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.     

不确定非线性系统的时滞依赖保性能控制

针对一类基于T-S模糊模型表示的具有时变状态时滞和范数有界不确定性非线性系统,研究了时滞依赖保性能模糊控制器设计问题。对于用T-S模糊模型表示的非线性时滞系统,已知系统的时变时滞本身及其变化率的上界,采用并行分散补偿技术,通过选取合适的Lyapunov函数,推导了依赖时滞上界及变化率上界的时滞保性能模糊控制器存在的充分条件,进而通过建立和求解LMI（线性矩阵不等式）约束的凸优化问题,给出了保性能控制律的设计方法。数值算例表明了该方法的有效性。     

A novel criterion for nonlinear time-delay systems using LMI fuzzy Lyapunov method

This study presents a kind of fuzzy robustness design for nonlinear time-delay systems based on the fuzzy Lyapunov method, which is defined in terms of fuzzy blending quadratic Lyapunov functions. The basic idea of the proposed approach is to construct a fuzzy controller for nonlinear dynamic systems with disturbances in which the delay-independent robust stability criterion is derived in terms of the fuzzy Lyapunov method. Based on the robustness design and parallel distributed compensation (PDC) scheme, the problems of modeling errors between nonlinear dynamic systems and Takagi–Sugeno (T–S) fuzzy models are solved. Furthermore, the presented delay-independent condition is transformed into linear matrix inequalities (LMIs) so that the fuzzy state feedback gain and common solutions are numerically feasible with swarm intelligence algorithms. The proposed method is illustrated on a nonlinear inverted pendulum system and the simulation results show that the robustness controller cannot only stabilize the nonlinear inverted pendulum system, but has the robustness against external disturbance.     

具有时延的网络控制系统控制器设计

针对一类存在随机时延的网络控制系统,传感器采用时间驱动,控制器和执行器采用事件驱动,提出了一种新的具有随机时延的网络控制系统的建模方法-离散模糊T-S模型,在此模型的基础上应用并行分布补偿（PDC）原理设计了模糊控制器。应用Lyapunov定理和线性矩阵不等式（LMI）方法,研究了系统的稳定性问题,给出基于LMI的状态反馈模糊控制器的设计方法。通过仿真实例验证控制方法能够保证系统稳定。     

基于T-S模型的非线性时滞系统非脆弱保性能模糊控制

针对一类用T-S模糊模型描述的非线性时滞系统,采用状态反馈的并行分布补偿方法,研究其保守性较小的非脆弱保性能模糊控制问题,使闭环系统在控制器存在加性摄动的情况下,其闭环性能指标值低于确定的上界.利用线性矩阵不等式处理方法,导出了非脆弱保性能模糊控制律的存在条件,通过建立和求解一个线性矩阵不等式问题,给出了非脆弱保性能模糊控制律的设计方法.仿真结果表明了所提出方法的有效性.     

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.     

Takagi-Sugeno模糊连续系统的模糊采样控制

研究了T-S模糊连续系统的模糊采样控制问题．利用广义系统的描述方法、Lyapunov-Krasovikii泛函以及线性矩阵不等式(LMI)方法,建立了LMIs形式的依赖于采样时间间隔的模糊采样镇定条件,同时给出了模糊采样控制律的设计方法．所设计的模糊采样控制律可以镇定T-S模糊系统．而且,当连续时间模糊控制律可以镇定T-S模糊系统时,对于足够小的采样时间间隔,带有同样增益矩阵的模糊采样控制律也可以镇定T-S模糊系统．最后,通过两个仿真实例说明了所给方法的有效性．     

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。     

一类非线性网络控制系统镇定的新方法

研究了一类非线性网络控制系统(Networked control systems, NCSs)的镇定问题. 在一般的网络环境中, 通过平行分布补偿技术, 将非线性NCS建模为包含一个稳定子系统和一个可能不稳定子系统的模糊时滞切换系统. 利用分段Lyapunov泛函方法和平均驻留时间方法, 得到了非线性NCS指数稳定的充分条件, 并以线性矩阵不等式(Linear matrix inequality, LMI) 形式给出了模糊控制器的设计方法. 最后通过数值例子说明了所给方法的有效性.     

