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Table 1 Type-2 fuzzy membership functions

From: A Type-2 Fuzzy Approach to Driver-Automation Shared Driving Lane Keeping Control of Semi-Autonomous Vehicles Under Imprecise Premise Variable

  Lower and upper membership functions
\(f_{1}\)   \(f_{2}\)
\(\Delta = - 10\%\) \(\overline{\alpha }_{{M_{1}^{1} }} = \overline{\alpha }_{{M_{2}^{1} }} = \frac{{f_{1max} - 0.9f_{1} }}{{f_{1max} - f_{1min} }}\) \(DS = 0.5\) \(\underline {\alpha }_{{M_{1}^{2} }} = \underline {\alpha }_{{M_{3}^{2} }} = \frac{{f_{2max} - f_{2} }}{{f_{2max} - f_{2min} }}\)
\(\underline {\alpha }_{{M_{3}^{1} }} = \underline {\alpha }_{{M_{4}^{1} }} = \frac{{0.9f_{1} - f_{1min} }}{{f_{1max} - f_{1min} }}\) \(\underline {\alpha }_{{M_{2}^{2} }} = \underline {\alpha }_{{M_{4}^{2} }} = \frac{{f_{2} - f_{2min} }}{{f_{2max} - f_{2min} }}\)
\(\Delta = 10\%\) \(\underline {\alpha }_{{M_{1}^{1} }} = \underline {\alpha }_{{M_{2}^{1} }} = \frac{{f_{1max} - 1.1f_{1} }}{{f_{1max} - f_{1min} }}\) \(DS = 1\) \(\overline{\alpha }_{{M_{1}^{2} }} = \overline{\alpha }_{{M_{3}^{2} }} = \frac{{f_{2max} - f_{2} }}{{f_{2max} - f_{2min} }}\)
\(\overline{\alpha }_{{M_{3}^{1} }} = \overline{\alpha }_{{M_{4}^{1} }} = \frac{{1.1f_{1} - f_{1min} }}{{f_{1max} - f_{1min} }}\) \(\overline{\alpha }_{{M_{2}^{2} }} = \overline{\alpha }_{{M_{4}^{2} }} = \frac{{f_{2} - f_{2min} }}{{f_{2max} - f_{2min} }}\)