Numerical Discretization Methods for the Extended Linear Quadratic Control Problem
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909169543544832 |
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| author | Zhang, Zhanhao Svensen, Jan Lorenz Kaysfeld, Morten Wahlgreen Christensen, Anders Hilmar Damm Hørsholt, Steen Jørgensen, John Bagterp |
| author_facet | Zhang, Zhanhao Svensen, Jan Lorenz Kaysfeld, Morten Wahlgreen Christensen, Anders Hilmar Damm Hørsholt, Steen Jørgensen, John Bagterp |
| contents | In this study, we introduce numerical methods for discretizing continuous-time linear-quadratic optimal control problems (LQ-OCPs). The discretization of continuous-time LQ-OCPs is formulated into differential equation systems, and we can obtain the discrete equivalent by solving these systems. We present the ordinary differential equation (ODE), matrix exponential, and a novel step-doubling method for the discretization of LQ-OCPs. Utilizing Euler-Maruyama discretization with a fine step, we reformulate the costs of continuous-time stochastic LQ-OCPs into a quadratic form, and show that the stochastic cost follows the $χ^2$ distribution. In the numerical experiment, we test and compare the proposed numerical methods. The results ensure that the discrete-time LQ-OCP derived using the proposed numerical methods is equivalent to the original problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09316 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical Discretization Methods for the Extended Linear Quadratic Control Problem Zhang, Zhanhao Svensen, Jan Lorenz Kaysfeld, Morten Wahlgreen Christensen, Anders Hilmar Damm Hørsholt, Steen Jørgensen, John Bagterp Systems and Control In this study, we introduce numerical methods for discretizing continuous-time linear-quadratic optimal control problems (LQ-OCPs). The discretization of continuous-time LQ-OCPs is formulated into differential equation systems, and we can obtain the discrete equivalent by solving these systems. We present the ordinary differential equation (ODE), matrix exponential, and a novel step-doubling method for the discretization of LQ-OCPs. Utilizing Euler-Maruyama discretization with a fine step, we reformulate the costs of continuous-time stochastic LQ-OCPs into a quadratic form, and show that the stochastic cost follows the $χ^2$ distribution. In the numerical experiment, we test and compare the proposed numerical methods. The results ensure that the discrete-time LQ-OCP derived using the proposed numerical methods is equivalent to the original problem. |
| title | Numerical Discretization Methods for the Extended Linear Quadratic Control Problem |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2404.09316 |