Robust adaptive fuzzy control of unknown chaotic systems

This paper presents a robust adaptive fuzzy control algorithm for controlling unknown chaotic systems. The control approach encompasses a fuzzy system and a robust controller. The fuzzy system is designed to mimic an ideal controller, based on sliding-mode control. The robust controller is designed to compensate for the difference between the fuzzy controller and the ideal controller. The parameters of the fuzzy system, as well as uncertainty bound of the robust controller, are tuned adaptively. The adaptive laws are derived in the Lyapunov sense to guarantee the stability of the controlled system. Numerical simulations show the effectiveness of the proposed approach.     

A robust stabilization problem of fuzzy control systems and itsapplication to backing up control of a truck-trailer

A robust stabilization problem for fuzzy systems is discussed in accordance with the definition of stability in the sense of Lyapunov. We consider two design problems: nonrobust controller design and robust controller design. The former is a design problem for fuzzy systems with no premise parameter uncertainty. The latter is a design problem for fuzzy systems with premise parameter uncertainty. To realize two design problems, we derive four stability conditions from a basic stability condition proposed by Tanaka and Sugeno: nonrobust condition, weak nonrobust condition, robust condition, and weak robust condition. We introduce concept of robust stability for fuzzy control systems with premise parameter uncertainty from the weak robust condition. To introduce robust stability, admissible region and variation region, which correspond to stability margin in the ordinary control theory, are defined. Furthermore, we develop a control system for backing up a computer simulated truck-trailer which is nonlinear and unstable. By approximating the truck-trailer by a fuzzy system with premise parameter uncertainty and by using concept of robust stability, we design a fuzzy controller which guarantees stability of the control system under a condition. The simulation results show that the designed fuzzy controller smoothly achieves backing up control of the truck-trailer from all initial positions     

Enhancement of robust performance for fuzzy parametric uncertain time‐delay systems using differential evolution algorithm

This paper proposes a systematic methodology for the enhancement of robust stability and performance of a fuzzy parametric uncertain time‐delay system. A fuzzy parametric uncertain time‐delay system is an example for a linear time‐invariant uncertain time‐delay system with fuzzy coefficients. By using the nearest approximation, these fuzzy coefficients are approximated into crisp sets called intervals to get an interval system. The proposed approach develops the necessary and sufficient stability conditions of interval polynomials for determining the robust stability. Then, by using these developed stability conditions, a set of inequalities in terms of controller parameters are obtained from the closed‐loop characteristic polynomial of fuzzy parametric uncertain time‐delay system. Finally, these inequalities are solved to obtain robust controller with the help of a differential evolution algorithm for an unstable fuzzy parametric uncertain time‐delay system. Consequently, a lead‐lag compensator is constructed based on the frequency domain approach to improve the performance of the fuzzy parametric uncertain time‐delay system. The proposed method has the advantage of less computational complexity and easy to implement on a digital computer. The viability of the proposed methodology is illustrated through a numerical example for its successful implementation. The efficacy of the proposed methodology is also evaluated against the available approach in the literature and the simulation results are successfully implemented for robust stability and performance of fuzzy parametric uncertain time‐delay systems.     

Robust fuzzy stabilisation of interval plants

The paper deals with the problem of stabilisation of interval systems. To this end, by using Takagi–Sugeno fuzzy mechanism, a Mamdani-type PID-like fuzzy controller is modified and extended to develop a new PID-like Takagi–Sugeno fuzzy stabilising controller for the plant described by an interval system. Indeed, a PID-like Takagi–Sugeno fuzzy controller and an interval plant are considered in the forward path of a unity feedback system, and parameters in Takagi–Sugeno fuzzy controller are determined so that the stability of the closed-loop system is assured. The closed-loop system has a multilinear uncertainty structure. Therefore, based on the Zero Exclusion Condition for multilinear uncertain systems, a new theorem presenting sufficient conditions for the Takagi–Sugeno fuzzy controller to be robust stability guaranteed is also derived. An example is given to illustrate the application and the effectiveness of the proposed controller.     

