Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinement of the polynomial Lyapunov function so as to make it share the fuzzy structure of the model proves advantageous. Conditions thus obtained are tested via available SOS software.     

Stable fuzzy logic design of point-to-point control for mechanical systems

An approach for the development of fuzzy point-to-point control laws for second-order mechanical systems is presented. Asymptotic stability of the resulting closed-loop system is proved using Lyapunov stability theory. Closed-loop performance and robustness are quantified in terms of the parameters of membership functions. As opposed to most existing fuzzy control laws, the closed-loop stability of the proposed controller does not depend on the knowledge of the entire dynamics. Moreover, the approach does not require the plant to be open-loop stable. The proposed approach is demonstrated on design and simulation study of a fuzzy controller for a two-link robotic arm.     

Sampled‐data reliable stabilization of T‐S fuzzy systems and its application

In this article, based on sampled‐data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input‐delay approach, the sampled‐data system is equivalently transformed into a continuous‐time system with a variable time delay. The main objective is to design a suitable reliable sampled‐data state feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger‐based double integral inequality, some new delay‐dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying system's stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models. © 2016 Wiley Periodicals, Inc. Complexity 21: 518–529, 2016     

Dual-Sampling-Rate Moving-Horizon Control of a Class of Linear Systems with Input Saturation and Plant Uncertainty

Moving-horizon control is a type of sampled-data feedback control in which the control over each sampling interval is determined by the solution of an open-loop optimal control problem. We develop a dual-sampling-rate moving-horizon control scheme for a class of linear, continuous-time plants with strict input saturation constraints in the presence of plant uncertainty and input disturbances. Our control scheme has two components: a slow-sampling moving-horizon controller for a nominal plant and a fast-sampling state-feedback controller whose function is to force the actual plant to emulate the nominal plant. The design of the moving-horizon controller takes into account the nonnegligible computation time required to compute the optimal control trajectory.We prove the local stability of the resulting feedback system and illustrate its performance with simulations. In these simulations, our dual-sampling-rate controller exhibits performance that is considerably superior to its single-sampling-rate moving-horizon controller counterpart.     

Indirect adaptive fuzzy observer and controller design based on interval type-2 T-S fuzzy model

This paper presents the design scheme of the indirect adaptive fuzzy observer and controller based on the interval type-2 (IT2) T-S fuzzy model. The nonlinear systems can be well approximated by IT2 T-S fuzzy model, in which the fuzzy rules’ antecedents are interval type-2 fuzzy sets and consequents are linear state equations. The proposed IT2 T-S fuzzy model is a combination of IT2 fuzzy system and T-S fuzzy model, and also inherits the benefits of type-2 fuzzy logic systems, which is able to directly handle uncertainties and can minimize the effects of uncertainties in rule-based fuzzy system. These characteristics can improve the accuracy of the system modeling and reduce the number of system rules. The proposed method using feedback control, adaptive laws, and on-line object parameters are adjusted to ensure observation error bounded. In addition, using Lyapunov synthesis approach and Lipschitz condition, the stability analysis is conducted. The simulation results show that the proposed method can handle unpredicted disturbance and data uncertainties very well in advantage of the effectiveness of observation and control.     

Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

具有量化误差的多步长非线性采样系统的镇定

该文研究一类非线性控制系统在采样器采样过程中产生量化误差的情况下多步长采样镇定问题. 运用近似DTD方法, 在非线性系统的近似离散时间模型上设计全局状态反馈镇定控制器. 当系统近似误差和采样量化误差被限制在一定的条件下, 可以得到含量化误差的多步长非线性采样系统是半全局实用渐近稳定. 最后, 仿真例子验证了所得结果的有效性.     

T–S fuzzy-model-based robust stabilization for a class of nonlinear discrete-time networked control systems

In this paper, the robust stabilization problem is investigated for a class of nonlinear discrete-time networked control systems (NCSs). To study the system stability and facilitate the design of fuzzy controller, Takagi–Sugeno (T–S) fuzzy models are employed to represent the system dynamics of the nonlinear discrete-time NCSs with effects of the approximation errors taken into account, and a unified model of NCSs in the T–S fuzzy model is proposed by modeling the approximation errors as norm-bounded uncertainties in system metrics, where non-ideal network Quality of Services (QoS), such as data dropout and network-induced delay, are coupled in a unified framework. Then, based on the Lyapunov–Krasovskii functional, sufficient conditions are derived for the existence of a fuzzy controller. By these criteria, two approaches to design a fuzzy controller are developed in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are provided to show the effectiveness of the proposed methods.     

