Application of evolving Takagi–Sugeno fuzzy model to nonlinear system identification

In this paper, a new encoding scheme is presented for learning the Takagi–Sugeno (T–S) fuzzy model from data by genetic algorithms (GAs). In the proposed encoding scheme, the rule structure (selection of rules and number of rules), the input structure (selection of inputs and number of inputs), and the antecedent membership function (MF) parameters of the T–S fuzzy model are all represented in one chromosome and evolved together such that the optimisation of rule structure, input structure, and MF parameters can be achieved simultaneously. The performance of the developed evolving T–S fuzzy model is first validated by studying the benchmark Box–Jenkins nonlinear system identification problem and nonlinear plant modelling problem, and comparing the obtained results with other existing results. Then, it is applied to approximate the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper of which identification problem is significantly difficult due to its inherently hysteretic and highly nonlinear dynamics. It is shown by the validation applications that the developed evolving T–S fuzzy model can identify the nonlinear system satisfactorily with acceptable number of rules and appropriate inputs.     

Differential evolution and quantum-inquired differential evolution for evolving Takagi–Sugeno fuzzy models

The differential evolution (DE) is a global optimization algorithm to solve numerical optimization problems. Recently the quantum-inquired differential evolution (QDE) has been proposed for binary optimization. This paper proposes DE/QDE to learn the Takagi–Sugeno (T–S) fuzzy model. DE/QDE can simultaneously optimize the structure and the parameters of the model. Moreover a new encoding scheme is given to allow DE/QDE to be easily performed. The two benchmark problems are used to validate the performance of DE/QDE. Compared to some existing methods, DE/QDE shows the competitive performance in terms of accuracy.     

Output feedback controller design of Takagi–Sugeno fuzzy systems

This paper presents new systematic design methods of two types of output feedback controllers for Takagi–Sugeno (T–S) fuzzy systems, one of which is constructed with a fuzzy regulator and a fuzzy observer, while the other is an output direct feedback controller. In order to use the structural information in the rule base to decrease the conservatism of the stability analysis, the standard fuzzy partition (SFP) is employed to the premise variables of fuzzy systems. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. The concept of parallel distributed compensation (PDC) is employed to design fuzzy regulators and fuzzy observers from the T–S fuzzy models. New stability analysis and design methods of output direct feedback controllers are also presented. The output feedback controllers design and simulation results for a nonlinear mass-spring-damper mechanical system show that these methods are effective.     

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 method.     

Stabilization for Takagi–Sugeno fuzzy systems based on partitioning the range of fuzzy weights

In this note, the state-feedback stabilization for continuous-time Takagi–Sugeno (T–S) fuzzy systems is addressed, where the fuzzy weighting functions are assumed to be differentiable and their ranges of variation are possibly bounded by some parameters. By utilizing the matrix elimination lemma and introducing a polyhedral partition to the range of the fuzzy weights, the quadratically parameterized condition is transformed to being piecewise linear in both the fuzzy weights and their derivatives. Then, a switching control based on the partition is considered, by utilizing the extreme points in each partition to address the constraints of the fuzzy weights and their derivatives. The simulation shows that finer subdivision in the partition leads to better stability and stabilization margins.     

Fault reconstruction for Takagi–Sugeno fuzzy systems via learning observers

This paper addresses the problem of observer-based fault reconstruction for Takagi–Sugeno fuzzy systems. Two types of fuzzy learning observers are constructed to achieve simultaneous reconstruction of system states and actuator faults. Stability and convergence of the proposed observers are proved using Lyapunov stability theory, and necessary conditions for the existence of the observers are further discussed. The design of fuzzy learning observers can be formulated in terms of a series of linear matrix inequalities that can be conveniently solved using convex optimisation technique. A single-link flexible manipulator is employed to verify the effectiveness of the proposed fault-reconstructing approaches.     

Non-fragile guaranteed cost control for Takagi–Sugeno fuzzy hyperbolic systems

This paper concerns the problems of non-fragile guaranteed cost control (GCC) for nonlinear systems with or without parameter uncertainties. The Takagi–Sugeno (T–S) fuzzy hyperbolic model is employed to represent the nonlinear system. The non-fragile controller is designed by parallel distributed compensation (PDC) method, and some sufficient conditions are formulated via linear matrix inequalities (LMIs) such that the system is asymptotically stable and the cost function satisfies an upper bound in the presence of the additive controller perturbations. The above approach is also extended to the non-fragile GCC of T–S fuzzy hyperbolic system with parameter uncertainties, and the robust non-fragile GCC scheme is obtained. The main advantage of the non-fragile GCC based on the T–S fuzzy hyperbolic model is that it can achieve small control amplitude via ‘soft’ constraint approach. Finally, a numerical example and the Van de Vusse example are given to illustrate the effectiveness and feasibility of the proposed approach.     

