# Series PIDA Controller Design for IPDT Processes

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## Abstract

**:**

## 1. Introduction

## 2. Tuning of PDA Controllers for IPDT Plant

#### 2.1. Quadruple Real Dominant Pole Controller Tuning

#### 2.2. Implementation and Noise Attenuation Low-Pass Filters

#### 2.3. Illustrative Examples of PDA Control

**Remark 1**

**Remark 2**

## 3. Analytical Tuning of Series PIDA Controller

#### 3.1. Quintuple Real Dominant Pole Controller Tuning

**Remark 3**

#### 3.2. Prefilter Design

#### 3.3. Illustrative Examples of PIDA Control

**Remark 4**

## 4. Calculation of the Series PIDA Controller for the IPDT System Using the Performance Portrait Method (PPM)

#### 4.1. Real-Time Temperature Control

#### 4.2. Organization of the Experiment

#### 4.3. Evaluation of the Dynamics of Transient Responses

## 5. Inspection of the Constrained Loops by Circle Criterion—The Saturation Nonlinearity

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

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

AVR | Automatic voltage regulator |

AW | Anti-windup |

$IAE$ | Integral of absolute error |

IMC | Internal model control |

IPDT | Integrator plus dead time |

ITAE | Integral of the time weighted absolute error |

LFC | Load frequency control |

MRDP | Multiple real dominant pole |

PDA | Proportional–derivative–accelerative |

PI | Proportional–integral |

PIDA | Proportional–integral–derivative–accelerative |

PIDD${}^{2}$ | Proportional–integral–derivative–second-order derivative |

PPM | Performance portrait method |

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

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**Figure 1.**IPDT process $\dot{y}\left(t\right)=u(t-1)+{d}_{i}$ and series PDA controller with different prefilter tuning corresponding to a setpoint step to $w=1$ at $t=0$ followed by a disturbance step ${d}_{i}=-0.2$ at $t=10$; ${T}_{e}=0.8$; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=1$.

**Figure 2.**IPDT process $\dot{y}\left(t\right)=u(t-1)+{d}_{i}$ and constrained series PDA controller with different prefilter tuning corresponding to a setpoint step to $w=1$ at $t=0$ followed by a disturbance step ${d}_{i}=-0.2$ at $t=10$; ${T}_{e}=0.8$; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=1$; ${U}_{max}=0.22;{U}_{min}=-0.02$.

**Figure 3.**Series PIDA controller (20) with two degrees of freedom: a third-order filter ${Q}_{n}\left(s\right),$ $n=3$ (16) of the measured output ${y}_{m}\left(t\right)$, the automatic reset with time constant ${T}_{i}$, with proportional, derivative and accelerative controller gains ${K}_{p}$, ${K}_{d}$ and ${K}_{a}$, with the reference setpoint $w\left(t\right)$ and a prefilter ${F}_{p}\left(s\right)$.

**Figure 4.**IPDT process $\dot{y}\left(t\right)=u(t-1)+{d}_{i}$ and series PIDA controller with different prefilter tuning corresponding to a setpoint step to $w=1$ at $t=0$ followed by a disturbance step ${d}_{i}=-0.2$ at $t=20$; ${T}_{e}=0.8$; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=1$.

**Figure 5.**IPDT process $\dot{y}\left(t\right)=u(t-1)+{d}_{i}$ and constrained series PIDA controller with different prefilter tuning corresponding to a setpoint step to $w=1$ at $t=0$ followed by a disturbance step ${d}_{i}=-0.2$ at $t=20$; ${T}_{e}=0.8$; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=1$; ${U}_{max}=0.22;{U}_{min}=-0.02$.

**Figure 6.**IPDT process $\dot{y}\left(t\right)=u(t-1)+{d}_{i}$ and modified constrained series PIDA controller with different prefilter tuning corresponding to a setpoint step to $w=1$ at $t=0$ followed by a disturbance step ${d}_{i}=-0.2$ at $t=20$; ${T}_{e}=0.8$; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=1$; ${U}_{max}=0.22;{U}_{min}=-0.02$.

**Figure 7.**IPDT process $\dot{y}\left(t\right)=u(t-1)+{d}_{i}$ and constrained series PIDA controller with MRDP tuning (24), modified approximative tuning (33), PPM tuning (40) and constrained parallel PIDA controller modified by conditional integration according to [14]: disturbance step ${d}_{i}=1$ at $t=0$ and ${d}_{i}=0$ at $t=30$; ${T}_{e}=0.8$; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=1$; ${U}_{max}=0.1;{U}_{min}=-1.1$.

