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A Finite-Time Convergent Neural Network for Solving Time-Varying Linear Equations with Inequality Constraints Applied to Redundant Manipulator

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Abstract

Zhang neural network (ZNN), a special recurrent neural network, has recently been established as an effective alternative for time-varying linear equations with inequality constraints (TLEIC) solving. Still, the convergent time produced by the ZNN model always tends to infinity. In contrast to ZNN, a finite-time convergent neural network (FCNN) is proposed for the TLEIC problem. By introducing a non-negative slack variable, the initial form of the TLEIC has been transformed into a system of time-varying linear equation. Afterwards, the stability and finite-time performance of the FCNN model is substantiated by the theoretical analysis. Then, simulation results further verify the effectiveness and superiority of the proposed FCNN model as compared with the ZNN model for solving TLEIC problem. Finally, the proposed FCNN model is successfully applied to the trajectory planning of redundant manipulators with joint limitations, thereby illustrating the applicability of the new neural network model.

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References

  1. Li J, Zhang Y, Mao M (2020) 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. https://doi.org/10.1109/TSMC.2018.2856266

    Article  Google Scholar 

  2. Zhang Y, Guo D (2015) Zhang functions and various models. Springer, Berlin

    Book  Google Scholar 

  3. Zhao YB (2013) New and improved conditions for uniqueness of sparsest solutions of underdetermined linear systems. Appl Math Comput 224:58–73

    MathSciNet  MATH  Google Scholar 

  4. Rump SM (2014) Improved componentwise verified error bounds for least squares problems and underdetermined linear systems. Numer Algorithms 66(2):309–322. https://doi.org/10.1007/s11075-013-9735-6

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen D, Zhang Y (2015) A hybrid multi-objective scheme applied to redundant robot manipulators. IEEE Trans Autom Sci Eng 14(3):1337–1350

    Article  Google Scholar 

  6. Pang LP, Spedicato E, Xia ZQ, Wang W (2007) A method for solving the system of linear equations and linear inequalities. Math Comput Model 46(5–6):823–836

    Article  MathSciNet  Google Scholar 

  7. Murav’eva OV, (2015) Consistency and inconsistency radii for solving systems of linear equations and inequalities. Comput Math Math Phys 55(3):366–377

  8. Golikov AI, Evtushenko YG (2015) Regularization and normal solutions of systems of linear equations and inequalities. Proc Steklov Inst Math 289(S1):102–110

    Article  MathSciNet  Google Scholar 

  9. Li H, Luo J, Wang Q (2014) Solvability and feasibility of interval linear equations and inequalities. Linear Algebra Appl 463:78–94

    Article  MathSciNet  Google Scholar 

  10. Esmaeili H, Spedicato E (2004) Explicit ABS solution of a class of linear inequality systems and LP problems. Bull Iran Math Soc 30

  11. Castillo E, Jubete F (2004) The \(\Gamma \)-algorithm and some applications. Int J Math Educ Sci Technol 35(3):369–389

    Article  MathSciNet  Google Scholar 

  12. Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci 79(8):2554–2558

    Article  MathSciNet  Google Scholar 

  13. Xia Y, Wang J, Hung DL (1999) Recurrent neural networks for solving linear inequalities and equations. IEEE Trans Circuits Syst I Fundam Theory Appl 46(4):452–462

    Article  MathSciNet  Google Scholar 

  14. Liang Xue-Bin, Shiu Kit Tso (2002) Improved upper bound on step-size parameters of discrete-time recurrent neural networks for linear inequality and equation system. IEEE Trans Circuits Syst I Fundam Theory Appl 49(5):695–698

    Article  Google Scholar 

  15. Cichocki A, Ramirez-Angulo J, Unbehauen R (1992) Architectures for analog VLSI implementation of neural networks for solving linear equations with inequality constraints

  16. Guo D, Zhang Y (2015) ZNN for solving online time-varying linear matrix-vector inequality via equality conversion. Appl Math Comput 259:327–338. https://doi.org/10.1016/j.amc.2015.02.060

    Article  MathSciNet  MATH  Google Scholar 

  17. Shao S, Li H, Qin S, Li G, Luo C (2020) An inverse-free Zhang neural dynamic for time-varying convex optimization problems with equality and affine inequality constraints. Neurocomputing 412:152–166

    Article  Google Scholar 

  18. Guo D, Zhang Y (2012) Novel recurrent neural network for time-varying problems solving [research Frontier]. IEEE Comput Intell Mag 7(4):61–65

    Article  Google Scholar 

  19. Stanimirović PS, Srivastava S, Gupta DK (2018) From Zhang Neural Network to scaled hyperpower iterations. J Comput Appl Math 331:133–155. https://doi.org/10.1016/j.cam.2017.09.048

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo D, Zhang Y (2014) 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. https://doi.org/10.1109/TNNLS.2013.2275011

