# Robust Stability Analysis of Filtered PI and PID Controllers for IPDT Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PI, PID, and PIDA Controller for the IPDT Plant Tuned with the MRDP Method

#### 2.1. Optimal “Ideal” Controller Design

#### 2.2. PID Controllers with Proper Transfer Functions and the Delay Equivalence

## 3. Robustness Issues

#### Robust Stability Analysis

## 4. Robust Stability of PI Controllers

#### 4.1. Unfiltered PI Controllers

**Definition 1**

#### 4.2. Filtered PI Controller with the First-Order Filter

#### 4.3. Filtered PI Controller with the Second-Order Filter

#### 4.4. Discussion on the Filtered PI Control

**Remark 1**

**Remark 2**

## 5. Robust Stability of Filtered PID Control

#### 5.1. PID with the First-Order Filters

#### 5.2. PID with the Second-Order Filter

#### 5.3. PID with the Third-Order Filter

#### 5.4. Discussion about Filtered PID Control

## 6. Robust Stability of PIDA Controller

#### 6.1. PIDA with the Second-Order Filters

#### 6.2. PIDA with the Third-Order Filters

#### 6.3. PIDA with the Fourth-Order Filters

#### 6.4. Discussion on the Filtered PIDA Control

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$BP$ | Boundary Point |

$IAE$ | Integral Absolute Error |

IPDT | Integrator Plus Dead-Time |

MRDP | Multiple Real Dominant Pole |

PI | Proportional-Integral |

PID | Proportional-Integral-Derivative |

PIDA | Proportional-Integral-Derivative-Accelerative |

## References

- Visioli, A. Practical PID Control; Springer: London, UK, 2006. [Google Scholar]
- Åström, K.J.; Hägglund, T. Advanced PID Control; ISA, Research Triangle Park: Raleigh, NC, USA, 2006. [Google Scholar]
- Isaksson, A.; Graebe, S. Derivative filter is an integral part of PID design. Control Theory Appl. IEE Proc.
**2002**, 149, 41–45. [Google Scholar] [CrossRef] - Ruel, P.E. Using filtering to improve performance. In Proceedings of the ISA Expo 2003, Houston, TX, USA, 21–23 October 2003. [Google Scholar]
- Hägglund, T. Signal Filtering in PID Control. IFAC Proc. Vol.
**2012**, 45, 1–10. [Google Scholar] [CrossRef][Green Version] - Micic, A.D.; Matausek, M.R. Optimization of PID controller with higher-order noise filter. J. Process Control
**2014**, 24, 694–700. [Google Scholar] [CrossRef] - Segovia, V.R.; Hägglund, T.; Åström, K. Measurement noise filtering for PID controllers. J. Process Control
**2014**, 24, 299–313. [Google Scholar] [CrossRef] - Fišer, J.; Zítek, P.; Vyhlídal, T. Dominant four-pole placement in filtered PID control loop with delay. IFAC-PapersOnLine
**2017**, 50, 6501–6506. [Google Scholar] [CrossRef] - Peker, F.; Kaya, I. Optimal integral-proportional derivative controller design for input load disturbance rejection of time delay integrating processes. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 2022; in press. [Google Scholar]
- Huba, M. Filter choice for an effective measurement noise attenuation in PI and PID controllers. In Proceedings of the ICM2015, Casablanca, Morocco, 20–23 December 2015. [Google Scholar]
- Huba, M.; Bisták, P.; Huba, T. Filtered PI and PID control of an Arduino based thermal plant. IFAC-PapersOnLine