**Figure 8.**Temperature control corresponding to the series PIDA controllers from previous example with the simplest prefilter (26) and (27), a setpoint step from $w=30$ °C to $w=40$ °C at $t=100$ followed by a setpoint step to $w=30$ °C at $t=200$, a setpoint step $w=35$ °C at $t=500$ and a disturbance step produced by a fan control ${u}_{f}=10$ applied at $t=600$; ${T}_{d}=1$ s; ${T}_{e}=0.6$ s; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=0.1$.

**Figure 9.**Detail of temperature control corresponding to the proposed series PIDA controllers with the simplest prefilter (26) and (27) and a setpoint step from $w=30$ °C to $w=40$ °C at $t=100$ ${T}_{d}=1$ s; ${T}_{e}=0.6$ s; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=0.1$.

**Figure 10.**Details of temperature control corresponding to the proposed series PIDA controllers with the simplest prefilter (26) and (27) and a setpoint step from $w=40$ °C to $w=30$ °C at $t=200$ ${T}_{d}=1$ s; ${T}_{e}=0.6$ s; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=0.1$.

**Figure 11.**Detail of temperature control corresponding to the proposed series PIDA controllers with the simplest prefilter (26) and (27) and a “small” setpoint step from $w=30$ °C to $w=35$ °C at $t=500$ ${T}_{d}=1$ s; ${T}_{e}=0.6$ s; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=0.1$.

**Figure 12.**Details of temperature control corresponding to the proposed series PIDA controllers with the simplest prefilter (26) and (27) and an external disturbance produced by switching on the fan voltage ${u}_{f}=10$ at $t=600$; ${T}_{d}=1$ s; ${T}_{e}=0.6$ s; $n=4$; ${P}_{n}\left(s\right)={(1+{T}_{f}s)}^{4}$; ${T}_{f}={T}_{e}/4$; ${K}_{sp}={K}_{s}=0.1$.

**Figure 13.**Standard nonlinear loop with saturation for analysis of the absolute stability by circle criterion (above) and Nyquist curve of the linear part ${L}_{s}\left(s\right)$ (below) drawn with the help of the average residence time equivalence.

**Table 1.**Performance measures $IAE$, $T{V}_{1}\left(y\right)$ and $T{V}_{1}\left(u\right)$ corresponding to the disturbance steps ${y}_{1}=y\left({t}_{1}\right),{u}_{1}=u\left({t}_{1}\right),{d}_{i1}=1,{t}_{1}\in [0,30)$ and ${y}_{2}=y\left({t}_{2}\right),{u}_{2}=u\left({t}_{2}\right),{d}_{i2}=0,{t}_{2}\in [30,60)$.

${\mathit{IAE}}_{1}$ | ${\mathit{TV}}_{1}\left({\mathit{y}}_{1}\right)$ | ${\mathit{TV}}_{1}\left({\mathit{u}}_{1}\right)$ | ${\mathit{IAE}}_{2}$ | ${\mathit{TV}}_{1}\left({\mathit{y}}_{2}\right)$ | ${\mathit{TV}}_{1}\left({\mathit{u}}_{2}\right)$ | |
---|---|---|---|---|---|---|

MRDP | 23.2146 | 0.2691 | 0.0570 | 23.2778 | 0.3081 | 0.2865 |

APR | 27.7230 | 1 $\times {10}^{-20}$ | 0.0056 | 27.58388 | 1 $\times {10}^{-12}$ | 0.1924 |

PPM | 14.3636 | 1 $\times {10}^{-20}$ | 1 $\times {10}^{-20}$ | 14.3527 | 5 $\times {10}^{-13}$ | 2 $\times {10}^{-16}$ |

AW | 23.6241 | 0.2648 | 0.0551 | 23.6981 | 0.3084 | 0.2886 |

**Table 2.**Performance measures $IA{E}_{1}$, $t{v}_{0}\left({y}_{1}\right)$, $\Delta \eta \left({y}_{1}\right)$ and $t{v}_{1}\left({u}_{1}\right)$ corresponding to the first setpoint step with ${y}_{1}=y\left({t}_{1}\right),{u}_{1}=u\left({t}_{1}\right),{t}_{1}\in [100,200)$ s, $\Delta \eta \left({y}_{1}\right)$—percentage overshoot over the setpoint value.