Robust stability analysis and design method for the fuzzy feedbacklinearization regulator

A robust stability analysis and design method for a fuzzy feedback linearization regulator is presented. The well-known Takagi-Sugeno fuzzy model is used as the nonlinear plant model. Uncertainties and disturbance are assumed to be included in the model structure with known bounds. For these structured uncertainties, stability robustness of the closed system is analyzed in both input-output sense and Lyapunov sense. The robust stability conditions are proposed using multivariable circle criterion and the relationship between input-output stability and Lyapunov stability. Also, based on the stability analysis, a systematic design procedure for the fuzzy feedback linearization regulator is provided. The effectiveness of the proposed analysis and design method is illustrated by a simple example     

Robust stability constraints for fuzzy model predictive control

This paper addresses the synthesis of a predictive controller for a nonlinear process based on a fuzzy model of the Takagi-Sugeno (T-S) type, resulting in a stable closed-loop control system. Conditions are given that guarantee closed-loop robust asymptotic stability for open-loop bounded-input-bounded-output (BIBO) stable processes with an additive l₁-norm bounded model uncertainty. The idea is closely related to (small-gain-based) l₁-control theory, but due to the time-varying approach, the resulting robust stability constraints are less conservative. Therefore the fuzzy model is viewed as a linear time-varying system rather than a nonlinear one. The goal is to obtain constraints on the control signal and its increment that guarantee robust stability. Robust global asymptotic stability and offset-free reference tracking are guaranteed for asymptotically constant reference trajectories and disturbances     

Robust fuzzy control for discrete perturbed time-delay affine Takagi-Sugeno fuzzy models

The purpose of this paper is to study the stability analysis and controller synthesis principles of Discrete Perturbed Time-Delay Affine (DPTDA) Takagi-Sugeno (T-S) fuzzy models. In general, the T-S fuzzy model is a weighted sum of some linear subsystems via fuzzy membership functions. This paper considers fuzzy rules include both linear nominal parts and uncertain parameters in the time-delay affine T-S fuzzy model. For DPTDA T-S fuzzy models, the T-S fuzzy control scheme is used to confront the H_∞ performance constraints. Some sufficient conditions are derived on robust H_∞ disturbance attenuation in which both robust stability and a prescribed performance are required to be achieved. In order to find suitable fuzzy controllers, the Iterative Linear Matrix Inequality (ILMI) algorithm is employed to solve these sufficient conditions. At last, a numerical simulation for the nonlinear truck-trailer system is given to show the applications of the present design approach.     

Robust fuzzy control of nonlinear systems with parametricuncertainties

Addresses the robust fuzzy control problem for nonlinear systems in the presence of parametric uncertainties. The Takagi-Sugeno (T-S) fuzzy model is adopted for fuzzy modeling of the nonlinear system. Two cases of the T-S fuzzy system with parametric uncertainties, both continuous-time and discrete-time cases are considered. In both continuous-time and discrete-time cases, sufficient conditions are derived for robust stabilization in the sense of Lyapunov asymptotic stability, for the T-S fuzzy system with parametric uncertainties. The sufficient conditions are formulated in the format of linear matrix inequalities. The T-S fuzzy model of the chaotic Lorenz system, which has complex nonlinearity, is developed as a test bed. The effectiveness of the proposed controller design methodology is finally demonstrated through numerical simulations on the chaotic Lorenz system     

Robust fuzzy control for uncertain discrete-time nonlinear Markovian jump systems without mode observations

This paper studies the robust fuzzy control problem of uncertain discrete-time nonlinear Markovian jump systems without mode observations. The Takagi and Sugeno (T-S) fuzzy model is employed to represent a discrete-time nonlinear system with norm-bounded parameter uncertainties and Markovian jump parameters. As a result, an uncertain Markovian jump fuzzy system (MJFS) is obtained. A stochastic fuzzy Lyapunov function (FLF) is employed to analyze the robust stability of the uncertain MJFS, which not only is dependent on the operation modes of the system, but also directly includes the membership functions. Then, based on this stochastic FLF and a non-parallel distributed compensation (non-PDC) scheme, a mode-independent state-feedback control design is developed to guarantee that the closed-loop MJFS is stochastically stable for all admissible parameter uncertainties. The proposed sufficient conditions for the robust stability and mode-independent robust stabilization are formulated as a set of coupled linear matrix inequalities (LMIs), which can be solved efficiently by using existing LMI optimization techniques. Finally, it is also demonstrated, via a simulation example, that the proposed design method is effective.     

