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Reliable and straightforward PID tuning rules for highly underdamped systems

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Abstract

Proportional-Integral-Derivative (PID) controllers reign absolute when automatic control is applied. There is an expressive number of tuning rules for these controllers in literature. However, for highly oscillatory (or highly underdamped) systems, such as the ones found in oil production and polymerization reactors, the available methods provide poor closed-loop performance and robustness. Besides, most of these tuning rules are developed for systems based on a first-order with pure time delay (FOPTD) transfer function and for parallel form PID controllers. Therefore, the focus of this paper is the development of appropriate tuning rules for highly underdamped systems through non-cancellation of dominant poles and easily adjustable robust performance, making them applicable for both series and parallel PID controllers, since the proposed tuning only places the controller zeros at the real axis. The new tuning rules were developed for these systems and were tested on 15,000 different transfer functions described by a second-order with pure time delay (SOPTD) expression. Additionally, a recommendation interval is also provided in which the controller gain can be varied online or by simulations to achieve the desired trade-off between performance and robustness. The proposed rules are also validated using two case studies: the suppression of slugging in oil production and the temperature control of an industrial gas phase polyethylene reactor.

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Acknowledgements

The authors would like to acknowledge the Coordination for the Improvement of Higher Education Personnel (CAPES)—Process 88882.181989/2018-01 for providing financial support to this work.

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Correspondence to Brício Ferreira Barreiros.

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Appendix A. Demonstration of how to obtain the adjustment of the controller using the new tuning rules

Appendix A. Demonstration of how to obtain the adjustment of the controller using the new tuning rules

The plant used for the demonstration is given by:

$${G}_{A}\left(s\right)=\frac{1}{{194.6}^{2}{s}^{2}+2\left(0.059\right)\left(194.6\right)s+1}exp\left(-4s\right)$$

The step-by-step application of the new tuning rules for GA(s) is exemplified below:

Step 1: Determination of \(\tau /\theta\)

$$\frac{\tau }{\theta }=\frac{194.6}{4}\underset{}{\Rightarrow }\frac{\tau }{\theta }=48.7$$

Step 2: Determination of the parameters Kp,1st, τI and τD from Table 1

$${K}_{p,1st}=\frac{1}{1}\mathrm{exp}\left[9.1\cdot 0.059+\left(2.2-2.7\cdot 0.059\right)\mathrm{ln}\left(\mathrm{min}\left(\mathrm{48.7,10}\right)\right)-3.4\right]$$
$$\underset{}{\Rightarrow }{K}_{p,1st}=6.3$$
$${\tau }_{I}=194.6\left[\left(0.2+0.5\cdot 0.059\right)\mathrm{min}\left(\mathrm{48.7,10}\right)+\mathrm{exp}\left(-33\cdot 0.059\right)+0.2\right]$$
$$\underset{}{\Rightarrow }{\tau }_{I}=513.3$$
$${\tau }_{D}=194.6\mathrm{exp}\left[1.3-0.2\mathrm{min}\left(\mathrm{48.7,10}\right)-2.9\cdot 0.059\right]$$
$$\underset{}{\Rightarrow }{\tau }_{D}=81.4$$

Step 3: Determination of the parameter Kp,2nd from Table 3

$$\frac{\tau }{\theta }=48.7\underset{}{\Rightarrow }10\le \frac{\tau }{\theta }\le 500$$
$${K}_{p,2nd}=\left\{14.7-14.5\mathrm{exp}\left[-0.009\cdot 48.7\right]\right\}6.3$$
$$\underset{}{\Rightarrow }{K}_{p,2nd}=33.7$$

where Kp,1st, and Kp,2nd provide a recommended interval for the controller gain.

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Barreiros, B.F., Trierweiler, J.O. & Farenzena, M. Reliable and straightforward PID tuning rules for highly underdamped systems. Braz. J. Chem. Eng. 38, 731–745 (2021). https://doi.org/10.1007/s43153-021-00127-0

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