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<p>The persistent excitation (PE) condition is one of the most important concepts for various adaptive algorithms. At present, more efforts have been made to study the PE condition for sufficiency. However, there are fewer studies on the excitation level, which commonly acts as a constant of convergence relation, affecting the performance of the algorithm. This paper establishes the PE level relations for radial basis function (RBF) neural networks under the discrete-time framework. From the standpoint of network parameters, the PE level is proportional to the separation distance and tends to diminish as the excitation distance grows larger. Furthermore, the PE level reaches its optimal value when the receptive field width is appropriately chosen. From the perspective of discrete signal properties, the PE level is observed to increase as the sampling period decreases or as the trajectory velocity slows down. As an illustration, based on the relations of PE level, the performance of deterministic learning under the sampled data framework is established for analysis. The results show that the existence of an optimal separation distance leads to maximum learning accuracy, caused by the trade-off between the approximation ability of the neural network and the excitation level. Moreover, the existence of an ill-conditioned interpolation matrix related to the interpolation method also yields the optimal receptive field width. A simulation example is used to demonstrate the effectiveness of the PE level relations obtained.</p>

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Relations of PE level for RBF neural networks under the discrete-time framework and their application to deterministic learning

  • 期刊论文

  • 发表日期：

    2026年7月31日

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  • Weiming Wu,
  • Jinyuan Zhang,
  • Chen Sun,
  • Qinghua Sun,
  • Juanjuan Xu,
  • Cong Wang

摘要

The persistent excitation (PE) condition is one of the most important concepts for various adaptive algorithms. At present, more efforts have been made to study the PE condition for sufficiency. However, there are fewer studies on the excitation level, which commonly acts as a constant of convergence relation, affecting the performance of the algorithm. This paper establishes the PE level relations for radial basis function (RBF) neural networks under the discrete-time framework. From the standpoint of network parameters, the PE level is proportional to the separation distance and tends to diminish as the excitation distance grows larger. Furthermore, the PE level reaches its optimal value when the receptive field width is appropriately chosen. From the perspective of discrete signal properties, the PE level is observed to increase as the sampling period decreases or as the trajectory velocity slows down. As an illustration, based on the relations of PE level, the performance of deterministic learning under the sampled data framework is established for analysis. The results show that the existence of an optimal separation distance leads to maximum learning accuracy, caused by the trade-off between the approximation ability of the neural network and the excitation level. Moreover, the existence of an ill-conditioned interpolation matrix related to the interpolation method also yields the optimal receptive field width. A simulation example is used to demonstrate the effectiveness of the PE level relations obtained.