Neural learning control for sampled-data nonlinear systems based on Euler approximation and first-order filter

The primary focus of this research paper is to explore the realm of dynamic learning in sampled-data strict-feedback nonlinear systems (SFNSs) by leveraging the capabilities of radial basis function (RBF) neural networks (NNs) under the framework of adaptive control. First, the exact discrete-time model of the continuous-time system is expressed as an Euler strict-feedback model with a sampling approximation error. We provide the consistency condition that establishes the relationship between the exact model and the Euler model with meticulous detail. Meanwhile, a novel lemma is derived to show the stability condition of a digital first-order filter. To address the non-causality issues of SFNSs with sampling approximation error and the input data dimension explosion of NNs, the auxiliary digital first-order filter and backstepping technology are combined to propose an adaptive neural dynamic surface control (ANDSC) scheme. Such a scheme avoids the $n$-step time delays associated with the existing NN updating laws derived by the common $n$-step predictor technology. A rigorous recursion method is employed to provide a comprehensive verification of the stability, guaranteeing its overall performance and dependability. Following that, the NN weight error systems are systematically decomposed into a sequence of linear time-varying subsystems, allowing for a more detailed analysis and understanding. In order to ensure the recurrent nature of the input variables, a recursive design is employed, thereby satisfying the partial persistent excitation condition specifically designed for the RBF NNs. Meanwhile, it can verify that the NN estimated weights converge to their ideal values. Compared with the common $n$-step predictor technology, there is no need to redesign the learning rules due to the designed NN weight updating laws without time delays. Subsequently, after capturing and storing the convergence weights, a novel neural learning dynamic surface control (NLDSC) scheme is specifically formulated by leveraging the acquired knowledge. The introduced methodology reduces computational complexity and facilitates practical implementation. Finally, empirical evidence obtained from simulation experiments validates the efficacy and viability of the proposed methodology.