Reachability for Interval Max-Plus Linear Systems
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- 英文论文题名：Reachability for Interval Max-Plus Linear Systems
- 论文作者：Cailu Wang ; Yuegang Tao ; Peng Yang
- 英文论文作者：Cailu Wang ; Yuegang Tao ; Peng Yang ; School of Control Science and Engineering ; Hebei University of Technology
- 年：2017
- 作者机构：School of Control Science and Engineering,Hebei University of Technology;
- 英文论文关键词：Interval max-plus linear system ; Reachability ; Interval reachability equation ; Strong solvability
- 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（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：TP13
- 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：5
- 文件大小：243k
- 原文格式：D
- 会议级别：全国
摘要
This paper proposes the interval max-plus linear system and introduces the reachability to such systems. A state of an interval max-plus linear system is said to be reachable if it can always be attainable from the initial state, no mater how the state and input matrices vary in a certain range. It is pointed out that an interval max-plus linear system is reachable if and only if the interval reachability equation is strongly solvable. By using the extremal subsystems of the interval reachability equation,the reachability of an interval system can be verified through testing the reachability of a finite number of subsystems. Some examples are presented to illustrate the results.
This paper proposes the interval max-plus linear system and introduces the reachability to such systems. A state of an interval max-plus linear system is said to be reachable if it can always be attainable from the initial state, no mater how the state and input matrices vary in a certain range. It is pointed out that an interval max-plus linear system is reachable if and only if the interval reachability equation is strongly solvable. By using the extremal subsystems of the interval reachability equation,the reachability of an interval system can be verified through testing the reachability of a finite number of subsystems. Some examples are presented to illustrate the results.
This paper proposes the interval max-plus linear system and introduces the reachability to such systems. A state of an interval max-plus linear system is said to be reachable if it can always be attainable from the initial state, no mater how the state and input matrices vary in a certain range. It is pointed out that an interval max-plus linear system is reachable if and only if the interval reachability equation is strongly solvable. By using the extremal subsystems of the interval reachability equation,the reachability of an interval system can be verified through testing the reachability of a finite number of subsystems. Some examples are presented to illustrate the results.
引文
[1]R.A.Cuninghame-Green.Minimax Algebra.SpringerVerlag,Berlin,1979.
[2]F.Baccelli,G.Cohen,G.J.Olsder,and J.P.Quadrat.Synchronization and Linearity:An Algebra for Discrete Event System.John Wiley and Sons,New York,1992.
[3]B.Heidergott,G.J.Olsder,and J.van der Woude.Max-Plus at Work:Modeling and Analysis of Synchronized Systems.Princeton University Press,New Jersey,2006.
[4]G.Cohen,D.Dubois,J.P.Quadrat,and M.Viot.Linear system theory for discrete event systems.In Proceedings of the23rd IEEE Conference on Decision and Control,Las Vegas,1984.
[5]G.Cohen,D.Dubois,J.P.Quadrat,and M.Viot.A linearsystem-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing.IEEE Transactions on Automatic Control,30(3):210-220,1985.
[6]G.J.Olsder and C.Roos.Cramer and Cayley-Hamilton in the max algebra.Linear Algebra and its Applications,101:87-108,1988.
[7]W.Chen,X.Qi,and S.Deng.The eigen-problem and period analysis of the discrete event system.Systems Science and Mathematical Sciences,3(3):243-260,1990.
[8]S.Gaubert.Théorie des syst`emes linéaires dans les dioides.Ph D thesis,L’Ecole Nationale Supérieure des Mines de Paris,1992.
[9]B.De Schutter and B.De Moor.A note on the characteristic equation in the max-plus algebra.Linear Algebra and its Applications,261(1):237-250,1997.
[10]Q.Zhao.A remark on inseparability of min-max systems.IEEE Transactions on Automatic Control,49(6):967-970,2004.
[11]G.A.D.Lopes,B.Kersbergen,B.De Schutter,T.van den Boom,and R.Babuˇska.Synchronization of a class of cyclic discrete-event systems describing legged locomotion.Discrete Event Dynamic Systems,26(2):225-261,2016.
[12]G.J.Olsder and R.E.de Vries.On an analogy of minimal realizations in conventional and discrete-event dynamic systems.Lecture Notes in Control and Information Sciences,103:149-161,1988.
[13]B.De Schutter and B.De Moor.Minimal realization problem in the max-algebra is an extended linear complementarity problem.Systems and Control Letters,25(2):103-111,1995.
