| 引用本文: | 邵立珍,张扬帆,胡广大.一种求解控制系统可达集的网格投影法[J].哈尔滨工业大学学报,2018,50(10):88.DOI:10.11918/j.issn.0367-6234.201710120 |
| SHAO Lizhen,ZHANG Yangfan,HU Guangda.A grid projection method for reachable sets of control systems[J].Journal of Harbin Institute of Technology,2018,50(10):88.DOI:10.11918/j.issn.0367-6234.201710120 |
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| 摘要: |
| 控制系统在某一时刻的前向可达集是指从初始状态出发，在该时刻能够达到的状态的集合.为了求解非线性控制系统的前向可达集，首先通过常微分方程数值方法对连续控制系统进行离散，转化为离散系统；其次由于连续控制系统的可达集可用其相应的离散系统的可达集来近似，针对离散控制系统，提出了基于最优化技术的网格投影法近似描述可达集，该方法布置了均匀分布的网格点，并将网格点向可达集边界投影，每个投影问题都对应一个最优化问题，通过求解这些优化问题得到可达集的近似描述；进一步地，理论分析证明了该方法布置的网格点间隔越小得到的可达集近似误差越小；最后通过数值仿真验证了该方法的有效性，并将其与文献中已有的DFOG(Distance fields on grids)方法进行对比.研究表明：网格投影法可以有效地处理具有非凸可达集的控制系统；相较于DFOG方法，该方法能够得到分布均匀的边界点，且求解的优化问题的数量少，计算时间短.
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| 关键词: 控制系统 非凸可达集 数值方法 网格投影 优化 |
| DOI：10.11918/j.issn.0367-6234.201710120 |
| 分类号:TP273 |
| 文献标识码:A |
| 基金项目:国家自然科学基金 (11371053); 北京市自然科学基金 (4152034); 中央高校基本科研业务费专项资金(FRF-BD-16-005A) |
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| A grid projection method for reachable sets of control systems |
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SHAO Lizhen1,ZHANG Yangfan1,HU Guangda2
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(1. School of Automation,University of Science and Technology Beijing, Beijing 100083, China; 2. School of Science, Shanghai University, Shanghai 200444, China)
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| Abstract: |
| The forward reachable set of a control system at a certain time is the set of states that can be reached at this time from the initial state. In order to solve the forward reachable set of a nonlinear control system, numerical methods for ordinary differential equations was used to discretize the continuous control system. Since the reachable set of a continuous system can be approximated by its discrete counterpart, a grid projection method based on optimization technology was proposed to approximate the reachable set. In this method, uniformly distributed grid points were placed and projected to the boundary of reachable set for a discrete control; each projection problem corresponded to an optimization problem; by solving these problems the approximated description of the reachable set was obtained. Theoretical analysis proves that the smaller the grid interval, the smaller the approximation error. Finally, numerical results verified the effectiveness of the proposed method, which were compared with those of the DFOG(distance fields on grids) method in literature. The research shows that the grid projection method can effectively handle control systems with non-convex reachable sets. Compared with the DFOG method, the method gets uniform distributed boundary points, solves less optimization problems and uses less computation time.
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| Key words: control system non-convex reachable set numerical method grid projection optimization |
