Computing Solutions to the Polynomial-Polynomial Regulator Problem
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910677519564800 |
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| author | Corbin, Nicholas A. Kramer, Boris |
| author_facet | Corbin, Nicholas A. Kramer, Boris |
| contents | We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas -- and a scalable, general purpose software implementation -- for computing the polynomial approximation to the value function that solves the optimal control problem. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of dimension $n=129$ of a partial differential equation, for which the PPR control reduces the control cost by approximately 75% compared to LQR for the initial condition of interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing Solutions to the Polynomial-Polynomial Regulator Problem Corbin, Nicholas A. Kramer, Boris Optimization and Control We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas -- and a scalable, general purpose software implementation -- for computing the polynomial approximation to the value function that solves the optimal control problem. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of dimension $n=129$ of a partial differential equation, for which the PPR control reduces the control cost by approximately 75% compared to LQR for the initial condition of interest. |
| title | Computing Solutions to the Polynomial-Polynomial Regulator Problem |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2410.22291 |