A support vector-based interval type-2 fuzzy system

In this paper, a new fuzzy regression model that is supported by support vector regression is presented. Type-2 fuzzy systems are able to tackle applications that have significant uncertainty. However general type-2 fuzzy systems are more complex than type-1 fuzzy systems. Support vector machines are similar to fuzzy systems in that they can also model systems that are non-linear in nature. In the proposed model the consequent parameters of type-2 fuzzy rules are learnt using support vector regression and an efficient closed-form type reduction strategy is used to simplify the computations. Support vector regression improved the generalisation performance of the fuzzy rule-based system in which the fuzzy rules were a set of interpretable IF-THEN rules. The performance of the proposed model was demonstrated by conducting case studies for the non-linear system approximation and prediction of chaotic time series. The model yielded promising results and the simulation results are compared to the results published in the area

Interval linear systems as a necessary step in fuzzy linear systems

International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

A support vector-based interval type-2 fuzzy system

In this paper, a new fuzzy regression model that is supported by support vector regression is presented. Type-2 fuzzy systems are able to tackle applications that have significant uncertainty. However general type-2 fuzzy systems are more complex than type-1 fuzzy systems. Support vector machines are similar to fuzzy systems in that they can also model systems that are non-linear in nature. In the proposed model the consequent parameters of type-2 fuzzy rules are learnt using support vector regression and an efficient closed-form type reduction strategy is used to simplify the computations. Support vector regression improved the generalisation performance of the fuzzy rule-based system in which the fuzzy rules were a set of interpretable IF-THEN rules. The performance of the proposed model was demonstrated by conducting case studies for the non-linear system approximation and prediction of chaotic time series. The model yielded promising results and the simulation results are compared to the results published in the area

General Dual Fuzzy Linear Systems

Abstract This paper mainly intends to discuss the solution of fully fuzzy linear systems (FFLS) Ax + b = Cx + d, where A and C are fuzzy matrices, b and d are fuzzy vectors. We transform the systems by using fuzzy numbers with a new parametric form for finding a fuzzy vector x that satisfies in the system

Expert supervision of conventional control systems

The objective of this paper is to outline a general concept for the design of supervising fuzzy controllers to back up or monitor a conventzonal control system. The use of fuzzy logic in an external, hierarchacal control structure provides a systematic approach to integrate heuristics in a conventional control loop. Supervising techniques become especially interesting, when the system to be controlled is highly non-linear (parameter variation, saturation of the control surfaces etc.). By the means of two application examples it will be shown, how this method can effectively be used to improve the performance of a conventional control system. Both examples are part of an extended research project that is being carried out at Akrospatiale and E.N.S.I.C.A. in France to study the role of fuzzy control for potential applications in aircraft control systems

Computational studies of some fuzzy mathematical problems

In modelling and optimizing real world systems and processes, one usually ends up with a linear or nonlinear programming problem, namely maximizing one or more objective functions subject to a set of constraint equations or inequalities. For many cases, the constraints do not need to be satisfied exactly, and the coefficients involved in the model are imprecise in nature and have to be described by fuzzy numbers to reflect the real world nature. The resulting mathematical programming problem is referred to as a fuzzy mathematical programming problem.Over the past decades, a great deal of work has been conducted to study fuzzy mathematical programming problems and a large volume of results have been obtained. However, many issues have not been resolved. This research is thus undertaken to study two types of fuzzy mathematical programming problems. The first type of problems is fuzzy linear programming in which the objective function contains fuzzy numbers. To solve this type of problems, we firstly introduce the concept of fuzzy max order and non-dominated optimal solution to fuzzy mathematical programming problems within the framework of fuzzy mathematics. Then, based on the new concept introduced, various theorems are developed, which involve converting the fuzzy linear programming problem to a four objective linear programming problem of non-fuzzy members. The theoretical results and methods developed are then validated and their applications for solving fuzzy linear problems are demonstrated through examples.The second type of problems which we tackle in this research is fuzzy linear programming in which the constraint equations or inequalities contain fuzzy numbers. For this work, we first introduce a new concept, the α-fuzzy max order. Based on this concept, the general framework of an α-fuzzy max order method is developed for solving fuzzy linear programming problems with fuzzy parameters in the constraints. For the special cases in which the constraints consist of inequalities containing fuzzy numbers with isosceles triangle or trapezoidal membership functions, we prove that the feasible solution space can be determined by the respective 3n or 4n non-fuzzy inequalities. For the general cases in which the constraints contain fuzzy numbers with any other form of membership functions, robust numerical algorithms have been developed for the determination of the feasible solution space and the optimal solution to the fuzzy linear programming problem in which the constraints contain fuzzy parameters. Further, by using the results for both the first and second types of problems, general algorithms have also been developed for the general fuzzy linear programming problems in which both the objective function and the constraint inequalities contain fuzzy numbers with any forms of membership functions. Some examples are then presented to validate the theoretical results and the algorithms developed, and to demonstrate their applications

Nonlinear Control and Estimation with General Performance Criteria

This dissertation is concerned with nonlinear systems control and estimation with general performance criteria. The purpose of this work is to propose general design methods to provide systematic and effective design frameworks for nonlinear system control and estimation problems. First, novel State Dependent Linear Matrix Inequality control approach is proposed, which is optimally robust for model uncertainties and resilient against control feedback gain perturbations in achieving general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. By solving a state dependent linear matrix inequality at each time step, the sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this dissertation unify existing results on nonlinear quadratic regulator, Hinfinity and positive real control. Secondly, an H2-Hinfinity State Dependent Riccati Equation controller is proposed in this dissertation. By solving the generalized State Dependent Riccati Equation, the optimal control solution not only achieves the optimal quadratic regulation performance, but also has the capability of external disturbance reduction. Numerically efficient algorithms are developed to facilitate effective computation. Thirdly, a robust multi-criteria optimal fuzzy control of nonlinear systems is proposed. To improve the optimality and robustness, optimal fuzzy control is proposed for nonlinear systems with general performance criteria. The Takagi-Sugeno fuzzy model is used as an effective tool to control nonlinear systems through fuzzy rule models. General performance criteria have been used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be achieved by solving the LMI at each time step. Lastly, since any type of controller and observer is subject to actuator failures and sensors failures respectively, novel robust and resilient controllers and estimators are also proposed for nonlinear stochastic systems to address these failure problems. The effectiveness of the proposed control and estimation techniques are demonstrated by simulations of nonlinear systems: the inverted pendulum on a cart and the Lorenz chaotic system, respectively

Zonotopic fault detection observer design for Takagi–Sugeno fuzzy systems

This paper considers zonotopic fault detection observer design in the finite-frequency domain for discrete-time Takagi–Sugeno fuzzy systems with unknown but bounded disturbances and measurement noise. We present a novel fault detection observer structure, which is more general than the commonly used Luenberger form. To make the generated residual sensitive to faults and robust against disturbances, we develop a finite-frequency fault detection observer based on generalised Kalman–Yakubovich–Popov lemma and P-radius criterion. The design conditions are expressed in terms of linear matrix inequalities. The major merit of the proposed method is that residual evaluation can be easily implemented via zonotopic approach. Numerical examples are conducted to demonstrate the proposed methodPeer ReviewedPostprint (author's final draft