Robust stability of impulsive Takagi-Sugeno fuzzy systems with parametric uncertainties

This paper presents a kind of time-varying impulsive Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties in which each subsystem of the model is time-varying. Several robust stabilities of time-varying systems with parametric uncertainties, such as general robust stability, robustly asymptotical stability and exponential stability, are studied using uniformly positive definite matrix functions and the Lyapunov method. Specifically, robust stability conditions of time-invariant impulsive T-S fuzzy systems are also derived in the formulation of quasi-linear matrix inequalities (QLMIs) and an iterative LMIs algorithm is designed for solving QLMIs. Finally, a unified chaotic system with continuous periodic switch and a unified time-invariant chaotic system are used for demonstrating the effectiveness of our respective results.     

一类不确定多输入模糊双线性系统的鲁棒H∞控制

针对一类带有参数不确定性和干扰的多输入模糊双线性系统(FBS)的鲁棒H_∞控制问题,使用并行分布补偿算法(PDC)设计了模糊控制器,得到了整个模糊控制系统鲁棒全局稳定的充分条件,控制器的设计可以通过求解一系列线性矩阵不等式(LMI)获得.仿真例子验证了方法的有效性.     

Robust fuzzy control for a plant with fuzzy linear model

A robust complexity reduced proportional-integral-derivative (PID)-like fuzzy controllers is designed for a plant with fuzzy linear model. The plant model is described with the expert's linguistic information involved. The linguistic information for the plant model is represented as fuzzy sets. In order to design a robust fuzzy controller for a plant model with fuzzy sets, an approach is developed to implement the best crisp approximation of fuzzy sets into intervals. Then, Kharitonov's Theorem is applied to construct a robust fuzzy controller for the fuzzy uncertain plant with interval model. With the linear combination of input variables as a new input variable, the complexity of the fuzzy mechanism of PID-like fuzzy controller is significantly reduced. The parameters in the robust fuzzy controller are determined to satisfy the stability conditions. The robustness of the designed fuzzy controller is discussed. Also, with the provided definition of relative robustness, the robustness of the complexity reduced fuzzy controller is compared to the classical PID controller for a second-order plant with fuzzy linear model. The simulation results are included to show the effectiveness of the designed PID-like robust fuzzy controller with the complexity reduced fuzzy mechanism.     

Robust fuzzy model-following control of robot manipulators

A robust fuzzy model-following control system is proposed for the control of robot manipulators. The application field to n-link robot manipulators with torque disturbance and measurement noise is addressed. The control objective is obtained by tailoring a nominal adaptation process of parameters to implement appropriate function approximation and facilitating a self-tuning mechanism on the consequent membership functions to overcome the equivalent uncertainty. A novel fuzzy system with self-tuning mechanism provides robust property and the rule-base in the form of “IF situation THEN the control input”. The proposed multilayer fuzzy logic controller can improve both transient and stability margins without a priori knowledge about the dynamic model or parameters of the robotic system. Using the Lyapunov stability method, the uniform ultimate boundedness of tracking error has been proved. The performance is demonstrated by simulating the control of a two-link robot in various situations     

Stability issues on Takagi-Sugeno fuzzy model-parametric approach

The main objective of this research is to provide a systematic stability analysis technique for fuzzy systems using well-established results from robust stability of a family of polynomials. The model-based Takagi-Sugeno (1985) fuzzy control system is considered and shown to have a dependent coefficient structure involving firing strengths. Two Kharitonov regions via overbounding are established to facilitate the extreme results for stability test in the space of polynomials as well as the complex plane. The criterion obtained is a sufficient condition for a fuzzy logic controller. Three nonlinear examples are demonstrated to show that the systematic approach proposed here can be used to solve fuzzy stability problems. Fuzzy users concerned over stability issues may find these analytic tools useful     

