## 4 Conclusions  
For the tracking control problem of the inverted pendulum on a cart, we proposed an adaptive fuzzy controller with a full-state feedback, which does not require any knowledge of the plant and uses the four states of the system: positions and velocities of both the pendulum and the cart. Stability and boundedness of solutions for the overall closed-loop system for this original controller were proven through a Lyapunov synthesis approach. The scheme was successfully implemented via simulation and real-time experiments, getting outstanding results and showing the feasibility of the controller, in addition of verifying the theoretical statements achieved here. Those are some of the contributions from this paper regarding previous works in which the problem is addressed as a second-order system based on feedback linearization dealing only with the rod dynamics and leading to the presence of unstable zero dynamics (the cart dynamics). This, consequently, might conduct to undesired behavior of the cart and infeasibility for experimental implementation. The previous works in which the four states of the system are taken into account do not present a study of the boundedness or stability of the signals of the closed-loop system.  
As an added value, an optimization tuning was investigated via evolutionary algorithms; specifically PSO, FFA and DE were used. Everyone tuned the parameters of the fuzzy systems in a model-based simulation, and then those optimized parameters were used in real-time experiments. The performance of the controller was notoriously improved with respect to a previous heuristic tuning. The exportation of the optimization results in simulation to physical experiments represents another key contribution since in control applications usually most of the works do not show a validation of the optimization procedures for the real physical system.  
Finally, despite the vastly documented fact that in general all EA are efficient and that each one can be more or less suitable for a given problem, in this work in particular, it is worth concluding that PSO had the best application based on its efficiency, ease implementation and fast convergence, without discrediting the results reached by DE and FFA as well as their capabilities.  
Conceptualization, M.L. and A.F.; data curation, M.L. and A.F.; formal analysis, M.L., A.F., and V.S.; funding acquisition, M.L., R.G.-H., and V.S.; investigation, M.L., A.F., R.G.-H., and V.S.; methodology, M.L., A.F., R.G.-H., and V.S.; project administration, M.L. and R.G.-H.; software, A.F.; supervision, M.L.; validation, M.L., A.F., and V.S.; visualization, R.G.-H.; writing--original draft, M.L.; writing--review and editing, R.G.-H. and V.S. All authors have read and agreed to the published version of the manuscript.  
This research work was funded by Tecnologico Nacional de Mexico (TecNM) grants.  
This work was done with the support of Tecnologico Nacional de Mexico (TecNM) projects.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} ACO-NM & Ant Colony Optimization method with a constrained Nelder-Mead algorithm \\ DC & Direct Current \\ DE & Differential Evolution \\ EA & Evolutionary Algorithms \\ FFA & Firefly Algorithm \\ ICP & Inverted Cart-Pendulum \\ I/O & Input-Output \\ LMI & Linear Matrix Inequality \\ LQR & Linear Quadratic Regulator \\ PD & Proportional-Derivative \\ PID & Proportional-Integral-Derivative \\ PSO & Particle Swarm Optimization \\ TSFSC & Takagi-Sugeno Fuzzy Servo Control \\ \end{tabular}