Generalized Flexible Fuzzy Inference Systems

We have constructively proved a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems to be universal approximators. The systems use any types of continuous fuzzy sets, fuzzy logic AND, fuzzy rules with linear rule consequent and the generalized defuzzifier. We first prove that the TS fuzzy systems can uniformly approximate any multivariate polynomial arbitrarily well, and then prove they can also uniformly approximate any multivariate continuous function arbitrarily well. We have derived a formula for computing the minimal upper bounds on the number of fuzzy sets and fuzzy rules necessary to achieve the prespecified approximation accuracy for any given bivariate function. A numerical example is furnished. Our results provide a solid-theoretical basis for fuzzy system applications, particularly as fuzzy controllers and models.

We focus on the analysis and design of two different sliding mode observers for dynamic Takagi-Sugeno (TS) fuzzy systems. A nonlinear system of this class is composed of multiple affine local linear models that are smoothly interpolated by weighting functions resulting from a fuzzy partitioning of the state space of a given nonlinear system subject to observation. The Takagi-Sugeno fuzzy system is then an accurate approximation of the original nonlinear system. Our approach to the analysis and design of observers for Takagi-Sugeno fuzzy systems is based on extending sliding mode observer schemes to the case of interpolated multiple local affine linear models. Thus, our main contribution is nonlinear observer analysis and design methods that can effectively deal with model/plant mismatches. Furthermore, we consider the difficult case when the weighting functions in the Takagi-Sugeno fuzzy system depend on the estimated state.

We prove that a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems are universal approximators. The fuzzy systems studied in this paper are general and typical: they use any types of continuous fuzzy sets, any types of fuzzy logic AND, fuzzy rules with linear TS rule consequent and the generalized defuzzifier containing the popular centroid defuzzifier as a special case. We first constructively prove that the general TS fuzzy systems can uniformly approximate any multivariate polynomial with arbitrarily small error and then prove that the general fuzzy systems can uniformly approximate any multivariate continuous function with arbitrarily high approximation accuracy.

The work treats the $H_{\infty}$ model reduction problem for two-dimensional (2D) discrete Takagi-Sugeno (T-S) fuzzy systems described by Fornasini-Marchesini local state-space (FM LSS) models. The problem to be solved in the paper concerns the design of a stable reduced-order modelables to approximate the original T-S fuzzy system with comparatively small and minimized $H_{\infty}$ performance index. Via the use of the Lyapunov functions, new design conditions guaranteeing the $H_{\infty}$ model reduction are established in terms of solutions to a set of linear matrix inequalities (LMIs). To highlight the importance of the suggested process, a practical application has been made.

This paper deals with the problem of stability and stabilization of Takagi-Sugeno (T-S) fuzzy systems with a fixed delay by linear programming (LP) while imposing positivity in closed-loop. The stabilization conditions are derived using the single Lyapunov-Krasovskii Functional (LKF). An example of a real plant is studied to show the advantages of the design procedure.

As explained in the literature, it is very hard to measure premise variables of a Takagi-Sugeno (TS) fuzzy system. Therefore, in this paper, a fault detection and prediction (FDP) scheme is designed for a class of TS fuzzy systems with immeasurable (unknown) premise variables and external disturbances. A fault detection (FD) observer is designed to approximate the system output and the premise variables. Subsequently, a FD residual is generated by comparing the observer output with respect to the system output. The FD residual is evaluated to detect any faults in the system. Further, time-to-failure (TTF) of the TS fuzzy system is obtained by using a mathematical equation. Note the parameter update law and TTF scheme utilize the approximated premise variables since they are not measurable. Stability of the fault detection and TTF prediction results are verified using Lyapunov theory. Finally, a simulation study using a truck-trailer system is presented to verify the theoretical claims.

Robot human interaction requires use of safe, compliant and light weight actuators. Conventional linear motors and pneumatic cylinders are normally used to actuate robots to assist and augment human motions. Lately it has been realized that these actuators are not suitable and safe for applications involving human actor. Their large weight, size and stiffer design raise concerns. Pneumatic muscle actuators (PMA) on the other hand are very light weight, compact and have inherent compliance which make them potential candidate for applications involving robot human interaction. Taking on the advantages, these actuators are now being experimented for a variety of medical and rehabilitation applications. However they are not very popular due to their highly nonlinear and time dependent behavior which poses control problems. In this paper, an attempt is being made to accurately predict the uncertain and ambiguous characteristics of PMA using Artificial Intelligence (AI). Conventional tools such as analytical and numerical methods can only model a nonlinear system which is time independent. Time varying nonlinear system characteristics can be best modeled using artificial intelligence-based regression models. In this research, Artificial Neural Network (ANN), Mamdani Fuzzy Inference System (FIS) and Takagi-Sugeno (TS)-based fuzzy system are developed after carefully analyzing the time series data obtained from a real system. To achieve higher accuracy from these models, their parameters are tuned. Parameters of ANN are tuned using back propagation algorithm whereas fuzzy parameters are tuned using three different methods, namely, gradient descent method (GD), genetic algorithms (GA) and Modified Genetic Algorithm (MGA). It was found that the TS fuzzy inference system tuned by MGA provides better accuracy and can also model the time dependent behavior of PMA. The proposed TS fuzzy system is found to perform better in terms of accuracy and maximum deviation when compared to the previous approaches in the literature.

