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

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## 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 |

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**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 |

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**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