# Compact Fuzzy Systems Based on Boolean Relations
## 8 Conclusions  
This work reviewed the methodology for designing fuzzy inference systems applied to specific cases to have compact fuzzy inference systems based on Boolean relations. The advantage of this methodology consists of presenting the rules of the system as a truth table and establishing the equations for system implementation using Boole and Kleene algebras.  
Possible application cases of the concepts used for the design of compact fuzzy systems based on Boolean relations were presented. These systems can be generalized to have more complex structures without losing interpretability. Through the cases under study, the equivalence is achievable to simplify expressions using Boole and Kleene algebras, which permits various configurations of compact inference systems based on Boolean relations.  
The application examples considered to identify and control nonlinear dynamic systems allow a straightforward application of fuzzy systems based on Boolean relations with interpretability and adaptability. Architecture I allowed for the identification of the system that presents multiplications between the variables that describe its behavior; meanwhile, architecture II allowed for the expansion of the adaptation capacity of a PID controller achieving the desired behavior of this control system considering the desired reference model.  
For the architecture I used to plant identification, the value of the objective function before training is 0.79851; the result after training is equal to \(9.3592\times 10^{-4}\) using 200 iterations. Meanwhile, for architecture II applied to control, the value before the optimization is 0.0876085 for the objective function; after the process, the value obtained is \(4.75608\times 10^{-5}\) using 100 iterations. These results show that the proposed architectures allow the adaptation of their parameters using optimization algorithms.  
The initial configuration to perform the optimization is essential; in this way, the proposal allows for the easy establishment of the initial configurations of the proposed systems. For architecture I, the initial configuration is obtained systematically, distributing the sets in the range of the inputs. For the control application, the sigmoidal fuzzy sets were configured considering the value of the desired output. With these initial configurations, an adequate optimization of the fuzzy systems used for identification and control is achieved.  
The researchers expect to employ compact fuzzy systems to design adaptive neuro-fuzzy control systems in further work. Given the compact structure, these systems make possible the use of Back Propagation and Dynamic Back Propagation algorithms for adaptive applications. Researchers will also consider carrying out MIMO applications using compact subsystems based on Boolean relationships in such a way that different MIMO configurations can be used for identification and control.  
Conceptualization, H.E. and J.S.; Methodology, H.E. and J.S.; Project administration, H.L.; Supervision, H.L.; Validation, I.M.; Writing--original draft, H.E.; Writing--review and editing, H.E. and I.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
This work does not use external data.  
The authors express gratitude to the Universidad de Oviedo, and the Universidad Distrial Francisco Jose de Caldas.  
The authors declare no conflict of interest.