# An Experimental Study of the Empirical Identification Method to Infer an Unknown System Transfer Function

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## Abstract

**:**

## 1. Introduction

## 2. Identification System and Evaluation Criteria

#### 2.1. System Identification

#### 2.2. Performance Evaluation

## 3. Experimentation

#### 3.1. Experimental Analysis Scheme

#### 3.2. Proving Ground

#### 3.3. Identification

#### 3.4. Performance Assessment of PID Controller Tuning

#### 3.5. Performance Evaluation on a Real System

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PID | Proportional–integral–derivative |

TF | Transfer function |

ISE | Integral squared error |

ITSE | Integral time squared error |

IAE | Integral absolute error |

ITAE | Integral of time-weighted absolute value of error |

DAS | Data acquisition system |

FPGA | Field programmable gate array |

RLS | Recursive least squares |

LMS | Least-mean squares |

KF | Kalman filter |

GA | Genetic algorithm |

PSO | Particle swarm optimization |

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**Figure 2.**(

**a**) System identification process and (

**b**) performance evaluation of the system recognition, by applying distinct excitation input signals.

**Figure 4.**System identification process. (

**a**) Stepped input, (

**b**) sinusoidal input, and (

**c**) random input stimuli. System output signals to the (

**d**) stepped, (

**e**) sinusoidal, and (

**f**) random inputs. System response comparison of the real output signal measured against the computationally obtained responses of the different-order TFs for the (

**g**) stepped, (

**h**) sinusoidal, and (

**i**) random input signals.

**Figure 5.**(

**a**) System response to the step function with perturbations utilizing different controllers. (

**b**) Real system response to a trajectory with a Gaussian acceleration profile, utilizing distinct controllers.

**Figure 6.**Repeatability tests for the three transfer functions identified through the stepped, sinusoidal, and random input stimuli, (

**a**) IAE, (

**b**) ISE, (

**c**) ITAE, and (

**d**) ITSE evaluation criteria.

STEPPED | SINE | RANDOM | |

$1s{t}_{order}$ | $G={\displaystyle \frac{82.541}{13.8225s+1}}$ | $G={\displaystyle \frac{\mathrm{22,402}}{139.42s+1}}$ | $G={\displaystyle \frac{-0.022836}{1+1{e}^{-6}}}$ |

$2n{d}_{order}$ | $G={\displaystyle \frac{45.665}{0.7604{s}^{2}+7.377s+1}}$ | $G={\displaystyle \frac{6930.6}{3.296{s}^{2}+29.42s+1}}$ | $G={\displaystyle \frac{1124.8}{12.69{s}^{2}+122.6s+1}}$ |

$3r{d}_{order}$ | $G={\displaystyle \frac{47.001}{0.018{s}^{3}+0.75{s}^{2}+7.7s+1}}$ | $G={\displaystyle \frac{\mathrm{573,100}}{75.2{s}^{3}+5578{s}^{2}+\mathrm{30,540}s+1}}$ | $G={\displaystyle \frac{1411.3}{0.3{s}^{3}+48.46{s}^{2}+204.86s+1}}$ |

**Table 2.**Performance evaluation through the simulation of a PID controller with stepped, sinusoidal, and random signal stimuli.

IAE | ISE | ITAE | ITSE | |

${C}_{step}$ | 0.0558 | 0.0235 | 0.0096 | 0.0022 |

${C}_{sine}$ | 0.0058 | 0.0016 | 0.0000138 | 0.000489 |

${C}_{rand}$ | 0.0386 | 0.0139 | 0.0081 | 0.0018 |

**Table 3.**Performance evaluation of a real system response for a PID controller with stepped, sinusoidal, and random signal stimuli.

IAE | ISE | ITAE | ITSE | |

${C}_{step}$ | 0.0196 | 0.000297 | 0.01846 | 0.000272 |

${C}_{sine}$ | 0.0031 | 0.0000079 | 0.002818 | 0.000007 |

${C}_{rand}$ | 0.0128 | 0.000134 | 0.01183 | 0.0001213 |

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## Share and Cite

**MDPI and ACS Style**

Gonzalez-Villagomez, J.; Gonzalez-Villagomez, E.; Rodriguez-Donate, C.; Cabal-Yepez, E.; Ledesma-Carrillo, L.M.; Hernández-Gómez, G.
An Experimental Study of the Empirical Identification Method to Infer an Unknown System Transfer Function. *Robotics* **2023**, *12*, 140.
https://doi.org/10.3390/robotics12050140

**AMA Style**

Gonzalez-Villagomez J, Gonzalez-Villagomez E, Rodriguez-Donate C, Cabal-Yepez E, Ledesma-Carrillo LM, Hernández-Gómez G.
An Experimental Study of the Empirical Identification Method to Infer an Unknown System Transfer Function. *Robotics*. 2023; 12(5):140.
https://doi.org/10.3390/robotics12050140

**Chicago/Turabian Style**

Gonzalez-Villagomez, Jacob, Esau Gonzalez-Villagomez, Carlos Rodriguez-Donate, Eduardo Cabal-Yepez, Luis Manuel Ledesma-Carrillo, and Geovanni Hernández-Gómez.
2023. "An Experimental Study of the Empirical Identification Method to Infer an Unknown System Transfer Function" *Robotics* 12, no. 5: 140.
https://doi.org/10.3390/robotics12050140