{pidIAE,
author={Grimholt, Chriss and Skogestad, Sigurd},
title={Improved Optimization-based Design of {PID} Controllers Using Exact Gradients},
booktitle={12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering},
address={Copenhagen, Denmark},
year=2015,
pages={1753--1756},
isbn={978-0-444-63445-0 (eBook)}
}
@inproceedings{pidIE,
author={Hast, Martin and {\AA}str{\"o}m, Karl Johan and Bernhardsson, Bo and Boyd, Stephen},
title={{PID} Design by Convex-Concave Optimization},
booktitle={{IEEE} European Control Conference ({ECC})},
address={Z\"{u}rish, Switzerland},
year=2013,
pages={4460--4465},
isbn={978-3-033-03962-9 (eBook)}
}
@book{astrom06,
author={{\AA}str{\"o}m, Karl Johan and H{\"a}gglund, Tore},
title={Advanced {PID} Control},
year=2006,
publisher={{ISA} - The Instrumentation, Systems and Automation Society},
isbn={978-1-55617-942-6}
}
@article{hast15,
author={Hast, Martin and H{\"a}gglund, Tore},
title={Optimal proportional-integral-derivative set-point weighting and tuning rules for proportional set-point weights},
journal={{IET Control Theory \& Applications}},
volume=9,
number=15,
year=2015,
pages={2266--2272},
doi={10.1049/iet-cta.2015.0171}
}
@phdthesis{garpinger15,
author={Garpinger, Olof},
title={Analysis and Design of Software-Based Optimal {PID} Controllers},
year=2015,
school={Department of Automatic Control, Lund University, Sweden},
number={{TFRT-1105}}
}
\end{filecontents}
\begin{filecontents}{steps.csv}
% FIXME: cvs data here
\end{filecontents}
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