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Algebraic state space approach to model and control combined automata

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Abstract

A new modeling tool, algebraic state space approach to logical dynamic systems, which is developed recently based on the theory of semi-tensor product of matrices (STP), is applied to the automata field. Using the STP, this paper investigates the modeling and controlling problems of combined automata constructed in the ways of parallel, serial and feedback. By representing the states, input and output symbols in vector forms, the transition and output functions are expressed as algebraic equations of the states and inputs. Based on such algebraic descriptions, the control problems of combined automata, including output control and state control, are considered, and two necessary and sufficient conditions are presented for the controllability, by which two algorithms are established to find out all the control strings that make a combined automaton go to a target state or produce a desired output. The results are quite different from existing methods and provide a new angle and means to understand and analyze the dynamics of combined automata.

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References

  1. Pighizzini G, Pisoni A. Limited automata and context-free languages. Fundamenta Informaticae, 2015, 136(1): 157–176

    MathSciNet  MATH  Google Scholar 

  2. Schluter N. Restarting automata with auxiliary symbols restricted by look ahead size. International Journal of Computer Mathematics, 2015, 92(5): 908–938

    Article  MathSciNet  MATH  Google Scholar 

  3. Gécseg F. Automata, Languages and Programming. New York: Springer, 1974

    MATH  Google Scholar 

  4. Henzinger T A. The theory of hybrid automata. In: Proceedings of Symposium on Logic in Computer Science. 1996, 278–292

    Chapter  Google Scholar 

  5. Kohavi Z, Jhan K. Switching and Finite Automata Theory. New York: Cambridge University Press, 2010

    Google Scholar 

  6. Holcombe M, Holcombe L. Algebraic Automata Theory. Cambridge: Cambridge University Press, 2004

    MATH  Google Scholar 

  7. Sokolova A, Devinke P. Validation of Stochastic Systems. New York: Springer, 2004

    Google Scholar 

  8. Olderog R, Swaminathan M. Formal Modeling and Analysis of Timed Systems. New York: Springer, 2010

    MATH  Google Scholar 

  9. Komenda J, Lahaye S, Boimond L. Synchronous composition of interval weighted automata. In: Proceedings of the International Workshop on Discrete Event Systems. 2010, 318–323

    Google Scholar 

  10. Dogruel M, Ozguner U. Controllability, reachability, stabilizability and state reduction in automata. In: Proceedings of the IEEE International Symposium on Intelligent Control. 1992, 192–197

    Google Scholar 

  11. Nerode A, Kohn W. Automata, Topologies, Controllability, Observability. New York: Springer, 1993

    Google Scholar 

  12. Sakarovitch J. Elements of Automata Theory. Cambridge: Cambridge University Press, 2009

    Book  MATH  Google Scholar 

  13. Charatonik W, Chorowska A. Parameterized complexity of basic decision problems for tree automata. International Journal of Computer Mathematics, 2013, 90(6): 348–356

    Article  MATH  Google Scholar 

  14. Czech L. On dynamics of automata with a stack. International Journal of Computer Mathematics, 2015, 92(3): 486–497

    Article  MathSciNet  MATH  Google Scholar 

  15. Cheng D Z. Semi-tensor product of matrices and its application toMorgen’s problem. Science in China Series F: Information Sciences, 2001, 44(3): 195–212

    MathSciNet  Google Scholar 

  16. Cheng D Z, Qi H S, Zhao Y. An Introduction to Semi-tensor Product of Matrices and Its Applications. Singapore: World Scientific Publishing, 2012

    Book  MATH  Google Scholar 

  17. Qi H S. On shift register via semi-tensor product approach. In: Proceedings of the 32nd Chinese Control Conference. 2013, 208–212

    Google Scholar 

  18. Cheng D Z. Disturbance decoupling of boolean control networks. IEEE Transactions on Automatic Control, 2011, 56(1): 2–10

    Article  MathSciNet  MATH  Google Scholar 

  19. Li F, Sun J. Controllability and optimal control of a temporal Boolean network. Neural Networks, 2012, 34(12): 10–17

    Article  MATH  Google Scholar 

  20. Wang Y Z, Zhang C H, Liu Z B. A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems. Automatica, 2012, 48(7): 1227–1236

    Article  MathSciNet  MATH  Google Scholar 

  21. Feng J E, Lv H L, Cheng D Z. Multiple fuzzy relation and its application to coupled fuzzy control. Asian Journal of Control, 2013, 15(5): 1313–1324

