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Leakage-type adaptive state and disturbance observers for uncertain nonlinear systems

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Abstract

This paper proposes a novel adaptive observer technique for estimating the state and disturbance of uncertain nonlinear systems. To remove the knowledge of the upper bounds of the disturbance and its derivative, a leakage-type (LT) algorithm is introduced to approximate the variations of their bounds. A state observer is first provided based on a conventional Walcott–Zak observer structure, and then, a disturbance observer is proposed by introducing an auxiliary dynamics. Due to the features of the LT adaptive law, the estimation error of the system state or the disturbance is bounded in a small neighborhood around zero in finite time. In addition, since the switching gain is automatically adapted to the disturbance change, the chattering in the estimation signal is effectively suppressed that is useful for the estimation precision in a practical system. Another important advantage of the proposed method lies in its simple structure compared to the existing finite-time observers. Lyapunov analysis demonstrates that for both types of observers, the estimation error is achieved to be globally uniformly ultimately bounded. To demonstrate the proposed method, simulation examples are separately carried out on a vehicle system and a linear motor system.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. U1813216 and No. 62003186) and the Natural Science Foundation of Guangdong Province (No. 2020A1515010334).

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Authors and Affiliations

  1. Center of Intelligent Control and Telescience, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, 518055, Guangdong, China

    Ke Shao & Xueqian Wang

  2. School of Software and Electrical Engineering, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Jinchuan Zheng

  3. Discipline of Engineering and Energy, Murdoch University, Murdoch, WA, 6150, Australia

    Hai Wang

  4. Research Institute of Tsinghua University in Shenzhen, Shenzhen, 518057, Guangdong, China

    Bin Liang

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  1. Ke Shao
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  2. Jinchuan Zheng
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  3. Hai Wang
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  4. Xueqian Wang
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Correspondence to Hai Wang.

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Appendix

Appendix

1.1 A Proof of Lemma 2.3

Inequality (3) can be rewritten as

$$\begin{aligned} \dot{V}(x)\le & {} -\kappa V(x)+\epsilon \nonumber \\= & {} -\kappa \theta V(x)-\kappa (1-\theta ) V(x)+\epsilon , \end{aligned}$$
(48)

where \(0<\theta <1\). It is obvious that if \(V(x)>\frac{\epsilon }{\kappa (1-\theta )}\), \(\dot{V}(x)\le -\kappa \theta V(x)\). Therefore, V(x) is bounded by

$$\begin{aligned} V(x)\le \frac{\epsilon }{\kappa (1-\theta )}:=\bar{b}. \end{aligned}$$
(49)

Furthermore, for \(\dot{V}(x)\le -\kappa \theta V(x)\), we obtain

$$\begin{aligned} \frac{1}{V(x)}d V(x)\le -\kappa \theta . \end{aligned}$$
(50)

Let \(t_V\) be the time required to reach the region \(V(x)=\bar{b}\). Integrating (50) between V(x) and \(\bar{b}\) leads to

$$\begin{aligned} \int _{V(0)}^{\bar{b}}\frac{1}{V(t)}d V(x)=\ln {\bar{b}}-\ln {V(0)}\le -\kappa \theta t_V. \end{aligned}$$
(51)

Therefore,

$$\begin{aligned} t_V\le \frac{1}{\kappa \theta }\ln {\frac{V(0)}{\bar{b}}}. \end{aligned}$$
(52)

This completes the proof.

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Shao, K., Zheng, J., Wang, H. et al. Leakage-type adaptive state and disturbance observers for uncertain nonlinear systems. Nonlinear Dyn 105, 2299–2311 (2021). https://doi.org/10.1007/s11071-021-06715-6

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