**7. Conclusions**

The heating behavior of the distillation column depends mostly on the boiler and its heating actuator, which must be monitored to guarantee its adequate operation.

In this work, a fuzzy model and two fuzzy observer to estimate the inductor current and capacitor voltage in a DC-DC Buck-Boosy power converter, used to regulate the heating power in a distillation column boiler, are presented.

The observer is based on the Takagi–Sugeno fuzzy model of four rules. The gains are calculated by means of LMIs in order to guarantee the stability for each of the closed loop linear subsystems.

The fuzzy model and fuzzy observers are validated in simulation. Different tests were carried out with ideal and different initial conditions between the nonlinear system and the fuzzy observer, as well as disturbances in the nonlinear system, in order to validate the fuzzy observer convergence.

The Takagi–Sugeno fuzzy observer has an adequate response as long as the disturbances occur in the operating points selected in its designing stage. By adding the sliding-mode term, the fuzzy observer increases its robustness under load and input voltage perturbations, making it suitable to be applied in different control strategies, such as Fault Detection and Isolation (FDI) and Fault Tolerant Control (FTC) systems.

**Author Contributions:** Conceptualization, J.A.-M. and A.d.C.T.-A.; methodology and formal analysis, M.H.-C.; writing—review and editing, A.d.C.T.-A. and M.H.-C.; supervision, E.E.-J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received funding from Scientific Research Coordination, UMSNH (Conv. 2016–2017) and Conacyt gran<sup>t</sup> number 401203.

**Acknowledgments:** Authors thank Juan Jesús Silva Romero for the experimentation support.

**Conflicts of Interest:** The authors declare no conflict of interest.
