This paper proposes two methods for training Takagi–Sugeno (T-S) fuzzy systems using batch least squares (BLS) and particle swarm optimization (PSO). The T-S system is considered with triangular and Gaussian membership functions in the antecedents and higher-order polynomials in the consequents of fuzzy rules. In the first method, the BLS determines the polynomials in a system in which the fuzzy sets are known. In the second method, the PSO algorithm determines the fuzzy sets, whereas the BLS determines the polynomials. In this paper, the ridge regression is used to stabilize the solution when the problem is close to the singularity. Thanks to this, the proposed methods can be applied when the number of observations is less than the number of predictors. Moreover, the leave-one-out cross-validation is used to avoid overfitting and this way to choose the structure of a fuzzy model. A method of obtaining piecewise linear regression by means of the zero-order T-S system is also presented.

This article discusses a novel type-2 fuzzy inference system with multiple variables in which no fuzzy rules are explicitly defined. By using a rule-free system, we avoid the serious disadvantage of rule-based systems, which are burdened with the curse of dimensionality. In the proposed system, Gaussian membership functions are used for its inputs, and linearly parameterized system functions are used to obtain its output. To obtain the system parameters, a genetic algorithm with multi-objective function is applied. In the presented method, the genetic algorithm is combined with a feature selection method and a regularized ridge regression. The objective functions consist of a pair in which one function is defined as the number of active features and the other as the validation error for regression models or the accuracy for classification models. In this way, the models are selected from the Pareto front considering some compromise between their quality and simplification. Compared to the author’s previous work on the regression-based fuzzy inference system, a new inference scheme with type-2 fuzzy sets has been proposed, and the quality has been improved compared to the system based on type-1 fuzzy sets. Four experiments involving the approximation of a function, the prediction of fuel consumption, the classification of breast tissue, and the prediction of concrete compressive strength confirmed the efficacy of the presented method.

This paper describes a design method that guarantees the stability of an adaptive fuzzy control system. The system consists of an unstable plant with an irrational transfer function, a Takagi-Sugeno fuzzy adaptive controller, a nonlinear reference model, and an adaptation mechanism. The gradient-based adaptation mechanism changes the consequents of the controller rules in such a way that the closed-loop system behaves like the reference model. The proposed design procedure utilizes frequency-domain methods in the forms of the Nyquist and circle criteria. It is assumed that the function of the fuzzy controller is a nonlinearity described by a sector condition. This condition means that the controller function lies between two lines passing through the origin. In this method, the function of the fuzzy controller is verified during online adaptation; therefore, it stays in the allowed sector. This method is illustrated by an example of a control system containing an unstable plant with a transport delay. The main contribution of this paper is to use frequency methods combined with adaptive fuzzy control. These methods have previously been used for nonadaptive fuzzy control, but in this paper, they are used for the first time for adaptive fuzzy control.

This paper proposes a design method for second-order systems that guarantees the stability of an adaptive fuzzy controller with state feedback. The system consists of a linear unstable plant, the Takagi-Sugeno controller, a nonlinear reference model, and an adaptation mechanism. In this paper, gradient-based adaptation is used to change the consequents of the controller rules so that the closed-loop system behaves like the reference model. The proposed method utilizes a frequency-domain criterion in the form of the modified circle theorem. The controller function is assumed to be a nonlinearity described by a sector condition, which means that the function lies between two planes. During the process of adaptation, this function is verified so it stays in the sector guaranteeing stability. The method described here is illustrated by an example of a control system containing an unstable plant with an unknown pole in the right half-plane. The motivation of this paper is to propose a frequency-domain method for the design of state feedback adaptive fuzzy controllers. Comparing with time-domain methods that require advanced software tools, the proposed method offers simple graphical interpretation on the Nyquist plane.

Simplifying fuzzy models, including those for predicting time series, is an important issue in terms of their interpretation and implementation. This simplification can involve both the number of inference rules (i.e., structure) and the number of parameters. This paper proposes novel hybrid methods for time series prediction that utilize Takagi–Sugeno fuzzy systems with reduced structure. The fuzzy sets are obtained using a global optimization algorithm (particle swarm optimization, simulated annealing, genetic algorithm, or pattern search). The polynomials are determined by elastic net regression, which is a sparse regression. The simplification is based on reducing the number of polynomial parameters in the then-part by using sparse regression and removing unnecessary rules by using labels. A new quality criterion is proposed to express a compromise between the model accuracy and its simplification. The experimental results show that the proposed methods can improve a fuzzy model while simplifying its structure.

This paper presents a method of monocular human motion tracking for estimation of hurdle clearance kinematic parameters. The analysis involved 10 image sequences of five hurdlers at various training levels. Recording of the sequences was carried out under simulated starting conditions of a 110 m hurdle race. The parameters were estimated using the particle swarm optimization algorithm and they are based on analysis of the images recorded with a 100 Hz camera. The proposed method does not involve using any special clothes, markers, inertial sensors, etc. As the quality criteria, the mean absolute error and mean relative error were used. The level of computed errors justifies the use of this method to estimate hurdle clearance parameters.