# A Data-Driven Approach of Takagi-Sugeno Fuzzy Control of Unknown Nonlinear Systems
## 4 Controller Synthesis for the T-S Fuzzy Model  
In order to design a global stabilizing controller for the nonlinear system using the identified T-S fuzzy model, the parallel distributed compensation (PDC) framework is adopted [25]. Local feedback rule is designed as a compensator for each local model and a global fuzzy controller is constructed by the aggregation of local compensators using the same fuzzy inference system in the T-S fuzzy model [6]: \(\mathbf{R}^{k}\): IF \(z_{l}\) is \(\mathbf{F_{1}}^{k}\), \(\cdots\), \(z_{v}\) is \(\mathbf{F_{v}}^{k}\), THEN \(\mathbf{u}=\mathbf{u}_{k}\), where \(k\in\{1,2,\cdots,p\}\). The fuzzy controller is aggregated as:  
\[\mathbf{u}=\sum_{k=1}^{p}\mu_{k}\mathbf{u}_{k}, \tag{11}\]  
where \(\mu_{k}\) is the normalized membership function same as in Equation (1). Assume that the fuzzy controller is designed to minimize the performance index:  
\[J=\int_{t_{0}}^{t_{f}}\left[(\mathbf{r}-\mathbf{x})^{T}\mathbf{Q}(\mathbf{r}- \mathbf{x})+\mathbf{u}^{T}\mathbf{R}\mathbf{u}\right]dt, \tag{12}\]  
where \(\mathbf{r}\) is the command input vector to be tracked, \(\mathbf{x}\) is the state vector, \(\mathbf{u}\) is the input vector, \(t_{0}\) is the initial time, \(t_{f}\) is the final time and \(\mathbf{Q}\) and \(\mathbf{R}\) are symmetric positive definite matrices to be determined by the designer, then for each local controller, the optimal control action \(\mathbf{u}_{k}\) can be derived as [26]:  
\[\mathbf{u}_{k}=\mathbf{K}_{k}(\mathbf{r}-\mathbf{x})+\left(\mathbf{B}_{k}^{T} \mathbf{B}_{k}\right)^{-1}\mathbf{B}_{k}^{T}\mathbf{d}_{k}, \tag{13}\]  
where \(\mathbf{K}_{k}\) is given by  
\[\mathbf{K}_{k}=\mathbf{R}^{-1}\mathbf{B}_{k}^{T}\mathbf{P}_{k} \tag{14}\]  
Figure 2: Evolutionary Strategy for selecting the operating points of linearization. \(\mathbf{P}_{k}\) is found by solving the Continuous Algebra Riccati Equation:  
\[\mathbf{A}_{k}^{T}\mathbf{P}_{k}+\mathbf{P}_{k}\mathbf{A}_{k}-\mathbf{P}_{k} \mathbf{B}_{k}\mathbf{R}^{-1}\mathbf{B}_{k}^{T}\mathbf{P}_{k}=-\mathbf{Q}. \tag{15}\]  
The stability condition of an affine T-S fuzzy control system based on a quadratic Lyapunov function is given in [27]. The equilibrium point (\(\mathbf{x}=\mathbf{x}_{k}\), \(\mathbf{u}=\mathbf{u}_{k}\)) of the control system is asymptotically stable in large if there exists a common positive definite matrix \(\mathbf{P}=\mathbf{P}^{T}>\mathbf{0}\) and scalars \(\tau_{ijq}\geq 0\) such that:  
\[ \begin{split}&\mathbf{G}_{i,j}{}^{T}\mathbf{P}+\mathbf{P}\mathbf{G }_{i,j}<0\text{ for fuzzy rules }i,j\in\{k|\mu_{k}(0)\
eq 0\}\\ &\left(\begin{array}{cc}\mathbf{G}_{i,j}{}^{T}\mathbf{P}+ \mathbf{P}\mathbf{G}_{i,j}-\sum\limits_{q=1}^{n}\tau_{ijq}\mathbf{T}_{ijq}& \mathbf{P}\mathbf{n}_{i,j}-\sum\limits_{q=1}^{n}\tau_{ijq}\mathbf{u}_{ijq}\\ \mathbf{n}_{i,j}{}^{T}\mathbf{P}-\sum\limits_{q=1}^{n}\tau_{ijq}\mathbf{u}_{ ijq}{}^{T}&-\sum\limits_{q=1}^{n}\tau_{ijq}\overline{v}_{ijq}\end{array}\right)<0 \text{ for other fuzzy rules}\end{split}  \tag{16}\]  
where the \(\mathbf{G}_{i,j}\), \(\mathbf{n}_{i,j}\), \(\mathbf{T}_{ijq}\), \(\mathbf{u}_{ijq}\) and \(v_{ijq}\) are defined as:  
\[ \begin{split}&\mathbf{G}_{i,j}=\frac{\left\{\mathbf{A}_{i}- \mathbf{B}_{i}\mathbf{K}_{j}\right\}+\left\{\mathbf{A}_{j}-\mathbf{B}_{j} \mathbf{K}_{j}\right\}}{2}\\ &\eta_{i,j}=\frac{\left\{\mathbf{d}_{i}-\mathbf{B}_{i}\sigma_{j} \right\}+\left\{\mathbf{d}_{j}-\mathbf{B}_{j}\sigma_{i}\right\}}{2}\\ &\mathbf{x}^{T}\overline{\mathbf{T}}_{ijq}\mathbf{x}+2\mathbf{u}_{ ijq}{}^{T}\mathbf{x}+v_{ijq}\leq 0\end{split}  \tag{17}\]  
Equation (16) belongs to bilinear matrix inequalities (BMIs) and can be solved in an iterative LMI manner. The details of the iterative linear matrix inequality (ILMI) algorithm can be found in [27]. The stability condition in Equation (16) can be integrated into the ES optimization procedure shown in Figure 2 as a constraint so that the fuzzy control system designed based on the optimized operating point set is guaranteed to be stable. The fuzzy controller is implemented in a full state-feedback manner.