Fuzzy Model-based Control Systems Research Articles

Event-triggered control design with varying gains for polynomial fuzzy systems against DoS attacks

This paper presents an innovative event-triggered control scheme for addressing the stabilization problem of polynomial fuzzy systems under the influence of Denial-of-Service (DoS) attacks. The proposed controller utilizes a sampling-based event-triggered mechanism to reduce communication resources and avoid Zeno behavior. Furthermore, a novel polynomial fuzzy model-based control system is developed to investigate the impact of periodic DoS attacks and the addressed event-triggered mechanism on system stability. To improve system performance, control gains are updated at each triggering instant. The exponential stability criteria are formulated in the form of sum-of-square constraints, supported by a triggering instant dependent piecewise Lyapunov-Krasovskii functional and an online asynchronous premise reconstruction approach. Finally, the efficiency and usefulness of the theoretical findings are validated through simulation examples.

An application of interval type-2 fuzzy model based control system for generic aircraft

This paper presents a design application of Interval Type-2 (IT2) Takagi–Sugeno (T–S)​ Fuzzy Model Based (FMB) control system for generic aircraft. The IT2 T–S FMB flight control system consists of an IT2 T–S fuzzy model and a fuzzy controller connected in a closed-loop. The IT2 T–S​ fuzzy model is obtained by linearizing the nonlinear aircraft dynamics about various representative points (equilibrium points) of the flight envelope with some fuzzy rules. The aircraft flight envelope parameters, i.e., operating altitude and aircraft speed are characterized as premise parameters and elements of stability and control derivative matrix are identified as consequent parameters of fuzzy model. Longitudinal dynamics of X-29A research aircraft is selected for design of IT2 T–S FMB control system. To achieve the optimum design flexibility, an imperfectly matched premise and membership function dependent (MFD) stability analysis is considered. In MFD stability conditions, the information of flight envelope parameters is included to capture the nonlinearity pertaining to variation in flight conditions. The closed-loop response of IT2 T–S FMB controller is presented at trim equilibrium points by taking four initial angle of attack flight conditions. The performance analysis of designed controller is also presented and discussed in comparison with Fuzzy LQR and LQR based optimal controllers. The simulation results reveal that proposed IT2 T–S FMB controller not only stabilizes the aircraft dynamics but also provides improved transient performance as compared with Fuzzy LQR and LQR based optimal controllers. This demonstrates the utility of IT2 T–S FMB control system for aircraft/ UAV’s related application.

Membership-Function-Dependent Control Design and Stability Analysis of Interval Type-2 Sampled-Data Fuzzy-Model-Based Control System

This article investigates the design and stability analysis of the interval type-2 (IT2) sampled-data (SD) fuzzy-model-based (FMB) control system with the optimal guaranteed cost performance. An IT2 Takagi–Sugeno (T-S) fuzzy model is applied to describe the dynamics of the nonlinear systems where the parameter uncertainties are captured by the lower and upper membership functions. To conduct the stability analysis for the SD FMB control system, a looped-functional approach taking the advantage of the information about the sampling periods is employed. Because of the SD control strategy, the state will be sampled at each sampling instant and the control signal generated by the IT2SD fuzzy controller will be kept by the zero-order holder during the sampling period, which will result in mismatched membership grades between IT2 T-S fuzzy model and IT2SD fuzzy controller that leads to the complexity in carrying out stability analysis. Thanks to the imperfect premise matching concept, which allows the difference on the number of rules and the premise membership functions between model and controller, the design of the IT2SD fuzzy controller can be more flexible. To further relax the stability conditions and minimize the upper bound of the guaranteed cost index, the membership-function-dependent stability analysis approach which can make use of the features of the IT2 membership functions is adopted. The performance of the control system can also be adjusted through the choice of the weighting matrices in the cost function. The stability conditions building on the Lyapunov stability theory and the performance conditions building on the concept of the guaranteed cost control in the shape of linear matrix inequalities are established to assure the system stability and acquire the optimal guaranteed cost performance. The proposed IT2SDFMB control design is tested on the inverted pendulum system and the simulation results verify the effectiveness of the proposed approach.

