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Geometrically minimal realizations of Boolean controlled systems

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Abstract

For the theory of realization over semirings, consideration was given to a new problem of generation, that of description of various systems with a given external behavior. A notion was introduced of geometric reduction and geometrically minimal realization based on transformations of arbitrary realizations of the given pulse response into each other, rather than on introducing some invariant of the dimensionality type. This notion was considered for the case of systems over the Boolean semiring: the geometrical reductions were classified, and various properties of the geometrically minimal realizations were studied. Some examples were discussed, as well as one class of non-geometric reductions which enabled us to explain the structure of some sets of geometrically minimal realizations.

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References

  1. Gilbert, E., Controllability and Observability in Multivariable Control Systems, SIAM J. Control, 1963, vol. 1, pp. 128–151.

    MATH  Google Scholar 

  2. Arbib, M.A. and Manes, E.G., Foundations of System Theory: Decomposable Systems, Automatica, 1974, vol. 10, pp. 285–302.

    Article  MATH  MathSciNet  Google Scholar 

  3. Anderson, B.D.O., Arbib, M.A., and Manes, E.G., Foundations of System Theory: Finitary and Infinitary Conditions. Lect. Notes in Economics and Math. Syst, 1976, vol. 115.

  4. Eilenberg, S., Automata, Languages and Machines, New York: Academic, 1974.

    MATH  Google Scholar 

  5. Baccelli, F., Cohen, G., Olsder, G., et al., Synchronization and Linearity, New York: Wiley, 1992.

    MATH  Google Scholar 

  6. Benvenuti, L. and Farina, G., A Tutorial on the Positive Realization Problem, IEEE Trans. Automat. Control, 2004, vol. 49, pp. 651–664.

    Article  MathSciNet  Google Scholar 

  7. Cohen, G., Mollier, P., Quadrat, J.-P., et al., Algebraic Tools for the Performance Evaluation of Discrete Event Systems, Proc. IEEE, 1989, vol. 77, pp. 39–85.

    Article  Google Scholar 

  8. Cohen, G., Gaubert, S., and Quadrat, J.P., Max-Plus Algebra and System Theory: Where We Are and Where to Go Now, Annu. Rev. Control, 1999, vol. 23, pp. 207–219.

    Article  Google Scholar 

  9. Gaubert, S., On Rational Series in one Variable over Certain Dioids, in INRIA Rapport de recherche, Jan. 1994, no. 2162.

    Google Scholar 

  10. Gaubert, S., Butkovic, P., and Cuninghame-Green, R., Minimal (max,+)-realization of Convex Sequences, SIAM J. Control Optim., 1998, vol. 36, pp. 137–147.

    Article  MATH  MathSciNet  Google Scholar 

  11. van den Hof, J., Realization of Positive Linear Systems, Linear Algebra Appl., 1997, vol. 256, pp. 287–308.

    Article  MATH  MathSciNet  Google Scholar 

  12. Maeda, H. and Kodama, S., Positive Realization of Difference Equation, IEEE Trans. Circuits Syst., 1981, vol. CAS-28, pp. 39–47.

    Article  MathSciNet  Google Scholar 

  13. Olsder, G.J. and De Schutter, B., The Minimal Realization Problem in the Max-Plus Algebra, in Open Problems in Mathematical Systems and Control Theory, Ch. 32, Blondel, V.D., Sontag, E.D., Vidyasagar, M., and Willems, J.C., Eds., London: Springer, 1999, pp. 157–162.

    Chapter  Google Scholar 

  14. Falb, P., Methods of Algebraic Geometry in Control Theory, pt 1: Scalar Linear Systems and Affine Algebraic Geometry, Basel: Birkhauser, 1990.

    Google Scholar 

  15. Falb, P., Methods of Algebraic Geometry in Control Theory, pt. 2: Multivariable Linear Systems and Projective Algebraic Geometry, Basel: Birkhauser, 1990.

    Google Scholar 

  16. Tannenbaum, A., Invariance and Systems Theory: Algebraic and Geometric Aspects. Lect. Notes in Mathematics, vol. 845, New York: Springer, 1981.

    Google Scholar 

  17. Osetinskii, N.I., Methods of Invariant Analysis for Linear Control Systems, J. Math. Sci., 1998, vol. 88, pp. 587–657.