Absolute stability analysis for a class of Takagi–Sugeno fuzzy control systems

This article presents absolute stability conditions for a particular class of Takagi–Sugeno fuzzy control systems. Initially, a Takagi–Sugeno fuzzy control system is transformed into a multivariable Lur’e type system. A simple algorithm for checking the absolute stability of this system is then proposed. Since the key of the proposed algorithm is to solve algebraic Riccati equations, software packages such as MATLAB provides a simple means to check the conditions. The proposed approach does not limit the methods of fuzzification and defuzzification. This article presents several analytical examples to verify the simplicity and efficiency of the proposed approach.     

H 8 control for Takagi?Sugeno fuzzy systems

In this paper the design problem of output feedback H      

Robust H ∞ reliable control of time delay nonlinear systems via Takagi–Sugeno fuzzy models

This article is focused on reliable fuzzy H _∞ controller design for a class of Takagi–Sugeno (T–S) fuzzy systems with state delay, actuator failures, disturbance input and norm bounded uncertainties. In the design, the H _∞ performance of the closed-loop system is optimised during normal operation (without failures) while the system satisfies a prescribed H _∞ performance level in the case of actuator failures. Two methods are presented in this study. In the first method, delay-dependent conditions are derived based on a single Lyapunov–Krasovskii function. This method improves delay-independent results existing in the literature. Next, to further reduce the conservatism, we use a parameter-dependent Lyapunov–Krasovskii function. The new sufficient conditions for the existence of the suboptimal robust reliable controller are shown in terms of linear matrix inequalities (LMIs), which can be solved by using LMI optimisation techniques. A simulation example shows the effectiveness of the proposed methods.     

A new evolving compact optimised Takagi–Sugeno fuzzy model and its application to nonlinear system identification

A new encoding scheme is presented for a fuzzy-based nonlinear system identification methodology, using the subtractive clustering and non-dominated sorting genetic algorithm. The proposed method consists of two parts. The first part is related to the selection of most relevant or influencing inputs to the system and the second one is related to the tuning of fuzzy rules and parameters of the membership functions. The main purpose of the proposed scheme is to reduce the complexity and increase the accuracy of the model. In particular, three objectives are considered in the process of optimisation, namely, the number of inputs, number of rules and the root mean square of the modelling error. The performance of the developed method is validated by identifying the Box–Jenkins nonlinear benchmark system, and to the modelling of the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper. The latter is also a challenging problem due to the inherent hysteretic and highly nonlinear dynamics of the MR damper. It is shown that the developed evolving Takagi–Sugeno (T–S) fuzzy model can identify and grasp the nonlinear dynamics of both systems very well, while a small number of inputs and fuzzy rules are required for this purpose.     

A fuzzy Lyapunov function approach to estimating the domain of attraction for continuous-time Takagi–Sugeno fuzzy systems

This paper deals with stability analysis and control design problems for continuous-time Takagi–Sugeno (T–S) fuzzy systems. The first aim is to present less conservative linear matrix inequality (LMI) conditions to design controllers and assess the stability. The second relevant contribution is to present a new strategy to find an inner estimate of the domain of attraction (DA) via LMIs. The results are based on the fuzzy Lyapunov functions (FLFs) and non-parallel distributed compensation (non-PDC) approaches. Finally, examples illustrate the effectiveness and merits of the proposed methods.     

AQM router design for TCP network via input constrained fuzzy control of time-delay affine Takagi–Sugeno fuzzy models

In this article, Takagi–Sugeno (T–S) fuzzy control theory is proposed as a key tool to design an effective active queue management (AQM) router for the transmission control protocol (TCP) networks. The probability control of packet marking in the TCP networks is characterised by an input constrained control problem in this article. By modelling the TCP network into a time-delay affine T–S fuzzy model, an input constrained fuzzy control methodology is developed in this article to serve the AQM router design. The proposed fuzzy control approach, which is developed based on the parallel distributed compensation technique, can provide smaller probability of dropping packets than previous AQM design schemes. Lastly, a numerical simulation is provided to illustrate the usefulness and effectiveness of the proposed design approach.     

Position control of a quadcopter drone using evolutionary algorithms-based self-tuning for first-order Takagi–Sugeno–Kang fuzzy logic autopilots

Trajectory tracking control of a quadcopter drone is a challenging work due to highly-nonlinear dynamics of the system, coupled with uncertainties in the flight environment (e.g. unpredictable wind gusts, measurement noise, modelling errors, etc). This paper addresses the aforementioned research challenges by proposing evolutionary algorithms-based self-tuning for first-order Takagi–Sugeno–Kang-type fuzzy logic controller (FLC). We consider three major state-of-the-art optimisation algorithms, namely, Genetic Algorithm, Particle Swarm Optimisation, and Artificial Bee Colony to facilitate automatic tuning. The effectiveness of the proposed control schemes is tested and compared under several different flight conditions, such as, constant, varying step and sine functions. The results show that the ABC-FLC outperforms the GA-FLC and PSO-FLC in terms of minimising the settling time in the absence of overshoots.     