Secure communications of chaotic systems with robust performancevia fuzzy observer-based design

This paper presents a systematic design methodology for fuzzy observer-based secure communications of chaotic systems with guaranteed robust performance. The Takagi-Sugeno fuzzy models are given to exactly represent chaotic systems. Then, the general fuzzy model of many well-known chaotic systems is constructed with only one premise variable in fuzzy rules and the same premise variable in the system output. Based on this general model, the fuzzy observer of chaotic system is given and leads the stability condition of a linear-matrix inequality problem. When taking the fuzzy observer-based design to applications on secure communications, the robust performance is presented by simultaneously considering the effects of parameter mismatch and external disturbances. Then, the error of the recovered message is stated in an H^∞ criterion. In addition, if the communication system is free of external disturbances, the asymptotic recovering of the message is obtained in the same framework. The main results also hold for applications on chaotic synchronization. Numerical simulations illustrate that this proposed scheme yields robust performance     

基于观测器的参数不确定非线性系统的模糊鲁棒控制

对一类非线性系统进行模糊建模及其模糊观测器设计，研究了在系统的状态不可测且存在参数不确定的模糊鲁捧控制问题，以线性矩阵不等式的形式给出了模糊控制系统具有李雅普诺夫意义下稳定的充分条件，最后把所提出的方法应用到倒立摆系统进行仿真，仿真结果验证了该控制方法的有效性。     

基于观测器的参数不确定非线性系统的模糊鲁棒控制

对一类非线性系统进行模糊建模及其模糊观测器设计, 研究了在系统的状态不可测且存在参数不确定的模糊鲁棒控制问题, 以线性矩阵不等式的形式给出了模糊控制系统具有李雅普诺夫意义下稳定的充分条件, 最后把所提出的方法应用到倒立摆系统进行仿真, 仿真结果验证了该控制方法的有效性.     

基于矩阵测度的非线性广义时滞系统鲁棒模糊控制

研究T-S模糊广义时滞系统的鲁棒控制问题.不同于传统的寻求公共正定矩阵的方法,基于矩阵测度给出保证系统鲁棒稳定的充分条件,并将此条件进一步转化为线性矩阵不等式.通过求解线性矩阵不等式,得到状态反馈控制器和静态输出反馈控制器.最后通过算例仿真验证了方法的有效性.     

On the robust regulation problem for nonlinear systems using fuzzy immersions

In this paper, a nonlinear robust regulator, based on fuzzy techniques, is proposed as an alternative procedure design to guarantee stability for a wider range of variation in both the exosystem and the immersion. This robust regulator is a fuzzy error feedback controller which relies on the existence of an internal model, obtained by finding, if possible, an observable immersion of the exosystem dynamics, which allows to generate all the possible steady state inputs for the admissible values of the system parameters in order to track and/or reject time-varying references and/or disturbances, while maintaining the stability conditions of the system. Finally, the proposed fuzzy controller scheme is compared with traditional techniques through a case study to test the robustness properties in the face of parameter uncertainties and changes on load disturbances.     

一类模糊时滞系统指定衰减度的鲁棒控制

研究了一类非线性时滞系统基于模糊T-S模型的鲁棒镇定问题,所考虑的不确定时滞系统含有时变未知但有界的状态时滞,首先利用Razumikhin定理和Lyapunov定理,得出了由模糊T-S模型描述的非线性时滞系统鲁棒稳定且具有指定衰减度的判据,其次得到了具有指定衰减度的无记忆状态反馈控制律存在的充分条件及相应的控制器设计方法,该条件被进一步等价地转化为一个线性矩阵不等式的可解性问题,所设计的控制器确保了闭环系统具有指定衰减度鲁棒稳定。