**2016**, 49, 336–341. [Google Scholar] [CrossRef] - Majhi, S.; Kotwal, V.; Mehta, U. FPAA-Based PI controller for DC servo position control system. IFAC Proc. Vol.
**2012**, 45, 247–251. [Google Scholar] [CrossRef][Green Version] - Hasler, J. Large-Scale Field-Programmable Analog Arrays. Proc. IEEE
**2020**, 108, 1283–1302. [Google Scholar] [CrossRef] - Valele, W.; Virambath, R.; Mehta, U.; Azid, S. Fractional analog scheme for efficient stabilization of a synchronous buck converter. J. Electr. Eng.
**2020**, 71, 116–121. [Google Scholar] [CrossRef] - Huba, M.; Vrančić, D.; Bisták, P. PID Control with Higher Order Derivative Degrees for IPDT Plant Models. IEEE Access
**2021**, 9, 2478–2495. [Google Scholar] [CrossRef] - Vrančić, D.; Huba, M. High-Order Filtered PID Controller Tuning Based on Magnitude Optimum. Mathematics
**2021**, 9, 1340. [Google Scholar] [CrossRef] - Oldenbourg, R.; Sartorius, H. Dynamik Selbsttätiger Regelungen; R.Oldenbourg-Verlag: München, Germany, 1944. [Google Scholar]
- Vítečková, M.; Víteček, A. Two-degree of Freedom Controller Tuning for Integral Plus Time Delay Plants. ICIC Express Lett. Int. J. Res. Surv. Jpn.
**2008**, 2, 225–229. [Google Scholar] - Vítečková, M.; Víteček, A. 2DOF PI and PID controllers tuning. In Proceedings of the 9th IFAC Workshop on Time Delay Systems, Guangzhou, China, 29 September–1 October 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 Int. Carpathian Control Conf. (ICCC), High Tatras, Slovakia, 29 May–1 June 2016; pp. 793–797. [Google Scholar]
- Víteček, A.; Vítečková, M. Series Two Degree of Freedom PID Controller for Integrating Plants with Time Delay. In Proceedings of the 2019 20th International Carpathian Control Conference (ICCC), Kraków-Wieliczka, Poland, 26–29 May 2019; pp. 1–4. [Google Scholar]
- Vítečková, M.; Víteček, A.; Janáčová, D. Robustness and Muliple Dominant Pole Method. In Proceedings of the 2020 21th ICCC, High Tatras, Slovakia, 27–29 October 2020; pp. 1–4. [Google Scholar]
- Huba, M. Designing Robust Controller Tuning for Dead Time Systems. In Int. Conf. System Structure and Control; IFAC: Ancona, Italy, 2010. [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] - Huba, M. Comparing 2DOF PI and Predictive Disturbance Observer Based Filtered PI Control. J. Process Control
**2013**, 23, 1379–1400. [Google Scholar] [CrossRef] - Oaxaca-Adams, G.; Villafuerte-Segura, R.; Aguirre-Hernández, B. On non-fragility of controllers for time delay systems: A numerical approach. J. Frankl. Inst.
**2021**, 358, 4671–4686. [Google Scholar] [CrossRef] - Oaxaca-Adams, G.; Villafuerte-Segura, R. On controllers performance for a class of time-delay systems: Maximum decay rate. Automatica
**2023**, 147, 110669. [Google Scholar] [CrossRef] - Huba, M.; Chamraz, S.; Bisták, P.; Vrančić, D. Making the PI and PID Controller Tuning Inspired by Ziegler and Nichols Precise and Reliable. Sensors
**2021**, 18, 6157. [Google Scholar] [CrossRef] [PubMed] - Huba, M.; Gao, Z. Uncovering Disturbance Observer and Ultra-Local Plant Models in Series PI Controllers. Symmetry
**2022**, 14, 640. [Google Scholar] [CrossRef] - Huba, M. Disturbance Observer in PID Controllers for First-Order Time-Delayed Systems. In Proceedings of the 13th IFAC Symposium Advances in Control Education, Hamburg, Germany, 24–27 July 2022. [Google Scholar]
- Huba, M.; Bisták, P. Should we forget the PID control? In Proceedings of the 2022 20th International Conference on Emerging eLearning Technologies and Applications (ICETA), High Tatras, Slovakia, 20–21 October 2022. [Google Scholar]
- Huba, M.; Vrančić, D. Comparing filtered PI, PID and PIDD