    Article  Google Scholar 

  21. Zhang Y, Wang Y, Jin L, Mu B, Zheng H (2013) Different ZFs leading to various ZNN models illustrated via online solution of time-varying underdetermined systems of linear equations with robotic application. Adv Neural Netw - ISNN 2013. Springer, Berlin Heidelberg, pp 481–488

  22. Xu F, Li Z, Nie Z, Shao H, Guo D (2019) New recurrent neural network for online solution of time-dependent underdetermined linear system with bound constraint. IEEE Trans Ind Inf 15(4):2167–2176

    Article  Google Scholar 

  23. Li S, Chen S, Liu B (2012) Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function. Neural Process Lett 37(2):189–205

    Article  Google Scholar 

  24. Guo D, Lin X (2020) Li-function activated Zhang neural network for online solution of time-varying linear matrix inequality. Neural Process Lett 52(1):713–726. https://doi.org/10.1007/s11063-020-10291-y

    Article  Google Scholar 

  25. Xiao L, Zhang Z, Li S (2019) 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

    Article  Google Scholar 

  26. Jin J, Zhao L, Li M, Yu F, Xi Z (2020) Improved zeroing neural networks for finite time solving nonlinear equations. Neural Comput Appl 32(9):4151–4160. https://doi.org/10.1007/s00521-019-04622-x

    Article  Google Scholar 

  27. Shen Y, Miao P, Huang Y, Shen Y (2015) Finite-time stability and its application for solving time-varying sylvester equation by recurrent neural network. Neural Process Lett 42(3):763–784

    Article  Google Scholar 

  28. Jin J, Gong J (2021) A noise-tolerant fast convergence ZNN for dynamic matrix inversion. Int J Comput Math. https://doi.org/10.1080/00207160.2021.1881498

    Article  MathSciNet  MATH  Google Scholar 

  29. Jin J (2021) An improved finite time convergence recurrent neural network with application to time-varying linear complex matrix equation solution. Neural Process Lett 53(1):777–786. https://doi.org/10.1007/s11063-021-10426-9

    Article  Google Scholar 

  30. Xiao L (2016) A new design formula exploited for accelerating Zhang neural network and its application to time-varying matrix inversion. Theor Comput Sci 647:50–58. https://doi.org/10.1016/j.tcs.2016.07.024

    Article  MathSciNet  MATH  Google Scholar 

  31. Xiao L (2017a) Accelerating a recurrent neural network to finite-time convergence using a new design formula and its application to time-varying matrix square root. J Franklin Inst 354(13):5667–5677. https://doi.org/10.1016/j.jfranklin.2017.06.012

    Article  MathSciNet  MATH  Google Scholar 

  32. Xiao L (2017b) A finite-time recurrent neural network for solving online time-varying Sylvester matrix equation based on a new evolution formula. Nonlinear Dyn 90(3):1581–1591. https://doi.org/10.1007/s11071-017-3750-4

    Article  MathSciNet  MATH  Google Scholar 

  33. Li S, Li Y, Wang Z (2013) A class of finite-time dual neural networks for solving quadratic programming problems and its k-winners-take-all application. Neural Netw 39:27–39

    Article  Google Scholar 

  34. Zhang Y, Yi C (2011) Zhang neural networks and neural-dynamic method. Nova Science Publishers Inc., Hauppauge

    Google Scholar 

  35. Kong Y, Jiang Y, Xia X (2020) Terminal recurrent neural networks for time-varying reciprocal solving with application to trajectory planning of redundant manipulators. IEEE Trans Syst Man Cybern Syst 1–13

  36. Zhang Y, Zhang Z (2014) Repetitive motion planning and control of redundant robot manipulators. Springer, Berlin

    MATH  Google Scholar 

Download references

Acknowledgements

This study was partly supported by the National Natural Science Foundation of China (61803338, 61972357, 61672337), the Zhejiang Key R&D Program (Grant No. 2019C03135) and the Youth Foundation of Zhejiang University of Science and Technology (2021QN001, 2021QN046).

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Authors and Affiliations

  1. Department of Information and Electronic Engineering, Zhejiang University of Science and Technology, Hangzhou, China

    Ying Kong, Tanglong Hu & Jingsheng Lei

  2. School of Science, Zhejiang University of Science and Technology, Hangzhou, China

    Renji Han

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  1. Ying Kong
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  2. Tanglong Hu
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  3. Jingsheng Lei
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  4. Renji Han
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Correspondence to Ying Kong.

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Kong, Y., Hu, T., Lei, J. et al. A Finite-Time Convergent Neural Network for Solving Time-Varying Linear Equations with Inequality Constraints Applied to Redundant Manipulator. Neural Process Lett 54, 125–144 (2022). https://doi.org/10.1007/s11063-021-10623-6

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