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Iterative Chebyshev approximation method for optimal control problems.

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Title: Iterative Chebyshev approximation method for optimal control problems.
Authors: Wu, Di1 (AUTHOR) rosemary_di@163.com, Yu, Changjun1 (AUTHOR) yuchangjun@126.com, Wang, Hailing1 (AUTHOR) wanghailingshu@163.com, Bai, Yanqin1 (AUTHOR) yqbai@t.shu.edu.cn, Teo, Kok-Lay2 (AUTHOR) K.L.Teo@curtin.edu.au, Toh, Kim-Chuan3 (AUTHOR) mattohkc@nus.edu.sg
Source: ISA Transactions. Sep2024, Vol. 152, p277-289. 13p.
Subjects: Chebyshev polynomials, Chebyshev systems, Chebyshev approximation, Dynamical systems, Collocation methods
Abstract: We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production. • A sequence of sub-problems is constructed by linearizing the dynamic system and constraints. • Each sub-problem can be transformed into an equivalent nonlinear optimization problem with linear equality constraints through Chebyshev polynomials. • Any feasible solution of the approximate sub-problem will satisfy its dynamic system and constraints over the entire time horizon. • Global convergence of the proposed method is established. [ABSTRACT FROM AUTHOR]
Copyright of ISA Transactions is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 179260953
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  Data: Iterative Chebyshev approximation method for optimal control problems.
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  Data: <searchLink fieldCode="AR" term="%22Wu%2C+Di%22">Wu, Di</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> rosemary_di@163.com</i><br /><searchLink fieldCode="AR" term="%22Yu%2C+Changjun%22">Yu, Changjun</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> yuchangjun@126.com</i><br /><searchLink fieldCode="AR" term="%22Wang%2C+Hailing%22">Wang, Hailing</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> wanghailingshu@163.com</i><br /><searchLink fieldCode="AR" term="%22Bai%2C+Yanqin%22">Bai, Yanqin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> yqbai@t.shu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Teo%2C+Kok-Lay%22">Teo, Kok-Lay</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> K.L.Teo@curtin.edu.au</i><br /><searchLink fieldCode="AR" term="%22Toh%2C+Kim-Chuan%22">Toh, Kim-Chuan</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> mattohkc@nus.edu.sg</i>
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  Data: <searchLink fieldCode="JN" term="%22ISA+Transactions%22">ISA Transactions</searchLink>. Sep2024, Vol. 152, p277-289. 13p.
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  Data: <searchLink fieldCode="DE" term="%22Chebyshev+polynomials%22">Chebyshev polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Chebyshev+systems%22">Chebyshev systems</searchLink><br /><searchLink fieldCode="DE" term="%22Chebyshev+approximation%22">Chebyshev approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamical+systems%22">Dynamical systems</searchLink><br /><searchLink fieldCode="DE" term="%22Collocation+methods%22">Collocation methods</searchLink>
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  Data: We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production. • A sequence of sub-problems is constructed by linearizing the dynamic system and constraints. • Each sub-problem can be transformed into an equivalent nonlinear optimization problem with linear equality constraints through Chebyshev polynomials. • Any feasible solution of the approximate sub-problem will satisfy its dynamic system and constraints over the entire time horizon. • Global convergence of the proposed method is established. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of ISA Transactions is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1016/j.isatra.2024.06.010
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      – Code: eng
        Text: English
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        PageCount: 13
        StartPage: 277
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        Type: general
      – SubjectFull: Chebyshev systems
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      – SubjectFull: Chebyshev approximation
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      – SubjectFull: Dynamical systems
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      – SubjectFull: Collocation methods
        Type: general
    Titles:
      – TitleFull: Iterative Chebyshev approximation method for optimal control problems.
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            NameFull: Wu, Di
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            NameFull: Yu, Changjun
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              Text: Sep2024
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              Value: 152
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