^{2}control for the FOTD plants. In Proceedings of the 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control, Ghent, Belgium, 9–11 May 2018; pp. 954–959. [Google Scholar] - Huba, M.; Vrančić, D. Introduction to the Discrete Time PID${}_{n}^{m}$ Control for the IPDT Plant. In Proceedings of the 15th IFAC Int. Conference on Programmable Devices and Embedded Systems, Ostrava, Czech Republic, 23–25 May 2018; pp. 119–124. [Google Scholar]
- 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-PapersOnLine
**2018**, 51, 25–30. [Google Scholar] [CrossRef] - Huba, M.; Vrančić, D.; Bisták, P. 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. 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]
- Huba, M.; Vrančić, D. Extending the Model-Based Controller Design to Higher-Order Plant Models and Measurement Noise. Symmetry
**2021**, 2021, 798. [Google Scholar] [CrossRef] - Huba, M.; Vrančić, D. Tuning of PID Control for the Double Integrator Plus Dead-Time Model by Modified Real Dominant Pole and Performance Portrait Methods. Mathematics
**2022**, 10, 971. [Google Scholar] [CrossRef] - Fortuna, L.; Frasca, M. Optimal and Robust Control: Advanced Topics with MATLAB, 1st ed.; CRC Press: Boca Raton, FL, USA, 2012. [Google Scholar]
- Ackermann, J. Robust Control: The Parameter Space Approach, 2nd ed.; Springer: Berlin, Germany, 2002. [Google Scholar]
- Zítek, P.; Fišer, J.; Vyhlídal, T. Dimensional analysis approach to dominant three-pole placement in delayed PID control loops. J. Process Control
**2013**, 23, 1063–1074. [Google Scholar] [CrossRef] - Huba, M.; Oliveira, P.M.; Bisták, P.; Vrančić, D. A Set of Active Disturbance Rejection Controllers Based on Integrator Plus Dead-Time Models. Appl. Sci.
**2021**, 2021, 1671. [Google Scholar] [CrossRef] - Ferrari, M.; Visioli, A. A software tool to understand the design of PIDA controllers. IFAC-PapersOnLine
**2022**, 55, 249–254. [Google Scholar] [CrossRef] - Visioli, A.; Sánchez-Moreno, J. A relay-feedback automatic tuning methodology of PIDA controllers for high-order processes. Int. J. Control, 2022; in press. [Google Scholar]
- Huba, M.; Vrančić, D. Delay Equivalences in Tuning PID Control for the Double Integrator Plus Dead-Time. Mathematics
**2021**, 9, 328. [Google Scholar] [CrossRef] - Ziegler, J.G.; Nichols, N.B. Optimum settings for automatic controllers. Trans. ASME
**1942**, 64, 759–768. [Google Scholar] [CrossRef] - Fliess, M.; Join, C. Model-free control. Int. J. Control
**2013**, 86, 2228–2252. [Google Scholar] [CrossRef][Green Version] - Gao, Z. On the centrality of disturbance rejection in automatic control. ISA Trans.
**2014**, 53, 850–857. [Google Scholar] [CrossRef][Green Version] - Mercader, P.; Banos, A. A PI tuning rule for integrating plus dead time processes with parametric uncertainty. ISA Trans.
**2017**, 67, 246–255. [Google Scholar] [CrossRef] - Huba, M.; Bélai, I. Limits of a Simplified Controller Design Based on IPDT models. ProcIMechE Part I J. Syst. Control Eng.
**2018**, 232, 728–741. [Google Scholar] [CrossRef] - Neimark, J.I. D-decomposition of the space of quasi-polynomials (on the stability of linearized distributive systems. Am. Math. Soc. Transl.
**1973**, 102, 95–131. [Google Scholar] - Householder, A.S. The Numerical Treatment of a Single Nonlinear Equation; McGraw-Hill: New York, NY, USA, 1970. [Google Scholar]

