This work presents a new design to dynamically adapt the proportional, the integral and the derivative (PID) controller’s gains using three interval type-1 non-singleton type-2 fuzzy logic systems (IT2 NSFLS-1), one fuzzy system for each gain of the PID, being the first main contribution of this proposal. This assembly is named as hybrid IT2 NSFLS-1 PID. Each IT2 NSFLS-1 system requires two non-singleton input values each period of discrete time k\documentclass[12pt]{minimal}
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\begin{document}$$ \left( \varvec{k} \right) $$\end{document}, (1) the error ek\documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{e}\left( \varvec{k} \right) $$\end{document} and its standard deviation σek\documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{\sigma e}\left( \varvec{k} \right) $$\end{document}, and (2) the change of error Δek\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta \varvec{e}\left( \varvec{k} \right) $$\end{document} and its standard deviation σΔek\documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{\sigma}\Delta \varvec{e}\left( \varvec{k} \right) $$\end{document}, to calculate the corresponding adjustment ΔKPk\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta \varvec{KP}\left( \varvec{k} \right) $$\end{document}, ΔKIk\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta \varvec{KI}\left( \varvec{k} \right) $$\end{document}, and ΔKDk\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta \varvec{KD}\left( \varvec{k} \right) $$\end{document} for the PID controller’s gains Kpk\documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{K}_{\varvec{p}} \left( \varvec{k} \right) $$\end{document}, Kik\documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{K}_{\varvec{i}} \left( \varvec{k} \right) $$\end{document}, and Kdk\documentclass[12pt]{minimal}
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\begin{document}$$ \varvec{K}_{\varvec{d}} \left( \varvec{k} \right) $$\end{document}. The second main contribution of this proposal is that the parameters of each IT2 NSFLS-1 system are tuned each period of discrete time k\documentclass[12pt]{minimal}
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\begin{document}$$ \left( \varvec{k} \right) $$\end{document} by the non-singleton backpropagation (BP) algorithm using the plant output error and its standard deviation, which are processed as non-singleton values together with its non-singleton partial derivatives with respect to each IT2 fuzzy system parameter. Then these updated gains are used by the PID controller to calculate the best control signal for the plant under control. The uncertainty and the mean value of the measurement are used to calculate the non-singleton error which is processed as (a) input and (b) as gradient vector by each of the three IT2 NSFLS-1 systems. Simulation results show that the proposed hybrid assembly presents the better performance than the next five benchmarking control systems (a) the classic Zeigler–Nichols PID controller, and (b) four hybrid assemblies using PID controller and fuzzy systems with fixed fuzzy rule bases (T1 SFLS, T1 NSFLS, IT2 SFLS, IT2 NSFLS-1). The proposed assembly produces the better performance in a shortest period of time and it maintains a stable behavior on the output of the second-order plant model subject to variations and noise.
