# Delay Equivalences in Tuning PID Control for the Double Integrator Plus Dead-Time

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Time and Shape Related Performance Measures

- (a)
**No connection to real-time control**: by evaluating data obtained from experiments on real processes, we cannot determine the actual values of the sensitivity functions;- (b)
**Unsuitability for unstable systems**: when controlling unstable systems, we are not content with the recommended ranges ${M}_{s}\in [1.2,2]$ (suitable only for controlling stable systems) [27], but the required values may be much higher (see, for example, [29], who recommends ${M}_{s}\approx 10$, or [30], who works even with ${M}_{s}\approx 20$);- (c)
**Potential counterproductivity**: in terms of robust control design, the use of sensitivity functions can lead to counterproductive results [10].

#### 2.1. Monotonicity-Based Shape Related Measures

**Definition**

**1**(nP function u(t))

**.**

**Remark**

**1**(Fundamental difference from traditional optimization)

**.**

#### 2.2. Optimization Problem

#### 2.3. Speed-Effort and Speed-Wobbling Characteristics

## 3. PID Controller According to the MRDP Method

**Definition**

**2**(Ideal PID controller)

**.**

**Theorem**

**1**(Optimal controller tuning)

**.**

**Proof.**

**Definition**

**3**(Low-pass filters)

**.**

## 4. Equivalent Delay Based Controller Tuning

- The ideal PID controller may not be realized—to be causal, it must be extended by at least a first-order low-pass filter (18);
- A more effective attenuation of the measurement noise can be achieved by the filter order $n>1$;
- The included filters ${Q}_{n}\left(s\right)$ modify the loop dynamics, which must be taken into account in the controller tuning.

- After identifying the system model parameters ${K}_{m}$ and ${T}_{m}$, select an appropriate value of the tuning parameter ${T}_{e}>0$ corresponding to the required degree of filtration;
- Select a filter order n and specify the filter time constant ${T}_{f}$ by a suitable delay equivalence described below, defined as$${T}_{f}/{T}_{e}=f\left(n\right)$$
- Check that the computed value ${T}_{f}$ satisfies the requirement ${T}_{f}>>{T}_{s}$ in (18), where ${T}_{s}$ represents the sampling period used for the quasi-steady control implementation.
- If not, either decrease ${T}_{s}$, or n, which must still fulfill the condition $n\ge 1$.
- By experimentally evaluating the noise attenuation characteristics for different n, choose an optimal controller that guarantees the optimal control loop performance.

#### 4.1. Half-Rule Equivalence (HRE)

#### 4.2. Average Residence Time Equivalence (ARTE)

#### 4.3. Dominant Poles Equivalence (DPE)

## 5. Evaluation of the Results

#### 5.1. Holistic Cost Functions versus Equivalent Delay

#### 5.2. Interpretation Is SE/SW Planes

**Remark**

**2**(Performance requirements in SE and SW planes)

**.**

## 6. Discussion

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1P | One-Pulse, response with 2 monotonic segments (1 extreme point) |

2P | Two-Pulse, response with 3 monotonic segments (2 extreme points) |

ADRC | Active Disturbance Rejection Control |

DIPDT | Double Integrator Plus Dead-Time |

FO | Fractional Order |

HO | Higher Order |

IAE | Integral Absolute Error |

IPDT | Integrator Plus Dead-Time |

ISE | Integral Square Error |

LESO | Linear Extended State Observer |

MRDP | Multiple Real Dominant Pole |

nP | n-Pulse, response with $n+1$ monotonic segments (n extreme points) |

PID | Proportional-Integrative-Derivative |

PID${}_{n}^{m}$ | generalized PID with mth order derivative action and nth order low-pass filter |

