Skip to main content
Springer Nature Link
Log in

Discrete generalized-Sylvester matrix equation solved by RNN with a novel direct discretization numerical method

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In the fields of artificial intelligence and control engineering, generalized-Sylvester matrix equation is considered as an important mathematic problem, and its solving process is usually viewed as a challenge that deserves particular attention. In this paper, a creative discrete-form recurrent neural network (RNN) model is developed, analyzed and investigated for solving discrete-form time-variant generalized-Sylvester matrix equation (DF-TV-GSME), which is derived by a direct discretization numerical method. Specifically, first of all, DF-TV-GSME, which includes the well-known Lyapunov matrix equation and Sylvester matrix equation, is presented as the target problem of this research. Secondly, for solving such problem, different from the traditional discrete-form RNN design philosophy, second-order Taylor expansion is applied to derive the discrete-form RNN model. This creative process avoids involving the continuous time-variant environment and continuous-form model. Then, theoretical properties analyses are presented, which present the convergence and precision of the discrete-form RNN model. Abundant numerical experiments are further carried out with different perspectives of DF-TV-GSME, which further confirm the excellent properties of discrete-form RNN model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Wang, X., Chen, J., Wei, Q., Richard, C.: Hyperspectral image super-resolution via deep prior regularization with parameter estimation. IEEE Trans. Circuits Syst. Video Technol. 32(4), 1708–1723 (2017)

    Google Scholar 

  2. Wu, A., Xu, Y.: On coprimeness of two polynomials in the framework of conjugate product. IET Control Theory Appl. 11(10), 1522–1529 (2017)

    MathSciNet  Google Scholar 

  3. Ke, Y., Ma, C.: The unified frame of alternating direction method of multipliers for three classes of matrix equations arising in control theory. Asian J. Control. 20(1), 437–454 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Sheng, X.: A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations. J. Franklin Inst. 355(10), 4282–4297 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Chen, J., Zhang, Y.: Continuous and discrete zeroing neural dynamics handling future unknown-transpose matrix inequality as well as scalar inequality of linear class. Numer Algor. 83, 529–547 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Jin, L., Zhang, Y.: Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization. Numer Algor. 73, 115–140 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Zhang, Y., Liu, X., Ling, Y., Yang, M., Huang, H.: Continuous and discrete zeroing dynamics models using JMP function array and design formula for solving time-varying Sylvester-transpose matrix inequality. Numer Algor. 86, 1591–1614 (2021)

    MathSciNet  MATH  Google Scholar 

  8. Wang, J.: Electronic realisation of recurrent neural network for solving simultaneous linear equations. Electron. Lett. 28(5), 493–495 (1992)

    Google Scholar 

  9. Zhang, Y., Jiang, D., Wang, J.: A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans. Neural Netw. 13(5), 1053–1063 (2002)

  10. Huang, B., Ma, C.: An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations. Numer Algor. 78, 1271–1301 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Qiu, B., Zhang, Y., Guo, J., Yang, Z., Li, X.: New five-step DTZD algorithm for future nonlinear minimization with quartic steady-state error pattern. Numer Algor. 81, 1043–1065 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Zhang, Y., Li, S., Geng, G.: Initialization-based k-winners-take-all neural network model using modified gradient descent (to be published. https://doi.org/10.1109/TNNLS.2021.3123240) (2021)

  13. Zhang, Y., Ge, S. S.: Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans. Neural Netw. 16 (6), 1477–1490 (2005)

    Google Scholar 

  14. Xiao, L., Dai, J., Lu, R., Li, S., Li, J., Wang, S.: Design and comprehensive analysis of a noise-tolerant ZNN model with limited-time convergence for time-dependent nonlinear minimization. IEEE Trans. Neural Netw. Learn. Syst. 31(12), 5339–5348 (2020)

    MathSciNet  Google Scholar 

  15. Xiao, L., Zhang, Y., Zuo, Q., Dai, J., Li, J., Tang, W.: A noise-tolerant zeroing neural network for time-dependent complex matrix inversion under various kinds of noises. IEEE Trans. Ind. Inform. 16(6), 3757–3766 (2020)

    Google Scholar 

  16. Xiao, L., Zhang, Z., Li, S.: Solving time-varying system of nonlinear equations by finite-time recurrent neural networks with application to motion tracking of robot manipulators. IEEE Trans. Syst. Man Cybern. Syst. 49(11), 2210–2220 (2019)

    Google Scholar 

  17. Yang, M., Zhang, Y., Hu, H.: Relationship between time-instant number and precision of ZeaD formulas with proofs. Numer Algor. 88, 883–902 (2021)