Evolving Takagi–Sugeno fuzzy model based on switching to neighboring models

In this paper, we propose a new online identification approach for evolving Takagi–Sugeno (TS) fuzzy models. Here, for a TS model, a certain number of models as neighboring models are defined and then the TS model switches to one of them at each stage of evolving. We define neighboring models for an in-progress (current) TS model as its fairly evolved versions, which are different with it just in two fuzzy rules. To generate neighboring models for the current model, we apply specially designed split and merge operations. By each split operation, a fuzzy rule is replaced with two rules; while by each merge operation, two fuzzy rules combine to one rule. Among neighboring models, the one with the minimum sum of squared errors – on certain time intervals – replaces the current model.To reduce the computational load of the proposed evolving TS model, straightforward relations between outputs of neighboring models and that of current model are established. Also, to reduce the number of rules, we define and use first-order TS fuzzy models whose generated local linear models can be localized in flexible fuzzy subspaces. To demonstrate the improved performance of the proposed identification approach, the efficiency of the evolving TS model is studied in prediction of monthly sunspot number and forecast of daily electrical power consumption. The prediction and modeling results are compared with that of some important existing evolving fuzzy systems.     

On reducing computational overhead in multi-objective genetic Takagi–Sugeno fuzzy systems

The use of multi-objective evolutionary algorithms (MOEAs) to generate a set of fuzzy rule-based systems (FRBSs) with different trade-offs between complexity and accuracy has gained more and more interest in the scientific community. The evolutionary process requires, however, a large number of FRBS generations and evaluations. When we deal with high dimensional datasets, these tasks can be very time-consuming, especially when we generate Takagi–Sugeno FRBSs, thus making a satisfactory exploration of the search space very awkward. In this paper, we first analyze the time complexity for both the generation and the evaluation of Takagi–Sugeno FRBSs. Then we introduce a simple but effective technique for speeding up the identification of the rule consequent parameters, one of the most time-consuming phases in Takagi–Sugeno FRBS generation. Finally, we highlight how the application of this technique produces as a side-effect a decoupling of the rules. This decoupling allows us to avoid re-computing consequent parameters of rules which are not directly modified during the evolutionary process, thus saving a considerable amount of time.In the experimental part we first test the correctness of the predicted asymptotical time complexity. Then we show the benefits in terms of computing time saving and improved search space exploration through an example of multi-objective genetic learning of Takagi–Sugeno FRBSs in the regression domain.     

A generalized Takagi–Sugeno–Kang recurrent fuzzy-neural filter for adaptive noise cancelation

This paper presents a recurrent fuzzy-neural filter for adaptive noise cancelation. The cancelation task is transformed to a system-identification problem, which is tackled by use of the dynamic neuron-based fuzzy neural network (DN-FNN). The fuzzy model is based on Takagi–Sugeno–Kang fuzzy rules, whose consequent parts consist of linear combinations of dynamic neurons. The orthogonal least squares method is employed to select the number of rules, along with the number and kind of dynamic neurons that participate in each rule. Extensive simulation results are given and performance comparison with a series of other dynamic fuzzy and neural models is conducted, underlining the effectiveness of the proposed filter and its superior performance over its competing rivals.     

Finite-time mixed H∞ and passive filtering for Takagi–Sugeno fuzzy nonhomogeneous Markovian jump systems

This paper is concerned with the finite-time mixed H_∞ and passivity performance analysis and filter design for a class of uncertain nonlinear discrete-time Markovian jump systems (MJSs) described by Takagi–Sugeno fuzzy model with nonhomogeneous jump processes. In this paper, the proposed MJSs fuzzy model is formulated with norm-bounded parameter uncertainties and time-varying jump transition probability matrices. In particular, the time-varying transition probability matrices are expressed in respect of a polytope. By constructing a suitable Lyapunov functional, a new set of sufficient conditions is derived in the form of linear matrix inequalities (LMIs) to ensure that the filtering error system is robustly stochastically finite-time bounded and a prescribed mixed H_∞ and passive performance index is achieved. Moreover, the robust mixed H_∞ and passivity filter design gain matrices can be computed from the obtained LMIs. Furthermore, the developed results unify H_∞ and passive filtering problems in a single framework. Finally, two numerical examples including an application-oriented example are provided to demonstrate the effectiveness of the proposed filter design technique.