T–S Fuzzy-Model-Based Output Feedback Tracking Control With Control Input Saturation

This paper investigates the output feedback tracking control for a fuzzy-model-based (FMB) control system when the control input is saturated, where the FMB is developed based on a T–S fuzzy model and a fuzzy controller. The controller is employed to close the feedback loop and generate the system to trace the trajectory of the states of a stable reference model subject to $H_\infty$ performance. To enhance the fuzzy controller design flexibility, the number of rules and premise membership functions can be adjusted. Stability analysis for the FMB control system is performed based on Lyapunov stability theory. To address the control input saturation problem, linear sectors are created by local linear upper and lower bounds to include the possible control area. Hence, the nonlinear saturation problem can be tackled by the stability analysis of linear sectors. The membership-functions-dependent technique is used to bring the information and address the nonlinearity of embedded membership functions into the stability analysis. The numerical simulation example demonstrates the effectiveness of the proposed approach and discusses the effect of $H_\infty$ performance and control input saturation rate according to the tracking result.

Robust-stable quadratic-optimal fuzzy-PDC controllers for systems with parametric uncertainties: A PSO based approach

The present paper shows how robust stable quadratic-optimal fuzzy controllers can be designed using particle swarm optimization (PSO) algorithm, a popular metaheuristic optimization technique. This controller is designed following the Parallel distributed compensation (PDC) technique, employed for Takagi–Sugeno (TS) fuzzy model based control systems, and is termed as a mixed H2/LMI controller. These controllers are designed to achieve two objectives simultaneously i.e. robust stabilization of the uncertain system with parametric uncertainties in hand and also to achieve desired transient response. The robust stabilization is guaranteed when a set of linear matrix inequalities (LMIs) formulated get satisfied. The controller simultaneously achieves the desired transient performance by minimizing a quadratic finite-horizon integral performance criterion for the nominal dynamical system. The proposed PSO based mixed H2/LMI controller has been employed for a benchmark dynamical system and it has been demonstrated to achieve better performance than a recently proposed hybrid Taguchi genetic algorithm (HTGA) based approach, implemented for the same problem.

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

This paper reviews the stability analysis of continuous-time fuzzy-model-based (FMB) control systems, with emphasis on state-feedback control techniques, which is an essential issue received a great deal of attention in the fuzzy control community. It gives an overview of the milestones and the trend of developments and achievements for the past decades. Focusing on the stability analysis of FMB control systems, it summarizes the issues in the four fundamental and essential aspects, namely, the types of membership-function matching, types of Lyapunov functions, types of stability analysis and the techniques of stability analysis are discussed. To start with systematic discussion, the FMB control systems are categorized into three types of membership-function matching, namely, perfectly, partially and imperfectly matched premises, regarding the premise membership functions and the number of rules used in the fuzzy model and fuzzy controller which forms the FMB control system. The features of each category are thoroughly discussed from theoretical to practical point of view. Various types of Lyapunov functions available in the literature for conducting stability analysis and their characteristics are then reviewed. Especially, the focus of this paper is to promote the concept of membership-function-dependent (MFD) stability analysis, which makes use of the information of membership functions aiming to relax the stability conditions compared with the dominant membership-function-independent (MFI) stability analysis in the literature. The techniques for MFI and MFD stability analysis are then discussed in details, which provide some solid ideas to analyze the stability of FMB control systems. More importantly, it sheds light on the fact that MFD stability analysis demonstrates a greater potential than the MFI one for relaxing the conservativeness of stability analysis results, which points a promising research direction for this topic. The purposes of this paper are to provide a comprehensive update for the stability analysis of FMB control systems to the researchers in the field and serve as a quick guide for the potential researchers who want to enter the field.

Design of Fuzzy Functional Observer-Controller via Higher Order Derivatives of Lyapunov Function for Nonlinear Systems.

In this paper, we investigate the stability of Takagi-Sugeno fuzzy-model-based (FMB) functional observer-control system. When system states are not measurable for state-feedback control, a fuzzy functional observer is designed to directly estimate the control input instead of the system states. Although the fuzzy functional observer can reduce the order of the observer, it leads to a number of observer gains to be determined. Therefore, a new form of fuzzy functional observer is proposed to facilitate the stability analysis such that the observer gains can be numerically obtained and the stability can be guaranteed simultaneously. The proposed form is also in favor of applying separation principle to separately design the fuzzy controller and the fuzzy functional observer. To design the fuzzy controller with the consideration of system stability, higher order derivatives of Lyapunov function (HODLF) are employed to reduce the conservativeness of stability conditions. The HODLF generalizes the commonly used first-order derivative. By exploiting the properties of membership functions and the dynamics of the FMB control system, convex and relaxed stability conditions can be derived. Simulation examples are provided to show the relaxation of the proposed stability conditions and the feasibility of designed fuzzy functional observer-controller.