**Figure 1.**Stability borders of the nonfiltered $PI{D}_{0}^{0}$ and filtered $PI{D}_{1}^{0}$ and $PI{D}_{2}^{0}$ control for several values of ${\tau}_{e}={T}_{e}/({T}_{dp}+{T}_{e}),{T}_{dp}=1$ and the average-residence-time tuning equivalence ${T}_{f}={T}_{e}/n$, $n=1,2$; + denoting the nominal tuning.

**Figure 2.**Calculating the gain margin ${\kappa}_{m}$ of the filtered $PI{D}_{1}^{0}$ and $PI{D}_{2}^{0}$ controllers for ${T}_{dp}=1$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=1,2$.

**Figure 3.**Calculating the dead-time margin ${\tau}_{m}$ of the $PI{D}_{1}^{0}$ and $PI{D}_{2}^{0}$ controllers for ${T}_{dp}=1$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=1,2$.

**Figure 4.**Verifying the dead-time and gain margins of the $PI{D}_{2}^{0}$ controllers for ${T}_{dp}=1$, ${T}_{e}=1$, and the ${T}_{f}={T}_{e}/2$ tuning equivalence.

**Figure 5.**Verifying the dead-time and gain margins of the $PI{D}_{2}^{0}$ controllers for ${T}_{dp}=1$, ${T}_{e}=1$, and the ${T}_{f}={T}_{e}/2$ tuning equivalence.

**Figure 6.**Stability borders of the nonfiltered PI ($PI{D}_{0}^{0}$) and filtered $PI{D}_{1}^{1}$, $PI{D}_{2}^{1}$, and $PI{D}_{3}^{1}$ control for several values of ${\tau}_{e}={T}_{e}/({T}_{dp}+{T}_{e}),{T}_{dp}=1$ and the average-residence-time tuning equivalence ${T}_{f}={T}_{e}/n$, $n=1,2,3$; + denoting the nominal tuning.

**Figure 7.**Calculating the gain margin ${\kappa}_{m}$ of the filtered $PI{D}_{1}^{1}$, $PI{D}_{2}^{1}$, and $PI{D}_{3}^{1}$ controllers for ${T}_{dp}=1$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=1,2,3$.

**Figure 8.**Calculating the dead-time margin ${\tau}_{m}$ of the $PI{D}_{1}^{1}$, $PI{D}_{2}^{1}$, and $PI{D}_{3}^{1}$ controllers for ${T}_{dp}=1$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=1,2,3$.

**Figure 9.**Disturbance responses of the nominal IPDT system with ${T}_{dp}=1$ and ${K}_{sp}=1$ for the unfiltered PI control ($PI{D}_{0}^{0}$) and $PI{D}_{n}^{1}$ controllers with three different values ${\tau}_{e}$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=1,2,3$.

**Figure 10.**Stability borders of the nonfiltered PI ($PI{D}_{0}^{0}$) and filtered PIDA controllers ($PI{D}_{2}^{2}$, $PI{D}_{3}^{2}$, and $PI{D}_{4}^{2}$) for several values of ${\tau}_{e}={T}_{e}/({T}_{dp}+{T}_{e}),{T}_{dp}=1$ and the average-residence-time tuning equivalence ${T}_{f}={T}_{e}/n$, $n=2,3,4$; + denoting the nominal tuning.

**Figure 11.**Calculating the gain margin ${\kappa}_{m}$ of the filtered $PI{D}_{2}^{2}$, $PI{D}_{3}^{2}$, and $PI{D}_{4}^{2}$ controllers for ${T}_{dp}=1$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=2,3,4$.

**Figure 12.**Calculating the dead-time margin ${\tau}_{m}$ of the $PI{D}_{2}^{2}$, $PI{D}_{3}^{2}$ and $PI{D}_{4}^{2}$ controllers for ${T}_{dp}=1$ and the ${T}_{f}={T}_{e}/n$ tuning equivalence, $n=2,3,4$.

$\mathit{m}=0$ | $\mathit{m}=1$ | $\mathit{m}=2$ | |
---|---|---|---|

K | 0.4612 | 0.78361 | 1.08268 |

${\tau}_{i}$ | 5.8284 | 3.73205 | 3.00000 |

${\tau}_{1}$ | 0 | 0.26289 | 0.37500 |

${\tau}_{2}$ | 0 | 0 | 0.04167 |

**Table 2.**Optimal normed ${\overline{IAE}}_{d}=IA{E}_{d}/\left({K}_{s}{T}_{d}^{2}\right)$ values corresponding to unit input disturbance step responses for PID${}^{m}$ from Table 1.

- | $\mathit{m}=0$ | $\mathit{m}=1$ | $\mathit{m}=2$ |
---|---|---|---|

${\overline{IAE}}_{d}$ | 12.639 | 4.763 | 2.771 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huba, M.; Bistak, P.; Vrancic, D.
Robust Stability Analysis of Filtered PI and PID Controllers for IPDT Processes. *Mathematics* **2023**, *11*, 30.
https://doi.org/10.3390/math11010030

**AMA Style**

Huba M, Bistak P, Vrancic D.
Robust Stability Analysis of Filtered PI and PID Controllers for IPDT Processes. *Mathematics*. 2023; 11(1):30.
https://doi.org/10.3390/math11010030

**Chicago/Turabian Style**

Huba, Mikulas, Pavol Bistak, and Damir Vrancic.
2023. "Robust Stability Analysis of Filtered PI and PID Controllers for IPDT Processes" *Mathematics* 11, no. 1: 30.
https://doi.org/10.3390/math11010030