Advertisement

Time-Varying Matrix Inverse

  • Yunong ZhangEmail author
  • Dongsheng Guo
Chapter
  • 550 Downloads

Abstract

In this chapter, focusing on time-varying matrix inversion, we propose, generalize, develop, and investigate different ZFs that lead to different ZD models. Meanwhile, a specific relationship between the proposed ZD model and others’ model/method [i.e., the Getz and Marsden (G-M) dynamic system] is presented. Eventually, the MATLAB Simulink modeling and simulative verifications with examples using such different ZD models are further researched. Both theoretical analysis and modeling results further substantiate the efficacy of the proposed ZD models which originate from different ZFs for time-varying matrix inversion.

Keywords

Matrix Inversion Sylvester Equation Linear Matrix Equation Matrix Vector Form Random Neural Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Gu C, Xue H (2009) A shift-splitting hierarchical identification method for solving Lyapunov matrix equations. Linear Algebra Appl 430(5–6):1517–1530CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Zhang Y, Jiang D, Wang J (2002) A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans Neural Netw 13(5):1053–1063CrossRefGoogle Scholar
  3. 3.
    Zhou B, Lam J, Duan GR (2009) On Smith-type iterative algorithms for the Stein matrix equation. Appl Math Lett 22(7):1038–1044CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Fuhrmann PA (2010) A functional approach to the Stein equation. Linear Algebra Appl 432(12):3031–3071CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Zhang Y, Yi C (2011) Zhang neural networks and neural-dynamic method. Nova Science Publishers, New YorkGoogle Scholar
  6. 6.
    Guo D, Zhang Y (2012) Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots’ kinematic control. Neurocomputing 97:22–32CrossRefGoogle Scholar
  7. 7.
    Guo D, Qiu B, Ke Z, Yang Z, Zhang Y (2014) Case study of Zhang matrix inverse for different ZFs leading to different nets. In: Proceedings of the international joint conference on neural networks, pp 2764–2769Google Scholar
  8. 8.
    Zhang Y, Guo D, Li Z (2013) Common nature of learning between back-propagation and Hopfield-type neural networks for generalized matrix inversion with simplified models. IEEE Trans Neural Netw Learn Syst 24(4):579–592CrossRefGoogle Scholar
  9. 9.
    Ben-Israel A, Greville TNE (2003) Generalized inverses: theory and applications, 2nd edn. Springer, New YorkGoogle Scholar
  10. 10.
    Zhang Y, Yang Y, Tan N, Cai B (2011) Zhang neural network solving for time-varying full-rank matrix Moore-Penrose inverse. Computing 92(2):97–121CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Yeung KS, Kumbi F (1988) Symbolic matrix inversion with application to electronic circuits. IEEE Trans Circuits Syst 35(2):235–238CrossRefzbMATHGoogle Scholar
  12. 12.
    El-Amawy A (1989) A systolic architecture for fast dense matrix inversion. IEEE Trans Comput 38(3):449–455CrossRefMathSciNetGoogle Scholar
  13. 13.
    Wang YQ, Gooi HB (1997) New ordering methods for space matrix inversion via diagonalization. IEEE Trans Power Syst 12(3):1298–1305CrossRefGoogle Scholar
  14. 14.
    Mathews JH, Fink KD (2004) Numerical methods using MATLAB. Prentice Hall, New JerseyGoogle Scholar
  15. 15.
    Koc CK, Chen G (1994) Inversion of all principal submatrices of a matrix. IEEE Trans Aerosp Electr Syst 30(1):280–281CrossRefGoogle Scholar
  16. 16.
    Gelenbe E, Hussain KF (2002) Learning in the multiple class random neural network. IEEE Trans Neural Netw 13(6):1257–1267CrossRefGoogle Scholar
  17. 17.
    Gelenbe E, Timotheou S (2008) Random neural networks with synchronized interactions. Neural Comput 20(9):2308–2324CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Wang J (1993) A recurrent neural network for real-time matrix inversion. Appl Math Comput 55(1):89–100CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Zhang Y, Shi Y, Chen K, Wang C (2009) Global exponential convergence and stability of gradient-based neural network for online matrix inversion. Appl Math Comput 215(3):1301–1306CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Zhang Y, Ma W, Cai B (2009) From Zhang neural network to Newton iteration for matrix inversion. IEEE Trans Circuits Syst I: Regul Pap 56(7):1405–1415CrossRefMathSciNetGoogle Scholar
  21. 21.
    Getz NH, Marsden JE (1995) A dynamic inverse for nonlinear maps. In: Proceedings of 34th IEEE conference on decision and control, pp 4218–4223Google Scholar
  22. 22.
    Getz NH, Marsden JE (1995) Joint-space tracking of workspace trajectories in continuous time. In: Proceedings of 34th IEEE conference on decision and control, pp 1001–1006Google Scholar
  23. 23.
    Getz NH, Marsden JE (1997) Dynamical methods for polar decomposition and inversion of matrices. Linear Algebra Appl 258:311–343CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Information Science and TechnologySun Yat-sen UniversityGuangzhouChina

Personalised recommendations