The BO Toolbox – Magnitude Optimum (undelayed+delayed input)

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

The BO Toolbox – Magnitude Optimum (Betragsoptimum)

The BO Toolbox offers the design of continuous controllers of the PID family (P-,I-,PD-,PI-, PID up to PID4) based on the general optimization equations of the Magnitude Optimum (MO) - without approximations, without calculation of characteristic areas, without restriction to PI controllers - but with consideration of the pre-filter, if applicable.

Definitions:

From today's point of view, it is reasonable to use the term’s Magnitude Optimum for undelayed input variables (classical Magnitude Optimum, without pre-filter, limited to one integrator in the open loop) and Magnitude Optimum for delayed input variables (classical Symmetric Optimum, with pre-filter, several integrators possible in the open loop). It is also important to emphasize the close relationship to the Naslin Polynomial Method (double ratios), because the first two optimization conditions are identical - see e.g. B. Ufnalski, 2014.

Topicality:

Controllers of the PID family are still used in many fields due to their clear structure. Over the past few years, the Magnitude Optimum has come back into focus as a possible option for optimizing such control structures and provides sufficient good control results in many cases. The present toolbox offers a number of advantages; in contrast to K.G. Papadopoulos 2015, there is no restriction to 3 degrees of freedom with undelayed inputs (see type I in Papadopoulos), no approximations are required, such as setting one side of the optimization equation to zero in case of several integrators in the open loop (see types II and III in Papadopoulos) and solutions are also available with one integrator in the open loop and delayed inputs (not dealt with in Papadopoulos); in contrast to D. Vrancic 2009 / 2021 no calculation of characteristic areas is necessary and solutions for controllers of higher orders with undelayed inputs are also available and in contrast to J. Cvejn 2022 there is no restriction to PI controllers, but properties and restrictions given there can be generalized accordingly.

Some more hints:

By designing continuous controllers based on this toolbox, the typical properties to be expected for the Magnitude Optimum (overshoot, rise time, disturbance behaviour ...) become apparent. In case of a desired reduction of the overshoot, pole compensation, optimization of the pre-filter or an additional low-pass filter at the controller output can lead to the goal. If, for whatever reason, pole compensation is preferred, it must be carried out before applying a function of the toolbox for designing a continuous controller and the controller must be assembled from the resulting two components. However, if the actual goal is the design of a digital controller, the application of the Digital Magnitude Optimum https://wwwpub.zih.tu-dresden.de/~geitner/ae_1_0e.htm

should be considered from the outset. As an essential advantage, the problem of discretization of a continuous controller is avoided.

Examples:

Some examples of the application of the toolbox functions are part of the toolbox, whereby one file for calculation and one file for simulation are prepared in each case, see also:

https://wwwpub.zih.tu-dresden.de/~geitner/ae_2_2e.htm

The examples typically do not use the possibility of pole compensation. If the implementation-related dynamic of the controller itself (D-terms) deviates too clearly from the ideal behaviour, it can be included in the design. Otherwise, the Simulink default setting in the PID controller block with N=1000 (factor 10^-3) is sufficient, or alternatively, a corresponding consideration of this fictitious time constant in the denominator of the transfer function of the controller. Both variants can be found in the collection of examples.

Historical background / Further information / References:

https://wwwpub.zih.tu-dresden.de/~geitner/ae_2e.htm

https://wwwpub.zih.tu-dresden.de/~geitner/ae_12_L.htm#BO-K_1