In this paper, two fuzzy classification functions of fuzzy c-means for data with tolerance are proposed. First, two clustering algorithms for data with tolerance are introduced. One is based on the standard method and the other is on the entropy-based one. Second, the fuzzy classification function for fuzzy c-means without tolerance is discussed as the solution of a certain optimization problem. Third, two optimization problems are shown so that the solutions are the fuzzy classification function values for fuzzy c-means algorithms with respect to data with tolerance, respectively. Fourth, Karush-Kuhn-Tucker conditions of two objective functions are considered, and two iterative algorithms are proposed for the optimization problems, respectively. Through some numerical examples, the proposed algorithms are discussed.
In general, the state feedback control gain can be obtained by solving certain linear matrix inequalities (LMIs) when using the Takagi-Sugeno (T-S) fuzzy model to develop a control system. In this paper, the reconstruction error between the real system to be controlled and its T-S fuzzy model, which consists of parameter uncertainties and external disturbance, is considered. As a result, we arrive at an adaptive controller that has two parts: one is obtained by solving certain LMIs (fixed part) and another one is acquired by an adaptive law (variable part). The proposed controller can guarantee the control state to converge and uniformly bounded while maintaining all the signals involved stable. Also, the convergence and boundedness in terms of relaxing the LMIs conservatism are discussed. An inverted pendulum is provided to demonstrate the effectiveness of the proposed adaptive fuzzy controller.