控制系统生存核的逼近算法

运筹学学报 ›› 2013, Vol. 17 ›› Issue (4): 24-32.

控制系统生存核的逼近算法

陈征^1,*, 高岩²

1. 宁波工程学院理学院, 浙江宁波 315016; 2. 上海理工大学管理学院, 上海 200093

出版日期:2013-12-15 发布日期:2013-12-15
通讯作者: 陈征 E-mail:chenzhengnb@tom.com
基金资助:
国家自然科学基金 (Nos. 11171221, 40901241), 上海市科委与地方院校能力建设 (10550500800), 上海市一流学科 (XTKX2012)

Approximating viability kernel for control systems

CHEN Zheng^1,*, GAO Yan²

1. School of Sciences, Ningbo University of Technology, Ningbo 315016, Zhengjiang, China; 2. Management School, University of Shanghai for Science and Technology, Shanghai 200093, China

Online:2013-12-15 Published:2013-12-15

摘要/Abstract

摘要： 生存核的计算是控制理论中的一个重要研究方向. 给出了一种计算一般离散控制系统生存核的新算法. 基于机器学习的方法，给出了逼近生存核的算法. 并在一定条件下，证明了此算法的收敛性. 此算法在一定程度上避免了计算量随控制空间的维数增长而指数增长的问题. 最后，给出具体的实际例子来说明算法的有效性.

关键词: 离散系统, 生存核, 机器学习

Abstract: The computation of the viability kernel is an important topic in control theory community. In this paper, we propose a new algorithm that computes the viability kernel of a discrete-time system. Based on the theory of machine learning, the algorithm of approximating viability kernel is presented. We give some conditions that guarantee the convergence of the approximations towards the actual viable kernel. This method avoids the exponential growth of the computing time with the dimension of the control space. Finally, examples are given to illustrate this result.

Key words: discrete-time systems, viability kernel, machine learning

中图分类号:

O231.2

陈征, 高岩. 控制系统生存核的逼近算法[J]. 运筹学学报, 2013, 17(4): 24-32.

CHEN Zheng, GAO Yan. Approximating viability kernel for control systems[J]. Operations Research Transactions, 2013, 17(4): 24-32.

参考文献

Aubin J P. Viability theory [M]. Boston: Birkhauser, 1991.

Gao Y, Lygeros J, Quincampoix M, et al. On the control uncertain impluse system: approximate stabilization and controlled invariance [J]. International Journal of Control, 2004, 77: 1393-1407.

Limon D, Alamo T, Camacho E F. Enlarging the domain of attraction of MPC controllers [J]. Automation, 2005, 41: 629-635.

Aubin J P，Bayen A，Patrick S P. Viability theory new directions [M]. Princeton: Springer-Verlag, 2011.

Aubin J P，Bayen A，Patrick S P. Viability theory new directions [M]. Princeton: Springer-Verlag, 2011.

陈征, 高岩. 人口控制的混杂模型与应用 [J]. 系统管理学报, 2009, 18: 107-110.

Bene C, Doyen L, Gabay D. A viability analysis for a bio-economic model [J]. Ecological economics, 2001, 36: 385-396.

Aubin J P，Bayen A，Patrick S P. Viability theory new directions [M]. Princeton: Springer-Verlag, 2011.

Aubin J P，Bayen A，Patrick S P. Viability theory new directions [M]. Princeton: Springer-Verlag, 2011.

高岩. 一类非线性控制系统关于非光滑区域生存性的判别 [J]. 控制与决策, 2006, 21: 923-925.

蒋卫华, 黄琳, 楚天广. 离散非线性时变凸多面体系统族的鲁棒正不变集 [J]. 自动化学报, 2001, 27: 631-636.

Rouge C, Mathias J D, Deffuant G. Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication [J]. Ecological Indicators, 2013, 29: 420-433.

Bernard C, Martin S. Building strategies to ensure language coexistence in presence of bilingualism [J]. Applied Mathematics and Computation, 2012, 218: 8825-8841.

Sicard M, Perrot N, Reuillon R, et al. A viability approach to control food processes: Application to a Camembert cheese ripening process [J]. Food Control, 2012, 23: 312-319.

Botkin N D, Ryazantseva E A. Structure of viability kernels for some linear differential games [J]. Journal of optimization theory and applications, 2010, 147: 42-57.

Maggiore M, Rawn B, Lehn P. Invariance kernels of single-input planar nonlinear systems [J]. SIAM Journal on Control and Optimization, 2012, 50: 1012-1037.

Bonneuil N. Computing the viability kernel in large state dimension [J]. Journal of Mathematical Analysis and Applications, 2006, 323: 1444-1454.

Doyena L, Martinetb V. Maximin, viability and sustainability [J]. Journal of Economic Dynamics and Control, 2012, 36: 1414-1430.

Rakovic S V, Kerrigan E C, Mayne D Q, et al. Reachability analysis of discrete-time systems with disturbances [J]. IEEE Transactions on Automatic Control, 2006, 51: 546-560.

高岩. 非光滑优化 [M]. 北京: 科学出版社, 2008.

Hastie T, Tibshirani R, Friedman J. The elements of statistical learning: data ming, Inference, and prediction [M]. Princeton: Springer-Verlag, 2009.

Jin W, Zhang J Q, Zhang X. Face recognition method based on support vector machine and particle swarm optimization [J]. Expert Systems with Applications, 2011, 38: 4390-4393.