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Fixed-time stabilization of high-order nonlinear systems with an asymmetric output constraint

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Abstract

This article studies the problem of fixed-time stabilization for a class of uncertain high-order nonlinear systems subjected to an asymmetric output constraint. A tangent-type barrier function is first developed as an intermediate design ingredient by subtly extracting and utilizing the inherent features of system nonlinearities. Next, the proposed barrier function along with the intrinsic attributes of signum functions is exploited to elegantly renovate the celebrated technique of adding a power integrator, thereby establishing a unified approach by which a tangent-type asymmetric barrier Lyapunov function together with a continuous state feedback fixed-time stabilizer can be constructed systematically while guaranteeing the achievement of pre-specified output constraints successfully. A technical novelty of the presented scheme is ascribed to the unified nature enabling us to design a fixed-time stabilizer simultaneously workable for the system subjected to or free from output constraints without needing to revamp the controller structure. A numerical example is provided to show the effectiveness and superiority of the developed method.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. It should be emphasized that for the stabilization problem of high-order nonlinear systems (e.g., system (1)) with output constraints, the work [35] is the very first result in the literature proposing a solution by using tangent-type BLFs.

  2. The task of fixed-time stabilization is to perform the finite-time stabilization with securing an upper bound of the settling time irrelative to initial states [41,42,43].

  3. Here, the asymmetric output constraint is the constraint with nonidentical (asymmetric) upper and lower bounds (constraints); e.g., \(-\varepsilon _L<y(t)=x_1(t)<\varepsilon _U\) for all \(t\ge t_0\) with \(\varepsilon _L>0\), \(\varepsilon _U>0\) and \(\varepsilon _L\ne \varepsilon _U\).

  4. Such an intermediate/unformed design ingredient in the design processes is named a barrier function rather than a BLF since extra requirements are necessary for a function to be a BLF.

  5. It can be found from the proof of Lemma 1 in [38] that the nonpositivity of the derivative of V(z) along system (2) is sufficient to ensure the boundedness of the solution z(t); thus, the condition (ii) here is simplified, compared to the one in [38].

  6. Because \(\phi (t,z)\) is continuous and probably not Lipschitz on \(\mathbb {R}_+\times \mathbb {R}^n\), the solution of system (2) satisfying a given initial state \(z(t_0)\in \mathbb {M}_n(\varepsilon _L,\varepsilon _U)\) is in general not unique [46].

  7. For the definition of forward completeness, please refer to [46].

  8. Because the functions on the right-hand side of (11) depend on \((t,x,u)\in \mathbb {R}_+\times \mathbb {M}_n(\varepsilon _L,\varepsilon _U)\times \mathbb {R}\), the valid region of the inequality (11) is explicitly presented.

  9. For any real constant \(1<\theta <2\), the existence of the smooth function \(\psi _1(x_1)\) is guaranteed by [49] due the continuity of \(|\xi _1(x_1)|^{(2\eta (\theta -1) - \omega _1\theta )/\mu }\) with regard to \(x_1\). Notably, it is shown in [49, Theorem 6.21, p. 136] that for any continuous function \(\varphi :\mathbb {R}^n\rightarrow \mathbb {R}\) one can always find a smooth function \(\overline{\varphi }:\mathbb {R}^n\rightarrow \mathbb {R}_+\) such that \(|\varphi (s)|\le \overline{\varphi }(s)\) for all \(s\in \mathbb {R}^n\). This truth will be used repeatedly in this article.

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Funding

This work was supported in part by the Ministry of Science and Technology (MOST), Taiwan, under Grants MOST 110-2221-E-006-173- and MOST 111-2221-E-006-204-MY2.

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Authors and Affiliations

  1. Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan, 70101, Taiwan

    Chih-Chiang Chen

  2. Institute of Automation, Qufu Normal University, Qufu, Shandong Province, 273165, China

    Zong-Yao Sun

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  1. Chih-Chiang Chen
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Correspondence to Chih-Chiang Chen.

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Chen, CC., Sun, ZY. Fixed-time stabilization of high-order nonlinear systems with an asymmetric output constraint. Nonlinear Dyn 111, 319–339 (2023). https://doi.org/10.1007/s11071-022-07839-z

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