    MathSciNet  MATH  Google Scholar 

  22. Li H T, Wang Y Z. Boolean derivative calculation with application to fault detection of combinational circuits via the semi-tensor product method. Automatica, 2012, 48(4): 688–693

    Article  MathSciNet  MATH  Google Scholar 

  23. Yan Y Y, Chen Z Q, Liu Z X. Semi-tensor product of matrices approach to reachability of finite automata with application to language recognition. Frontiers of Computer Science, 2014, 8(6): 948–957

    Article  MathSciNet  Google Scholar 

  24. Xu X R, Hong Y G. Observability analysis and observer design for finite automata via matrix approach. Control Theory & Applications, 2013, 7(12): 1609–1615

    Article  MathSciNet  Google Scholar 

  25. Xu X R, Hong Y G. Matrix expression and reachability analysis of finite automata. Journal of Control Theory and Applications, 2012, 10(2): 210–215

    Article  MathSciNet  Google Scholar 

  26. Yan Y Y, Chen Z Q, Liu Z X. Semi-tensor product approach to controlliability and stabilizability of finite automata. Journal of Systems Engineering and Electronics, 2015, 26(1): 134–141

    Article  Google Scholar 

  27. Cheng D Z, Li Z Q, Qiao Y, Qi H S, Liu J B. Stability of switched polynomial systems. Journal of Systems Science and Complexity, 2008, 21(3): 362–377

    Article  MathSciNet  MATH  Google Scholar 

  28. Yan Y Y, Chen Z Q, Yue J M, Fu Z M. STP approach to model controlled automata with application to reachability analysis of discrete event dynamic systems. Asian Journal of Control, 2016, doi: 10.1002/asjc.1294

    Google Scholar 

  29. Zhang G Q. Automata, Boolean matrices, and ultimate periodicity. Information and Computatioin, 1999, 152(1): 138–154

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang Y Q, Xu X R, Hong Y G. Bi-decomposition analysis and algorithm of automata based on semi-tensor product. In: Proceedings of the 31st Chinese Control Conference. 2012, 2151–2156

    Google Scholar 

  31. He C. Automata Theory and Its Applications. Beijing: Science Press, 1990

    Google Scholar 

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Acknowledgements

This work was supported by Key Scientific Research Program of the Higher Education Institutions of He’nan Educational Committee (15A416005), the 2015 Science Foundation of Henan University of Science and Technology for Youths (2015QN016), and the National Natural Science Foundation of China (Grant No. 61573199, 61473115, and U1404610). The authors would like to express their thanks to Prof. Y G Hong for his helpful suggestions.

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Authors and Affiliations

  1. College of Information Engineering, Henan University of Science and Technology, Luoyang, 471023, China

    Yongyi Yan

  2. College of Computer and Control Engineering, Nankai University, Tianjin, 300071, China

    Yongyi Yan & Zengqiang Chen

  3. College of Agricultural Engineering, Henan University of Science and Technology, Luoyang, 471003, China

    Jumei Yue

Authors
  1. Yongyi Yan
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  2. Zengqiang Chen
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  3. Jumei Yue
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Corresponding author

Correspondence to Zengqiang Chen.

Additional information

Yongyi Yan received his BS and MS in mathematics from Luoyang Normal University, China and Xidian University, China in 2005 and 2008, respectively, and is currently pursuing his PhD in control theory and engineering at Nankai University, China. His current research interests are in the fields of modeling and optimization of complex systems, fuzzy control, and intelligent predictive control.

Zengqiang Chen received his BS in mathematics, and his MS and DE in control theory and engeering from Nankai University (NKU), China in 1987, 1990, and 1997, respectively. He is currently a professor of control theory and engineering of NKU, and deputy director of the Institute of Robotics and Information Automation. His current research interests include intelligent predictive control, chaotic systems and complex dynamic networks, and multi-agent system control.

Jumei Yue received her PhD from Nankai University, China in 2013. She is currently a lecturer of College of Agricultural Engineering, Henan University of Science and Technology, China. Her research interest is fuzzy logic systems.

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Yan, Y., Chen, Z. & Yue, J. Algebraic state space approach to model and control combined automata. Front. Comput. Sci. 11, 874–886 (2017). https://doi.org/10.1007/s11704-016-5128-z

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