# Modeling and Discrete-Time Terminal Sliding Mode Control of a DEAP Actuator with Rate-Dependent Hysteresis Nonlinearity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hammerstein Model for DEAP Actuator

#### 2.1. MPI Model

#### 2.2. ARX Model

#### 2.3. Parameters Identification

#### 2.4. Model Validation

## 3. Compound Controller Design

#### 3.1. Design of the Novel DTSMC

**Assumption**

**1.**

**Assumption**

**2.**

**Property**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.2. Stability Analysis

**Assumption**

**3.**

**Proposition**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 4. Experimental Results and Discussion

#### 4.1. Experimental Setup

#### 4.2. Experimental Results

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The working principle of dielectric electro-active polymer (DEAP) materials. (

**a**) Power cut: Initial state of DEAP materials. (

**b**) Power supply: Actuation direction of DEAP materials.

**Figure 6.**The hysteresis loops of the Hammerstein model under different frequencies. (

**a**) 0.1 Hz. (

**b**) 0.2 Hz. (

**c**) 0.3 Hz. (

**d**) 0.4 Hz. (

**e**) 0.5 Hz. (

**f**) 0.1/0.2/0.3/0.4/0.5 Hz. Red dotted line: output of the Hammerstein model. Blue solid line: output of the DEAP actuator.

**Figure 10.**The tracking results under 0.1 Hz input signal. (

**a**) Displacement output. (

**b**) Tracking error. (

**c**) Desired displacement versus actual displacement of the DSMC. (

**d**) Desired displacement versus actual displacement of the DTSMC.

**Figure 11.**The tracking results under 0.2 Hz input signal. (

**a**) Displacement output. (

**b**) Tracking error. (

**c**) Desired displacement versus actual displacement of the DSMC. (

**d**) Desired displacement versus actual displacement of the DTSMC.

**Figure 12.**The tracking results under 0.3 Hz input signal. (

**a**) Displacement output. (

**b**) Tracking error. (

**c**) Desired displacement versus actual displacement of the DSMC. (

**d**) Desired displacement versus actual displacement of the DTSMC.

**Figure 13.**The tracking results under 0.4 Hz input signal. (

**a**) Displacement output. (

**b**) Tracking error. (

**c**) Desired displacement versus actual displacement of the DSMC. (

**d**) Desired displacement versus actual displacement of the DTSMC.

**Figure 14.**The tracking results under 0.5 Hz input signal. (

**a**) Displacement output. (

**b**) Tracking error. (

**c**) Desired displacement versus actual displacement of the DSMC. (

**d**) Desired displacement versus actual displacement of the DTSMC.

**Figure 15.**The tracking results under the hybrid frequency input signal. (

**a**) Displacement output. (

**b**) Tracking error. (

**c**) Desired displacement versus actual displacement of the DSMC. (

**d**) Desired displacement versus actual displacement of the DTSMC.

Index n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

${\vartheta}_{n}$ | −1.3188 | 1.9548 | −0.9392 | 0.8491 | 1.3619 | 0.3145 | 0.2648 | 0.2226 |

${r}_{hn}$ | 0.0007 | 0.0058 | 0.0196 | 0.0466 | 0.0909 | 0.1571 | 0.2495 | 0.3724 |

Index n | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

${\vartheta}_{n}$ | 0.0456 | 0.2584 | 1.5619 | −0.2742 | −0.3569 | −0.3978 | −0.4230 | |

${r}_{hn}$ | 0.5303 | 0.7274 | 0.9682 | 1.2570 | 1.5982 | 1.9961 | 2.4551 |

Index m | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

${\theta}_{m}$ | −17.5517 | −15.7191 | −7.8020 | 13.5245 | 58.4796 | −30.1681 | −0.9978 | 0.4027 |

${r}_{dm}$ | 0.0001 | 0.0013 | 0.0064 | 0.0201 | 0.0491 | 0.1019 | 0.1888 | 0.3220 |

Index m | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

${\theta}_{m}$ | −0.0577 | 0.1685 | 0.2633 | 0.1806 | 0.2844 | 0.2620 | 0.5739 | |

${r}_{dm}$ | 0.5158 | 0.7862 | 1.1510 | 1.6302 | 2.2453 | 3.0201 | 3.9799 |

${\mathbf{a}}_{1}$ | ${\mathbf{a}}_{2}$ | ${\mathbf{b}}_{1}$ | ${\mathbf{b}}_{2}$ |
---|---|---|---|

−1.9560 | 0.9545 | −0.3095 | 0.3086 |

Frequency/Hz | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.1/0.2/0.3/0.4/0.5 |
---|---|---|---|---|---|---|

RMSE (mm) | 0.0052 | 0.0098 | 0.0145 | 0.0190 | 0.0231 | 0.0271 |

MAE (mm) | 0.0040 | 0.0076 | 0.0113 | 0.0149 | 0.0182 | 0.0083 |

$\mathbf{Controller}$ | ${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | $\mathit{\alpha}$ | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | $\mathit{\beta}$ | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|

DSMC | 3.50 | - | 7/11 | 0.25 | 0.50 | 1/2 | 0.01 |

DTSMC | 3.50 | 0.55 | 7/11 | 0.25 | 0.50 | 1/2 | 0.01 |

Frequency/Hz | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.1/0.2/0.3/0.4/0.5 | |
---|---|---|---|---|---|---|---|

RMSE (mm) | DSMC | 0.0565 | 0.0942 | 0.1054 | 0.1514 | 0.2226 | 0.1536 |

DTSMC | 0.0536 | 0.0612 | 0.0750 | 0.0761 | 0.0847 | 0.0500 | |

MAE (mm) | DSMC | 0.0207 | 0.0504 | 0.0788 | 0.1255 | 0.1834 | 0.0990 |

DTSMC | 0.0122 | 0.0215 | 0.0314 | 0.0389 | 0.0456 | 0.0242 |

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**MDPI and ACS Style**

Li, M.; Wang, Q.; Li, Y.; Jiang, Z.
Modeling and Discrete-Time Terminal Sliding Mode Control of a DEAP Actuator with Rate-Dependent Hysteresis Nonlinearity. *Appl. Sci.* **2019**, *9*, 2625.
https://doi.org/10.3390/app9132625

**AMA Style**

Li M, Wang Q, Li Y, Jiang Z.
Modeling and Discrete-Time Terminal Sliding Mode Control of a DEAP Actuator with Rate-Dependent Hysteresis Nonlinearity. *Applied Sciences*. 2019; 9(13):2625.
https://doi.org/10.3390/app9132625

**Chicago/Turabian Style**

Li, Mengmeng, Qinglin Wang, Yuan Li, and Zhaoguo Jiang.
2019. "Modeling and Discrete-Time Terminal Sliding Mode Control of a DEAP Actuator with Rate-Dependent Hysteresis Nonlinearity" *Applied Sciences* 9, no. 13: 2625.
https://doi.org/10.3390/app9132625