Robust H ∞ fuzzy control of nonlinear systems with multiple time delays

A robustness design of fuzzy control via model-based approach is proposed in this article to overcome the effect of approximation error between multiple time-delay nonlinear systems and Takagi--Sugeno (T-S) fuzzy models. A stability criterion is derived based on Lyapunov's direct method to ensure the stability of nonlinear multiple time-delay systems especially for the resonant and chaotic systems. Positive definite matrices P and R_k of the criterion are obtained by using linear matrix inequality (LMI) optimization algorithms to solve the robust fuzzy control problem. In terms of the control scheme and this criterion, a fuzzy controller is then designed via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay system and the H ^∞ control performance is achieved at the same time. Finally, two numerical examples of the chaotic and resonant systems are demonstrated to show the concepts of the proposed approach.     

Delay-dependent T–S fuzzy control for nonlinear interconnected systems: Using dithers as auxiliaries

In this study, a novel approach via the composite of fuzzy controllers and dithers is presented. According to this approach, we can synthesize a set of fuzzy controllers and find appropriate dithers to stabilize nonlinear multiple time-delay (NMTD) interconnected systems. A robustness design of model-based fuzzy control is first proposed to overcome the effect of modeling errors between the NMTD interconnected subsystems and Takagi–Sugeno (T–S) fuzzy models. In terms of Lyapunov's direct method, a delay-dependent stability criterion is then derived to guarantee the asymptotic stability of NMTD interconnected systems. Based on this criterion and the decentralized control scheme, a set of model-based fuzzy controllers is synthesized via the technique of parallel distributed compensation (PDC) to stabilize the NMTD interconnected system. When the designed fuzzy controllers cannot stabilize the NMTD interconnected systems, a batch of high-frequency signals (commonly referred to as dithers) is simultaneously introduced to stabilize it. If the frequencies of dithers are high enough, the outputs of the dithered interconnected system and those of its corresponding mathematical model–the relaxed interconnected system can be made as close as desired. This makes it possible to obtain a rigorous prediction of the stability of the dithered interconnected system based on the one of the relaxed interconnected system. Finally, a numerical example is given to illustrate the feasibility of our approach.     

Stability analysis of T–S fuzzy models for nonlinear multiple time-delay interconnected systems

In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the stability analysis for nonlinear interconnected systems with multiple time-delays using linear matrix inequality (LMI) theory. In terms of Lyapunov’s direct method for multiple time-delay fuzzy interconnected systems, a novel LMI-based stability criterion which can be solved numerically is proposed. Then, the common P matrix of the criterion is obtained by LMI optimization algorithms to guarantee the asymptotic stability of nonlinear interconnect systems with multiple time-delay. Finally, the proposed stability conditions are demonstrated with simulations throughout this paper.     

Adaptive fuzzy semi‐decentralized control for a class of large‐scale nonlinear systems with unknown interconnections

In this paper, a general method is developed to generate a stable adaptive fuzzy semi‐decentralized control for a class of large‐scale interconnected nonlinear systems with unknown nonlinear subsystems and unknown nonlinear interconnections. In the developed control algorithms, fuzzy logic systems, using fuzzy basis functions (FBF), are employed to approximate the unknown subsystems and interconnection functions without imposing any constraints or assumptions about the interconnections. The proposed controller consists of primary and auxiliary parts, where both direct and indirect adaptive approaches for the primary control part are aiming to maintain the closed‐loop stability, whereas the auxiliary control part is designed to attenuate the fuzzy approximation errors. By using Lyapunov stability method, the proposed semi‐decentralized adaptive fuzzy control system is proved to be globally stable, with converging tracking errors to a desired performance. Simulation examples are presented to illustrate the effectiveness of the proposed controller. Copyright © 2006 John Wiley & Sons, Ltd.     

一类离散非线性不确定互联系统的模糊分散控制

利用模糊控制方法研究一类离散非线性互联系统的分散控制问题.首先采用模糊(T-S)模型对离散非线性不确定互联系统进行模糊建模,应用并行分布补偿算法(PDC)给出状态反馈分散模糊控制方案,并基于李亚普诺夫函数方法证明了闭环系统的稳定性.然后当系统的状态不完全可测时,设计模糊分散观测器来估计各子系统的状态,从而给出基于观测器的状态反馈分散模糊控制设计的方法.因为该分散模糊控制设计问题是以线性矩阵不等式的形式给出,所以很容易用凸优化方法求解.仿真结果验证了所提出控制方法的有效性.     

String stability of infinite-dimension stochastic interconnected large-scale systems with time-varying delay

The exponential string stability for a class of nonlinear interconnected large-scale systems with time-varying delay is analysed by using the box theory and constructing a vector Lyapunov function. Under the assumption that the time delay is bounded and continuous, a criterion for exponential string stability of the systems is obtained by analysing the stability of differential inequalities with time-varying delay. The large-scale system is exponential string stable when the conditions associating with the coefficient matrices of the system and the solutions of the Lyapunov equations, interconnected with the system, are satisfied. Since it is independent of the delays and simplifies the calculation, the criterion is easy to apply.     

