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Active fault tolerant control for polynomial nonlinear systems with asymmetric state constraints and measurement noise

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Abstract

In this paper, the problem of active fault tolerant control (AFTC) is studied for polynomial nonlinear systems subject to asymmetric state constraints and measurement noise. The AFTC scheme is composed of the fault detection and isolation unit and the fault tolerant control unit. By using polynomial filters, the fault detection (FD) module and the fault isolation (FI) module are designed to obtain the accurate estimations of states and faults in the presence of measurement noise. The FD is realized by the residual evaluation approach, while the FI is achieved through the residual matching method. When a fault is detected and isolated, the controller switches from a nominal controller (NC) to a reconfigured controller (RC). To ensure that the asymmetric state constraint is not violated, a universal barrier Lyapunov function is introduced in the design of the NC and the RC. Moreover, the boundedness of tracking errors and estimation errors is analysed. Finally, the effectiveness of the proposed AFTC method is verified by a simulation for a dynamic point-the-bit rotary steerable drilling tool system.

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Data Availability Statement

The simulation data that support the findings of this study are available within the article.

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Funding

This work was supported by the National Natural Science Foundation of China under Grants 62073339, 62173343, 62033008, the Natural Science Foundation of Shandong Province of China under Grants ZR2020YQ49, ZR2022ZD34, and the Research Fund for the Taishan Scholar Project of Shandong Province of China.

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Authors and Affiliations

  1. College of Control Science and Engineering, China University of Petroleum (East China), Qingdao, 266580, China

    Yuhan Ma, Ming Gao, Li Sheng & Yongli Wei

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  1. Yuhan Ma
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  2. Ming Gao
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  3. Li Sheng
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Correspondence to Li Sheng.

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Appendix

Appendix

1.1 Proof of Theorem 1

Proof

Letting \(\vartheta =[\xi ^T,{\bar{v}}^T,\xi ^T_y]^T\), it has

$$\begin{aligned} {\dot{V}}_e\le&e^T\bigg (P({\bar{A}}+H_g(\hat{{\bar{x}}})-L(\hat{{\bar{x}}})H_y(\hat{{\bar{x}}}))\nonumber \\ {}&\quad +({\bar{A}}+H_g(\hat{{\bar{x}}}) -L(\hat{{\bar{x}}})H_y(\hat{{\bar{x}}}))^TP\bigg )e\nonumber \\&\quad +2e^TP(\varTheta (\hat{{\bar{x}}})-{\bar{L}}(\hat{{\bar{x}}}){\tilde{G}}(\hat{{\bar{x}}}))\vartheta . \end{aligned}$$
(52)

Letting \({\bar{\vartheta }}=[e^T,\vartheta ^T]^T\), one has

$$\begin{aligned} {\dot{V}}_e&+\beta (\gamma _1e^Te-\gamma _2\vartheta ^T\vartheta )\le {\bar{\vartheta }}^T\varPi (\hat{{\bar{x}}}){\bar{\vartheta }}. \end{aligned}$$
(53)

Considering condition (10), it has

$$\begin{aligned} {\dot{V}}_e&\le -\beta \gamma _1e^Te+\beta \gamma _2\vartheta ^T\vartheta \le -p_eV_e+q_e. \end{aligned}$$
(54)

Moreover, for \(t\in [0,\infty )\), inequality (54) implies that

$$\begin{aligned}&V_e(t) \le \left( V_e(0)-\frac{q_e}{p_e}\right) \textrm{exp}(-p_et)+\frac{q_e}{p_e} \le V_e(0)\nonumber \\&\quad +\frac{q_e}{p_e}={\bar{q}}_e. \end{aligned}$$
(55)

By using (55), we know \(V_e=e^TPe\le {\bar{q}}_e\), then it is clear that e converges to \(\varOmega _e\). The proof is complete. \(\square \)

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Ma, Y., Gao, M., Sheng, L. et al. Active fault tolerant control for polynomial nonlinear systems with asymmetric state constraints and measurement noise. Nonlinear Dyn 111, 14157–14175 (2023). https://doi.org/10.1007/s11071-023-08601-9

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