## 5 Further Discussion on the Tracking Control Algorithm  
### The Proposed Controller versus Other Control Schemes  
This section discusses the differences between the proposed control algorithm (PCA) and other selected schemes in recent years. The selected methods are classified into the following groups:  
* Control strategies based on combinations of neural networks with other methods [31; 35; 36; 37; 38; 49; 50; 52; 69; 70];
* Sliding mode control based approaches [18] or combined with backstepping [27];
* Control schemes with guaranteed prescribed performance [41; 42; 71; 72];
* Extended observer based control algorithms [28; 57; 58; 59; 60];
* Fuzzy logic based control [40];
* Modified dynamic inversion [46];
* Input-output linearization [56]. In [31], the neural shunt model method was applied in order to solve the problem arising from virtual control law. The adaptive control strategy used a combination of backstepping sliding mode. Numerical simulation of trajectory tracking performed on a benchmark prototype and for circular trajectory showed the superiority of the algorithm compared to a known sliding control type method (with model uncertainty and environmental disturbances). In [35], an adaptive trajectory tracking control strategy for an underactuated surface vehicle subject to unknown dynamics and time-varying external disturbances was presented. It consisted of first-order sliding mode, second-order sliding mode and the neural network minimum learning parameter method. The latter method was introduced into the design of controller. Numerical simulations were performed for straight and circular trajectories and the results obtained showed the effectiveness of tracking and were slightly better than for the sliding mode tracking control method selected from the literature (chattering was avoided). Paper [36] addressed the control problem of trajectory tracking by underactuated autonomous surface vehicles with parameter uncertainty and nonlinear external disturbances. The control strategy included backstepping method, neural network and sliding mode control. The radial basis function neural network (RBFNN) was applied for estimation and approximation of unknown functions representing uncertainties due to its simplicityand linear parameterization. Simulation studies were carried out on a ship model taken from the literature and a desired sinusoidal trajectory and with sinusoidal external disturbances. The proposed neural-based command filtered backstepping (NBCFB) approach was compared with the traditional backstepping method and dynamic surface control (DSC) tracking based on neural network observer. Lower trajectory tracking errors were achieved and oscillations of position and velocity error signals were reduced. In paper [37], a control algorithm for underactuated autonomous underwater vehicle is proposed which is a combination of Radial Basis Function (RBF) neural network algorithm and state prediction using sliding mode control method and backstepping. A sinusoidal trajectory was used for the simulation. The study found that the vehicle under the controller could track the desired trajectory at the highest speed to achieve accurate tracking of the desired trajectory. However, the velocity and input signal errors contained oscillations, which was a significant shortcoming of the algorithm. In addition, there was no simulation comparison study with another controller. The problem of marine surface vessels (MSVs) tracking control in the presence of uncertain dynamics and external disturbances was investigated in [38]. The facts that actuators are subject to undesirable faults and input saturation are taken into consideration. Moreover, the event-triggered control (ETC) technique was applied. The proposed ETC was compared using simulations with the continuous control scheme for a ship model called CyberShip II as an example. The results were for only one trajectory and the tracking errors were not significantly different in both cases. [PERSON] et al.