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BO - Magnitude Optimum (Betragsoptimum) |
The Magnitude Optimum (Betragsoptimum) by Kessler offers significantly
more potential than the often discussed limitations on simple systems
with first-order time-delay elements would suggest. The key to this can
be found in two considerations.
In a first step, systems of equations
for calculating the controller parameters must be defined in a general
form without approximations or restrictions for controller order. It
turns out that two systems of equations are sufficient. In the case of
undelayed inputs a linear system of equations results. This is known
as classical Magnitude Optimum (Betragsoptimum). As is well known, no
pre-filter for the set-point branch is defined in this case. For
delayed inputs (Symmetrical Optimum), a nonlinear system of equations
follows. A pre-filter with a denominator polynomial is defined. This
first step already offers greater application potential for the
Magnitude Optimum.
However, a second step leads to the essential generalization. For this
purpose, weighting factors are introduced for the controller parameters
in the set-point branch, in contrast to the feedback branch. As is well
known, the integral term must be excluded. This generalization allows
the two results from step one to be combined into a single, unified
system of equations.
The two original optimization variants, the
classical Magnitude Optimum (Betragsoptimum) and the Symmetrical
Optimum, then represent special cases of the weighting factors in this
single generalized and unified solution.
The >>>current BO Toolbox Version 2.0<<<
offers functions for calculating controllers with the Magnitude Optimum
in accordance with the explanations in step 2. A template file is
available for determining meaningful weighting factors. Possible
criteria include ITAE (integral of time and error) or overshoot.
Numerous examples with calculations and simulations illustrate the ease
of use and usefulness of the results. The corresponding equations and
conversions are provided in
"The Magnitude Optimum – the merged
general solution for the PID controller family".
A table summarizes
the removal of restrictions in recent works on the Magnitude Optimum
known from the current literature
(Papadopoulos 2015, Vrancic 2009/2012/2021,
Cvejn 2022, Kos 2020/2021, Mandic 2024)
by the solution presented here in step 2.
Detailed explanations of
previous step 1, including the previous version 1.2 of the BO Toolbox,
can be found >>>here<<<.