Passive fuzzy controller design via observer feedback for stochastic Takagi-Sugeno fuzzy models with multiplicative noises

This paper investigates the fuzzy control problem of a class of nonlinear continuous-time stochastic systems with achieving the passivity performance. A model-based observer feedback fuzzy control utilizing the concept of so-called parallel distributed compensation (PDC) is employed to stabilize the class of nonlinear stochastic systems that are represented by the Takagi-Sugeno (T-S) fuzzy models. Based on the Lyapunov criteria, the Linear Matrix Inequality (LMI) technique is used to synthesize the observer feedback fuzzy controller design such that the closed-loop system satisfies stability and passivity constraints, simultaneously. Finally, a numerical example is given to demonstrate the applicability and effectiveness of the proposed design method.     

一类非匹配不确定性非线性系统的鲁棒镇定

研究了非线性系统存在非匹配不确定性时控制器的鲁棒镇定问题. 基于对象的模糊动态模型, 提出了一种状态反馈控制器的设计, 给出控制器在建模不确定性等各种非匹配不确定性存在下仍能够镇定非线性系统的一个充分条件. 仿真结果表明了设计方法的正确性.     

Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation

Takagi-Sugeno (TS) fuzzy models can provide an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a set of linear input-output submodels. In this paper, the TS fuzzy modeling approach is utilized to carry out the stability analysis and control design for nonlinear systems with actuator saturation. The TS fuzzy representation of a nonlinear system subject to actuator saturation is presented. In our TS fuzzy representation, the modeling error is also captured by norm-bounded uncertainties. A set invariance condition for the system in the TS fuzzy representation is first established. Based on this set invariance condition, the problem of estimating the domain of attraction of a TS fuzzy system under a constant state feedback law is formulated and solved as a linear matrix inequality (LMI) optimization problem. By viewing the state feedback gain as an extra free parameter in the LMI optimization problem, we arrive at a method for designing state feedback gain that maximizes the domain of attraction. A fuzzy scheduling control design method is also introduced to further enlarge the domain of attraction. An inverted pendulum is used to show the effectiveness of the proposed fuzzy controller.     

Robustness design of nonlinear dynamic systems via fuzzy linearcontrol

This study introduces a fuzzy linear control design method for nonlinear systems with optimal H^∞ robustness performance. First, the Takagi and Sugeno fuzzy linear model (1985) is employed to approximate a nonlinear system. Next, based on the fuzzy linear model, a fuzzy controller is developed to stabilize the nonlinear system, and at the same time the effect of external disturbance on control performance is attenuated to a minimum level. Thus based on the fuzzy linear model, H^∞ performance design can be achieved in nonlinear control systems. In the proposed fuzzy linear control method, the fuzzy linear model provides rough control to approximate the nonlinear control system, while the H^∞ scheme provides precise control to achieve the optimal robustness performance. Linear matrix inequality (LMI) techniques are employed to solve this robust fuzzy control problem. In the case that state variables are unavailable, a fuzzy observer-based H^∞ control is also proposed to achieve a robust optimization design for nonlinear systems. A simulation example is given to illustrate the performance of the proposed design method     

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.     

基于模糊模型的时滞不确定系统的模糊H∞鲁棒反馈控制

讨论了一类具有状态和控制时滞的不确定非线性系统的模糊H_∞ 状态反馈控制问题. 采用具有时滞的不确定Takagi-Sugeno(T-S)模糊模型对非线性系统进行建模, 提出了一套基于LMI的模糊鲁棒控制器的系统设计方法, 给出了模糊H_∞状态反馈控制器存在的充分条件, 以保证闭环模糊系统渐近稳定并满足从干扰输入到控制输出的H_∞范数界约束. 示例仿真表明了该方法的有效性.     

