FSE-RBFNN-based LPF-AILC of finite time complete tracking for a class of time-varying NPNL systems with initial state errors

The paper proposes a low-pass filter adaptive iterative learning control (LPF-AILC) strategy for unmatched, uncertain, time-varying, non-parameterized nonlinear systems (NPNL systems). To address the difficulty of nonlinear parameterization terms in system models, a new function approximator (FSE-RBFNN), which combines the radial basis function neural network (RBFNN) and Fourier series expansion (FSE), is introduced to model each time-varying nonlinear parameterized function. The adaptive backstepping method is used to design control laws and parameter adaptive laws. In the process of controller design, we may encounter the problem of too many derivatives, which can cause parameter explosions after derivatives. Therefore, we introduce a first-order low-pass filter to solve this problem and simplify the structure of the controller. As the number of iterations increases, the maximum tracking error gradually decreases until it converges to the nearby region, approaching zero within the entire given interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math> , according to the Lyapunov-like synthesis. To mitigate the impact of initial state errors, a dynamically changing boundary layer is introduced, along with a series to deal with the unknown error upper bounds. Finally, two simulation examples prove the correctness of the proposed control method.