A Convex Data-Driven Approach for Nonlinear Control Synthesis
We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based st...
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| Format: | Article |
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MDPI AG
2021-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/19/2445 |
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| author | Hyungjin Choi Umesh Vaidya Yongxin Chen |
| author_facet | Hyungjin Choi Umesh Vaidya Yongxin Chen |
| author_sort | Hyungjin Choi |
| collection | DOAJ |
| description | We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples. |
| first_indexed | 2024-03-10T06:54:56Z |
| format | Article |
| id | doaj.art-5757820f7f854be3bb0783c905bfe17a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T06:54:56Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5757820f7f854be3bb0783c905bfe17a2023-11-22T16:30:26ZengMDPI AGMathematics2227-73902021-10-01919244510.3390/math9192445A Convex Data-Driven Approach for Nonlinear Control SynthesisHyungjin Choi0Umesh Vaidya1Yongxin Chen2Energy Storage Technology and Systems at Sandia National Laboratories, Albuquerque, NM 87185, USADepartment of Mechanical Enginerring, Clemson University, Clemson, SC 29634, USASchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAWe consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.https://www.mdpi.com/2227-7390/9/19/2445nonlinear controlKoopman operatorsum of squares |
| spellingShingle | Hyungjin Choi Umesh Vaidya Yongxin Chen A Convex Data-Driven Approach for Nonlinear Control Synthesis Mathematics nonlinear control Koopman operator sum of squares |
| title | A Convex Data-Driven Approach for Nonlinear Control Synthesis |
| title_full | A Convex Data-Driven Approach for Nonlinear Control Synthesis |
| title_fullStr | A Convex Data-Driven Approach for Nonlinear Control Synthesis |
| title_full_unstemmed | A Convex Data-Driven Approach for Nonlinear Control Synthesis |
| title_short | A Convex Data-Driven Approach for Nonlinear Control Synthesis |
| title_sort | convex data driven approach for nonlinear control synthesis |
| topic | nonlinear control Koopman operator sum of squares |
| url | https://www.mdpi.com/2227-7390/9/19/2445 |
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