# Arm Angle Tracking Control with Pole Balancing Using Equivalent Input Disturbance Rejection for a Rotational Inverted Pendulum

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

## 3. EID Estimator Design

**Theorem**

**1.**

**Proof.**

## 4. LQR Based Tracking Controller Design

**Theorem**

**2.**

**Proof.**

## 5. Experimental Results

#### 5.1. Performances of Arm Angle Tracking Control and Pole Balancing

- Case 1: Conventional proportional-derivative (PD) controller, $u={k}_{p\theta}{e}_{\theta}+{k}_{d\theta}{\dot{e}}_{\theta}+{k}_{p\alpha}{e}_{\alpha}+{k}_{d\alpha}{\dot{e}}_{\alpha}$
- Case 2: Proposed method without EID compensation, $u={u}^{d}+Ke$
- Case 3: Proposed method with EID compensation, $u={u}^{d}+Ke-{\widehat{d}}_{ei{d}_{f}}$.
- Case 3: Proposed method with EID compensation under the parameter uncertainties (at most, ±20%), $u={u}^{d}+Ke-{\widehat{d}}_{ei{d}_{f}}$.

#### 5.2. Robustness against External Disturbance

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Control performance in case 1. (

**a**) Arm position (case 1). (

**b**) Pendulum pole position (case 1). (

**c**) Arm position error (case 1).

**Figure 4.**Control performance in case 2. (

**a**) Arm position (case 2). (

**b**) Pendulum pole position (case 2). (

**c**) Arm position error (case 2).

**Figure 5.**Control performance in case 3. (

**a**) Arm position (case 3). (

**b**) Pendulum pole position (case 3). (

**c**) Arm position error (case 3). (

**d**) Estimated EID (case 3).

**Figure 6.**Control performance in case 4. (

**a**) Arm position (case 4). (

**b**) Pendulum pole position (case 4). (

**c**) Arm position error (case 4). (

**d**) Estimated EID (case 4).

**Figure 8.**Control performance of the proposed method under the external disturbance. (

**a**) Arm position (case 4). (

**b**) Pendulum pole position (case 4). (

**c**) Arm position error (case 4). (

**d**) Estimated EID (case 4).

Symbol | Description | Value |
---|---|---|

${k}_{m}$ | DC motor torque constant | 0.042 N·m/A |

R | Terminal resistance | 8.4 $\mathsf{\Omega}$ |

${J}_{m}$ | Rotor inertia | 4.0 × 10${}^{-6}$ kg·m${}^{2}$ |

${m}_{r}$ | Rotary arm mass | 0.095 kg |

${L}_{r}$ | Rotary arm length | 0.085 m |

${m}_{p}$ | Pendulum mass | 0.024 kg |

${L}_{p}$ | Pendulum length | 0.129 m |

${J}_{r}$ | Rotary arm inertia | 5.72 × 10${}^{-5}$ kg·m${}^{2}$ |

${J}_{p}$ | Pendulum inertia | 5.72 × 10${}^{-5}$ kg·m${}^{2}$ |

g | Gravitational acceleration | 9.81 m/s${}^{2}$ |

Case | ASE |
---|---|

Case 1 | 0.0817 |

Case 2 | 0.0122 |

Case 3 | 0.061 |

Case 4 | 0.062 |

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**MDPI and ACS Style**

Lee, H.; Gil, J.; You, S.; Gui, Y.; Kim, W.
Arm Angle Tracking Control with Pole Balancing Using Equivalent Input Disturbance Rejection for a Rotational Inverted Pendulum. *Mathematics* **2021**, *9*, 2745.
https://doi.org/10.3390/math9212745

**AMA Style**

Lee H, Gil J, You S, Gui Y, Kim W.
Arm Angle Tracking Control with Pole Balancing Using Equivalent Input Disturbance Rejection for a Rotational Inverted Pendulum. *Mathematics*. 2021; 9(21):2745.
https://doi.org/10.3390/math9212745

**Chicago/Turabian Style**

Lee, Hojin, Jeonghwan Gil, Sesun You, Yonghao Gui, and Wonhee Kim.
2021. "Arm Angle Tracking Control with Pole Balancing Using Equivalent Input Disturbance Rejection for a Rotational Inverted Pendulum" *Mathematics* 9, no. 21: 2745.
https://doi.org/10.3390/math9212745