A fuzzy state-dependent Riccati equation control: adaptive tuning of the state weighting matrix

The tuning of the state-dependent Riccati equation (SDRE) is achieved by selecting/adjusting weighting matrices for states, Q(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {Q(x)}}$$\end{document}, and inputs, R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {R(x)}}$$\end{document}. The tuning results in a trade-off between the performance of the control system and energy consumption. An increase in Q(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {Q(x)}}$$\end{document} or a decrease in R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {R(x)}}$$\end{document} enhances the amplitude of the input signals and also reduces the error. In order to adjust the SDRE control law with high precision, the input signal may face saturation at the beginning of the regulation (point-to-point motion) since the error is at the highest value there. The proposed fuzzy tuning method adapts the weighting matrix of states, Q(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {Q(x)}}$$\end{document}, according to the amplitude of the error and avoids saturation of the input signal. It also increases the precision during the steady-state phase of regulation without significantly increasing energy consumption. To assess the performance of the proposed closed-loop system, a blood glucose control case study is modeled and simulated to apply this approach as a challenging and important application; besides, a planar manipulator is simulated as an illustrative example to show the effectiveness of the fuzzy mechanism in tuning and error. The results illustrated that the proposed fuzzy-tuned SDRE controller obtained less error in point-to-point control with a smoother input signal.