The stability of an oceanic structure with T–S fuzzy models

In this study we construct and derive analytical solutions for a mathematical model of an oceanic environment in which wave-induced flow fields cause structural surge motion after which a fuzzy control technique is developed to alleviate structural vibration. Specifically the Takagi–Sugeno (T–S) fuzzy model is employed to approximate the oceanic structure and a parallel-distributed-compensation (PDC) scheme is utilized in a control procedure designed to reduce the structural response. All local state feedback controllers are integrated to construct a global fuzzy logic controller. The Lyapunov method is used to achieve structural stability. The interaction between the wave motion and the structural response is explained using the separation of variables method. The surge motion is related to the characteristics of the wave and the structure. A parametric approach is utilized to show these effects. Other parameters remain constant. In an oceanic structural system, platform migration is often caused by the wave force. The stability of an oceanic structure can be proven theoretically based on stability analysis. The decay of the displacement and velocity due to the use of the proposed fuzzy controllers is demonstrated by a numerical simulation.     

Controller synthesis for uncertain fuzzy systems with integrating multiple time-varying delays

This article mainly investigates the controller synthesis for uncertain fuzzy systems with integrating multiple time-varying delays, which can describe a much larger class of nonlinear systems or uncertain time-delay systems. First, a sufficient stability condition for the unforced Takagi–Sugeno (T-S) fuzzy models with multiple time-varying delays and uncertainties are derived. By involving the parallel distributed compensator (PDC), some design conditions for the resulting closed-loop fuzzy systems are further presented. These proposed conditions are all expressed in terms of linear matrix inequalities (LMIs), and we can readily perform the PDC synthesis from current LMI solvers. Numerical examples are given to demonstrate the validity and superiority of the proposed approach.     

Stable adaptive fuzzy control of nonlinear systems using small-gain theorem and LMI approach

A new design scheme of stable adaptive fuzzy control for a class of nonlinear systems is proposed in this paper. The T-S fuzzy model is employed to represent the systems. First, the concept of the so-called parallel distributed compensation (PDC) and linear matrix inequality (LMI) approach are employed to design the state feedback controller without considering the error caused by fuzzy modeling. Sufficient conditions with respect to decay rate α are derived in the sense of Lyapunov asymptotic stability. Finally, the error caused by fuzzy modeling is considered and the input-tostate stable (ISS) method is used to design the adaptive compensation term to reduce the effect of the modeling error. By the small-gain theorem, the resulting closed-loop system is proved to be input-to-state stable. Theoretical analysis verifies that the state converges to zero and all signals of the closed-loop systems are bounded. The effectiveness of the proposed controller design methodology is demonstrated through numerical simulation on the chaotic Henon system.     

模糊时滞系统的输出反馈控制及其稳定性分析

利用模糊T－S模型对一类不确定非线性时滞系统进行建模，在此基础上，提出了模糊不确定时滞系统的状态反馈控制及其输出反馈控制的设计，并给出了模糊闭环系统渐近稳定的充分条件，基于李亚普诺夫函数和线性矩阵不等式方法，证明了模糊系统的渐近稳定性。     

Controller design for electric power steering system using T-S fuzzy model approach

Pressure ripples in electric power steering （EPS） systems can be caused by the phase lag between the driver s steering torque and steer angle, the nonlinear frictions, and the disturbances from road and sensor noise especially during high-frequency maneuvers. This paper investigates the use of the robust fuzzy control method for actively reducing pressure ripples for EPS systems. Remarkable progress on steering maneuverability is achieved. The EPS dynamics is described with an eight-order nonlinear state-space model and approximated by a Takagi-Sugeno （T-S） fuzzy model with time-varying delays and external disturbances. A stabilization approach is then presented for nonlinear time-delay systems through fuzzy state feedback controller in parallel distributed compensation （PDC） structure. The closed-loop stability conditions of EPS system with the fuzzy controller are parameterized in terms of the linear matrix inequality （LMI） problem. Simulations and experiments using the proposed robust fuzzy controller and traditional PID controller have been carried out for EPS systems. Both the simulation and experiment results show that the proposed fuzzy controller can reduce the torque ripples and allow us to have a good steering feeling and stable driving.