[14]S.Gaubert,P.Butkoviˇc,and R.A.Cuninghame-Green.Minimal(max,+)realization of convex sequences.SIAM Journal on Control and Optimization,36(1):137-147,1998.
[15]G.J.Olsder and B.De Schutter.The minimal realization problem in the max-plus algebra.In Open Problem in Mathematical Systems and Control Theory,pages 157-162,Springer Verlag,London,1999.
[16]B.De Schutter and T.van den Boom.Model predictive control for max-plus-linear discrete event systems.Automatica,37(7):1049-1056,2001.
[17]P.Butkoviˇc and M.Mac Caig.On the integer max-linear programming problem.Discrete Applied Mathematics,162:128-141,2014.
[18]A.N′u?nez,C.E.Cortés,D.S′aez,B.De Schutter,and M.Gendreau.Multiobjective model predictive control for dynamic pickup and delivery problems.Control Engineering Practice,32:73-86,2014.
[19]D.Adzkiya,B.De Schutter,and A.Abate.Computational techniques for reachability analysis of max-plus-linear systems.Automatica,53:293-302,2015.
[20]P.J.van Overloop,J.M.Maestre,A.D.Sadowska,E.F.Camacho,and B.De Schutter.Human-in-the-loop model predictive control of an irrigation canal.IEEE Control Systems Magazine,35(4):19-29,2015.
[21]L.Hardouin,Y.Shang,C.A.Maia,and B.Cottenceau.Observer-based controllers for max-plus linear systems.doi:10.1109/TAC.2016.2604562,2016.
[22]L.Wang and D.Zheng.On the reachability of linear discrete event dynamic systems.Applied Mathematics A Journal of Chinese Universities,5(2):292-301,1990.
[23]W.Chen,X.Qi,and D.Li.The reachability,observability and period merging assignment of DEDS.In Procccdings of IFAC Symposi Large Scale System:Theory and Applications,Beijing,1992.
[24]G.Cohen,S.Gaubert,and J.-P.Quadrat.Max-plus algebra and system theory:where we are and where to go now.Annual Reviews in Control,23:207-219,1999.
[25]M.J.Gazarik and E.W.Kamen.Reachability and observability of linear systems over max-plus.Kybernetika,35(1):2-22,1999.
[26]S.Gaubert and R.Katz.Reachability and invariance problems in max-plus algebra.Positive systems,294:15-22,2003.
[27]K.Cechl′arov′a and R.A.Cuninghame-Green.Interval systems of max-separable linear equations.Linear Algebra and its Applications,340:215-224,2002.
[28]H.Zhang,Y.Tao,and Z.Zhang.Strong solvalibility of interval max-plus systems and applications to optimal control.Systems&Control Letters,96:88-94,2016.
[29]H.Myˇskov′a.Interval systems of max-separable linear equations.Linear Algebra and its Applications,423:263-272,2005.
[30]H.Myˇskov′a.Control solvability of interval systems of maxseparable linear equations.Linear Algebra and its Applications,416:215-223,2006.
[31]H.Myˇskov′a.On the algorithm for testing T4 solvability of max-plus interval systems.Kybernetika,48(5):924-938,2012.
[32]H.Myˇskov′a.Interval max-plus systems of linear equations.Linear Algebra and its Applications,437(8):1992-2000,2012.
[33]G.L.Litvinov and A.N.Sobolevskiˇ?.Idempotent interval analysis and optimization problems.Reliable Computing,7(5):353-377,2001.
[2]F.Baccelli,G.Cohen,G.J.Olsder,and J.P.Quadrat.Synchronization and Linearity:An Algebra for Discrete Event System.John Wiley and Sons,New York,1992.
[3]B.Heidergott,G.J.Olsder,and J.van der Woude.Max-Plus at Work:Modeling and Analysis of Synchronized Systems.Princeton University Press,New Jersey,2006.
[4]G.Cohen,D.Dubois,J.P.Quadrat,and M.Viot.Linear system theory for discrete event systems.In Proceedings of the23rd IEEE Conference on Decision and Control,Las Vegas,1984.
[5]G.Cohen,D.Dubois,J.P.Quadrat,and M.Viot.A linearsystem-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing.IEEE Transactions on Automatic Control,30(3):210-220,1985.
[6]G.J.Olsder and C.Roos.Cramer and Cayley-Hamilton in the max algebra.Linear Algebra and its Applications,101:87-108,1988.
[7]W.Chen,X.Qi,and S.Deng.The eigen-problem and period analysis of the discrete event system.Systems Science and Mathematical Sciences,3(3):243-260,1990.
