# Zhang Neural Dynamics Approximated by Backward Difference Rules in Form of Time-Delay Differential Equation

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## Abstract

Time-varying matrix inversion (TVMI) and tracking control of vehicular inverted pendulum (VIP) system are usually considered as two important benchmarking examples in mathematical science and control fields. In this paper, Zhang neural dynamics (ZND) method is used for solving these two problems. Besides, in order to investigate the effect of time delay (especially the time delay caused by the derivative approximation) on the ZND method when solving the above-mentioned two problems, backward difference rules in the form of time-delay differential equation are elegantly applied to approximating the ZND method, which is simply termed time-delay ZND method. Moreover, two theorems about time-delay ZND method are presented together with theoretical proofs. Finally, simulative tests of TVMI and tracking control of VIP system are conducted to show the effectiveness and accuracy of non-time-delay and time-delay ZND methods.

## Keywords

Zhang neural dynamics (ZND) Time delay Backward difference rules Time-varying matrix inversion (TVMI) Tracking control## Notes

### Acknowledgements

The work is supported by the National Natural Science Foundation of China (with number 61473323). Kindly note that all authors of the paper are jointly of the first authorship.

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