# Fuzzy Adaptive Asymptotic Control for a Class of Large-Scale High-Order Unknown Nonlinear Systems
## 1 Introduction  
The nonlinear control problem has been widely studied because many practice systems are usually described as nonlinear models [1; 2; 3], such as permanent magnet synchronous motors [4], induction motor systems [5; 6], and four-tank systems [7]. To achieve a better control effect, a series of nonlinear control techniques, such as adaptive control and the backstepping technique, are developed [8; 9; 10; 11; 12]. An adaptive backstepping method, which comes from the combination of adaptive control and the backstepping technique, is an important technique to handle the parameters uncertainty of high-order nonlinear systems [13; 14; 15; 16]. However, the above methods were proposed for the system with the accurate nonlinear model or the nonlinear system with unknown parameters. In addition, at each step of backstepping, a virtual control law should be differentiated. The repeated differentiation can lead to an \"explosion of complexity\" problem such that the traditional backstepping method cannot be used in practice. Therefore, this paper focuses on a class of high-order nonlinear systems control problems with unmodeled nonlinear term, low complexity, and asymptotic tracking performance. A large-scale system is modeled to describe a group of interrelated subsystems, thereby reducing control complexity and improving the application feasible for multiple nonlinear systems [17; 18]. Therefore, a decentralized control strategy has been applied to many large-scale nonlinear systems in the existing research, and it has received constant attention from different perspectives [19; 20; 21]. To obtain the prescribed tracking performance, a nonlinear feedback function was constructed and used instead of the linear feedback function to improve the dynamic performance for uncertain MIMO nonlinear systems in [22]. To ensure tracking performance, a modified DSC method was proposed by introducing a differentiator in [23]. An adaptive DSC algorithm is developed to optimize the convergence speed of the tracking error in [24]. For large-scale systems, a decentralized stabilizer is designed, and the transient performance is also analyzed and evaluated by the \(L_{2}\) boundof he tracking error in [25]. However, the above research is based on the nonlinear system that can be modeled. In order to realize the necessary accuracy for a system model, neural networks or fuzzy logic are used to parameterize nonlinear terms or approximate the control input directly in recent research. In [26], a neural network is directly used to design a controller such that the unknown nonlinear term was not used in the controller design procedure. For an LSHONS with unknown interconnections and non-affine nonlinear subsystems, RBF NNs are used to develop a controller whose stability is analyzed by the Lyapunov stability theory in [27]. For a large-scale non-affine nonlinear system, an adaptive fuzzy method, where only the system state is used for feedback, is developed to achieve the stable control in [28]. For strong interconnections in large-scale nonlinear systems, a nonlinearly parameterized NNs is used to design a controller such that the a priori knowledge for the nonlinear term is released in [29]. The above researchers handle the nonlinear control problem without an accurate system model, but all of these only achieve the bounded tracking performance. The main reasons are (1) the approximation error caused by neural networks and fuzzy logic and (2) the boundary layer error caused by the DSC technique for LSHONS control. How to design an asymptotic tracking controller without an accurate system model for LSHONS plays an important role in theory and practice. This paper develops a fuzzy adaptive DSC method for LSHONS.