SE | Speed-Effort |

SW | Speed-Wobbling |

TV | Total Variation |

TV${}_{0}$ | Deviation from monotonicity (0P shape) |

TV${}_{1}$ | Deviation from 1P shape |

TV${}_{2}$ | Deviation from 2P shape |

## References

- Richard, J.P. Time-delay systems: An overview of some recent advances and open problems. Automatica
**2003**, 39, 1667–1694. [Google Scholar] [CrossRef] - O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules, 3rd ed.; Imperial College Press: London, UK, 2009. [Google Scholar]
- O’Dwyer, A. An Overview of Tuning Rules for the PI and PID Continuous-Time Control of Time-Delayed Single-Input, Single-Output (SISO) Processes. In PID Control in the Third Millennium. Lessons Learned and New Approaches; Vilanova, R., Visioli, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Chen, S.; Xue, W.; Zhong, S.; Huang, Y. On comparison of modified ADRCs for nonlinear uncertain systems with time delay. Sci. China Inf. Sci.
**2018**, 61, 70223. [Google Scholar] [CrossRef] [Green Version] - Zhao, S.; Gao, Z. Modified active disturbance rejection control for time-delay systems. ISA Trans.
**2014**, 53, 882–888. [Google Scholar] [CrossRef] [Green Version] - Pekař, L. On Simple Algebraic Control Design and Possible Controller Tuning for Linear Systems with Delays. Int. J. Mech.
**2018**, 12, 178–191. [Google Scholar] - Pekař, L.; Gao, Q. Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results. IEEE Access
**2018**, 6, 35457–35491. [Google Scholar] [CrossRef] - Tepljakov, A.; Alagoz, B.B.; Yeroglu, C.; Gonzalez, E.; HosseinNia, S.H.; Petlenkov, E. FOPID Controllers and Their Industrial Applications: A Survey of Recent Results. IFAC Pap.
**2018**, 51, 25–30. [Google Scholar] [CrossRef] - Huba, M.; Vrančič, D.; Bisták, P. ${\mathrm{PID}}_{n}^{m}$ Control for IPDT Plants. Part 1: Disturbance Response. In Proceedings of the 26th Mediterranean Conference on Control and Automation (MED), Zadar, Croatia, 19–22 June 2018. [Google Scholar]
- Huba, M.; Vrančić, D.; Bisták, P. PID Control with Higher Order Derivative Degrees for IPDT Plant Models. IEEE Access
**2020**, 9, 2478–2495. [Google Scholar] [CrossRef] - Jung, S.; Dorf, R.C. Novel Analytic Technique for PID and PIDA Controller Design. IFAC Proc. Vol.
**1996**, 29, 1146–1151. [Google Scholar] [CrossRef] - Ukakimaparn, P.; Pannil, P.; Boonchuay, P.; Trisuwannawat, T. PIDA Controller designed by Kitti’s Method. In Proceedings of the 2009 ICCAS-SICE, Fukuoka City, Japan, 18–21 August 2009; pp. 1547–1550. [Google Scholar]
- Guha, D.; Roy, P.K.; Banerjee, S. Multi-verse optimisation: A novel method for solution of load frequency control problem in power system. IET Gener. Transm. Distrib.
**2017**, 11, 3601–3611. [Google Scholar] [CrossRef] - Kumar, M.; Hote, Y.V. Robust CDA-PIDA Control Scheme for Load Frequency Control of Interconnected Power Systems. IFAC Pap.
**2018**, 51, 616–621. [Google Scholar] [CrossRef] - Kumar, M.; Hote, Y.V. Robust PIDD2 Controller Design for Perturbed Load Frequency Control of an Interconnected Time-Delayed Power Systems. IEEE Trans. Control. Syst. Technol.
**2020**, 1–8. [Google Scholar] [CrossRef] - Huba, M. Filtered PIDA Controller for the Double Integrator Plus Dead Time. In Proceedings of the 16th IFAC International Conference on Programmable Devices and Embedded Systems, High Tatras, Slovakia, 29–31 October 2019. [Google Scholar]
- Huba, M.; Bisták, P.; Skachová, Z.; Žáková, K. P- and PD-Controllers for I
_{1}and I_{2}Models with Dead Time. In Proceedings of the 6th IEEE Mediterranean Conference on Control and Automation, Sardinia, Italy, 9–11 June 1998; Volume 11, pp. 514–519. [Google Scholar] - Huba, M.; P.Bisták, Z.; Žáková, K. Predictive Antiwindup PI and PID-Controllers Based on I
_{1}and I_{2}Models with Dead Time. IEEE Mediterr. Conf.**1998**, 11, 532–535. [Google Scholar] - Grimholt, C.; Skogestad, S. Optimal PID control of double integrating processes. IFAC Pap.
**2016**, 49, 127–132. [Google Scholar] [CrossRef] - Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process. Control.
**2003**, 13, 291–309. [Google Scholar] [CrossRef] [Green Version] - Fliess, M.; Join, C. Model-free control. Int. J. Control.
**2013**, 86, 2228–2252. [Google Scholar] [CrossRef] [Green Version] - Han, J. From PID to Active Disturbance Rejection Control. Ind. Electron. IEEE Trans.
**2009**, 56, 900–906. [Google Scholar] [CrossRef] - Gao, Z. Active disturbance rejection control: A paradigm shift in feedback control system design. In Proceedings of the American Control Conference, Minneapolis, MN, USA, 14–16 June 2006; pp. 2399–2405. [Google Scholar]
- Gao, Z. On the centrality of disturbance rejection in automatic control. ISA Trans.
