切换系统多面体区域的生存性判别
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摘要
本文研究了切换系统关于多面体区域的生存性判别问题.考虑多面体由有限点集凸包来表示,利用非光滑分析理论,得到一个切换系统生存性的充分条件.该条件只需检验在极点处是否满足特定条件,而不需要对每个边界点进行验证.其优点在于将生存性的判别转化为向量内积与切锥的计算.这种生存性判别方法简便易行.最后通过实例阐述了算法的有效性.
This paper is devoted to the viability criterion for a polytope under a switched system. A sufficient viability criterion for a polytope, which is expressed by a convex hull of finite number of points, is proposed by using nonsmooth analysis. Based on this criterion, instead of all boundary points, just several vertices are needed to be verified whether satisfying some conditions. The advantage of the proposed methods is that determining the viability is transformed into computing vectors inner product and tangents. This method of determining the viability is easy to be complemented.Finally, two examples are listed to illustrate the effectiveness of main results of this paper.
This paper is devoted to the viability criterion for a polytope under a switched system. A sufficient viability criterion for a polytope, which is expressed by a convex hull of finite number of points, is proposed by using nonsmooth analysis. Based on this criterion, instead of all boundary points, just several vertices are needed to be verified whether satisfying some conditions. The advantage of the proposed methods is that determining the viability is transformed into computing vectors inner product and tangents. This method of determining the viability is easy to be complemented.Finally, two examples are listed to illustrate the effectiveness of main results of this paper.
引文
[1]AUBIN J P.Viability Theory[M].Berlin:Springer-Verlag,2011.
[2]GAO Yan.Nonsmooth Optimization[M].Beijing:Science Press,2008.(高岩.非光滑优化[M].北京:科学出版社,2008.)
[3]XIONG Jiandong,LIU Yongqi,SHEN Zhiping,et al.Stabilizing slow-switching strategy for continuous-time switched linear systems[J].Control and Decision,2016,31(5):797–804.(熊建栋,刘永奇,沈志萍,等.连续线性切换系统的镇定慢切换设计[J].控制与决策,2016,31(5):797–804.)
[4]WU J,SUN Z D.Observer-driven switching stabilization of switched linear systems[J].Automatica,2013,49(8):2556–2560.
[5]HU T S,MA L Q,LIN Z L.Stabilization of switched systems via composite quadratic functions[J].IEEE Transaction on Automatic Control,2008,53(11):2571–2585.
[6]GAO Yan.Determining viability of polytopic set for a linear control system[J].Control and Decision,2016,31(9):1720–1722.(高岩.线性控制系统多面体区域的生存性判别[J].控制与决策,2016,31(9):1720–1722.)
[7]GAO Yan.On viable set for a linear control system[J].Control and Decision,2007,22(7):833–835.(高岩.线性控制系统的生存域[J].控制与决策,2007,22(7):833–835.)
[8]ZHANG Xia,GAO Yan,XIA Zunquan.Unbounded polyhedral invariant sets for linear control systems[J].Control Theory&Applications,2011,28(6):874–880.(张霞,高岩,夏尊铨.线性控制系统的无界多面体不变集[J].控制理论与应用,2011,28(6):874–880.)
[9]CHEN Zheng,GAO Yan.Computation of approximate viable sets for linear systems[J].Control Theory&Applications,2013,30(11):1473–1478.(陈征,高岩.线性系统近似生存域的计算[J].控制理论与应用,2013,30(11):1473–1478.)
[10]CHEN Z,GAO Y.Determining the viable unbounded polyhedron under linear control systems[J].Asian Journal of Control,2014,16(5):1561–1567.
[11]GAO Y.Viability criteria for differential inclusions[J].Journal of Systems Science and Complexity,2011,24(5):825–834.
[12]KOMEI F,VERA R.Combinatorial face enumeration in convex polytopes[J].Computational Geometry,1994,4(4):191–198.
[2]GAO Yan.Nonsmooth Optimization[M].Beijing:Science Press,2008.(高岩.非光滑优化[M].北京:科学出版社,2008.)
[3]XIONG Jiandong,LIU Yongqi,SHEN Zhiping,et al.Stabilizing slow-switching strategy for continuous-time switched linear systems[J].Control and Decision,2016,31(5):797–804.(熊建栋,刘永奇,沈志萍,等.连续线性切换系统的镇定慢切换设计[J].控制与决策,2016,31(5):797–804.)
[4]WU J,SUN Z D.Observer-driven switching stabilization of switched linear systems[J].Automatica,2013,49(8):2556–2560.
[5]HU T S,MA L Q,LIN Z L.Stabilization of switched systems via composite quadratic functions[J].IEEE Transaction on Automatic Control,2008,53(11):2571–2585.
[6]GAO Yan.Determining viability of polytopic set for a linear control system[J].Control and Decision,2016,31(9):1720–1722.(高岩.线性控制系统多面体区域的生存性判别[J].控制与决策,2016,31(9):1720–1722.)
[7]GAO Yan.On viable set for a linear control system[J].Control and Decision,2007,22(7):833–835.(高岩.线性控制系统的生存域[J].控制与决策,2007,22(7):833–835.)
[8]ZHANG Xia,GAO Yan,XIA Zunquan.Unbounded polyhedral invariant sets for linear control systems[J].Control Theory&Applications,2011,28(6):874–880.(张霞,高岩,夏尊铨.线性控制系统的无界多面体不变集[J].控制理论与应用,2011,28(6):874–880.)
[9]CHEN Zheng,GAO Yan.Computation of approximate viable sets for linear systems[J].Control Theory&Applications,2013,30(11):1473–1478.(陈征,高岩.线性系统近似生存域的计算[J].控制理论与应用,2013,30(11):1473–1478.)
[10]CHEN Z,GAO Y.Determining the viable unbounded polyhedron under linear control systems[J].Asian Journal of Control,2014,16(5):1561–1567.
[11]GAO Y.Viability criteria for differential inclusions[J].Journal of Systems Science and Complexity,2011,24(5):825–834.
[12]KOMEI F,VERA R.Combinatorial face enumeration in convex polytopes[J].Computational Geometry,1994,4(4):191–198.