[8]S.Gaubert.Théorie des syst`emes linéaires dans les dioides.Ph D thesis,L’Ecole Nationale Supérieure des Mines de Paris,1992.
[9]B.De Schutter and B.De Moor.A note on the characteristic equation in the max-plus algebra.Linear Algebra and its Applications,261(1):237-250,1997.
[10]Q.Zhao.A remark on inseparability of min-max systems.IEEE Transactions on Automatic Control,49(6):967-970,2004.
[11]G.A.D.Lopes,B.Kersbergen,B.De Schutter,T.van den Boom,and R.Babuˇska.Synchronization of a class of cyclic discrete-event systems describing legged locomotion.Discrete Event Dynamic Systems,26(2):225-261,2016.
[12]G.J.Olsder and R.E.de Vries.On an analogy of minimal realizations in conventional and discrete-event dynamic systems.Lecture Notes in Control and Information Sciences,103:149-161,1988.
[13]B.De Schutter and B.De Moor.Minimal realization problem in the max-algebra is an extended linear complementarity problem.Systems and Control Letters,25(2):103-111,1995.
[14]S.Gaubert,P.Butkoviˇc,and R.A.Cuninghame-Green.Minimal(max,+)realization of convex sequences.SIAM Journal on Control and Optimization,36(1):137-147,1998.
[15]G.J.Olsder and B.De Schutter.The minimal realization problem in the max-plus algebra.In Open Problem in Mathematical Systems and Control Theory,pages 157-162,Springer Verlag,London,1999.
[16]B.De Schutter and T.van den Boom.Model predictive control for max-plus-linear discrete event systems.Automatica,37(7):1049-1056,2001.
[17]P.Butkoviˇc and M.Mac Caig.On the integer max-linear programming problem.Discrete Applied Mathematics,162:128-141,2014.
[18]A.N′u?nez,C.E.Cortés,D.S′aez,B.De Schutter,and M.Gendreau.Multiobjective model predictive control for dynamic pickup and delivery problems.Control Engineering Practice,32:73-86,2014.
[19]D.Adzkiya,B.De Schutter,and A.Abate.Computational techniques for reachability analysis of max-plus-linear systems.Automatica,53:293-302,2015.
[20]P.J.van Overloop,J.M.Maestre,A.D.Sadowska,E.F.Camacho,and B.De Schutter.Human-in-the-loop model predictive control of an irrigation canal.IEEE Control Systems Magazine,35(4):19-29,2015.
[21]L.Hardouin,Y.Shang,C.A.Maia,and B.Cottenceau.Observer-based controllers for max-plus linear systems.doi:10.1109/TAC.2016.2604562,2016.
[22]L.Wang and D.Zheng.On the reachability of linear discrete event dynamic systems.Applied Mathematics A Journal of Chinese Universities,5(2):292-301,1990.
[23]W.Chen,X.Qi,and D.Li.The reachability,observability and period merging assignment of DEDS.In Procccdings of IFAC Symposi Large Scale System:Theory and Applications,Beijing,1992.
[24]G.Cohen,S.Gaubert,and J.-P.Quadrat.Max-plus algebra and system theory:where we are and where to go now.Annual Reviews in Control,23:207-219,1999.
[25]M.J.Gazarik and E.W.Kamen.Reachability and observability of linear systems over max-plus.Kybernetika,35(1):2-22,1999.
[26]S.Gaubert and R.Katz.Reachability and invariance problems in max-plus algebra.Positive systems,294:15-22,2003.
[27]K.Cechl′arov′a and R.A.Cuninghame-Green.Interval systems of max-separable linear equations.Linear Algebra and its Applications,340:215-224,2002.
[28]H.Zhang,Y.Tao,and Z.Zhang.Strong solvalibility of interval max-plus systems and applications to optimal control.Systems&Control Letters,96:88-94,2016.
[29]H.Myˇskov′a.Interval systems of max-separable linear equations.Linear Algebra and its Applications,423:263-272,2005.
[30]H.Myˇskov′a.Control solvability of interval systems of maxseparable linear equations.Linear Algebra and its Applications,416:215-223,2006.
[31]H.Myˇskov′a.On the algorithm for testing T4 solvability of max-plus interval systems.Kybernetika,48(5):924-938,2012.
[32]H.Myˇskov′a.Interval max-plus systems of linear equations.Linear Algebra and its Applications,437(8):1992-2000,2012.
[33]G.L.Litvinov and A.N.Sobolevskiˇ?.Idempotent interval analysis and optimization problems.Reliable Computing,7(5):353-377,2001.
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