# An Adaptive Controller Based on Interconnection and Damping Assignment Passivity-Based Control for Underactuated Mechanical Systems: Application to the Ball and Beam System

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Review of IDA-PBC Design

#### 2.2. Possible Uncertainties

## 3. Controller Design and Stability Analysis

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 4. Example: The Ball and Beam System

#### 4.1. System Model

#### 4.2. Controller Design

#### 4.3. Stability Analysis

**Assumption**

**1.**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

#### 4.4. Numerical Simulation Results

^{2}, ${J}_{b}=2\times {10}^{-6}$ kg·m

^{2}and $g=9.81$ m/s

^{2}, which are taken from [28]. The simulation results are shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13.

^{2}in $0.2$ s, and then decays to 0 gradually. In addition, it can be observed from Figure 6 that the control signal of the proposed adaptive controller is much smaller and settles down faster than for the backstepping method. As depicted in Figure 7, the estimated values of parameters are all bounded.

**Remark**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Liu, X.; Shao, H.; Liu, C.; Li, N.; Guo, X.; Zheng, F.; Sun, L.
An Adaptive Controller Based on Interconnection and Damping Assignment Passivity-Based Control for Underactuated Mechanical Systems: Application to the Ball and Beam System. *Actuators* **2023**, *12*, 408.
https://doi.org/10.3390/act12110408

**AMA Style**

Liu X, Shao H, Liu C, Li N, Guo X, Zheng F, Sun L.
An Adaptive Controller Based on Interconnection and Damping Assignment Passivity-Based Control for Underactuated Mechanical Systems: Application to the Ball and Beam System. *Actuators*. 2023; 12(11):408.
https://doi.org/10.3390/act12110408

**Chicago/Turabian Style**

Liu, Xiaoping, Huaizhi Shao, Cungen Liu, Ning Li, Xinpeng Guo, Fei Zheng, and Lijun Sun.
2023. "An Adaptive Controller Based on Interconnection and Damping Assignment Passivity-Based Control for Underactuated Mechanical Systems: Application to the Ball and Beam System" *Actuators* 12, no. 11: 408.
https://doi.org/10.3390/act12110408