BO Toolbox – Magnitude Optimum (Betragsoptimum) by Kessler

Continuous PID family (P-,I-,PD-,PI-, PID up to PID4) controller design including pre-filter - either approximations or characteristic areas

BO Toolbox – Magnitude Optimum (Betragsoptimum) by Kessler

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 in this paper 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<<<.