# A Data-Driven Approach of Takagi-Sugeno Fuzzy Control of Unknown Nonlinear Systems
## 3 Identification of Takagi-Sugeno Fuzzy Model  
The T-S fuzzy model is expected to attain a good approximation of not only the local dynamics of the underlying system to facilitate local compensator design, but also the global dynamics to guarantee the overall control performance. However, considering them together is difficult because building constituent local linear models from input-output data is not always straightforward and the tradeoff between the local and global accuracy should be carefully addressed [2]. In this paper, a novel approach that utilizes the neuro-fuzzy model to obtain a T-S fuzzy model is proposed. The neuro-fuzzy models are based on the fusion of fuzzy inference systems and neural networks. Among diverse neuro-fuzzy models, the one proposed by [PERSON] and [PERSON] [20], which is also known as fuzzy basis function network (FBFN), has gained much attention as it has a similar structure to radial basis function networks (RBFNs) and thus can adopt the training methods that are already established for the RBFN. It has been proved that the FBFNs can uniformly approximate any real continuous nonlinear functions on a compact set to arbitrary accuracy [21]. First, from the input-output data, a set of multi-input-single-output FBFNs with product fuzzy inference, Gaussian membership functions (MFs), and centroid defuzzifier are trained by the least square algorithm [22] to approximate the nonlinear system whose analytical mathematical model is not available. Each FBFN can be represented as a fuzzy system in layered network form (see Figure 1), and is used to approximate the dynamics of one state variable. For example, the FBFN for the \(p\)-th state variable is constructed using fuzzy rules as:  
\[\mathbf{R}_{p}^{l}:\text{ IF }x_{1}\text{ is }\mathbf{A}_{1p}^{l},\text{ }\cdots,\text{ }x_{n}\text{ is }\mathbf{A}_{np}^{l}\text{ and }u_{1}\text{ is }\mathbf{B}_{1p}^{l},\text{ }\cdots,\text{ }u_{m}\text{ is }\mathbf{B}_{mp}^{l},\text{ THEN}\dot{x}_{p}=w_{p}^{l},\]  
where \(l\in\{1,\text{ }2,\text{ }\cdots,\text{ }M_{p}\}\), and \(Mp\) is the number of fuzzy rules for the \(p\)-th FBFN. Through fuzzy inference and defuzzification, the FBFN based on the above fuzzy rules can be written as:  
\[\dot{x}_{p}=f_{p}(\mathbf{x},\mathbf{u})=\frac{\sum\limits_{l=1}^{M_{p}}w_{p} ^{l}\prod\limits_{i=1}^{n}e^{-\frac{2}{2}(\frac{\dot{x}_{i}-l_{i}^{l}}{x_{ip}^ {l}})}\sum\limits_{j=1}^{m}e^{-\frac{1}{2}(\frac{\dot{u}_{j}-l_{j}^{l}}{x_{ip}^  
\[ \begin{array}{c}\text{Gaussian MFs }e^{-\frac{1}{2}(\frac{x_{i}-x_{ip}^{l}}{\sigma_{ip} ^{l}})^{2}}\\ \text{of Gaussian MFs }e^{-\frac{1}{2}(\frac{x_{i}-x_{ip}^{l}}{\sigma_{ip}^{l}})^{2}} \end{array} \]  
\[ \begin{array}{c}\text{Gaussian MFs }e^{-\frac{1}{2}(\frac{x_{i}-x_{ip}^{l}}{ \sigma_{ip}^{l}})^{2}}\\ \text{of Gaussian MFs }e^{-\frac{1}{2}(\frac{x_{i}-x_{ip}^{l}}{\sigma_{ip}^{l}})^{2}} \end{array} \]  
Next, the T-S fuzzy model is derived from the neuro-fuzzy model by linearizing Equation (2). Linearization about an operating point (\(\textbf{x}_{k}\), \(\textbf{u}_{k}\)) results in  
\[\dot{\textbf{x}}=\textbf{A}_{k}(\textbf{x}-\textbf{x}_{k})+\textbf{B}_{k}( \textbf{u}-\textbf{u}_{k})+\textbf{f}(\textbf{x}_{k},\textbf{u}_{k})+H.O.T, \tag{3}\]  
where \(H.O.T\) denotes the higher order terms in the local model, which will be neglected in the following analysis, \(\textbf{f}(\textbf{x}_{k},\textbf{u}_{k})=[f_{1}(\textbf{x}_{k},\textbf{u}_{ k}),\ldots,f_{n}(\textbf{x}_{k},\textbf{u}_{k})]^{T}\), and  
\[\textbf{A}_{k}=\left[ \begin{array}{ccc}\frac{\partial f_{1}(\textbf{x}, \textbf{u})}{\partial x_{1}}&\ldots&\frac{\partial f_{1}(\textbf{x},\textbf{u} )}{\partial x_{n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{n}(\textbf{x},\textbf{u})}{\partial x_{1}}&\ldots&\frac{ \partial f_{n}(\textbf{x},\textbf{u})}{\partial x_{n}}\end{array}