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Generalised semi-tensor product of matrices

Generalised semi-tensor product of matrices

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By introducing matrix multiplier and vector multiplier two kinds of semi-tensor products (STPs), called matrix–matrix (MM) STPs and matrix–vector (MV) STPs, are introduced. They are generalisations of conventional matrix product, and contain standard STP as a particular case. Certain properties are revealed. Using consistent MM-STP and MV-STP, a cross-dimensional linear system is proposed. It is shown that the cross-dimensional linear system is a linear semi-group (S) system. Next, the quotient matrix space based on matrix multiplier and the quotient vector space based on vector multiplier are introduced, and the S-system structure is extended to quotient matrix and quotient vector spaces. Finally, an inner-product is introduced, which poses a topology on the cross-dimensional state space and hence turns the linear S-system into a dynamic linear system.

References

    1. 1)
      • 1. Cheng, D.: ‘Semi-tensor product of matrices and its application to Morgan's problem’, Sci. China, F, Inf. Sci., 2001, 44, (3), pp. 195212.
    2. 2)
      • 2. Cheng, D., Ma, J., Lu, Q., et al: ‘Quadratic form of stable sub-manifold for power systems’, Int. J. Robost Nonlinear Control, 2004, 14, pp. 773788.
    3. 3)
      • 3. Mei, S., Liu, F., Xe, A.: ‘Semi-tensor product approach to transient process analysis of power systems’ (Tsinghua Universtiy Press, Beijing, 2010) (in Chinese).
    4. 4)
      • 4. Xe, A., Wu, F., Lu, Q., et al: ‘Power system dynamic security region and its approximations’, IEEE Trans. Circuits Syst. I, 2006, 53, (12), pp. 28492859.
    5. 5)
      • 5. Cheng, D., Qi, H., Li, Z.: ‘Analysis and control of Boolean networks – a semi-tensor product approach’ (Springer, London, 2011).
    6. 6)
      • 6. Fornasini, E., Valcher, M.E.: ‘Observability, reconstructibility and state observers of Boolean control networks,’, IEEE Trans. Autom. Control, 2013, 58, (6), pp. 13901401.
    7. 7)
      • 7. Laschov, D., Margaliot, M.: ‘A maximum principle for single-input Booolean control notworks’, IEEE Trans. Autom. Control, 2011, 56, (4), pp. 913917.
    8. 8)
      • 8. Cheng, D.: ‘On finite potential games’, Automatica, 2014, 50, (7), pp. 17931801.
    9. 9)
      • 9. Cheng, D., He, F., Qi, H., et al: ‘Modeling, analysis and control of networked evolutionary games’, IEEE Trans. Autom. Control, 2015, 60, (9), pp. 24022415.
    10. 10)
      • 10. Guo, P., Wang, Y., Li, H.: ‘Algebraic formulation and strategy optimization for a class of evolutionary networks via semi-tensor product method’, Automatica, 2013, 49, pp. 33843389.
    11. 11)
      • 11. Cheng, D., Qi, H., Zhao, Y.: ‘An introduction to semi-tensor product of matrices and its applications’ (World Scientific, Singapore, 2012).
    12. 12)
      • 12. Fornasini, E., Valcher, M.E.: ‘Recent developments in Boolean networks control’, J. Control Decis., 2016, 3, (1), pp. 118.
    13. 13)
      • 13. Lu, J., Li, H., Liu, Y., et al: ‘Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems’, IET Control Theory Applic., 2017, 11, (13), pp. 20402047.
    14. 14)
      • 14. Muhammad, A., Rushdi, A., Ghaleb, F.A.M.: ‘A tutorial exposition of semi-tensor products of matrices with a stress on their representation of Boolean function’, JKAU Comput. Sci., 2016, 5, pp. 330.
    15. 15)
      • 15. Hua, G.: ‘Fundation of number theory’ (Science Press, Beijing, 1957), (in Chinese).
    16. 16)
      • 16. Cheng, D., Xu, Z., Shen, T.: ‘Equivalence-based model of dimension-varying linear systems’, IEEE TAC, under revision, (preprint: https://arxiv.org/submit/2414749).
    17. 17)
      • 17. Hungerford, T.W.: ‘Algebra’ (Springer-Verlag, New York, 1974).
    18. 18)
      • 18. Ahsan, J.: ‘Monoids characterized by their quasi-injective S-systems’, Semigroup Fourm, 1987, 36, pp. 285292.
    19. 19)
      • 19. Liu, Z., Qiao, H.: ‘S-system theory of semigroup’ (Science Press, Beijing, 2008, 2nd edn.), (in Chinese).
    20. 20)
      • 20. Cheng, D.: ‘On equivalence of matrices’, Asian J. Math., 2019, 23, (2), pp. 257348.
    21. 21)
      • 21. Cheng, D.: ‘From dimension-free matrix theory to cross dimensional dynamic systems’ (Elsevier, London, 2019).
    22. 22)
      • 22. Burris, S., Sankappanavar, H.P.: ‘A course in universal algebra’ (Springer, New York, 1981).
    23. 23)
      • 23. Greub, W.: ‘Linear algebra’ (Springer-Verlag, New York, 1981, 4th edn.).
    24. 24)
      • 24. Abraham, R., Marsden, J.: ‘Foundations of mechanics’ (Benjamin/Cummings Pub., London, 1978, 2nd edn.).
    25. 25)
      • 25. Jamich, K.: ‘Topology’ (Springer-Verlag, New York, 1984).
    26. 26)
      • 26. Horn, R.A., Johnson, C.R.: ‘Matrix analysis’ (Cambridge University Press, Cambridge, 1985).
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