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Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

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Abstract

This paper focuses on linear-quadratic (LQ) optimal control for a class of systems governed by first-order hyperbolic partial differential equations (PDEs). Different from most of the previous works, an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems. The contributions of this paper consist of the following aspects: 1) The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method. 2) A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed. Meanwhile, the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations (FBPDEs). 3) The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented. It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case, but also support the existing operator results and give a more convenient form of computation.

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Acknowledgements

The authors would like to thank Prof. Xu Zhang, Sichuan University, and Prof. Long Hu, Shandong University, for their valuable discussions.

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Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Xiaomin Xue & Juanjuan Xu

  2. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266000, China

    Huanshui Zhang

Authors
  1. Xiaomin Xue
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  2. Juanjuan Xu
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  3. Huanshui Zhang
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Correspondence to Huanshui Zhang.

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The authors declare no conflict of interest.

Additional information

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61821004 and 62250056, and the Natural Science Foundation of Shandong Province under Grant Nos. ZR2021ZD14 and ZR2021JQ24, and Science and Technology Project of Qingdao West Coast New Area under Grant Nos. 2019–32, 2020–20, 2020-1-4, High-level Talent Team Project of Qingdao West Coast New Area under Grant No. RCTD-JC-2019-05, Key Research and Development Program of Shandong Province under Grant No. 2020CXGC01208.

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Xue, X., Xu, J. & Zhang, H. Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations. J Syst Sci Complex 37, 230–252 (2024). https://doi.org/10.1007/s11424-024-3324-8

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  • DOI: https://doi.org/10.1007/s11424-024-3324-8

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