## 1 Introduction
An iterative algorithm based on the Fourier series approximation generates the optimal feedforward to force the state trajectory to converge to a stable sliding surface. Using a Fourier series approximation of tracking error, both the magnitudes and phases of the error can be compensated. Therefore, time delay can be easily handled. The iterative learning controller based on the Fourier series compensates for the deterministic uncertainties, and the feedback controller is responsible for stabilizing the system [15; 16; 17; 18]. Without prior knowledge of the system model, the nonlinearities and uncertainties of the dynamical system that are lumped together with no restrictions on their relationship can be compensated online by an Adaptive Fourier Neural Network (AFNN) [19]. There is another nonlinearity that causes extreme difficulty in system modeling. Flexibility is a potential source of uncertainty that can degrade the control performance and even destabilize the system in some cases. To solve this, a Fourier Neural Network (FNN)-based output feedback learning control scheme was proposed. This learning algorithm was carried out in the Fourier space, and it can guarantee the stability of the closed-loop system and the convergence of the tracking error [20]. Nowadays, the Fourier series in control areas have been applied for aerial robots. The use of a frequency-domain iterative learning scheme for periodic quadrocopter flight, parameterizing the non-causal tracking error compensation as Fourier series, was presented by [21]. Uncertainties, such as measurement noise, inaccuracies of the approximate transfer function or model uncertainty, can be accounted for in the learning step size, and the optimal step size for each frequency component can be computed from the statistical properties thereof [21]. There are complex cases where the controllers require help from Fourier series expansions, like the control of autonomous underwater vehicles (AUV). The bottom-following or seabed tracking of AUV, traversing at constant altitudes from the sea bottom, is a big challenge for controllers. However, using the tools of nonlinear output regulation theory combined with Fourier series expansion and pseudo-spectral methods, it is possible to solve the bottom-following problem without requiring knowledge in advance of the seabed profile and using only one single echo sounder to measure the altitude from the seabed [22]. One of the most important contributions of the expansion in Fourier series is helping as a function approximation (FA) when it is difficult to establish an accurate dynamic model. For example, the approach of estimated Fourier coefficients can be used by a nonlinear controller, which simultaneously achieves accurate and ripple-free torque control and minimizes copper losses in brushless motors [23]. In piezoelectricly-actuated systems, this approach can estimate the system dynamics and uncertainties on-line, and no prior knowledge of system dynamics is required [24; 25]. An adaptive iterative learning controller can be designed for nonlinear systems with unknown time-varying parameters to realize the non-uniform trajectory tracking perfectly when the time varying parameters are expanded into Fourier series with bounded remainder terms [26]. Moreover, the problem of modeling and tracking control for a high speed ball screw drive, which includes time-varying parametric uncertainties and disturbances, can be solved when a function approximation technique based on Fourier series expansion is applied to estimate the total uncertainties containing inertia uncertainties, stiffness uncertainties and external disturbances [27]. The Fourier series have been complementary tools for conventional controllers, such as Proportional Integral Derivative (PID). However, the Fourier series convergence depends on the conventional controller stability. Furthermore, to obtain the coefficients of the Fourier series, the continuous Fourier transform has been used, which depends on the period of the function, and it is normally unknown.