# Modeling and Control of IPMC Actuators Based on LSSVM-NARX Paradigm

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## Abstract

**:**

## 1. Introduction

#### 1.1. Modeling

#### 1.2. Control

## 2. Testing of Hysteresis Characteristics of the IPMC Actuator

#### 2.1. Testing Platform

#### 2.2. Testing of Hysteresis Characteristics

## 3. Modeling of IPMC Actuator Based on Hysteresis Characteristics

#### 3.1. IPMC Actuator Modeling Based on Prandtl-Ishlinskii Method

#### 3.2. Modeling of IPMC Actuators Based on the LSSVM-NARX Method

_{y}and d

_{u}are the output and input delays, respectively. In NARX, the current output is determined by not only the current input but also the delay d

_{u}of the current input and delay d

_{u}of the output. The weight parameters of the neural network can be continuously adjusted by learning the nonlinearity of these data so that the prediction of future parameters can be achieved.

#### 3.3. Results of the LSSVM-NARX Model

#### 3.4. LSSVM-NARX Model Optimization Based on Artificial Colony Algorithm

#### 3.5. Verification of Optimized LSSVM-NARX Model

## 4. Design of IPMC Actuator Control Method Based on Inverse Controller

#### 4.1. Inverse Controller Based on the LSSVM-NARX Model

_{l}and u

_{l}are the input displacement and output voltage at moment l, respectively, and ${h}_{1}(x)={\omega}_{1}{}^{\mathrm{T}}{\phi}_{1}(x)+{b}_{1}$, the input space is mapped to a high-dimensional space by the nonlinear function ${\phi}_{1}(x)$. Furthermore, ${\omega}_{1}$ and ${b}_{1}$ are the weight vector and error, while $K(\cdot )$ denotes the kernel function in Equation (13).

^{6}and 7.1077×10

^{3}, respectively. Figure 40 shows the structure of the LSSVM-NARX inverse controller. In this inverse controller, the input and output of the inverse LSSVM-NARX model are the expected displacement and actuating voltage, respectively. Three delays were involved in both input and output. The output voltage of the inverse model was regarded as the actuating voltage of the IPMC so that the expected displacement could easily be reached by compensation.

#### 4.2. Simulation of Inverse Controller

#### 4.3. Design of Hybrid PID Control System

#### 4.4. Experimental Results and Analysis

## 5. Conclusions and Prospects

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 16.**Results of the Prandtl-Ishlinskii model at sinusoidal actuating voltage with amplitude of 2V and frequency of 1/2π Hz.

**Figure 17.**Results of the Prandtl-Ishlinskii model at sinusoidal actuating voltage with amplitude of 3V and frequency of 5/2π Hz.

**Figure 18.**Errorof the Prandtl-Ishlinskii model at sinusoidal actuating voltage with amplitude of 2V and frequency of 1/2π Hz (RMSE = 0.5685).

**Figure 19.**Error of the Prandtl-Ishlinskii model at sinusoidal actuating voltage with amplitude of 3V and frequency of 5/2π Hz (RMSE = 0.8345).

**Figure 22.**Results of the LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 2V and frequency of 1/2π Hz.

**Figure 23.**Results of the LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 3V and frequency of 5/2π Hz.

**Figure 24.**Error of the LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 2V and frequency of 1/2π Hz (RMSE = 0.5147).

**Figure 25.**Error of the LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 3V and frequency of 5/2π Hz (RMSE = 0.3042).

**Figure 30.**The displacement of the tip by the Random sinusoidal drive voltage I: (

**a**) Random sinusoidal drive voltage I; (

**b**) Random sinusoidal tip displacement I.

**Figure 31.**The displacement of tip by the Random sinusoidal drive voltage II: (

**a**) Random sinusoidal drive voltage II; (

**b**) Random sinusoidal tip displacement II.

**Figure 32.**Results of the optimized LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 2V and frequency of 1/2π Hz.

**Figure 33.**Results of the optimized LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 3V and frequency of 5/2π Hz.

**Figure 34.**Error of the optimized LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 2V and frequency of 1/2π Hz (RMSE = 0.1308).

**Figure 35.**Error of the optimized LSSVM-NARX model at sinusoidal actuating voltage with amplitude of 3V and frequency of 5/2π Hz (RMSE = 0.1261).

**Figure 38.**Error of the optimized LSSVM-NARX model at random sinusoidal actuating voltage I (RMSE = 0.1169).

**Figure 39.**Error of the optimized LSSVM-NARX model at random sinusoidal actuating voltage II(RMSE = 0.0941).

**Figure 48.**Input displacement is the control result at a frequency of 1/2π Hz and an amplitude of 2.5 mm.

**Figure 49.**Input displacement is the control error at a frequency of 1/2π Hz and an amplitude of 2.5 mm.

Amplitude of Actuating Voltage (V) | 1 | 2 | 3 | ||||||
---|---|---|---|---|---|---|---|---|---|

Frequency of actuating voltage (Hz) | 0.5/2π | 1/2π | 5/2π | 0.5/2π | 1/2π | 5/2π | 0.5/2π | 1/2π | 5/2π |

Lower limit of tip displacements (mm) | −0.9873 | −1.1276 | −1.1993 | −2.5604 | −3.9327 | −2.487 | −4.7787 | −8.7983 | −6.1326 |

Upper limit of tip displacements (mm) | 0.8256 | 0.8923 | 0.8134 | 1.988 | 2.3796 | 2.7003 | 2.6328 | 2.5967 | 2.7528 |

Range of tip displacements (mm) | 1.8129 | 2.0199 | 2.0127 | 4.5484 | 6.3123 | 5.1873 | 7.4115 | 11.395 | 8.8854 |

a1 | a2 | a3 | a4 | a5 | a6 | a7 | a8 | a9 | a10 | |

Value | 2.446 | −8.8831 | −2.5468 | −5.3785 | 7.4586 | −8.13 | 3.711 | 4.986 | 2.3852 | 10 |

r1 | r2 | r3 | r4 | r5 | r6 | r7 | r8 | r9 | r10 | |

Value | 4.2268 | −10 | −4.2276 | 2.6812 | 6.3775 | 10 | −7.9211 | 10 | −10 | −3.8301 |

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**MDPI and ACS Style**

Huang, L.; Hu, Y.; Zhao, Y.; Li, Y.
Modeling and Control of IPMC Actuators Based on LSSVM-NARX Paradigm. *Mathematics* **2019**, *7*, 741.
https://doi.org/10.3390/math7080741

**AMA Style**

Huang L, Hu Y, Zhao Y, Li Y.
Modeling and Control of IPMC Actuators Based on LSSVM-NARX Paradigm. *Mathematics*. 2019; 7(8):741.
https://doi.org/10.3390/math7080741

**Chicago/Turabian Style**

Huang, Liangsong, Yu Hu, Yun Zhao, and Yuxia Li.
2019. "Modeling and Control of IPMC Actuators Based on LSSVM-NARX Paradigm" *Mathematics* 7, no. 8: 741.
https://doi.org/10.3390/math7080741