# Experimental Evaluation of a Takagi-Sugeno Fuzzy Controller for an EV3 Ballbot System +
## 5 Conclusions  
From the experimental evaluation results, we conclude that the proposal for the control of the EV3 BRS performs well. So it can be seen that the control objective has been reached considering just two fuzzy rules for the TSFM with membership functions differing from that of the Gaussian type and taking the absolute value for the angular displacement of the EV3 BRS's body as the premise variable for the fuzzy rules. Consequently, the EV3 BRS's TSFM has been validated experimentally, as well as its original nonlinear dynamical model. It has also been confirmed that when following the controller design in a decentralized way,  
Figure 14: Control signal for the actuator aligned with the \(x\)-axis.  
Figure 12: Dynamics of the rotor shaft around the \(x\)-axis corresponding to the displacement of the EV3 BRS on the plane of motion \(y-z\).  
Figure 13: Control signal for the actuator aligned with the \(y\)-axis.  
the control task can be carried out successfully, although for one of the planes of motion, the dynamics are more demanding. Moreover, a disadvantage of the TSFC technique is the lack of approaches for the improvement in the transient dynamics if one tries to reach stability in a fixed time. Furthermore, global asymptotic stability is achieved via a conservative stability condition. In future work, other control design methodologies may be explored in order to evaluate their respective performance in the achievement of the control task in real time.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14104103/s1](https://www.mdpi.com/article/10.3390/app14104103/s1)]([https://www.mdpi.com/article/10.3390/app14104103/s1](https://www.mdpi.com/article/10.3390/app14104103/s1)), Video S1: Real-time closed-loop performance of the TSFC on the EV3 BRS.  
Conceptualization, F.J.; investigation, R.E., F.J. and J.V.-T.; methodology, R.E. and F.J.; formal analysis, R.E.; validation, R.E., F.J. and J.V.-T.; software, R.E.; data curation, R.E.; visualization, R.E., F.J. and J.V.-T.; writing--original draft preparation, R.E. and F.J.; writing--review and editing, F.J.; supervision, F.J. and J.V.-T.; funding acquisition, F.J.; resources, F.J.; project administration, F.J. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Tecnologico Nacional de Mexico (TecNM) projects and, partially, under grant 43433 from the EDD 2023 program. This work was supported by CONACYT, Mexico, under grant 759631.  
Not applicable.  
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} BEMF & Back Electromotive Force \\ BRS & Ball Robot System \\ BS & Ball Segway \\ DC & Direct Current \\ DOF & Degree-of-Freedom \\ DTLQR & Discrete-Time Linear Quadratic Regulator \\ EV3 BRS & EV3 Ballbot Robotic System \\ HSMC & Hierarchical Sliding Mode Control \\ LQR & Linear Quadratic Regulator \\ LMIs & Linear Matrix Inequalities \\ MIMO & Multi-Input Multi-Output \\ NXTBRS & NXT Ballbot Robotic System \\ ODW & Omnidirectional Wheel \\ PDC & Parallel Distributed Compensation \\ PLDI & Polytopic Linear Differential Inclusion \\ PC & Polytopic Controller \\ PM & Polytopic Model \\ TS & Takagi-Sugeno \\ TSFC & Takagi-Sugeno Fuzzy Controller \\ TSFM & Takagi-Sugeno Fuzzy Model \\ \end{tabular}