# Performance Portrait Method: An Intelligent PID Controller Design Based on a Database of Relevant Systems Behaviors

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. PD and PID Controller Schemes for Analysis and Implementation

#### 2.1. Double-Integrator and Double-Integrator-Plus-Dead-Time Models

#### 2.2. Stabilizing PD Controller Tuning by the Triple Real Dominant Pole Method

#### 2.3. From the PD Controller to the Series and Parallel PID Controller Design

#### 2.4. Optimal Parallel PID Tuning by the Quadruple Real Dominant Pole Method

## 3. Basic Limitations of Analytical Tuning of PD and PID Controller

#### 3.1. Evaluation of the Speed of Transient Response

**Remark**

**1**

**Remark**

**2**

#### 3.2. Evaluating the Excessive Controller Effort

#### 3.3. Example 1: Performance Evaluation of PD and PID Controllers

- The QRDP PID controller (24) (${K}_{p}=0.08;\phantom{\rule{0.277778em}{0ex}}{K}_{d}=0.4043;\phantom{\rule{0.277778em}{0ex}}{T}_{D}=5.0537;\phantom{\rule{0.277778em}{0ex}}{T}_{i}=12.905;$ ${T}_{o}=3.0062$) and the prefilter (25) with the numerator tuning ${c}_{1}=0,\phantom{\rule{0.277778em}{0ex}}{b}_{1}={T}_{o}$ (QRDP);

## 4. Performance Portrait Method

**Definition**

**1**

#### 4.1. Example 2: Generating PP of the Parallel PID Controller

#### 4.2. Example 3: Optimal Nominal Tuning of the Parallel PID Controller

**Theorem**

**1.**

**Remark**

**3**

#### 4.3. Parallel PID Controller Tuning Optimized under Consideration of Sensitivity Constraints

#### 4.4. Example 4: Optimal Nominal Tuning of the Series PID Controller

**Remark**

**4**

**Remark**

**5**

#### 4.5. Example 5: Considering the Pareto-Front in Setpoint Tracking and Disturbance Rejection

## 5. Discussion

## 6. 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) |

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

3P | Three-Pulse, response with 4 monotonic segments (3 extreme points) |

3D | Three-Dimensional |

ADRC | Active Disturbance Rejection Control |

AI | Artificial Intelligence |

DIPDT | Double Integrator Plus Dead-Time |

$IAE$ | Integral Absolute Error |

IPDT | Integrator Plus Dead-Time |

MFC | Model-Free Control |

MFRP | Modified sets of Four Real Poles |

MRDP | Multiple Real Dominant Pole |

PD | Proportional-Derivative |

PID | Proportional-Integral-Derivative |

PP | Performance Portrait |

PPM | Performance Portrait Method |

QRDP | Quadruple Real Dominant Pole |

TRDP | Triple Real Dominant Pole |

$TV$ | Total Variation |

$T{V}_{0}$ | Deviation from Monotonicity |

$T{V}_{1}$ | Deviation from 1P Shape |

$T{V}_{2}$ | Deviation from 2P Shape |

$T{V}_{3}$ | Deviation from 3P Shape |

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**Figure 3.**A “useful” control signal contribution of a more complex plant input (controller output) calculated as: $T{V}_{2}\left(u\right)={u}_{m1}-{u}_{0}+{u}_{m1}-{u}_{m2}+{u}_{\infty}-{u}_{m2}=2{u}_{m1}-2{u}_{m2}+{u}_{\infty}-{u}_{0}$.

**Figure 4.**Setpoint step responses of the QRDP PID controller (24) with the prefilter (25), ${c}_{1}=0,$${b}_{1}={T}_{o}$, TRDP PD controller (14) with the prefilter (15), $b={T}_{0}$ (TRD Pb) and $b=0$ (TRDP0) and the PD controller according to Gerov and Jovanovic [45] with the prefilter (15), $b=0$ (GJ).

**Figure 6.**Unit setpoint step responses (

**left**) and unit input disturbance step responses (

**right**) achieved with the QRDP PID controller (black), responses corresponding to POI-PID controllers [44] with a prescribed ${M}_{s}$ constraints and responses corresponding to optimal tuning calculated by the PPM with the parameter grid (41) under the performance specifications (46) and (44); ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{s}=0.001$.

