Koopman-Hopf Hamilton-Jacobi Reachability and Control
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915350246850560 |
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| author | Sharpless, Will Shinde, Nikhil Kim, Matthew Chow, Yat Tin Herbert, Sylvia |
| author_facet | Sharpless, Will Shinde, Nikhil Kim, Matthew Chow, Yat Tin Herbert, Sylvia |
| contents | The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional differential games, producing the set of initial states and corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable method, the Hopf formula avoids the curse of dimensionality that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf solution with Koopman theory for global linearization. By first lifting a nonlinear system to a linear space and then solving the Hopf formula, approximate reachable sets can be efficiently computed that are much more accurate than local linearizations. Furthermore, we construct a Koopman-Hopf disturbance-rejecting controller, and test its ability to drive a 10-dimensional nonlinear glycolysis model. We find that it significantly out-competes expectation-minimizing and game-theoretic model predictive controllers with the same Koopman linearization in the presence of bounded stochastic disturbance. In summary, we demonstrate a dimension-robust method to approximately solve HJR, allowing novel application to analyze and control high-dimensional, nonlinear systems with disturbance. An open-source toolbox in Julia is introduced for both Hopf and Koopman-Hopf reachability and control. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_11590 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Koopman-Hopf Hamilton-Jacobi Reachability and Control Sharpless, Will Shinde, Nikhil Kim, Matthew Chow, Yat Tin Herbert, Sylvia Systems and Control Optimization and Control The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional differential games, producing the set of initial states and corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable method, the Hopf formula avoids the curse of dimensionality that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf solution with Koopman theory for global linearization. By first lifting a nonlinear system to a linear space and then solving the Hopf formula, approximate reachable sets can be efficiently computed that are much more accurate than local linearizations. Furthermore, we construct a Koopman-Hopf disturbance-rejecting controller, and test its ability to drive a 10-dimensional nonlinear glycolysis model. We find that it significantly out-competes expectation-minimizing and game-theoretic model predictive controllers with the same Koopman linearization in the presence of bounded stochastic disturbance. In summary, we demonstrate a dimension-robust method to approximately solve HJR, allowing novel application to analyze and control high-dimensional, nonlinear systems with disturbance. An open-source toolbox in Julia is introduced for both Hopf and Koopman-Hopf reachability and control. |
| title | Koopman-Hopf Hamilton-Jacobi Reachability and Control |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2303.11590 |