Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality
In this paper, we propose a novel state-feedback backstepping control design approach for a single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunov functions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adop...
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| Format: | Article |
| Language: | English |
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IEEE
2020-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9026968/ |
| _version_ | 1819159142541033472 |
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| author | Zhengru Ren Bo Zhao Dong Trong Nguyen |
| author_facet | Zhengru Ren Bo Zhao Dong Trong Nguyen |
| author_sort | Zhengru Ren |
| collection | DOAJ |
| description | In this paper, we propose a novel state-feedback backstepping control design approach for a single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunov functions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adopted in the stability proof. Semiglobal practical finite-time stability, or global asymptotically stability, is guaranteed by a continuous control law using a commonly used recursive backstepping-like approach. Unlike the inductive design of typical finite-time backstepping controllers, the proposed method has the advantage of reduced design complexity. The virtual control laws are designed by directly canceling the nonlinear terms in the derivative of the specific Lyapunov functions. The terms with exponents are transformed into linear forms as their bases. The stability proof is simplified by applying several inequalities in the final proof, instead of in each step. Furthermore, the singularity problem no longer exists. The weakness of the concept of practical finite-time stability is discussed. The method can be applied to smoothly extend numerous design methodologies with asymptotic stability with a higher convergence rate near the equilibrium. Two numerical case studies are provided to present the performance of the proposed control. |
| first_indexed | 2024-12-22T16:35:51Z |
| format | Article |
| id | doaj.art-c19a1d4401ad4a0889ca6f1829cd12be |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-12-22T16:35:51Z |
| publishDate | 2020-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-c19a1d4401ad4a0889ca6f1829cd12be2022-12-21T18:19:58ZengIEEEIEEE Access2169-35362020-01-018477684777510.1109/ACCESS.2020.29790689026968Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli InequalityZhengru Ren0https://orcid.org/0000-0001-5522-8094Bo Zhao1https://orcid.org/0000-0002-7428-9850Dong Trong Nguyen2https://orcid.org/0000-0002-8026-3761Centre for Research-based Innovation of Marine Operations (SFI MOVE), Trondheim, NorwaySchool of Automation, Harbin Engineering University, Harbin, ChinaDepartment of Marine Technology, Norwegian University of Science and Technology (NTNU), Trondheim, NorwayIn this paper, we propose a novel state-feedback backstepping control design approach for a single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunov functions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adopted in the stability proof. Semiglobal practical finite-time stability, or global asymptotically stability, is guaranteed by a continuous control law using a commonly used recursive backstepping-like approach. Unlike the inductive design of typical finite-time backstepping controllers, the proposed method has the advantage of reduced design complexity. The virtual control laws are designed by directly canceling the nonlinear terms in the derivative of the specific Lyapunov functions. The terms with exponents are transformed into linear forms as their bases. The stability proof is simplified by applying several inequalities in the final proof, instead of in each step. Furthermore, the singularity problem no longer exists. The weakness of the concept of practical finite-time stability is discussed. The method can be applied to smoothly extend numerous design methodologies with asymptotic stability with a higher convergence rate near the equilibrium. Two numerical case studies are provided to present the performance of the proposed control.https://ieeexplore.ieee.org/document/9026968/Finite-time stabilityrational-exponent Lyapunov functionbacksteppingLyapunov methodsBernoulli inequality |
| spellingShingle | Zhengru Ren Bo Zhao Dong Trong Nguyen Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality IEEE Access Finite-time stability rational-exponent Lyapunov function backstepping Lyapunov methods Bernoulli inequality |
| title | Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality |
| title_full | Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality |
| title_fullStr | Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality |
| title_full_unstemmed | Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality |
| title_short | Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality |
| title_sort | finite time backstepping of a nonlinear system in strict feedback form proved by bernoulli inequality |
| topic | Finite-time stability rational-exponent Lyapunov function backstepping Lyapunov methods Bernoulli inequality |
| url | https://ieeexplore.ieee.org/document/9026968/ |
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