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Backward Reachability Analysis for Nonlinear Dynamical Systems via Pseudospectral Method

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  • Control Theory and Applications
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Abstract

In this paper, we propose a new approach to solving the backward reachability problem for nonlinear dynamical systems. Previously, this class of problems has been studied within frameworks of optimal control and zero-sum differential games, where a backward reachable set can be expressed as the zero sublevel set of the value function that can be characterized by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). In many cases, however, a high computational cost is incurred in numerically solving such HJB PDEs due to the curse of dimensionality. We use the pseudospectral method to convert the associated optimal control problem into nonlinear programs (NLPs). We then show that the zero sublevel set obtained by the optimal cost of the NLP is the corresponding backward reachable set. Note that our approach does not require solving complex HJB PDEs. Therefore, it can reduce computation time and handle high-dimensional dynamical systems, compared with the numerical software package developed by I. Mitchell, which has been used widely in the literature to obtain backward reachable sets by solving HJB equations. We provide several examples to validate the effectiveness of the proposed approach.

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Authors and Affiliations

  1. School of Electrical and Computer Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan, 44919, Korea

    Myoung Hoon Lee

  2. Department of Electrical Engineering, Hanyang University, Seoul, 04763, Korea

    Jun Moon

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  1. Myoung Hoon Lee
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Correspondence to Jun Moon.

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Recommended by Associate Editor Niket Kaisare under the direction of Editor Jessie (Ju H.) Park. This research was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT, South Korea (NRF-2017R1A5A1015311) and in part by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2020-0-01373, Artificial Intelligence Graduate School Program (Hanyang University)).

Myoung Hoon Lee received his B.S. degree in electrical and computer engineering from Kyungpook National University, Daegu, Korea, in 2016. He is currently working towards a Ph.D. degree in the School of Electrical and Computer Engineering, Ulsan National Institute of Science and Technology (UNIST), Korea. His research interests include decentralized optimal control, mean field games, and optimal control for nonlinear systems, and their applications.

Jun Moon is currently an Associate Professor in the Department of Electrical Engineering at Hanyang University, Seoul, Korea. He obtained his Ph.D. degree in electrical and computer engineering from the University of Illinois at Urbana-Champaign, USA, in 2015. He received B.S. degree in electrical and computer engineering and M.S. degree in electrical engineering from Hanyang University, Seoul, Korea, in 2006 and 2008, respectively. From Feb. 2008 to Jun. 2011, he was a Researcher at the Agency for Defense Development (ADD) in Korea. From Feb. 2016 to Feb. 2019, he was an Assistant Professor in the School of Electrical and Computer Engineering at Ulsan National Institute of Science and Technology (UNIST), Korea. From Mar. 2019 to Aug. 2020, he was an Associate Professor in the School of Electrical and Computer Engineering at University of Seoul, Korea. He is the recipient of the Fulbright Graduate Study Award 2011. His research interests include stochastic games, control and estimation, mean field games, distributed optimal control, networked control systems, and control of unmanned vehicles.

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Lee, M.H., Moon, J. Backward Reachability Analysis for Nonlinear Dynamical Systems via Pseudospectral Method. Int. J. Control Autom. Syst. 19, 575–586 (2021). https://doi.org/10.1007/s12555-019-0705-1

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