**Author details**

finally the classical integration preserving zero steady - state closed - loop system

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

*Remark 4.1*. In the Numerical example proposed here the VFOPID and the classical PID controllers maximal control signal values are the same reaching

*SSE KP*, *KI* ½ ,*KD*, 1, �1Þ� ¼ 1*:*3312

One should emphasize that the proposed solution of the VFOPID controller do not guarantee the absolute optimum of the closed-loop control system synthesis. It proves that the proposal of a physically realizable VFOPID controller by microcontroller (with finite memory) leads to better results due to the assumed perfor-

The main idea of the proposed method is to assume *a priori* the order functions

Here, it is worth mentioning that there are still open problems of the VFOPID

• For evaluation of the VFOPID controller parameters one can apply another optimization methods. It seems that optimization methods based on the

• Another performance index may be applied. Some penalty functions may be introduced to SSE as well a term taking into account the minimal value of the

The work was supported by the National Science Center Poland by Grant no.

FOPID fractional-Order proportional, fractional-order integral and differen-

with unknown parameters. In the VFOPID controller synthesis essential is an assumption that the summation order equals 1 One can express the action as the assumption of skeleton order functions with unknown parameters evaluated fur-

• One should define a program evaluating the order functions.

artificial intelligence will be very effective.

PID proportional-integral-derivative controller FOBD fractional-order backward difference FOBS fractional-order backward sum

VFOPID variable, fractional-order PID controller

tial controller

SSE squared sum of the error

*SSE KP*, *KI* <sup>½</sup> ,*KD*, *<sup>ν</sup>*ð Þ *kh* , *<sup>μ</sup>*ð Þ *kh* � ¼ <sup>1</sup>*:*<sup>2899</sup> (41)

assumed bounding value max ½ � *uI*ð Þ *kl* , max ½ �¼ *uF*ð Þ *kh* 2.

*Remark* 4.2. In the Numerical example

ther in an SSE optimization algorithm.

error.

**5. Conclusions**

mance criterion.

controllers tuning.

error signal.

**Acknowledgments**

2016/23/B/ ST7/03686.

**Abbreviations**

**12**

Piotr Ostalczyk<sup>1</sup> \*† and Piotr Duch2†

1 Institute of Applied Computer Sciences, Lodz University of Technology No. 1, Lodz, Poland

2 Institute of Applied Computer Sciences, Lodz University of Technology No. 2, Lodz, Poland

\*Address all correspondence to: postalcz@p.lodz.pl

† These authors contributed equally.

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