- | ${\mathit{IAE}}_{1}$ | ${\mathit{tv}}_{0}\left({\mathit{y}}_{1}\right)$ | $\mathbf{\Delta}\mathit{\eta}\left({\mathit{y}}_{1}\right)$ | ${\mathit{tv}}_{1}\left({\mathit{u}}_{1}\right)$ |
---|---|---|---|---|

MRDP-PIDA | 76.2942 | 0.5854 | 3.2934 | 0.5257 |

APR-PIDA | 75.4978 | 0.6154 | 0.5000 | 0.5317 |

PPM-PIDA | 73.8782 | 0.6633 | 0.2994 | 1.4753 |

**Table 3.**Performance measures $IA{E}_{2}$, $t{v}_{0}\left({y}_{2}\right)$, $\Delta \eta \left({y}_{2}\right)$ and $t{v}_{1}\left({u}_{2}\right)$ corresponding to the second setpoint step with ${y}_{2}=y\left({t}_{2}\right),{u}_{2}=u\left({t}_{2}\right),{t}_{2}\in [200,500)$ s, $\Delta \eta \left({y}_{2}\right)$—percentage undershoot below the setpoint value.

- | ${\mathit{IAE}}_{2}$ | ${\mathit{tv}}_{0}\left({\mathit{y}}_{2}\right)$ | $\mathbf{\Delta}\mathit{\eta}\left({\mathit{y}}_{2}\right)$ | ${\mathit{tv}}_{1}\left({\mathit{u}}_{2}\right)$ |
---|---|---|---|---|

MRDP-PIDA | 548.3830 | 1.3706 | 0.3996 | 3.2381 |

APR-PIDA | 659.6626 | 1.0840 | 0.3000 | 1.7996 |

PPM-PIDA | 671.9482 | 1.0808 | 0.2991 | 4.2985 |

**Table 4.**Performance measures $IA{E}_{3}$, $t{v}_{0}\left({y}_{3}\right)$, $\Delta \eta \left({y}_{3}\right)$ and $t{v}_{1}\left({u}_{3}\right)$ corresponding to the third setpoint step with ${y}_{3}=y\left({t}_{3}\right),{u}_{3}=u\left({t}_{3}\right),{t}_{3}\in [500,600)$ s, $\Delta \eta \left({y}_{3}\right)$—percentage overshoot over the setpoint value.

- | ${\mathit{IAE}}_{3}$ | ${\mathit{tv}}_{0}\left({\mathit{y}}_{3}\right)$ | $\mathbf{\Delta}\mathit{\eta}\left({\mathit{y}}_{3}\right)$ | ${\mathit{tv}}_{1}\left({\mathit{u}}_{3}\right)$ |
---|---|---|---|---|

MRDP-PIDA | 26.9294 | 1.2760 | 0.6000 | 0.5285 |

APR-PIDA | 27.2956 | 1.4480 | 0.6000 | 0.6453 |

PPM-PIDA | 26.5804 | 1.3333 | 0.5988 | 1.4719 |

**Table 5.**Performance measures $IA{E}_{4}$, $t{v}_{0}\left({y}_{4}\right)$, $\eta \left({y}_{4}\right)$ and $t{v}_{1}\left({u}_{4}\right)$ corresponding to the fourth (disturbance) step with ${y}_{4}=y\left({t}_{4}\right),{u}_{4}=u\left({t}_{4}\right),{t}_{4}\in [600,700)$ s, $\eta \left({y}_{4}\right)$—output decrease below the setpoint value.

- | ${\mathit{IAE}}_{4}$ | ${\mathit{tv}}_{0}\left({\mathit{y}}_{4}\right)$ | $\mathit{\eta}\left({\mathit{y}}_{4}\right)$ [°C] | ${\mathit{tv}}_{1}\left({\mathit{u}}_{4}\right)$ |
---|---|---|---|---|

MRDP-PIDA | 9.7366 | 5.5057 | −0.8800 | 6.7714 |

APR-PIDA | 9.9496 | 4.4651 | −1.0500 | 8.4301 |

PPM-PIDA | 7.7930 | 6.2360 | −0.7800 | 16.4729 |

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**MDPI and ACS Style**

Huba, M.; Bistak, P.; Vrancic, D.
Series PIDA Controller Design for IPDT Processes. *Appl. Sci.* **2023**, *13*, 2040.
https://doi.org/10.3390/app13042040

**AMA Style**

Huba M, Bistak P, Vrancic D.
Series PIDA Controller Design for IPDT Processes. *Applied Sciences*. 2023; 13(4):2040.
https://doi.org/10.3390/app13042040

**Chicago/Turabian Style**

Huba, Mikulas, Pavol Bistak, and Damir Vrancic.
2023. "Series PIDA Controller Design for IPDT Processes" *Applied Sciences* 13, no. 4: 2040.
https://doi.org/10.3390/app13042040