    MathSciNet  MATH  Google Scholar 

  18. Shi, Y., Zhang, Y.: New discrete-time models of zeroing neural network solving systems of time-variant linear and nonlinear inequalities. IEEE Trans. Syst. Man Cybern. Syst. 50(2), 565–576 (2020)

    MathSciNet  Google Scholar 

  19. Shi, Y., Zhang, Y.: Solving future equation systems using integral-type error function and using twice ZNN formula with disturbances suppressed. J. Franklin Inst. 365(4), 2130–2152 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Shi, Y., Qiu, B., Chen, D., Li, J., Zhang, Y.: Proposing and validation of a new four-point finite-difference formula with manipulator application. IEEE Trans. Ind. Inf. 14(4), 1323–1333 (2018)

    Google Scholar 

  21. Duan, G.: Generalized Sylvester Equations: Unified Parametric Solutions. CRC Press, Boca Raton (2003)

    Google Scholar 

  22. Jin, L., Yan, J., Du, X., Xiao, X., Fu, D.: A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals. IEEE Trans. Industr. Inform. 16(10), 6359–6369 (2020)

    Google Scholar 

  23. Lv, L., Zhang, Z.: A numerical approach to generalized periodic Sylvester matrix equation. Asian J Control. 21(5), 2468–2475 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Li, S., Li, Y.: Nonlinearly activated neural network for solving time-varying complex Sylvester equation. IEEE Trans Cybern. 44(8), 1397–1407 (2014)

    Google Scholar 

  25. Liao, B., Wang, Y., Li, W., Peng, C., Xiang, Q.: Prescribed-time convergent and noise-tolerant Z-type neural dynamics for calculating time-dependent quadratic programming. Neural Comput. Appl. 33(10), 5327–5337 (2021)

    Google Scholar 

  26. Liao, B., Xiang, Q., Li, S.: Bounded Z-type neurodynamics with limited-time convergence and noise tolerance for calculating time-dependent Lyapunov equation. Neurocomputing. 325, 234–241 (2019)

    Google Scholar 

  27. Li, W., Han, L., Xiao, X., Liao, B., Peng, C.: A gradient-based neural network accelerated for vision-based control of an RCM-constrained surgical endoscope robot. Neural Comput. Appl. 34, 1329–1343 (2022)

    Google Scholar 

  28. Shi, Y., Zhao, W., Li, S., Li, B., Sun, X.: Novel discrete-time recurrent neural network for robot manipulator: A direct discretization technical route. IEEE Trans. Neural Netw. Learn. Syst. (to be published. https://doi.org/10.1109/TNNLS.2021.3108050) (2022)

  29. Shi, Y., Jin, L., Li, S., Li, J., Qiang, J., Gerontitis, D.K.: Novel discrete-time recurrent neural networks handling discrete-form time-variant multi-augmented Sylvester matrix problems and manipulator application. IEEE Trans. Neural Netw. Learn. Syst. 33(2), 587–599 (2022)

    MathSciNet  Google Scholar 

  30. Shi, Y., Mou, C., Qi, Y., Li, B., Li, S., Yang, B.: Design, analysis and verification of recurrent neural dynamics for handling time-variant augmented Sylvester linear system. Neurocomputing 426, 274–284 (2021)

    Google Scholar 

  31. Li, J., Shi, Y., Xuan, H.: Unified model solving nine types of time-varying problems in the frame of zeroing neural network. IEEE Trans. Neural Netw. Learn. Syst. 32(5), 1896–1905 (2021)

    MathSciNet  Google Scholar 

  32. Shi, Y., Pan, Z., Li, J., Li, B., Sun, X.: Recurrent neural dynamics for handling linear equation system with rank-deficient coefficient and disturbance existence. IEEE Trans. Neural Netw. Learn. Syst. 359, 3090–3102 (2022)

    MathSciNet  MATH  Google Scholar 

  33. Qiu, B., Guo, J., Li, X., Zhang, Z., Zhang, Y.: Discrete-time advanced zeroing neurodynamic algorithm applied to future equality-constrained nonlinear optimization with various noises. IEEE Trans. Cybern. 52(5), 3539–3552 (2022)

    Google Scholar 

  34. Zhang, Y., Li, S., Weng, J.: Learning and near-optimal control of underactuated surface vessels with periodic disturbances. IEEE Trans. Cybern. 52(8), 7453–7463 (2022)

