Applications of fuzzy theories to multi-objective system optimization

Most of the computer aided design techniques developed so far deal with the optimization of a single objective function over the feasible design space. However, there often exist several engineering design problems which require a simultaneous consideration of several objective functions. This work presents several techniques of multiobjective optimization. In addition, a new formulation, based on fuzzy theories, is also introduced for the solution of multiobjective system optimization problems. The fuzzy formulation is useful in dealing with systems which are described imprecisely using fuzzy terms such as, 'sufficiently large', 'very strong', or 'satisfactory'. The proposed theory translates the imprecise linguistic statements and multiple objectives into equivalent crisp mathematical statements using fuzzy logic. The effectiveness of all the methodologies and theories presented is illustrated by formulating and solving two different engineering design problems. The first one involves the flight trajectory optimization and the main rotor design of helicopters. The second one is concerned with the integrated kinematic-dynamic synthesis of planar mechanisms. The use and effectiveness of nonlinear membership functions in fuzzy formulation is also demonstrated. The numerical results indicate that the fuzzy formulation could yield results which are qualitatively different from those provided by the crisp formulation. It is felt that the fuzzy formulation will handle real life design problems on a more rational basis

Sum Normal Optimization of Fuzzy Membership Functions

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a certain shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a small number of variables and the membership optimization problem can be reduced to a parameter optimization problem. This is the approach that is typically taken, but it results in membership functions that are not (in general) sum normal. That is, the resulting membership function values do not add up to one at each point in the domain. This optimization approach is modified in this paper so that the resulting membership functions are sum normal. Sum normality is desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The sum normal constraint is applied in this paper to both gradient descent optimization and Kalman filter optimization of fuzzy membership functions. The methods are illustrated on a fuzzy automotive cruise controller

H-infinity Estimation for Fuzzy Membership Function Optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization

A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretability of a Neuro-Fuzzy Controller

In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binary-real Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The  approach  is  tested  on  two  control examples:  a single  input  single  output (SISO) system  and  a  multivariable (MIMO) system

Aggregated fuzzy answer set programming

Fuzzy Answer Set programming (FASP) is an extension of answer set programming (ASP), based on fuzzy logic. It allows to encode continuous optimization problems in the same concise manner as ASP allows to model combinatorial problems. As a result of its inherent continuity, rules in FASP may be satisfied or violated to certain degrees. Rather than insisting that all rules are fully satisfied, we may only require that they are satisfied partially, to the best extent possible. However, most approaches that feature partial rule satisfaction limit themselves to attaching predefined weights to rules, which is not sufficiently flexible for most real-life applications. In this paper, we develop an alternative, based on aggregator functions that specify which (combination of) rules are most important to satisfy. We extend upon previous work by allowing aggregator expressions to define partially ordered preferences, and by the use of a fixpoint semantics

Design and Rule Base Reduction of a Fuzzy Filter for the Estimation of Motor Currents

Fuzzy systems have been used extensively and successfully in control systems over the past few decades, but have been applied much less often to filtering problems. This is somewhat surprising in view of the dual relationship between control and estimation. This paper discusses and demonstrates the application of fuzzy filtering to motor winding current estimation in permanent magnet synchronous motors. Motor winding current estimation is an important problem because in order to implement effective closed-loop control, a good estimation of the current is needed. Motor winding currents are notoriously noisy because of electrical noise in the motor drive. We use a fuzzy system with correlation-product inference and centroid defuzzification for motor winding current estimation, With the assumption that the membership functions are triangular (but not necessarily symmetric), we then optimize the membership functions using gradient descent. Next we use singular value decomposition to reduce the rule base for the fuzzy filter. Rule base reduction can be important for fuzzy systems in those cases where the fuzzy system needs to be implemented in real time. This is especially true with regard to fuzzy filtering in a real time motor controller. The methods discussed in this paper are demonstrated on real motor winding currents that were collected with a digital oscilloscope. It is demonstrated that fuzzy techniques provide a feasible approach to motor current estimation, that gradient descent optimization improves the performance of the filter, and that rule base reduction results in a relatively small degradation of filter performance. (C) 2000 Elsevier Science Inc. All rights reserved

Design and Rule Base Reduction of a Fuzzy Filter for the Estimation of Motor Currents

Fuzzy systems have been used extensively and successfully in control systems over the past few decades, but have been applied much less often to filtering problems. This is somewhat surprising in view of the dual relationship between control and estimation. This paper discusses and demonstrates the application of fuzzy filtering to motor winding current estimation in permanent magnet synchronous motors. Motor winding current estimation is an important problem because in order to implement effective closed-loop control, a good estimation of the current is needed. Motor winding currents are notoriously noisy because of electrical noise in the motor drive. We use a fuzzy system with correlation-product inference and centroid defuzzification for motor winding current estimation, With the assumption that the membership functions are triangular (but not necessarily symmetric), we then optimize the membership functions using gradient descent. Next we use singular value decomposition to reduce the rule base for the fuzzy filter. Rule base reduction can be important for fuzzy systems in those cases where the fuzzy system needs to be implemented in real time. This is especially true with regard to fuzzy filtering in a real time motor controller. The methods discussed in this paper are demonstrated on real motor winding currents that were collected with a digital oscilloscope. It is demonstrated that fuzzy techniques provide a feasible approach to motor current estimation, that gradient descent optimization improves the performance of the filter, and that rule base reduction results in a relatively small degradation of filter performance. (C) 2000 Elsevier Science Inc. All rights reserved