Robust stabilization of nonlinear multiple time-delay large-scale systems via decentralized fuzzy control

To overcome the effect of modeling errors between nonlinear multiple time-delay subsystems and Takagi-Sugeno (T-S) fuzzy models with multiple time delays, a robustness design of fuzzy control is proposed in This work. In terms of Lyapunov's direct method, a delay-dependent stability criterion is hence derived to guarantee the asymptotic stability of nonlinear multiple time-delay large-scale systems. Based on this criterion and the decentralized control scheme, a set of model-based fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay large-scale system. Finally, a numerical example with simulations is given to demonstrate the concepts discussed throughout This work.     

Robust fuzzy control for continuous perturbed time‐delay affine takagi–sugeno fuzzy models

The stability analysis and controller synthesis methodology for a continuous perturbed time‐delay affine (CPTDA) Takagi–Sugeno (T‐S) fuzzy model is proposed in this paper. The CPTDA T‐S fuzzy models include both linear nominal parts and uncertain parameters in each fuzzy rule. The proposed fuzzy control approach is developed based on an iterative linear matrix inequality (ILMI) algorithm to cope with the stability criteria and H_∞ performance constraints for the CPTDA T‐S fuzzy models. Finally, a numerical simulation for the nonlinear inverted pendulum system is given to show the application and availability of the present design approach. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new discretized Lyapunov–Krasovskii functional for stability analysis and control design of time‐delayed TS fuzzy systems

This paper proposes a new Lyapunov–Krasovskii functional to cope with stability analysis and control design for time‐delay nonlinear systems modeled in the Takagi–Sugeno (TS) fuzzy form. The delay‐dependent conditions are formulated as linear matrix inequalities (LMIs), solvable through several numerical tools. By using the Gu's discretization technique and by employing an appropriated fuzzy functional, less conservative conditions are obtained. Numerical results illustrate the efficiency of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

New delay‐dependent stability analysis and synthesis of T–S fuzzy systems with time‐varying delay

In this paper, a new method is proposed for stability analysis and synthesis of Takagi–Sugeno (T–S) fuzzy systems with time‐varying delay. Based on a new Lyapunov–Krasovskii functional (LKF), some less conservative delay‐dependent stability criteria are established. In the derivation process, some additional useful terms, ignored in previous methods, are considered and new free‐weighting matrices are introduced to estimate the upper bound of the derivative of LKF for T–S fuzzy systems with time‐varying delay. The proposed stability criterion and stabilization condition are represented in terms of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive Finite‐Time Robust Control of Nonlinear Delay Hamiltonian Systems Via Lyapunov‐Krasovskii Method

This paper investigates the adaptive finite‐time robust control problem of a class of nonlinear time‐delay Hamiltonian systems via the Lyapunov‐Krasovskii (L‐K) method, and proposes some delay‐dependent results on the issue. Different from existing works, this paper first presents a time‐varying finite‐time stability (FTS) criterion via an L‐K functional approach, and obtains two FTS conditions by constructing specific L‐K functionals. Then, the adaptive finite‐time robust control problem is investigated for nonlinear time‐delay port‐controlled Hamiltonian (PCH) systems, and a control design procedure is presented. Finally, the effectiveness of the results is demonstrated by an illustrative example.     

Stability analysis and H∞ control design for a class of nonlinear time‐delay systems

This paper investigates the stability and H_∞ control problem for a class of nonlinear time‐delay systems with a nonsingular Jacobian matrix, and provides a number of new results regarding stability analysis and control design. Firstly, an equivalent form is obtained for this class of systems by means of coordinate transformation and/or orthogonal decomposition of vector fields. Then, based on the equivalent form and free‐weighting matrix method, several sufficient conditions, in terms of nonlinear matrix inequalities, are derived for the stability analysis of the time‐delay systems by constructing suitable Lyapunov functionals. Finally, we use the equivalent form and the obtained stability results to investigate the H_∞ control problem, and present a control design procedure for this class of time‐delay systems. A study of illustrative examples shows that the results obtained in this paper have less conservatism, and work very well in the stability analysis and control design of some nonlinear time‐delay systems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

不确定时滞关联大系统的全局稳定模糊容错控制

研究了一类带有时变时滞的不确定非线性关联大系统的自适应模糊容错控制问题.用有界的参考信号代换模糊逼近器输入中的未知时滞信号,使得控制器的设计与应用不再依赖于时滞假设条件,使得控制器的设计和控制方法的应用更为方便.容错反推控制技术和自适应技术相结合来处理代换误差和逼近误差.所提出的方案能有效补偿所有4种类型的执行器故障,同时还可保证闭环系统的全局稳定性.仿真结果进一步验证了本文方法的有效性.     

