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Reachable Set Estimation and Safety Verification of Nonlinear Systems via Iterative Sums of Squares Programming

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Abstract

In this paper, the problems of forward reachable set estimation and safety verification of uncertain nonlinear systems with polynomial dynamics are addressed. First, an iterative sums of squares (SOS) programming approach is developed for reachable set estimation. It characterizes the over-approximations of the forward reachable sets by sub-level sets of time-varying Lyapunov-like functions that satisfy an invariance condition, and formulates the problem of searching for the Lyapunov-like functions as a bilinear SOS program, which can be solved via an iterative algorithm. To make the over-approximation tight, the proposed approach seeks to minimize the volume of the over-approximation set with a desired shape. Then, the reachable set estimation approach is extended for safety verification, via explicitly encoding the safety constraint such that the Lyapunov-like functions guarantee both reaching and avoidance. The efficiency of the presented method is illustrated by some numerical examples.

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Author information

Authors and Affiliations

  1. School of Information Science and Technology, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Wang Lin & Zuohua Ding

  2. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Wang Lin & Zhengfeng Yang

Authors
  1. Wang Lin
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  2. Zhengfeng Yang
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  3. Zuohua Ding
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Corresponding author

Correspondence to Zhengfeng Yang.

Additional information

This research was supported in part by the National Natural Science Foundation of China under Grant Nos. 12171159 and 61772203, and in part by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY20F020020.

This paper was recommended for publication by Editor FENG Ruyong.

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Lin, W., Yang, Z. & Ding, Z. Reachable Set Estimation and Safety Verification of Nonlinear Systems via Iterative Sums of Squares Programming. J Syst Sci Complex 35, 1154–1172 (2022). https://doi.org/10.1007/s11424-022-1121-9

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  • DOI: https://doi.org/10.1007/s11424-022-1121-9

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