@@ -148,14 +149,14 @@ The filter is implemented with relative damping $\zeta=1/\sqrt{2}$. The role of

\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{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.

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{astrom06}, when $M_s=M_t=1.5$ is used.

\begin{figure}[t]

\centering

\begin{tikzpicture}[scale=1]

\begin{axis}[

xlabel=$t$,

ylabel=$e(t)$]

% FIXME: plot with axis middle, narrow and no tick marks

\begin{axis}[xlabel=$t$,ylabel=$e(t)$,axis lines=middle,xmax=17,ticks=none,axis line style={-narrow},

y label style={at=(current axis.above origin),anchor=north east},

x label style={at=(current axis.right of origin),anchor=north},

@@ -168,7 +169,7 @@ A disadvantage with \pidIE, is that it does not provide low-pass filtering of th

The \pidIAE and \pidfIAE methods rely on gradient-based methods to find a (local optimum) to the constrained optimization design problem. Since most practical applications result in problems with unique optima, the methods are broadly applicable. They both rely on computing the sensitivities of objective and constraints with respect to the controller (and filter) parameters, in order to obtain the Jacobian matrix by the optimization tool. They execute slower than \pidIE, and since the two often result in similar performance, \pidIE is almost always preferred to \pidIAE for practical purposes.

In addition to the PID controller, \pidfIAE, synthesizes a low-pass filter \eqref{eq:filter}, which allows for a controller with high-frequency roll-off, and thus good noise attenuation properties. This makes constraining $K_{ks}$ practically feasible. An alternative is to manually iterate between filter and controller design, and incorporate the filter as a series connected component of $P$ when running either \pidIE or \pidIAE. In this context, it can can be mentioned that it is common in academic work on PID control to consider minimization of IAE or IE under constraints on sensitivity (and complementary sensitivity), while disregarding noise sensitivity. One reason for this might be that it is complicated to decide on a reasonable constraint level, as it depends both on the gain of the process and on the spectral density of the measurement noise. However, disregarding noise sensitivity can result in very aggressive controllers, not suitable for practical use.

In addition to the PID controller, \pidfIAE, synthesizes a low-pass filter \eqref{eq:filter}, which allows for a controller with high-frequency roll-off, and thus good noise attenuation properties. This makes constraining $M_{ks}$ practically feasible. An alternative is to manually iterate between filter and controller design, and incorporate the filter as a series connected component of $P$ when running either \pidIE or \pidIAE. In this context, it can can be mentioned that it is common in academic work on PID control to consider minimization of IAE or IE under constraints on sensitivity (and complementary sensitivity), while disregarding noise sensitivity. One reason for this might be that it is complicated to decide on a reasonable constraint level, as it depends both on the gain of the process and on the spectral density of the measurement noise. However, disregarding noise sensitivity can result in very aggressive controllers, not suitable for practical use.

\section{Practical aspects\label{sec:practical}}

Below is an unsorted list of practical aspects, which could be worth considering.