Stability analysis of fuzzy large-scale systems with time delays in interconnections

The stability problem of fuzzy large-scale systems with time delays in interconnections is considered in this paper. The fuzzy large-scale system consists of J interconnected subsystems, which are represented by Takagi-Sugeno (T-S) fuzzy models. The stability conditions are derived using the Lyapunov-Krasovskii functional approach. This condition is independent of time delay and does not need the solution of a Lyapunov equation or Riccati equation. A stabilization approach for the delayed fuzzy large-scale systems through fuzzy state feedback-based controller is also presented in this paper. Finally, an example is given to demonstrate the result.     

Robustness design of fuzzy control for nonlinear multiple time-delay large-scale systems via neural-network-based approach

The stabilization problem is considered in this correspondence for a nonlinear multiple time-delay large-scale system. First, the neural-network (NN) model is employed to approximate each subsystem. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of each NN model. According to the LDI state-space representation, a robustness design of fuzzy control is proposed to overcome the effect of modeling errors between subsystems and NN models. Next, in terms of Lyapunov's direct method, a delay-dependent stability criterion is derived to guarantee the asymptotic stability of nonlinear multiple time-delay large-scale systems. Finally, based on this criterion and the decentralized control scheme, a set of fuzzy controllers is synthesized to stabilize the nonlinear multiple time-delay large-scale system.     

Stability analysis and control for switched stochastic delayed systems

This paper investigates almost sure exponential stability and feedback stabilization for switched time‐delay systems with nonlinear stochastic perturbations. The main contributions of this paper are threefold: (i) based on the non‐convolution type multiple Lyapunov functionals and the mathematical induction approach, a mean‐square exponential stability condition for nonlinear stochastic switched systems is first established, such that the obtained average dwell time does not rely on any given decay rate; (ii) by using the method developed in part (i) and the stochastic analysis techniques and limit methods in probability, an almost sure exponential stability criterion for switched delayed systems with nonlinear stochastic uncertainties is presented, and then a state feedback controller for the systems under consideration is designed; and (iii) when certain assumptions are made on the nonlinear stochastic perturbations, the results in this paper are further improved by relaxing some conditions. The effectiveness of the proposed method is demonstrated by three illustrative examples. Copyright © 2015 John Wiley & Sons, Ltd.     

H∞ decentralized fuzzy model reference tracking controldesign for nonlinear interconnected systems

In general, due to the interactions among subsystems, it is difficult to design an H_∞ decentralized controller for nonlinear interconnected systems. The model reference tracking control problem of nonlinear interconnected systems is studied via H_∞ decentralized fuzzy control method. First, the nonlinear interconnected system is represented by an equivalent Takagi-Sugeno type fuzzy model. A state feedback decentralized fuzzy control scheme is developed to override the external disturbances such that the H∞ model reference tracking performance is achieved. Furthermore, the stability of the nonlinear interconnected systems is also guaranteed. If states are not all available, a decentralized fuzzy observer is proposed to estimate the states of each subsystem for decentralized control. Consequently, a fuzzy observer-based state feedback decentralized fuzzy controller is proposed to solve the H_∞ tracking control design problem for nonlinear interconnected systems. The problem of H _∞ decentralized fuzzy tracking control design for nonlinear interconnected systems is characterized in terms of solving an eigenvalue problem (EVP). The EVP can be solved very efficiently using convex optimization techniques. Finally, simulation examples are given to illustrate the tracking performance of the proposed methods     

Robust fuzzy tracking control for nonlinear networked control systems with integral quadratic constraints

This paper investigates the robust tracking control problem for a class of nonlinear networked control systems (NCSs) using the Takagi-Sugeno (T-S) fuzzy model approach. Based on a time-varying delay system transformed from the NCSs, an augmented Lyapunov function containing more useful information is constructed. A less conservative sufficient condition is established such that the closed-loop systems stability and time-domain integral quadratic constraints (IQCs) are satisfied while both time-varying network-induced delays and packet losses are taken into account. The fuzzy tracking controllers design scheme is derived in terms of linear matrix inequalities (LMIs) and parallel distributed compensation (PDC). Furthermore, robust stabilization criterion for nonlinear NCSs is given as an extension of the tracking control result. Finally, numerical simulations are provided to illustrate the effectiveness and merits of the proposed method.     

非线性时延网络控制系统的模糊建模与控制

针对时变网络诱导时延小于一个采样周期的非线性时延网络控制系统,讨论系统的稳定性及控制器的设计方法.利用基于“IF-THEN”规则的模糊模型近似系统中的非线性,将时延的不确定性转化为系统参数的不确定性,从而将此类非线性网络控制系统建模为一类具有参数不确定性的离散Takagi-Sugeno（T-S）模糊模型.基于建立的模型,利用Lyapunov方法和线性矩阵不等式方法,分析了系统的稳定性及模糊状态反馈控制器的设计方法,最后通过仿真实例验证了所提出方法的有效性.