# Online Adaptive Parameter Estimation for Quadrotors

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A parameter estimation is developed to obtain the parameter error information by using system dynamics and the estimated parameters with simple filter operations.
- (2)
- Novel parameter-error-based adaptive parameter estimation algorithms are suggested for quadrotors to guarantee fast convergence.
- (3)
- A constructive method is suggested to validate the standard persistent excitation (PE) condition online.

## 2. Adaptive Parameter Estimation

**Lemma**

**1 ([13]).**

**Theorem**

**1 ([22,23]).**

**Proof of Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Simulation

## 4. Experimental Verification

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Huang, W.; Ding, F. Coupled Least Squares Identification Algorithms for Multivariate Output-Error Systems. Algorithms
**2017**, 10, 12. [Google Scholar] [CrossRef] - Liu, S.; Xu, L.; Ding, F. Iterative Parameter Estimation Algorithms for Dual-Frequency Signal Models. Algorithms
**2017**, 10, 118. [Google Scholar] [CrossRef] - Sastry, S.; Bodson, M.; Bartram, J.F. Adaptive Control: Stability, Convergence, and Robustness. J. Acoust. Soc. Am.
**1990**, 88, 588–589. [Google Scholar] [CrossRef] - Ioannou, P.; Baldi, S. Robust Adaptive Control. In Proceedings of the 1984 American Control Conference, San Diego, CA, USA, 6–8 June 1984; pp. 333–335. [Google Scholar]
- Li, Z.; Ring, P.; Macrae, K.; Hinsch, A. Applied Nonlinear Control; China Machine Press: Beijing, China, 1991. [Google Scholar]
- Adetola, V.; Guay, M. Finite-Time Parameter Estimation in Adaptive Control of Nonlinear Systems. IEEE Trans. Autom. Control
**2008**, 53, 807–811. [Google Scholar] [CrossRef] - Atassi, A.N.; Khalil, H.K. A separation principle for the stabilization of a class of nonlinear systems. Autom. Control IEEE Trans.
**1999**, 44, 1672–1687. [Google Scholar] [CrossRef] - Ahmed-Ali, T.; Kenné, G.; Lamnabhi-Lagarrigue, F. Identification of nonlinear systems with time-varying parameters using a sliding-neural network observer. Neurocomputing
**2009**, 72, 1611–1620. [Google Scholar] [CrossRef] - Hornik, K.; Stinchcombe, M.; White, H. Multilayer Feedforward Networks are Universal Approxmations Neural Networks. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - Adetola, V.; Guay, M.; Lehrer, D. Adaptive Estimation for a Class of Nonlinearly Parameterized Dynamical Systems. IEEE Trans. Autom. Control
**2014**, 59, 2818–2824. [Google Scholar] [CrossRef] - Na, J.; Herrmann, G.; Burke, R.; Brace, C. Adaptive input and parameter estimation with application to engine torque estimation. In Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, 15–18 December 2015; pp. 3687–3692. [Google Scholar]
- Na, J.; Mahyuddin, M.N.; Herrmann, G.; Ren, X. Robust adaptive finite-time parameter estimation for linearly parameterized nonlinear systems. In Proceedings of the 32nd Chinese Control Conference, Xi’an, China, 26–28 July 2013; pp. 1735–1741. [Google Scholar]
- Na, J.; Mahyuddin, M.N.; Herrmann, G.; Ren, X.; Barber, P. Robust adaptive finite-time parameter estimation and control for robotic systems. Int. J. Robust Nonlinear Control
**2015**, 25, 3045–3071. [Google Scholar] [CrossRef] [Green Version] - Yang, J.; Na, J.; Guo, Y.; Wu, X. Adaptive estimation of road gradient and vehicle parameters for vehicular systems. IET Control Theory Appl.
**2015**, 9, 935–943. [Google Scholar] [CrossRef] - Liu, H.; Li, D.; Xi, J.; Zhong, Y. Robust attitude controller design for miniature quadrotors. Int. J. Robust Nonlinear Control
**2016**, 26, 681–696. [Google Scholar] [CrossRef] - Esteban, S.; Gordillo, F.; Aracil, J. Three-time scale singular perturbation control and stability analysis for an autonomous helicopter on a platform. Int. J. Robust Nonlinear Control
**2013**, 23, 1360–1392. [Google Scholar] [CrossRef] - Hoffmann, G.M.; Huang, H.; Waslander, S.L.; Tomlin, C.J. Precision flight control for a multi-vehicle quadrotor helicopter testbed. Control Eng. Pract.
**2011**, 19, 1023–1036. [Google Scholar] [CrossRef] - Isidori, A.; Marconi, L.; Serrani, A. Robust nonlinear motion control of a helicopter. IEEE Trans. Autom. Control
**2003**, 48, 413–426. [Google Scholar] [CrossRef] - Mahony, R.; Hamel, T. Robust trajectory tracking for a scale model autonomous helicopter. Int. J. Robust Nonlinear Control
**2004**, 14, 1035–1059. [Google Scholar] [CrossRef] - Luo, Z.; Zhang, X.; Liu, Z. Research of Modeling and Sliding Mode Advanced Control for Four Rotor Aircraft. In Proceedings of the Fifth International Conference on Instrumentation and Measurement, Computer, Communication and Control, Qinhuangdao, China, 18–20 September 2015; pp. 992–996. [Google Scholar]
- Jing, N.; Herrmann, G. Online Adaptive Approximate Optimal Tracking Control with Simplified Dual Approximation Structure for Continuous-time Unknown Nonlinear Systems. IEEE/CAA J. Autom. Sin.
**2014**, 1, 412–422. [Google Scholar] - Lv, Y.; Na, J.; Ren, X. Online H∞ control for completely unknown nonlinear systems via an identifier–critic-based ADP structure. Int. J. Control
**2017**. [Google Scholar] [CrossRef] - Na, J.; Lv, Y.; Wu, X.; Guo, Y.; Chen, Q. Approximate optimal tracking control for continuous-time unknown nonlinear systems. In Proceedings of the 33rd Chinese Control Conference, Nanjin, China, 28–30 July 2014; pp. 8990–8995. [Google Scholar]

**Figure 1.**The schematic diagram of the ground coordinate system and the body axes coordinate system.

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

m | 0.25 | Kg | J_{z} | 0.061 | Kg·m^{2} |

l | 0.25 | m | g | 9.8 | m/s^{2} |

J_{x} | 0.033 | Kg·m^{2} | ------- | ------- | ------- |

J_{y} | 0.033 | Kg·m^{2} | ------- | ------- | ------- |

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## Share and Cite

**MDPI and ACS Style**

Zhao, J.; Wang, X.; Gao, G.; Na, J.; Liu, H.; Luan, F.
Online Adaptive Parameter Estimation for Quadrotors. *Algorithms* **2018**, *11*, 167.
https://doi.org/10.3390/a11110167

**AMA Style**

Zhao J, Wang X, Gao G, Na J, Liu H, Luan F.
Online Adaptive Parameter Estimation for Quadrotors. *Algorithms*. 2018; 11(11):167.
https://doi.org/10.3390/a11110167

**Chicago/Turabian Style**

Zhao, Jun, Xian Wang, Guanbin Gao, Jing Na, Hongping Liu, and Fujin Luan.
2018. "Online Adaptive Parameter Estimation for Quadrotors" *Algorithms* 11, no. 11: 167.
https://doi.org/10.3390/a11110167