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Independent cycle time assignment for min-max systems

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Abstract

A variety of problems in digital circuits, computer networks, automated manufacturing plants, etc., can be modeled as min-max systems. The cycle time is an important performance metric of such systems. In this paper, we focus on the cycle time assignment of min-max systems which corresponds to the pole assignment problem in traditional linear control systems. For the minmax system with max-plus inputs and outputs, we show that the cycle time can be assigned disjointedly by a state feedback, if and only if the system is reachable. Furthermore, a necessary and sufficient condition for the cycle time to be assigned independently by a state feedback is given. The methods are constructive, and some numerical examples are given to illustrate how the methods work in practice.

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References

  1. F. L. Baccelli, G. Cohen, G. J. Olsder, J. P. Quadrat. Synchronization and Linearity, New York, USA: John Wiley and Sons, 1992.

    MATH  Google Scholar 

  2. C. G. Cassandras, S. Lafortune. Introduction to Discrete Event Systems, Boston, USA: Kluwer Academic Publishers, 1999.

    MATH  Google Scholar 

  3. G. Cohen, P. Moller, J. P. Quadrat, M. Viot. Linear system theory for discrete event systems. In Proceedings of the 23rd Conference on Decision and Control, IEEE, vol. 23, pp. 539–544, 1984.

    Article  Google Scholar 

  4. G. Cohen, P. Moller, J. P. Quadrat, M. Viot. Algebraic tools for the performance evaluation of discrete event systems. Proceedings of the IEEE, vol. 77, no. 1, pp. 39–58, 1989.

    Article  Google Scholar 

  5. R. A. Cuninghame-Green. Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, vol. 16, Berlin, Germay: Springer, 1979.

    MATH  Google Scholar 

  6. G. J. Olsder. Eigenvalues of dynamic max-min systems. Discrete Event Dynamic Systems, vol. 1, no. 2, pp. 177–207, 1991.

    Article  MATH  Google Scholar 

  7. J. Gunawardena. Cycle times and fixed points of min-max functions. In Proceedings of the 11th International Conference on Analysis and Optimization of Systems Discrete Event Systems, Lecture Notes in Control and Information Sciences, Springer, vol. 199, pp. 266–272, 1994.

    Article  Google Scholar 

  8. S. Gaubert, J. Gunawardena. The duality theorem for minmax functions. Comptes Rendus de l’Académie des Sciences — Series I — Mathematics, vol. 326, no. 1, pp. 43–48, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Cochet-Terrasson, S. Gaubert, J. Gunawardena. A constructive fixed point theorem for min-max functions. Dynamics and Stability of Systems, vol. 14, no. 4, pp. 407–433, 1999.

    MathSciNet  Google Scholar 

  10. J. van der Woude. Conditions for the structural existence of an eigenvalue of bipartite (min, max, +)-system. Theoretical Computer Science, vol. 293, no. 1, pp. 13–24, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. De Schutter, T. van den Boom. Model predictive control for max-min-plus systems. Discrete event systems: Analysis and Control, R. Boel, G. Stremersch, Eds., Boston, USA: Kluwer Academic, pp. 201–208, 2000.

    Google Scholar 

  12. I. Necoara, B. De Schutter, T. van den Boom, H. Hellendoorn. Model predictive control for uncertain maxmin-plus-scaling systems. International Journal of Control, vol. 81, no. 5, pp. 701–713, 2007.

    Article  Google Scholar 

  13. W. D. Chen. Cycle time assignment of nonlinear DEDS. Systems Science and Mathematical Science, vol. 13, no. 2, pp. 213–218, 2000. (in Chinese)

    MATH  Google Scholar 

  14. W. D. Chen. Cycle time assignment of nonlinear discrete event dynamic systems and coloring condensation graph. Control and Decision, vol. 18, no. 5, pp. 517–521, 2003. (in Chinese)

    MathSciNet  Google Scholar 

  15. Y. G. Tao, W. D. Chen. Cycle time assignment of min-max systems. International Journal of Control, vol. 76, no. 18, pp. 1790–1799, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. D. Chen, Y. G. Tao. Observability and reachability for nonlinear discrete event dynamic systems and colored graphs. Chinese Science Bulletin, vol. 45, no. 22, pp. 2457–2461, 2000.

