# An Inverse Neural Controller Based on the Applicability Domain of RBF Network Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. RBF Networks

#### The PSO-NSFM Algorithm

## 3. Inverse Controller Design

#### 3.1. RBF-Based Inverse Controllers and BIBS Stability

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Incorporating the AD Concept

#### 3.3. Robustifying Term

## 4. Case Studies

#### 4.1. Control of an Experimental DC Motor

^{2}indices on the validation dataset, and the training time. Based on the values for RMSE and R

^{2}achieved by the best network, it can be seen that the PSO-NSFM algorithm manages to develop a satisfactory inverse model of the system, especially taking into account that training is based on data from a real system, which inevitably includes noise. It should be noted that some of the remaining top-performing networks present a smaller number of RBF kernel centers, compared to the best network found. However, it was found experimentally that incorporating these models to the resulting control scheme produced inferior results in terms of MAE, and for this reason, the best performing network in terms of RMSE and R

^{2}was selected.

#### 4.2. Control of a Simulated Inverted Pendulum

^{2}values. The best network in terms of RMSE and R

^{2}was found to produce better controller performance, as far as MAE is concerned, and thus it was used in the control problems that follow.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Calculating the bounds on the value of ω(k) that guarantee that extrapolation is avoided. The 3-D surface represents the AD of the RBF controller.

**Figure 4.**Closed loop with the RBF INNER control scheme, taking into account the applicability domain and the robustifying term.

**Figure 6.**Armature-controlled experimental DC motor: (

**a**) controller responses; (

**b**) controller actions.

**Figure 9.**Inverted pendulum, (

**a**) M = 1.4 kg: controller responses; (

**b**) M = 2.0 kg: controller responses.

Parameter | Symbol | Description/Value |
---|---|---|

Rotor angular velocity | $\dot{\theta}$ | State variable (RPM) |

Armature current | i_{a} | State variable (A) |

Armature voltage | V_{a} | Manipulated variable (V) |

Armature resistance | R_{a} | 3.2 Ω |

Armature inductance | L_{a} | 8.6 × 10^{−3} H |

Back-EMF constant of motor | K_{e} | 100 × 10^{−3} V/rad/s |

Torque constant of motor | K_{t} | 3.3 × 10^{−3} N∙m/A |

Total moment of inertia | J | 32 × 10^{−6} kg∙m^{2} |

Motor time constant | T_{m} | 250 × 10^{−3} s |

Viscous friction coefficient of motor shaft | B_{L} | 128 × 10^{−6} N∙m∙s |

Parameter | Fuzzy Partition | RBF Kernel Centers | RMSE Validation | R^{2} Validation | Training Time ^{1} (s) | |
---|---|---|---|---|---|---|

System | ||||||

DC Motor | [18 23 32] | 242 | 9.7 | 0.93 | 398 | |

[16 23 30] | 210 | 9.8 | 0.93 | |||

[21 23 25] | 227 | 9.8 | 0.92 | |||

[21 27 28] | 270 | 10.0 | 0.90 | |||

[15 24 18] | 199 | 10.3 | 0.89 | |||

Inverted Pendulum | [35 32 40 40] | 189 | 0.48 | 0.98 | 912 | |

[32 30 40 35] | 173 | 0.50 | 0.97 | |||

[30 32 38 36] | 179 | 0.50 | 0.97 | |||

[36 32 37 36] | 180 | 0.52 | 0.96 | |||

[31 27 32 35] | 151 | 0.56 | 0.91 |

^{1}training was performed on a PC with an Intel i7 processor at 2.10 GHz and 8 GBs of memory.

Controller | MAE | |||
---|---|---|---|---|

DC Motor | Inverted Pendulum | |||

Setpoint Tracking | Stabilization M = 1 kg | Stabilization M = 1.4 kg | Stabilization M = 2.0 kg | |

IN | 0.595 | 0.315 | 0.676 | 0.8909 |

INNER | 0.262 | 0.270 | 0.288 | 0.3201 |

PID | 0.461 | 0.500 | 0.533 | 0.5581 |

Parameter | Symbol | Description/Value |
---|---|---|

Position of the wagon | p | State variable |

Velocity of the wagon | v | State variable |

Angle of the pendulum | θ | State variable |

Angular velocity of the pendulum | $\dot{\theta}$ | State variable |

Force applied on the cart | F | Manipulated variable |

Mass of the wagon | M | 1 kg |

Mass of the pendulum | m | 0.5 kg |

Gravitational constant | g | 9.8 m/s |

Length of the pendulum | L | 0.3 m |

Friction coefficient of the link | f_{θ} | 0.3 N/(m/s) |

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**MDPI and ACS Style**

Alexandridis, A.; Stogiannos, M.; Papaioannou, N.; Zois, E.; Sarimveis, H.
An Inverse Neural Controller Based on the Applicability Domain of RBF Network Models. *Sensors* **2018**, *18*, 315.
https://doi.org/10.3390/s18010315

**AMA Style**

Alexandridis A, Stogiannos M, Papaioannou N, Zois E, Sarimveis H.
An Inverse Neural Controller Based on the Applicability Domain of RBF Network Models. *Sensors*. 2018; 18(1):315.
https://doi.org/10.3390/s18010315

**Chicago/Turabian Style**

Alexandridis, Alex, Marios Stogiannos, Nikolaos Papaioannou, Elias Zois, and Haralambos Sarimveis.
2018. "An Inverse Neural Controller Based on the Applicability Domain of RBF Network Models" *Sensors* 18, no. 1: 315.
https://doi.org/10.3390/s18010315