Measures of uncertainty for a fuzzy probabilistic approximation space

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

Incorporating model predictive control with fuzzy approximation for robot manipulation under remote center of motion constraint

Remote center of motion (RCM) constraint has attracted many research interests as one of the key challenges for robot-assisted minimally invasive surgery (RAMIS). Although it has been addressed by many studies, few of them treated the motion constraint with an independent workspace solution, which means they rely on the kinematics of the robot manipulator. This makes it difficult to replicate the solutions on other manipulators, which limits their population. In this paper, we propose a novel control framework by incorporating model predictive control (MPC) with the fuzzy approximation to improve the accuracy under the motion constraint. The fuzzy approximation is introduced to manage the kinematic uncertainties existing in the MPC control. Finally, simulations were performed and analyzed to validate the proposed algorithm. By comparison, the results prove that the proposed algorithm achieved success and satisfying performance in the presence of external disturbances.

Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

Numerous research papers and several engineering applications have proved that the fuzzy set theory is an intelligent effective tool to represent complex uncertain information. In fuzzy multi-criteria decision-making (fuzzy MCDM) methods, intelligent information system and fuzzy control-theoretic models, complex qualitative information are extracted from expert’s knowledge as linguistic variables and are modeled by linear/non-linear fuzzy numbers. In numerical computations and experiments, the information/data are fitted by nonlinear functions for better accuracy which may be little hard for further processing to apply in real-life problems. Hence, the study of non-linear fuzzy numbers through triangular and trapezoidal fuzzy numbers is very natural and various researchers have attempted to transform non-linear fuzzy numbers into piecewise linear functions of interval/triangular/trapezoidal in nature by different methods in the past years. But it is noted that the triangular/trapezoidal approximation of nonlinear fuzzy numbers has more loss of information. Therefore, there is a natural need for a better piecewise linear approximation of a given nonlinear fuzzy number without losing much information for better intelligent information modeling. On coincidence, a new notion of Generalized Hexagonal Fuzzy Number has been introduced and its applications on Multi-Criteria Decision-Making problem (MCDM) and Generalized Hexagonal Fully Fuzzy Linear System (GHXFFLS) of equations have been studied by Lakshmana et al. in 2020. Therefore, in this paper, approximation of nonlinear fuzzy numbers into the hexagonal fuzzy numbers which includes trapezoidal, triangular and interval fuzzy numbers as special cases of Hexagonal fuzzy numbers with less loss/gain of information than other existing methods is attempted. Since any fuzzy information is satisfied fully by its modal value/core of that concept, any approximation of that concept is expected to be preserved with same modal value/core. Therefore, in this paper, a stepwise procedure for approximating a non-linear fuzzy number into a new Hexagonal Fuzzy Number that preserves the core of the given fuzzy number is proposed using constrained nonlinear programming model and is illustrated numerically by considering a parabolic fuzzy number. Furthermore, the proposed method is compared for its efficiency on accuracy in terms of loss of information. Finally, some properties of the new hexagonal fuzzy approximation are studied and the applicability of the proposed method is illustrated through the Group MCDM problem using an index matrix (IM).

Adaptive Fuzzy Approximation Control of PV Grid-Connected Inverters

Three-phase inverters are widely used in grid-connected renewable energy systems. This paper presents a new control methodology for grid-connected inverters using an adaptive fuzzy control (AFC) technique. The implementation of the proposed controller does not need prior knowledge of the system mathematical model. The capabilities of the fuzzy system in approximating the nonlinear functions of the grid-connected inverter system are exploited to design the controller. The proposed controller is capable to achieve the control objectives in the presence of both parametric and modelling uncertainties. The control objectives are to regulate the grid power factor and the dc output voltage of the photovoltaic systems. The closed-loop system stability and the updating laws of the controller parameters are determined via Lyapunov analysis. The proposed controller is simulated under different system disturbances, parameters, and modelling uncertainties to validate the effectiveness of the designed controller. For evaluation, the proposed controller is compared with conventional proportional-integral (PI) controller and Takagi–Sugeno–Kang-type probabilistic fuzzy neural network controller (TSKPFNN). The results demonstrated that the proposed AFC showed better performance in terms of response and reduced fluctuations compared to conventional PI controllers and TSKPFNN controllers.