# A Data-Driven Approach of Takagi-Sugeno Fuzzy Control of Unknown Nonlinear Systems
###### Abstract  
A novel approach to build a Takagi-Sugeno (T-S) fuzzy model of an unknown nonlinear system from experimental data is presented in the paper. The neuro-fuzzy models or, more specifically, fuzzy basis function networks (FBFNs) are trained from input-output data to approximate the nonlinear systems for which analytical mathematical models are not available. Then, the T-S fuzzy models are derived from the direct linearization of the neuro-fuzzy models. The operating points for linearization are chosen using the evolutionary strategy to minimize the global approximation error so that the T-S fuzzy models can closely approximate the original unknown nonlinear system with a reduced number of linearizations. Based on T-S fuzzy models, optimal controllers are designed and implemented for a nonlinear two-link flexible joint robot, which demonstrates the possibility of implementing the well-established model-based optimal control method onto unknown nonlinear dynamic systems. Takagi-Sugeno model; data-driven system identification; neuro-fuzzy model; optimal control; flexible robot +
Footnote †: 2020  
rules based on clustering [13; 14]. With the determined antecedent structural parameters, the T-S fuzzy model transforms into a set of linear models, of which the parameters are obtained by the recursive least square method [15; 16], genetic algorithm [17] or particle swarm optimization [18]. The objective of data-driven approaches is to minimize the global prediction error of a T-S fuzzy model. However, they may result in constituent linear models significantly different from the local linearization of nonlinear systems, though they may offer good global performance [19]. Hence, the T-S models obtained by these methods might not be satisfactory for the controller design in the Parallel Distributed Compensation (PDC) scheme, since the local compensators need to be designed based on local linear models. The authors in [2] studied various identification algorithms and concluded that the constrained and regularized identification method can improve the interpretability of constituent local models as local linearization, and the locally weighted least square technique may facilitate the compromise between the local and global accuracy of T-S models. However, the effectiveness and practicality of this method were demonstrated only by very simple examples. When considering more complicated higher-order and multivariate problems, the issues related to interpretability and identifiability will be more pronounced and difficult to address. An alternative method to circumvent these difficulties is presented in this paper. A neuro-fuzzy model referred to as fuzzy basis function networks (FBFN) is adopted to approximate the unknown nonlinear systems [20]. The Stone-Weierstrass theorem proves that this kind of neuro-fuzzy model could approximate any real continuous function on a compact domain arbitrarily well [21]. Then, the T-S fuzzy model is derived from the linearization of the neuro-fuzzy model at a series of operating points. Therefore, each local model is close to the local linearization of the original system and, thus, suitable for local compensator design. In addition, the positions of operating points for linearization are optimized by the evolutionary strategy to minimize global approximation error so that the entire T-S fuzzy model is a good global approximation of the original system. A fuzzy control scheme can be applied to the consequent T-S fuzzy model in which a local optimal compensator is designed for each of the local affine models and the overall control action is derived from the fuzzy inferencing of local control actions. The contribution of this paper is that it presents a practical way to build T-S fuzzy models with both good local and global approximations by deriving the direct linearization of FBFN models and introducing the evolutionary strategy for fuzzy rule location selection.