    MathSciNet  Google Scholar 

  17. Y. Tao, G. P. Liu. Cycle time assignability and feedback design for min-max-plus systems. In Proceedings of IEEE Conference on Decision and Control, and European Control Conference, IEEE, Seville, Spain, pp. 7810–7815, 2005.

    Google Scholar 

  18. Y. Tao, G. P. Liu. State feedback stabilization and majorizing achievement of min-max-plus systems. IEEE Transactions on Automatic Control, vol. 50, no. 12, pp. 2027–2033, 2005.

    Article  MathSciNet  Google Scholar 

  19. Y. Zhu, Y. Tao, G. P. Liu. Output feedback stabilization for a class of nonlinear time-evolution systems. Nonlinear Analysis Series A: Theory, Methods & Applications, vol. 70, no. 7, pp. 2654–2664, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  20. W. D. Chen, X. D. Qi. The period assignment of the discrete event dynamic system. Science in China A, vol. 23, no. 1, pp. 1–7, 1993.

    Google Scholar 

  21. W. D. Chen, X. D. Qi. The Discrete Event Dynamic System, Max-algebra Methods, Beijing, PRC: Science Press, 1994.

    Google Scholar 

  22. J. Gunawardena. Min-max functions. Discrete Event Dynamic Systems, vol. 4, no. 4, pp. 377–407, 1994.

    Article  MATH  Google Scholar 

  23. K. Alimhan, H. Inaba. Output feedback control for a class of nonlinear systems. International Journal of Automation and Computing, vol. 3, no. 3, pp. 215–221, 2006.

    Article  Google Scholar 

  24. L. Yang, X. P. Guan, C. N. Long, X. Y. Luo. Feedback stabilization over wireless network using adaptive coded modulation. International Journal of Automation and Computing, vol. 5, no. 4, pp. 381–388, 2008.

    Article  Google Scholar 

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, PRC

    Wen-De Chen

  2. Department of Mathematics, Zhengzhou University, Henan, 450001, PRC

    Yue-Gang Tao

  3. Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, PRC

    Yue-Gang Tao

  4. Faculty of Computing, Engineering and Technology, Staffordshire University, Stafford, ST16 9DG, UK

    Hong-Nian Yu

Authors
  1. Wen-De Chen
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  2. Yue-Gang Tao
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  3. Hong-Nian Yu
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Corresponding author

Correspondence to Wen-De Chen.

Additional information

This work was supported by National Natural Science Foundation of China (No. 60774007) and the Royal Society of UK.

Wen-De Chen graduated from University of Science and Technology of PRC in 1964. He is a full professor of the Institute of Systems Science, which is now a part of the Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, PRC.

His research interests include discrete event systems and coding theory.

Yue-Gang Tao received the Ph.D. degree from the Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, PRC in 2002. He is currently a professor at the Department of Mathematics, Zhengzhou University, PRC, and working at Institute of Automation, the Chinese Academy of Sciences.

His research interest includes algebra, optimization and control, and their applications in discrete-event systems.

Hong-Nian Yu is currently a professor of computer science and the head of the Mobile Computing and Distributed Control Systems Research Group at Staffordshire University, UK. He is the general co-chair IEEE International Conference on Networking, Sensing and Control in 2009, general chair UK EPSRC Postgraduate Workshop on Human Adaptive Mechatronics in 2009, International Program Committee (IPC) co-chair of the United Kingdom Automatic Control Council (UKACC) in 2008, program chair of IEEE Conference on Networking, Sensing and Control in 2007, general chair of International conference on Software Knowledge Information Management and Applications in 2006, and is serving on various other conferences and academic societies.

His research interests include experience in neural networks, mobile computing, modelling, control of robot manipulators, and modelling, scheduling, planning, and simulations of large discrete event dynamic systems with applications to manufacturing systems, supply chains, transportation networks, and computer networks.

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Chen, WD., Tao, YG. & Yu, HN. Independent cycle time assignment for min-max systems. Int. J. Autom. Comput. 7, 254–260 (2010). https://doi.org/10.1007/s11633-010-0254-9

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  • DOI: https://doi.org/10.1007/s11633-010-0254-9

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