Abstract
This paper concerns with the time-variant neural computing in a semi-global sense, taking into account initial conditions located within a region with a definitely finite radius. Both the conventional single and double power-rate RNN models are characterized and the closed-form expressions of the settling time functions are presented for given initial conditions, by which the fixed/predefined-time convergence can be assured in the semi-global sense. Despite asymptotic convergence behavior, the conventional linear RNN model is examined for comparison purposes. Modified RNN models adopt the inverse of the bound, according to the fixed-time convergence results, and the prescribed time can be an adjustable parameter. A novel two-phase RNN model with the pre-specified transition state is proposed, which has not only semi-global fixed/predefined-time stability but also a faster convergence rate than that of the conventional models. The proposed models are applied and compared, through numerical simulation, for time-variant matrix inversion, linear equation solving, and repeatable motion planning of a redundant manipulator in the presence of initial errors.
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This work was supported by the National Natural Science Foundation of China under Grant 62073291.
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Appendix A. The proof for Corollary 1
Appendix A. The proof for Corollary 1
For model (8), it follows that
From \(\Lambda _{ij}(0) > 1\) to \(\Lambda _{ij}(t_{1})=1\), we have
due to that \(\Lambda _{ij}^{\alpha } \le \Lambda _{ij}\). Defining \(y =\Lambda _{ij}^{1-\beta }\) yields
Solving the above linear differential inequality, we obtain
Since \(y(t_1)=1\),
From \(\Lambda _{ij}(t_1)=1\) to \(\Lambda _{ij}(t_2)=0\), we have
due to that \(\Lambda _{ij}^{\beta } \le \Lambda _{ij}\). Defining \(y =\Lambda _{ij}^{1-\alpha }\) leads to
Solving the above differential inequality for \(t \ge t_1\), we obtain
Since \(y(t_1) = 1\) and \(y(t_2)=0\),
Hence, the settling time function satisfies
This completes the proof.
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Sun, M., Li, X. & Zhong, G. Semi-global fixed/predefined-time RNN models with comprehensive comparisons for time-variant neural computing. Neural Comput & Applic 35, 1675–1693 (2023). https://doi.org/10.1007/s00521-022-07820-2
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DOI: https://doi.org/10.1007/s00521-022-07820-2