    Article  MathSciNet  Google Scholar 

  18. Vainstein, F.S. and Osetinskii, N.I., Classification of Systems under State and Output Transformations, J. Math. Sci., 1998, vol. 88, pp. 659–674.

    Article  MATH  MathSciNet  Google Scholar 

  19. Rouchaleau, Y. and Sontag, E., On the Existence of Minimal Realizations of Linear Dynamical Systems over Noetherian Integral Domains, J. Comput. Syst. Sci., 1979, vol. 18, pp. 65–75.

    Article  MATH  MathSciNet  Google Scholar 

  20. Brewer, J.W., Bunce, J.W., and van Vleck, F.S., Linear Systems over Commutative Rings, New York: Marcel Dekker, 1986.

    MATH  Google Scholar 

  21. Blondel, V., Gaubert, S., and Portier, N., The Set of Realizations of a Max-Plus Linear Sequence is Semi-Polyhedral, J. Comput. Syst. Sci., 2011, vol. 77, pp. 820–833.

    Article  MATH  MathSciNet  Google Scholar 

  22. Le Bruyn, L. and Reineke, M., Canonical Systems and Non-Commutative Geometry, arXiv:math/0303304v2. http://arxiv.org/abs/math/0303304.

  23. Sontag, E., The Lattice of Minimal Realizations of Response Maps over Rings, Math. Syst. Theory, 1977, vol. 11, pp. 169–175.

    Article  MATH  MathSciNet  Google Scholar 

  24. Silverman, L., Representation and Realization of Time-Variable Linear Systems, Tech. Rep. Dept. Electrical Engin., Columbia Univ., 1966, vol. 94.

  25. Silverman, L., Realization of Linear Dynamical Systems, IEEE Trans. Automat. Control, 1971, vol. 16, pp. 554–567.

    Article  Google Scholar 

  26. Gaubert, S., Theorie des systemes lineaires dans les dioides, PhD Dissertation, Ecole Normale Superieure des Mines des Paris, 1992.

    Google Scholar 

  27. De Schutter, B., Max-Algebraic System Theory for Discrete-Event Systems, PhD Dissertation, Faculty of Applied Sciences, K.U. Leuven, 1996.

    Google Scholar 

  28. De Schutter, B., Blondel, V., and de Moor, B., On the Complexity of the Boolean Minimal Realization Problem in the Max-Plus Algebra, Technical Report 98-41, Department of Electrical Engineering, K.U. Leuven.

  29. De Schutter, B., Blondel, V., de Vries, R., et al., On the Boolean Minimal Realization Problem in the Max-Plus Algebra, Syst. Control Lett., 1998, vol. 35, pp. 69–78.

    Article  MATH  Google Scholar 

  30. Golan, J.S., Semirings and Their Applications, Dordrecht: Kluwer, 1999.

    Book  MATH  Google Scholar 

  31. Mac Lane, S., Categories for the Working Mathematician, New York: Springer, 1998, 2nd ed.

    MATH  Google Scholar 

  32. Kalman, R., Falb, P., and Arbib, M., Topics in Mathematical System Theory, New York: McGraw-Hill, 1969.

    MATH  Google Scholar 

  33. Vasil’ev, O.O., Osetinskii, N.I., and Vainshtein, F.S., Linear Stationary Controlled Systems over Boolean Semiring: Geometrical Properties and the Problem of Isomorphism, Differ. Uravn., 2009, vol. 45, no. 12, pp. 1748–1755.

    MathSciNet  Google Scholar 

  34. Luce, R.D., A Note in Boolean Matrix Theory, Proc. AMS, 1952, vol. 3, pp. 382–388.

    Article  MATH  MathSciNet  Google Scholar 

  35. Kim, K.H., Boolean Matrix Theory and Applications, New York: Marcel Dekker, 1982.

    MATH  Google Scholar 

  36. Lang, S., Algebra, Boston: Addison-Wesley, 1965.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    O. O. Vasil’ev

  2. Gubkin State University of Oil and Gas, Moscow, Russia

    O. O. Vasil’ev

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  1. O. O. Vasil’ev
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Correspondence to O. O. Vasil’ev.

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Original Russian Text © O.O. Vasil’ev, 2014, published in Avtomatika i Telemekhanika, 2014, No. 3, pp. 114–143.

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Vasil’ev, O.O. Geometrically minimal realizations of Boolean controlled systems. Autom Remote Control 75, 503–525 (2014). https://doi.org/10.1134/S0005117914030084

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