author={Hast, Martin and {\AA}str\"{o}m, Karl Johan and Bernhardsson, Bo and Boyd, Stephen},

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},

...

...

@@ -43,6 +43,33 @@ 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}

\newcommand{\figref}[1]{Figure~\ref{fig:#1}}

...

...

@@ -173,7 +200,7 @@ F(s) = \dfrac{1}{T^2s^2+2\zeta T s +1}.

The filter is implemented with relative damping $\zeta=1/\sqrt{2}$. The role of the filter is to ensure high-frequency roll-off. In order to achieve this also with \pidIE and \pidIAE, simply design a low-pass filter and include it as a series connected component of $P$ prior to conducting the design.

\subsection{Which method to use?}

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{garpinger}, although it is possible to construct pathological examples, for which this is not true.

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]

...

...

@@ -209,13 +236,12 @@ 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{hast} for a discussion on the topic.

\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.

\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

%\nobibliography{\jobname} % FIXME: add references here