Quadratic Optimal Control with Disturbance Attenuation for Uncertain Continuous-Time T-S Fuzzy Systems

ABSTRACTIn this study, an algebraically computational method is proposed to synthesize non-parallel-distributed-compensation (non-PDC) fuzzy controller such that (1) the prescribed disturbance attenuation level for the uncertain continuous-time Takagi-Sugeno (T-S) fuzzy system can be achieved, and (2) a quadratic integral performance index for nominal T-S fuzzy model-based control system can be minimized. We first derive relaxed linear matrix inequality (LMI) conditions by non-quadratic Lyapunov function and non-PDC fuzzy controller to meet prescribed disturbance attenuation level for the uncertain T-S systems. Then by using LMIs and orthogonal function array, the robust quadratic optimal control with disturbance attenuation for uncertain T-S fuzzy system is transformed into constrained-optimization problem represented by algebraic equations and LMI constraints. For static constrained-optimization problem, the HTGA is employed to search the gains non-PDC controllers. Therefore, the robust optimal controller design problem can be greatly simplified with the proposed method. The design example is given to demonstrate the applicability of the proposed approach.

Stable and Optimal Controller Design for Takagi-Sugeno Fuzzy Model Based Control Systems via Linear Matrix Inequalities

AbstractThis paper deals with a systematic design procedure that guarantees the stability and optimal performance of the nonlinear systems described by Takagi-Sugeno fuzzy models. Takagi-Sugeno fuzzy model allows us to represent a nonlinear system by linear models in different state space regions. The overall fuzzy model is obtained by fuzzy blending of these linear models. Then based on this model, linear controllers are designed for each linear model using parallel distributed compensation. Stability and optimal performance conditions for Takagi-Sugeno fuzzy control systems can be represented by a set of linear matrix inequalities which can be solved using software packages such as MATLAB’s LMI Toolbox. This design procedure is illustrated for a nonlinear system which is described by a two-rule Takagi-Sugeno fuzzy model. The fuzzy model was built in MATLAB Simulink and a code was written in LMI Toolbox to determine the controller gains subject to the proposed design approach.

Membership-Function-Dependent Stability Analysis and Control Synthesis of Guaranteed Cost Fuzzy-Model-Based Control Systems

This paper focuses on the guaranteed cost stability analysis of fuzzy-model-based (FMB) control systems. Representing the nonlinear plant using a Takagi–Sugeno (T–S) fuzzy model, a fuzzy controller is employed to close the feedback loop. A weighted linear quadratic cost function is considered as the cost index to measure the performance of the closed-loop fuzzy system in terms of the system states, system outputs, and control signals. The stability of the FMB control system is investigated by the Lyapunov stability theory subject to the minimization of cost index for performance realization. A membership-function-dependent approach using the piecewise-linear membership functions is employed to include the information of membership functions into the stability analysis. Membership-function-dependent stability conditions in terms of linear matrix inequalities are obtained to determine the system stability and feedback gains with the consideration of the system performance measured by the cost function. A simulation example is provided to illustrate the effectiveness and merits of the proposed approach.

Linear matrix inequalities‐based membership‐function‐dependent stability analysis for non‐parallel distributed compensation fuzzy‐model‐based control systems

This study investigates the stability of fuzzy-model-based (FMB) control system, which is formed by a Takagi–Sugeno (T–S) fuzzy model representing a non-linear plant and a fuzzy controller connected in a closed loop. The proposed fuzzy controller considers a higher design flexibility that the premise membership functions and the number of fuzzy rules are allowed to be not the same as those of the T–S fuzzy model. By employing simple membership functions and a smaller number of fuzzy rules, the complexity of the fuzzy controller can be reduced to facilitate its implementation. Based on the Lyapunov stability theory, a membership-function-dependent (MFD) stability analysis is proposed to develop stability conditions in the form of linear matrix inequalities (LMIs) to determine the system stability and facilitate the control synthesis. The information of the membership functions is brought to the MFD LMI-based stability conditions, which are dedicated to the FMB control system under consideration. It is thus the proposed MFD stability analysis result is more relaxed compared with the existing membership-function-independent stability analysis results. A simulation example is given to illustrate the effectiveness of the proposed approach.

