# Fuzzy Adaptive Asymptotic Control for a Class of Large-Scale High-Order Unknown Nonlinear Systems
## 2 Preliminaries and Problem Formulation  
The LSHONS is modeled as follows:  
\[\left\{ \begin{array}{ll}\hat{x}_{k,r_{k}}&=x_{k,(k_{r}+1)}+f_{k,r_{k}}(\hat{ x}_{k,r_{k}}),\\ \hat{x}_{k,r_{k}}&=u_{k}+f_{k,r_{k}}(\hat{x}_{k,r_{k}}),k=1,\ldots,N,r_{k}=1, \ldots,n_{k}-1\end{array} \right. \tag{1}\]  
where \(\hat{x}_{k,r_{k}}=[x_{k,1},\ldots,x_{k,r_{k}}]^{T},\hat{x}_{k,r_{k}}=[x_{k,1}, \ldots,x_{k,r_{k}}]^{T}\in\Re^{n},n=n_{1}+\ldots+n_{N}\) denote the system states; \(f_{k,r_{k}}(\cdot):\Re^{n_{k}}\rightarrow\Re^{n_{k}}\) denotes the unknown smooth functions; \(n_{k}\) describes the order of the \(k\)th subsystem; \(u_{k}\in\Re^{N}\) and \(y_{k}\in\Re^{N}\) describe the control input and output; \(N\) represents the number of subsystems. Due to the nonlinear terms \(f_{k,r_{k}}(\cdot):\Re^{n_{k}}\rightarrow\Re^{n_{k}}\) are unknown functions, which cannot be used to design a controller directly. In this paper, \(f_{k,r_{k}}(\cdot)\) is approximated and parameterized by FLSs, and its detailed definition is introduced in Lemma 1. [[30]] If function \(g(z)\) defined in a compact set \(\Omega_{z}\) is continuous, there exists an FLS \(Q^{\mathrm{T}}\Psi(z)\) that causes  
\[\sup_{z\in\Omega_{z}}\Bigl{|}g(z)-Q^{\mathrm{T}}\Psi(z)\Bigr{|}\leq\tilde{ \epsilon}^{*} \tag{2}\]  
where \(z=[z_{1}(t),z_{2}(t),\cdots,z_{n}(t)]^{\mathrm{T}}\in\Re^{n}\) is an input vector; and \(\tilde{\epsilon}^{*}\) denotes the bounds of the approximation error. \(Q=[q_{1},\ldots,q_{m}]^{\mathrm{T}}\) describes the fuzzy basic function, and \(m\) describes the number of fuzzy rules. \(q_{p}(p=1,\ldots,m)\) can be expressed as \(q_{p}=\frac{\text{exp}\left(\frac{-(z-p_{p})^{\mathrm{T}}(z_{-p})}{\tilde{x}_{ p}^{\mathrm{T}}}\right)}{\sum_{p=1}^{m}\text{exp}\left(\frac{-(z-p_{p})^{ \mathrm{T}}(z_{-p})}{\tilde{x}_{p}^{\mathrm{T}}}\right)}\), \(1\leq p\leq m\) with center vector \(\mu_{p}=[\mu_{p1},\mu_{p2},\ldots,\mu_{pn}]^{\mathrm{T}}\) if a Gaussian function is chosen as the fuzzy membership function. _If an optimal vector \(Q^{*}\) exists, which means \(Q^{*}:=\arg\min_{Q\in\Omega_{Q}}\Big{\{}\sup_{z\in\Omega_{z}}\big{|}g(z)-Q^{T} \Psi(z)\big{|}\Big{\}}\), \(g(z)\) is described as_  
\[g(z)=Q^{*}\Psi(z)+\varepsilon^{*}(z) \tag{3}\]  
_where \(Q^{*}\in\Omega_{Q^{*}}\), \(z\in\Omega_{z}\); \(\Omega_{Q^{*}}\) and \(\Omega_{z}\) are compact sets; \(\varepsilon^{*}(z)\) satisfies \(|\varepsilon^{*}(z)|\leq\tilde{\varepsilon}^{*}\)._  
Leveraging Lemma 1, nonlinear terms are approximated by FLSs, and (1) is written as:  
\[\left\{ \begin{array}{ll}\dot{x}_{k_{rR}}&=x_{k_{rR}+1}+{Q^{*}_{k_{rR}}}^{T} \Psi_{k_{rR}}(\tilde{x}_{k_{rR}})+\varepsilon_{k_{rR}}(\tilde{x}_{k_{rR}}),\\ \dot{x}_{k_{rR}}&=u_{k}+{Q^{*}_{k_{rR}}}^{T}\Psi_{k_{rR}}(\tilde{x}_{k_{rR}})+ \varepsilon_{k_{rR}}(\tilde{x}_{k_{rR}})\end{array} \right. \tag{4}\]  
where \(|\varepsilon_{k_{rR}}(\tilde{x}_{k_{rR}})|\leq\tilde{\varepsilon}_{k_{rR}}\leq\infty\) with \(\tilde{\varepsilon}_{k_{rR}}\) is an unknown small positive constant. This paper aims to design \(u_{k}\) for each subsystem without an accurate nonlinear model such that the following conditions are met: \(u_{k}\) only contains system states; \(u_{k}\) can avoid the \"explosion of complexity\" problem of the backstepping technique for a high-order system using the DSC technique; and the approximation error caused by FLSs and the boundary layer error caused by DSC can be compensated such that \(y_{k}\) can track the reference trajectory \(y_{k,d}\), asymptotically. Thus, the boundedness of all states in the closed-loop system can be guaranteed. To design the controller and achieve the above objectives, the following Assumption and Lemmas are given. **Assumption 1**. : _We assume \(y_{k,d}\) and its derivatives \(y^{1}_{k,d},y^{2}_{k,d},\ldots,y^{n}_{k,d}\) are bounded and continuous._  
**Lemma 2** ([31]). : _For any \(\eta\in\Re\), inequality \(0\leq|\eta|-\eta\tanh(\frac{\eta}{\sigma_{0}})\leq\alpha\sigma_{0}\) is satisfied, where \(\sigma_{0}>0\), \(\alpha=e^{-(\alpha+1)},\alpha=0.2785\)._  
**Lemma 3** ([32]).