Continuity of the Polytope Generated by a Set of Matrices
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- 英文论文题名：Continuity of the Polytope Generated by a Set of Matrices
- 论文作者：Lingxin Meng ; Cong Lin ; Xiushan Cai
- 英文论文作者：Lingxin Meng ; Cong Lin ; Xiushan Cai ; College of Mathematics ; Physics and Information Engineering Zhejaing Normal University ; Xingzhi College ; Zhejiang Normal University
- 年：2017
- 作者机构：College of Mathematics, Physics and Information Engineering Zhejiang Normal University;Xingzhi College, Zhejiang Normal University;
- 英文论文关键词：Set-valued mappings ; polytopes ; matrices ; compactness ; continuity
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：英文
- 分类号：O151.21
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：196k
- 原文格式：D
- 会议级别：全国
摘要
Continuity of the polytope generated by a set of matrices is dealt with in this paper. We have defined a norm for the polytope, and verified when these matrices belong to a compact set in R~(n×n) the norm is bounded and the polytope is compact.Moreover, we have verified if the polytope is treated as a set-valued mapping from R~n to R~n, it is continuous and with convex and compact values.
Continuity of the polytope generated by a set of matrices is dealt with in this paper. We have defined a norm for the polytope, and verified when these matrices belong to a compact set in R~(n×n) the norm is bounded and the polytope is compact.Moreover, we have verified if the polytope is treated as a set-valued mapping from R~n to R~n, it is continuous and with convex and compact values.
Continuity of the polytope generated by a set of matrices is dealt with in this paper. We have defined a norm for the polytope, and verified when these matrices belong to a compact set in R~(n×n) the norm is bounded and the polytope is compact.Moreover, we have verified if the polytope is treated as a set-valued mapping from R~n to R~n, it is continuous and with convex and compact values.
引文
[1]J.P.Aubin,H.Frankowska,and C.Olech,Controllability of convex processes,SIAM Journal on Control and Optimization,24(6):1192-1211,1986.
[2]R.Goebel,Linear systems with conical constraints and convex Lyapunov functions,in Proceedings of 53th IEEE Conference on Decision and Control,2014:6329-6334.
[3]G.V.Smirnov,Introduction to the theory of differential inclusions.Graduate Studies in Mathematics,Philadelphia:SIAM,2002.
[4]Z.Z.Han,X.S.Cai,and J.Huang,The theory for control systems described by differential inclusions.Shanghai:Shanghai Jiaotong University press,2013.
[5]R.H.Gielen,S.Olaru,M.Lazar,W.P.M.H.Heemels,N.van de Wouw,and S.I.Niculescu,On polytopic inclusions as a modeling framework for systems with time-varying delays,Automatica,2010,46(3):615-619,2010.
[6]A.Karimi,H.Khatibi,and R.Longchamp,Robust control of polytopic systems by convex optimization,Automatica,43(8):1395-1402,2007.
[7]T.Hu,Z.Lin,Properties of the composite quadratic Lyapunov fuctions,IEEE Transactions on Automatic Control,49(7):1162-1167,2004.
[8]T.Hu,A.Teel,and L.Zaccarian.Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov function,IEEE Transactions on Automatic Control,51(11):1770-1780,2006.
[9]T.Hu,Nonlinear control design for linear differential inclusions via convex hull of quadratics.Automatica,43:685-692,2007.
[10]X.Cai,L.Liu,and W.Zhang,Saturated control design for linear differential inclusions subject to disturbance,Nonlinear Dynamics,58(3):487-496,2009.
[11]X.Cai,J.Huang,and L.Liu,Stability analysis of linear timedelay differential inclusion systems subject to input saturation,IET Control and Applications,4(11):2592-2602,2010.
[12]X.Cai,L.Liu,J.Huang,and W.Zhang,Globally stabilizable for a class of feedback linearizable differential inclusion,IET Control and Applications,5(14):1586-1596,2011.
[13]N.Vlassis,R.M.Jungers,Polytopic uncertainty for linear systems:New and old complexity results.Systems and Control Letters,67:9-13,2014.
[14]J.P.Aubin,H.Frankowska,Set-valued Analysis.Berlin:Birkhauser,1990.
1The symbol A(1s)1 is not regular.The formal notation is A(ks)1 .But it will lead to a complicated notation for the subseries of A(ks)2 .Hence we adopt a somewhat simple notation.
[2]R.Goebel,Linear systems with conical constraints and convex Lyapunov functions,in Proceedings of 53th IEEE Conference on Decision and Control,2014:6329-6334.
[3]G.V.Smirnov,Introduction to the theory of differential inclusions.Graduate Studies in Mathematics,Philadelphia:SIAM,2002.
[4]Z.Z.Han,X.S.Cai,and J.Huang,The theory for control systems described by differential inclusions.Shanghai:Shanghai Jiaotong University press,2013.
[5]R.H.Gielen,S.Olaru,M.Lazar,W.P.M.H.Heemels,N.van de Wouw,and S.I.Niculescu,On polytopic inclusions as a modeling framework for systems with time-varying delays,Automatica,2010,46(3):615-619,2010.
[6]A.Karimi,H.Khatibi,and R.Longchamp,Robust control of polytopic systems by convex optimization,Automatica,43(8):1395-1402,2007.
[7]T.Hu,Z.Lin,Properties of the composite quadratic Lyapunov fuctions,IEEE Transactions on Automatic Control,49(7):1162-1167,2004.
[8]T.Hu,A.Teel,and L.Zaccarian.Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov function,IEEE Transactions on Automatic Control,51(11):1770-1780,2006.
[9]T.Hu,Nonlinear control design for linear differential inclusions via convex hull of quadratics.Automatica,43:685-692,2007.
[10]X.Cai,L.Liu,and W.Zhang,Saturated control design for linear differential inclusions subject to disturbance,Nonlinear Dynamics,58(3):487-496,2009.
[11]X.Cai,J.Huang,and L.Liu,Stability analysis of linear timedelay differential inclusion systems subject to input saturation,IET Control and Applications,4(11):2592-2602,2010.
[12]X.Cai,L.Liu,J.Huang,and W.Zhang,Globally stabilizable for a class of feedback linearizable differential inclusion,IET Control and Applications,5(14):1586-1596,2011.
[13]N.Vlassis,R.M.Jungers,Polytopic uncertainty for linear systems:New and old complexity results.Systems and Control Letters,67:9-13,2014.
[14]J.P.Aubin,H.Frankowska,Set-valued Analysis.Berlin:Birkhauser,1990.
1The symbol A(1s)1 is not regular.The formal notation is A(ks)1 .But it will lead to a complicated notation for the subseries of A(ks)2 .Hence we adopt a somewhat simple notation.