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ADRC in output and error form: connection, equivalence, performance

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Abstract

In this work, we investigate two specific linear ADRC structures, namely output- and error-based. The former is considered a “standard” version of ADRC, a title obtained primarily thanks to its simplicity and effectiveness, which have spurred its adoption across multiple industries. The latter is found to be especially appealing to practitioners as its feedback error-driven structure bares similarities to conventional control solutions, like PI and PID. In this paper, we describe newly found connections between the two considered ADRC structures, which allowed us to formally establish conditions for their equivalence. Furthermore, the conducted comprehensive performance comparison between output- and error-based ADRCs has facilitated the identification of specific modules within them, which can now be conveniently used as building blocks, thus aiding the control designers in customizing ADRC-based solutions and making them most suitable for their applications.

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Notes

  1. The details of the used notation “output-based” and “error-based” in the context of ADRC are explained at the beginning of Sect. 2.

  2. https://www.mathworks.com/matlabcentral/fileexchange/102249-active-disturbance-rejection-control-adrc-toolbox.

  3. Throughout the paper, we assume, without loss of generality, that the initial values that appear when making Laplace transformation are equal to zero.

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Correspondence to Rafal Madonski.

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The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

The work of Dr. R. Madonski was supported by the Fundamental Research Funds for the Central Universities (Project No. 21620335). The work of Dr. M. Stankovic was supported by the International Foreign Expert Project Fund of Jinan University (Project No. G2021199027L, coordinator: Dr. Hui Deng).

Appendix: Influence of reference derivatives

Appendix: Influence of reference derivatives

In order to show the influence of various levels of knowledge about the reference derivatives on the tracking quality in oADRC, we introduce a third-order (\(n=3\)) plant model:

$$\begin{aligned} G_{\textrm{P3}}(s)=\frac{1}{s^3+s^2+s+1}, \end{aligned}$$
(35)

that does not represent a physical plant and is used here solely as a toy example which happens to be useful to us in supporting the idea we want to convey here.

Table 4 Considered oADRC cases for \(G_{\textrm{P3}}(s)\) (cf. Table 1)

Next, a numerical simulation is performed with oADRC controllers (in previously defined cases A and B) as well as two intermediate cases, which include terms related to the first and second reference derivative in (3). Those intermediate cases of oADRC are denoted as A1 and A2 and are formally defined in Table 4. The controllers considered in the simulation are all tuned using the “bandwidth parameterization” approach with the same \(\omega _\textrm{CL}=2.5\) rad/s and \(k_\textrm{ESO}=5\). The tracking performances obtained for a reference sinusoidal signal \(r(t)=\sin t\) are shown in Fig. 11.

From Fig. 11, it is evident that, by including more derivatives of the reference signal to the oADRC control law, the reference tracking performance is improved. This makes oADRC (case B) variant especially suitable for tracking control of dynamic signals. However, it should also be noted that adding more reference derivatives during the controller synthesis also makes the final controller more complex and that in industrial practice the reference derivatives are not always available. Therefore, the bottom line is that the selection of a specific ADRC controller structure should be done on a case-by-case basis.

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Madonski, R., Herbst, G. & Stankovic, M. ADRC in output and error form: connection, equivalence, performance. Control Theory Technol. 21, 56–71 (2023). https://doi.org/10.1007/s11768-023-00129-y

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