In this paper, a novel model-based fault detection (FD) and prediction scheme is developed for a class of Takagi-Sugeno (T-S) fuzzy systems. Unlike other FD schemes, in the proposed design, an FD observer with online fault learning capability is utilized to generate a residual which is obtained by comparing the system output with respect to the observer output. A fault is declared active if the generated residual exceeds an a priori chosen threshold. Subsequently, the fault magnitude is estimated online by using a suitable parameter update law. Upon detection, the online estimate of the fault magnitude is used in a mathematical equation to determine time-to-failure (TTF) or remaining useful life. TTF is determined by projecting the estimated fault magnitude at the current time instant against a failure threshold. Note that the previously reported FD schemes could neither estimate the magnitude of a growing fault in real time nor were they able to predict the remaining useful life of the fuzzy system. In this paper, the stability of the proposed FD and prognostics scheme is verified using the Lyapunov theory. Finally, two different simulation case studies are considered to verify the theoretical conjectures presented in this paper.

Several non-linear systems show complex behaviors. For example, some of those plants present a high degree of oscillations throughout the time. Adaptive algorithms used to approximate such difficult neural behaviors can show important definciencies. The differential network is not an exception. Indeed, when just one neural network is applied to get an adequate approximation, the identification error could be not so close to zero. One possible suggestion to solve this problem is to define a set of neuronal networks that works together. The members of such set will work each one on well defined trajectories subspaces of the uncertain system. In this paper, it is discussed how to combine the identification properties offered by the continuous neural network and the characteristic decision capabilities arised by fuzzy methods. The selection of which neural network is activated depends on decision achieved by a takagi-sugeno fuzzy system. The Chen circuit will be used to demonstrate the superior performance achieved by the suggested class of mixed neural network and fuzzy system, usually so-called neuro-fuzzy system.

This paper proposes a new robust stability condition for Takagi-Sugeno (T-S) fuzzy control system with parametric uncertainties. The condition is represented in the form of linear matrix inequalities (LMIs) and is less conservative than some relaxed quadratic stabilization conditions published recently in the literature by collecting the interrelations of fuzzy subsystems into a set of matrices, not just into a single matrix. Based on the LMI-based conditions derived, one can easily synthesize controllers for state feedback robust stabilization for T-S fuzzy control system with parametric uncertainties. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, the validity of the proposed approach is successfully demonstrated in the the control of simulation table.

Based on the properties analysis of Takagi-Sugeno (T-S) fuzzy systems with two-overlapped fuzzy partition (TFP) inputs, an analytical design approach of T-S fuzzy control systems is proposed in this paper. This approach includes two parts: (1) a new method to check the stability of a T-S fuzzy control system which only requires to find a common positive-definite matrix in each maximal overlapped-rules group (MORG). This method can reduce the conservatism and difficulty of the former stability analysis methods. (2) The systematic design of T-S fuzzy controllers by means of the parallel distributed compensation (PDC) and linear matrix inequalities (LMI). The simulation results of a nonlinear mass-spring-damper system show that this approach is effective.

In this paper, the stabilization of Takagi-Sugeno (T-S) fuzzy control systems whose premise variables employ two-overlapped fuzzy partition is investigated. Based on the piecewise fuzzy Lyapunov function and the nonparallel-distributed-compensation (non-PDC) control laws, sufficient conditions for stabilization of T-S fuzzy systems with parametric uncertainties are derived in the form of linear matrix inequalities (LMIs). The piecewise fuzzy Lyapunov function approach avoid the disadvantages of the fuzzy Lyapunov function approach who is valid only when the time derivatives of the membership functions exist in the whole fuzzy region. Moreover, compared to the PDC fuzzy controller, the non-PDC control laws shows greater potential to relax the stability conditions. An illustrative example is given to validate the effectiveness of the proposed approach.

This paper is concerned with control design for a generalized discrete-time Takagi-Sugeno fuzzy system. The Takagi-Sugeno fuzzy system generally describes nonlinear systems by employing local linear system representations. It is a great tool to analyze a nonlinear system and to design a stabilizing controller for it. However, there is a case where a controller design for a nonlinear system based on the Takagi-Sugeno fuzzy system becomes difficult due to its representation. A generalized fuzzy system to be considered in this paper describes even a wider class of nonlinear systems by representing locally nonlinear systems, and gives a different representation from the standard Takagi-Sugeno fuzzy system, which leads to a rather easy controller design. For such a generalized system, a stabilizing controller design method is proposed by introducing a new class of nonlinear controllers. Then, the resulting closed-loop system becomes a fuzzy system with linear system structures locally, which makes the control design easier. Stabilizing controller design conditions are given in terms of a set of linear matrix inequalities(LMIs), which are easily numerically solvable. A relaxation method is used to reduce the conservatism of design conditions. Finally, numerical examples are given to illustrate our nonlinear control design and to show the effectiveness over other existing results.

A robust controller design method based on the concept of fuzzy region is presented for Takagi-Sugeno (T-S) fuzzy systems with parametric uncertainties. The uncertain T-S fuzzy model is converted into uncertain T-S fuzzy region model. A relaxed sufficient condition is derived for robust stabilization by using Lyapunov function approach in the form of linear matrix inequalities (LMIs). Comparing with other similar conditions, the difficulty of solving LMIs is greatly reduced in this method. The efficiency of the method is illustrated through the simulation of a numerical uncertain fuzzy system.

Robust reliable H∞ control for fuzzy systems with random delays and linear fractional uncertainties