**2014**, 53, 850–857. [Google Scholar] [CrossRef] [Green Version] - Huba, M.; Škrinárová, J.; Dudáš, A.; Bisták, P. Optimal PID controller tuning for the time delayed double integrator. In Proceedings of the 21st International Carpathian Control Conference—ICCC, Starý Smokovec, Slovak Republic, 27–29 May 2020. [Google Scholar]
- Huba, M. Designing Robust Controller Tuning for Dead Time Systems. In International Conference on System Structure and Control; IFAC: Ancona, Italy, 2010. [Google Scholar]
- Huba, M. Performance Measures and the Robust and Optimal Control Design. In Proceedings of the 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control, Ghent, Belgium, 9–11 May 2018; pp. 960–965. [Google Scholar]
- Shinskey, F. How good are Our Controllers in Absolute Performance and Robustness. Meas. Control.
**1990**, 23, 114–121. [Google Scholar] [CrossRef] - Begum, K.G.; Rao, A.S.; Radhakrishnan, T. Maximum sensitivity based analytical tuning rules for PID controllers for unstable dead time processes. Chem. Eng. Res. Des.
**2016**, 109, 593–606. [Google Scholar] [CrossRef] - Boskovic, M.C.; Sekara, T.B.; Rapaic, M.R. Novel tuning rules for PIDC and PID load frequency controllers considering robustness and sensitivity to measurement noise. Int. J. Electr. Power Energy Syst.
**2020**, 114, 105416. [Google Scholar] [CrossRef] - Ziegler, J.G.; Nichols, N.B. Optimum settings for automatic controllers. Trans. ASME
**1942**, 11, 759–768. [Google Scholar] [CrossRef] - Feldbaum, A. Optimal Control Systems; Academic Press: New York, NY, USA, 1965. [Google Scholar]
- Pontrjagin, L.; Boltjanskij, V.; Gamkrelidze, R.; Miščenko, J. The Mathematical Theory of Optimal Processes; Interscience: New York, NY, USA, 1962. [Google Scholar]
- Föllinger, O. Regelungstechnik. 8. Auflage; Hüthig Buch Verlag: Heidelberg, Germany, 1994. [Google Scholar]
- Kuo, B. Discrete-Data Control Systems; Prentice-Hall: Upper Saddle River, NY, USA, 1970. [Google Scholar]
- Huba, M. Performance measures, performance limits and optimal PI control for the IPDT plant. J. Process. Control.
**2013**, 23, 500–515. [Google Scholar] [CrossRef] - Chen, W.H.; Yang, J.; Guo, L.; Li, S. Disturbance-Observer-Based Control and Related Methods—An Overview. IEEE Trans. Ind. Electron.
**2016**, 63, 1083–1095. [Google Scholar] [CrossRef] [Green Version] - Huba, M.; Šimunek, M. Modular Approach to Teaching PID Control. IEEE Trans. Ind. Electr.
**2007**, 54, 3112–3121. [Google Scholar] [CrossRef] - Huba, M. Open flexible PD-controller design for different filtering properties. In Proceedings of the 39th Annual Conference of the IEEE Industrial Electronics Society (IECON), Vienna, Austria, 10–13 November 2013. [Google Scholar]
- Oldenbourg, R.; Sartorius, H. Dynamik Selbsttätiger Regelungen, 2nd ed.; R.Oldenbourg-Verlag: München, Germany, 1951. [Google Scholar]
- Vítečková, M.; Víteček, A. 2DOF PI and PID controllers tuning. In Proceedings of the 9th IFAC Workshop on Time Delay Systems, Prague, Czech Republic, 7–9 June 2010; Volume 9, pp. 343–348. [Google Scholar]
- Vítečková, M.; Víteček, A. 2DOF PID controller tuning for integrating plants. In Proceedings of the 2016 17th International Carpathian Control Conference (ICCC), High Tatras, Slovakia, 29 May–1 June 2016; pp. 793–797. [Google Scholar]
- Viteckova, M.; Vitecek, A.; Janacova, D. Robustness and Muliple Dominant Pole Method. In Proceedings of the 2020 21th International Carpathian Control Conference (ICCC), Star Smokovec, Slovak Republic, 27–29 October 2020; pp. 1–4. [Google Scholar]
- Huba, M. Comparing 2DOF PI and Predictive Disturbance Observer Based Filtered PI Control. J. Process. Control.
**2013**, 23, 1379–1400. [Google Scholar] [CrossRef] - Bélai, I.; Huba, M.; Burn, K.; Cox, C. PID and filtered PID control design with application to a positional servo drive. Kybernetika
**2019**, 55, 540–560. [Google Scholar] [CrossRef] - Huba, M. Filter choice for an effective measurement noise attenuation in PI and PID controllers. In Proceedings of the ICM2015, Nagoya, Japan, 6–8 March 2015. [Google Scholar]
- Åström, K.J.; Hägglund, T. Advanced PID Control; ISA: Research Triangle Park, NC, USA, 2006. [Google Scholar]