Control Design for Interval Type-2 Fuzzy Systems Under Imperfect Premise Matching

This paper focuses on designing interval type-2 (IT2) control for nonlinear systems subject to parameter uncertainties. To facilitate the stability analysis and control synthesis, an IT2 Takagi-Sugeno (T-S) fuzzy model is employed to represent the dynamics of nonlinear systems of which the parameter uncertainties are captured by IT2 membership functions characterized by the lower and upper membership functions. A novel IT2 fuzzy controller is proposed to perform the control process, where the membership functions and number of rules can be freely chosen and different from those of the IT2 T-S fuzzy model. Consequently, the IT2 fuzzy-model-based (FMB) control system is with imperfectly matched membership functions, which hinders the stability analysis. To relax the stability analysis for this class of IT2 FMB control systems, the information of footprint of uncertainties and the lower and upper membership functions are taken into account for the stability analysis. Based on the Lyapunov stability theory, some stability conditions in terms of linear matrix inequalities are obtained to determine the system stability and achieve the control design. Finally, simulation and experimental examples are provided to demonstrate the effectiveness and the merit of the proposed approach.

H∞ fault detection for a class of T-S fuzzy model-based nonlinear networked control systems

In this paper, a robust H ∞ fault detection filter is designed for a class of discrete-time nonlinear networked control systems via T-S fuzzy model with multiple bounded state delay and random packet dropout induced by the limited bandwidth of communication networks. Our aim in this paper is to analyze and design a robust H ∞ full-order fault detection filter, such that the filtering error dynamics is asymptotically mean-square stable with a prescribed H ∞ performance level. Sufficient conditions for the existence of the desired filter are presented. Finally, an example is given to illustrate the effectiveness and applicability of the proposed new design method.

Analysis of stability and robust stability of polynomial fuzzy model-based control systems using a sum-of-squares approach

The well known Takagi---Sugeno (T---S) fuzzy model can be extended in different ways including the polynomial fuzzy model, whose consequent parts are polynomial sub-systems. Compared with the traditional T---S fuzzy model, the polynomial fuzzy model can represent a nonlinear system more accurately with a smaller number of fuzzy logic rules. It is worth emphasizing that the stability analysis and controller design of polynomial fuzzy model-based (PFMB) control systems are not based on the linear matrix inequalities but the recently developed sum-of-squares decompositions. In this paper, based on an existing result for traditional fuzzy control systems, we propose a new stability condition for the stability analysis of PFMB control systems. Furthermore, the stability of PFMB control systems with parameter uncertainties is investigated. The popular inverted pendulum and an unstable nonlinear system are employed to demonstrate the quality of the proposed stability conditions.

Membership-function-dependent stability analysis of fuzzy-model-based control systems using fuzzy Lyapunov functions

This paper investigates the stability of fuzzy-model-based (FMB) control system, formed by a T–S fuzzy model and a fuzzy controller connected in a close loop, based on a fuzzy-Lyapunov function. A general FMB control system that the T–S fuzzy model and fuzzy controller not sharing the same premise membership functions and/or the same number of fuzzy rules is considered. A membership-function-dependent stability analysis approach is proposed to consider the membership functions of both the T–S fuzzy model and fuzzy controller in the stability analysis and incorporate them in the stability conditions in the form of linear matrix inequalities. As the stability conditions are membership-function dependent, they are dedicated to the FMB control system with the specified membership functions under consideration. It is thus the membership-function-dependent stability conditions are more relaxed compared to the existing membership-function-independent stability conditions. A fuzzy-Lyapunov function is a weighted sum of quadratic functions which are required to be positive definite in most of the existing work. In this paper, this criterion is not required and thus the stability conditions can be further relaxed. Some simulation examples are given to demonstrate the merits of the proposed approach.

Stability analysis of nonlinear-function fuzzy-model-based control systems

This paper presents the stability analysis of nonlinear-function fuzzy-model-based (FMB) control systems. A fuzzy model with nonlinear function terms is proposed to represent nonlinear plants. A fuzzy controller with the nonlinear function terms is then proposed to close the feedback loop. The nonlinear function terms enhance the capability of fuzzy modelling and feedback compensation. Following the linear-matrix-inequality (LMI) based stability analysis approach for the traditional FMB control systems, it will lead to stability conditions involving the nonlinear terms for the proposed nonlinear-function FMB control systems. As a result, the stability conditions will have an infinite number of LMIs, which make it impractical to find a feasible solution numerically using convex programming techniques. A stability analysis approach is proposed to obtain effective LMI-based stability conditions for the proposed nonlinear-function FMB control systems. Simulation examples are given to illustrate the merits of the proposed fuzzy control approach.

Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model‐based (FMB) systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA) is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy modeling. To validate the model, a controller based on proposed polynomial fuzzy systems is designed and then applied to both original nonlinear plant and fuzzy model for comparison. Additionally, stability analysis for the proposed polynomial FMB control system is investigated employing Lyapunov theory and a sum of squares (SOS) approach. Moreover, the form of the membership functions is considered in stability analysis. The SOS‐based stability conditions are attained using SOSTOOLS. Simulation results are also given to demonstrate the effectiveness of the proposed method.

LMI-Based Stability Analysis for Fuzzy-Model-Based Control Systems Using Artificial T–S Fuzzy Model

This paper investigates the stability of fuzzy-model-based (FMB) control systems. An alternative stability-analysis approach using an artificial fuzzy system based on the Lyapunov stability theory is proposed. To facilitate the stability analysis, the continuous membership functions of the Takagi-Sugeno (T-S) fuzzy model are represented by the staircase ones. With the nice property of the staircase membership functions, it turns the set of infinite number of linear-matrix-inequality (LMI) based stability conditions into a finite one. Furthermore, the staircase membership functions carrying system information can be brought to the stability conditions to relax the stability conditions. The stability of the original FMB control systems is guaranteed by the satisfaction of the LMI-based stability conditions. The proposed stability analysis is applied to the FMB control systems of which the T-S fuzzy model and fuzzy controller do not share the same premise membership functions and, thus, is able to enhance the design flexibility of the fuzzy controller. A simulation example is given to illustrate the merits of the proposed approach.

Design of stable fuzzy controller for non-linear systems subject to imperfect premise matching based on grid-point approach

This study investigates the systems stability of fuzzy-model-based (FMB) control systems. Based on the T-S fuzzy model representing the non-linear system, a fuzzy controller using grid-point (GP) technique is proposed to close the feedback loop. A GP is defined as the sub-operating domain of the non-linear system. For each GP, a corresponding GP fuzzy controller is employed to control the system. As the non-linearity in each GP is lower compared to that of the full operating domain, it is in favour of yielding relaxed stability analysis result using the GP control technique of which the nature of the membership functions and operating domain are taken into account. Furthermore, unlike most of the fuzzy control approaches, the proposed one can be applied to FMB control systems subject to imperfect premise matching that the fuzzy model and fuzzy controller do not share the same premise membership functions. As a result, some simple membership functions can be employed for the fuzzy controllers to lower the implementation cost. Based on the Lyapunov stability theory, stability conditions in terms of linear matrix inequalities are derived to guarantee the system stability and facilitate the controller synthesis. Simulation examples are given to demonstrate the merits of the proposed FMB control scheme using the proposed GP technique.

SOS-Based Stability Analysis of Polynomial Fuzzy-Model-Based Control Systems Via Polynomial Membership Functions

This paper presents stability analysis of polynomial fuzzy-model-based (FMB) control systems using the sum-of-squares (SOS) approach. Recently, stability analysis of the polynomial fuzzy-control systems, which is a generalized form of the well-known Takagi-Sugeno (T-S) FMB control systems, has been reported in the form of SOS-based stability conditions. Lack of information on the relations between membership functions and premise variables, in the existing stability analysis approaches, causes conservatism of their results. In this paper, to derive relaxed stability conditions for polynomial FMB control systems, membership functions which are approximated with polynomials and carrying relations between membership functions and premise variables are brought into the stability analysis. Considering a polynomial FMB control system and based on the Lyapunov stability theory, stability conditions in the form of fuzzy summations are derived, where each term contains product of membership functions of the polynomial fuzzy model and polynomial fuzzy controller. Each product term is approximated by a polynomial. In order to obtain better approximation, the operating domain of membership functions is partitioned to subregions. Then, SOS-based stability conditions for all subregions are derived. Unlike some published stability-analysis approaches, the proposed one can be employed for stability analysis of polynomial fuzzy-control systems under imperfect premise matching of which the fuzzy model and fuzzy controller do not share the same membership functions. The solution of the SOS-based stability conditions can be found numerically using SOSTOOLS, which is a free third-party MATLAB Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed stability conditions.