**Figure 8.**Unit setpoint step responses (

**left**) and unit input disturbance step responses (

**right**) achieved with the parallel QRDP PID controller (black) and responses corresponding to series PID controller (see Figure 9) with the optimal tuning calculated by the PPM over the parameter grid (41) under the performance specifications (53) and (44); ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{s}=0.001$.

**Figure 11.**Unit setpoint step responses (

**left**) and unit input disturbance step responses (

**right**) achieved with the parallel QRDP PID controller (black) and responses corresponding to series PID controller (see Figure 9) with the optimal tuning calculated by the PPM over the parameter grid (41) under the output performance specifications (53) and (44) and the input admissible deviations (55); ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{s}=0.001$.

**Figure 12.**Unit setpoint step responses (

**left**) and unit input disturbance step responses (

**right**) achieved with the parallel (p) and series (s) PID controllers (black) with the optimal tuning calculated by the PPM over the parameter grid (41) under the output performance specifications (55) and (44) and the cost function weights ${s}_{w}=1$ and ${s}_{w}=0$, ${s}_{i}=1-{s}_{w}$; ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{s}=0.001$.

**Table 1.**Performance measures corresponding to the setpoint step responses in Figure 4.

- | QRDP | TRDPb | TRDP0 | Gerov-Jovanovic [45] |
---|---|---|---|---|

$IA{E}_{w}$ | 9.8868 | 5.1608 | 7.2944 | 5.4013 |

$T{V}_{0}\left({y}_{w}\right)$ | 0.0000 | 0.0000 | 0.0000 | 0.0457 |

$T{V}_{2}\left({u}_{w}\right)$ | 0.0000 | 0.0000 | 0.0000 | 0.0033 |

**Table 2.**Parameters of parallel (pPID) and series PID controllers (sPID) found for ${\u03f5}_{yw}={\u03f5}_{yi}=0.001$ and ${\u03f5}_{uw}={\u03f5}_{ui}=0.3$ in PP (41); ${\kappa}_{p}={K}_{s}{K}_{cp}{T}_{d}^{2}$, ${\delta}_{p}={K}_{s}{K}_{dp}{T}_{d}$, ${\tau}_{ip}={T}_{ip}/{T}_{d}$, ${b}_{p}$, ${\kappa}_{s}={\kappa}_{p}/2$, ${\tau}_{is}={\tau}_{ip}/2$ and ${\tau}_{Ds}={\tau}_{ip}/2$.

$2\mathit{DoF}\phantom{\rule{0.277778em}{0ex}}\mathit{PID}$ | ${\mathit{s}}_{\mathit{w}}$ | ${\mathit{\kappa}}_{\mathit{p}}$ | ${\mathit{\delta}}_{\mathit{p}}$ | ${\mathit{\tau}}_{\mathit{ip}}$ | ${\mathit{b}}_{\mathit{p}}$ | ${\mathit{\kappa}}_{\mathit{s}}$ | ${\mathit{\tau}}_{\mathit{is}}$ | ${\mathit{\tau}}_{\mathit{Ds}}$ |
---|---|---|---|---|---|---|---|---|

Parallel | 1 | 0.1700 | 0.6300 | 22 | 0.8 | - | - | - |

0 | 0.2700 | 0.7200 | 8 | 0.1 | - | - | - | |

Series | 1 | 0.2100 | 0.7200 | 14 | 0.6 | 0.1050 | 7 | 7 |

0 | 0.2500 | 0.7500 | 12 | 0.1 | 0.1250 | 6 | 6 |

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**MDPI and ACS Style**

Huba, M.; Vrancic, D.
Performance Portrait Method: An Intelligent PID Controller Design Based on a Database of Relevant Systems Behaviors. *Sensors* **2022**, *22*, 3753.
https://doi.org/10.3390/s22103753

**AMA Style**

Huba M, Vrancic D.
Performance Portrait Method: An Intelligent PID Controller Design Based on a Database of Relevant Systems Behaviors. *Sensors*. 2022; 22(10):3753.
https://doi.org/10.3390/s22103753

**Chicago/Turabian Style**

Huba, Mikulas, and Damir Vrancic.
2022. "Performance Portrait Method: An Intelligent PID Controller Design Based on a Database of Relevant Systems Behaviors" *Sensors* 22, no. 10: 3753.
https://doi.org/10.3390/s22103753