**Figure 1.**Considered control loop with an output y, a reference setpoint r, an input disturbance ${d}_{i}$ and a measurement noise $\delta $. The possibly improper PID controller with transfer function $C\left(s\right)$ is combined with a low-pass filter ${Q}_{n}\left(s\right)$ and the plant model $F\left(s\right)$.

**Figure 2.**Dominant pole equivalence: input disturbance step responses corresponding to parameters (25), no noise.

**Figure 3.**Half rule equivalence: input disturbance step responses corresponding to parameters (25), no noise.

**Figure 4.**Average residence time equivalence: input disturbance step responses corresponding to parameters (25), no noise.

**Figure 6.**Speed-effort (SE) (left) and speed-wobbling (SW) characteristics (right) corresponding for ${T}_{e}$ (25) to DPE (full curves), HRE (dashed) and ARTE (dotted); noise $\left|\delta \right|<0.01$ (above) and $\left|\delta \right|<0.2$ (below); $k=1$.

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

${f}_{HRE}$ | 2 | 1 | 0.667 | 0.5 | 0.4 | 0.333 | 0.286 |

${f}_{ARTE}$ | 1 | 0.5 | 0.333 | 0.25 | 0.2 | 0.167 | 0.143 |

${f}_{DPE}$ | 0.601 | 0.373 | 0.271 | 0.213 | 0.176 | 0.149 | 0.130 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huba, M.; Vrancic, D.
Delay Equivalences in Tuning PID Control for the Double Integrator Plus Dead-Time. *Mathematics* **2021**, *9*, 328.
https://doi.org/10.3390/math9040328

**AMA Style**

Huba M, Vrancic D.
Delay Equivalences in Tuning PID Control for the Double Integrator Plus Dead-Time. *Mathematics*. 2021; 9(4):328.
https://doi.org/10.3390/math9040328

**Chicago/Turabian Style**

Huba, Mikulas, and Damir Vrancic.
2021. "Delay Equivalences in Tuning PID Control for the Double Integrator Plus Dead-Time" *Mathematics* 9, no. 4: 328.
https://doi.org/10.3390/math9040328