  35. Jin, L., Li, S., Liao, B., Zhang, Z.: Zeroing neural networks: a survey. Neurocomputing 267(6), 597–604 (2017)

    Google Scholar 

  36. Shi, Y., Jin, L., Li, S., Qiang, J.: Proposing, developing and verification of a novel discrete-time zeroing neural network for solving future augmented Sylvester matrix equation. J. Franklin Inst. 357(6), 3636–3655 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Guo, D., Zhang, Y.: Zhang neural network for online solution of time-varying linear matrix inequality aided with an equality conversion. IEEE Trans. Neural Netw. Learn. Syst. 25(2), 370–382 (2014)

    Google Scholar 

  38. Horn, R. A., Johnson, C. R.: Matrix Analysis. Cambridge, Cambridge University Press (2012)

    Google Scholar 

  39. David, F. G., Desmond, J. H.: Numerical Methods for Ordinary Differential Equations: Initial Value Problems. Springer, Berlin (2010)

    MATH  Google Scholar 

  40. Li, J., Zhang, Y., Li, S., Mao, M.: New discretization-formula based zeroing dynamics for real-time tracking control of serial and parallel manipulators. IEEE Trans. Ind. Informat. 14(8), 3416–3425 (2018)

    Google Scholar 

  41. Li, J., Zhang, Y., Mao, M.: Five-instant type discrete-time ZND solving discrete time-varying linear system, division and quadratic programming. Neurocomputing 331, 323–335 (2019)

    Google Scholar 

  42. Li, J., Zhang, Y., Mao, M.: Continuous and discrete zeroing neural network for different-level dynamic linear system with robot manipulator control. IEEE Trans. Syst. Man Cybern. Syst. 50(11), 4633–4642 (2020)

    Google Scholar 

  43. Qiu, B., Zhang, Y., Yang, Z.: New discrete-time ZNN models for least-squares solution of dynamic linear equation system with time varying rank-deficient coefficient. IEEE Trans. Neural Netw. Learn. Syst. 29(11), 5767–5776 (2018)

    MathSciNet  Google Scholar 

  44. Qiu, B., Zhang, Y.: Two new discrete-time neurodynamic algorithms applied to online future matrix inversion with nonsingular or sometimes-singular coefficient. IEEE Trans. Cybern. 49(6), 2032–2045 (2019)

    Google Scholar 

  45. Guo, D., Nie, Z., Yan, L.: Novel discrete-time Zhang neural network for time-varying matrix inversion. IEEE Trans. Syst. Man Cybern. Syst. 47(8), 2301–2310 (2017)

    Google Scholar 

  46. Li, W., Xiao, L., Liao, B.: A finite-time convergent and noiserejection recurrent neural network and its discretization for dynamic nonlinear equations solving. IEEE Trans. Cybern. 50(7), 3195–3207 (2020)

    Google Scholar 

  47. Xiao, L., Zhang, Z., Zhang, Z., Li, W., Li, S.: Design, verification and robotic application of a novel recurrent neural network for computing dynamic Sylvester equation. Neural Netw. 105, 185–196 (2018)

    MATH  Google Scholar 

Download references

Funding

This work was supported in part by the National Natural Science Foundation of China (with numbers 61906164 and 61972335), in part by the Natural Science Foundation of Jiangsu Province of China (with number BK20190875), in part by the Six Talent Peaks Project in Jiangsu Province (with number RJFW-053), in part by Jiangsu “333” Project, in part by Qinglan project of Yangzhou University, in part by the Cross-Disciplinary Project of the Animal Science Special Discipline of Yangzhou University, in part by the Yangzhou University Interdisciplinary Research Foundation for Animal Husbandry Discipline of Targeted Support (with number yzuxk202015), in part by the Yangzhou City-Yangzhou University Science and Technology Cooperation Fund Project (with number YZ2021157), in part by the Yangzhou University Top-level Talents Support Program (2021, 2019) by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (with numbers KYCX21_3234 and SJCX22_1709).

Author information

Authors and Affiliations

  1. School of Information Engineering, Yangzhou University, Yangzhou, 225127, China

    Yang Shi, Chenling Ding, Bin Li & Xiaobing Sun

  2. Jiangsu Province Engineering Research Center of Knowledge Management and Intelligent Service, Yangzhou University, Yangzhou, 225127, China

    Yang Shi, Chenling Ding, Bin Li & Xiaobing Sun

  3. College of Engineering, Swansea University, Fabian Way, Swansea, UK

    Shuai Li

Authors
  1. Yang Shi
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Chenling Ding
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Shuai Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Bin Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  5. Xiaobing Sun
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding authors

Correspondence to Yang Shi or Shuai Li.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, Y., Ding, C., Li, S. et al. Discrete generalized-Sylvester matrix equation solved by RNN with a novel direct discretization numerical method. Numer Algor 93, 971–992 (2023). https://doi.org/10.1007/s11075-022-01449-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01449-x

Keywords