Robust design of rule-based fuzzy controllers

This paper deals with the design of stable and robust rule-based fuzzy control systems. New expressions to compute indices which provide a measure of the stability and robustness of fuzzy control systems are presented. The relation between the modification of the rules and the stability is studied through the so-called sensitivity indices. The paper presents procedures that make use of these indices to improve the design of fuzzy control systems, including the modification of the rules to obtain the global stability of an unstable system with multiple attractors, and to improve the dynamic behavior or the robustness of a non-linear plant. An example with a fuzzy controller for a system with non-linear damping and saturation in the actuation is presented to illustrate the design procedure.     

Stochastic stability and robust control for sampled-data systems with Markovian jump parameters

In this paper, the problems of stochastic stability and robust control for a class of uncertain sampled-data systems are studied. The systems consist of random jumping parameters described by finite-state semi-Markov process. Sufficient conditions for stochastic stability or exponential mean square stability of the systems are presented. The conditions for the existence of a sampled-data feedback control and a multirate sampled-data optimal control for the continuous-time uncertain Markovian jump systems are also obtained. The design procedure for robust multirate sampled-data control is formulated as linear matrix inequalities (LMIs), which can be solved efficiently by available software toolboxes. Finally, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed techniques.     

Adaptive fuzzy sliding-mode control for multi-input multi-output chaotic systems

This paper describes an adaptive fuzzy sliding-mode control algorithm for controlling unknown or uncertain, multi-input multi-output (MIMO), possibly chaotic, dynamical systems. The control approach encompasses a fuzzy system and a robust controller. The fuzzy system is designed to mimic an ideal sliding-mode controller, and the robust controller compensates the difference between the fuzzy controller and the ideal one. The parameters of the fuzzy system, as well as the uncertainty bound of the robust controller, are tuned adaptively. The adaptive laws are derived in the Lyapunov sense to guarantee the asymptotic stability and tracking of the controlled system. The effectiveness of the proposed method is shown by applying it to some well-known chaotic systems.     

A general fuzzy control scheme for nonlinear processes with stability analysis

In this paper we develop a general fuzzy control scheme for nonlinear processes. Assuming little knowledge about the dynamics of the controlled process, the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation. A simple local controller is then designed at each fuzzy locality. A fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm. By employing a newly developed structure-adaptive fuzzy modelling scheme, the appropriate fuzzy rule-base for the inference mechanism can be extracted stably and efficiently. The conditions for the stability of the global controller are rigourously established. Simulation results are presented to illustrate the effectiveness of the scheme.     

Adaptive synchronization of T–S fuzzy chaotic systems with unknown parameters

This paper presents a fuzzy model-based adaptive approach for synchronization of chaotic systems which consist of the drive and response systems. Takagi–Sugeno (T–S) fuzzy model is employed to represent the chaotic drive and response systems. Since the parameters of the drive system are assumed unknown, we design the response system that estimates the parameters of the drive system by adaptive strategy. The adaptive law is derived to estimate the unknown parameters and its stability is guaranteed by Lyapunov stability theory. In addition, the controller in the response system contains two parts: one part that can stabilize the synchronization error dynamics and the other part that estimates the unknown parameters. Numerical examples, including Duffing oscillator and Lorenz attractor, are given to demonstrate the validity of the proposed adaptive synchronization approach.     

Robust digital controllers for uncertain chaotic systems: A digital redesign approach

In this paper, a new and systematic method for designing robust digital controllers for uncertain nonlinear systems with structured uncertainties is presented. In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs. Applying conventional robust control techniques, continuous-time robust controllers are first designed to eliminate the effects of the uncertainties on the underlying system. Then, a robust digital controller is obtained as the result of a digital redesign of the designed continuous-time robust controller using the state-matching technique. The effectiveness of the proposed controller design method is illustrated through some numerical examples on complex nonlinear systems––chaotic systems.     

Model reference adaptive synchronization of T–S fuzzy discrete chaotic systems using output tracking control

This paper presents a model reference adaptive control approach for the synchronization of a discrete-time chaotic systems using output tracking control. The reference model system is chosen using the output of master system and Takagi–Sugeno (T–S) fuzzy model is employed to represent the discrete-time chaotic slave system. Design the control input so that the controlled slave system achieves asymptotic synchronization with the reference system given that two systems start from different initial conditions, different parameters and/or different type of model. Using a gradient algorithm, the ideal controller gains which can stabilize the error equation are estimated. Simulation examples of two cases are given to demonstrate the validity of our proposed adaptive method.     

一类不确定非线性系统的执行器故障模糊容错控制

针对一类状态不完全可测的不确定非线性系统,研究了带有执行器故障的容错控制问题.采用 T-S模型对非线性系统进行模糊建模,利用并行分布补偿(PDC)算法设计了状态现潮器和基于状态现 潮器的客错控制,给出了保证该模糊容错控制系统稳定的充分条件.根据李雅普诺夫稳定性理论和线性 矩阵不等式(LMI),证明了所提出的模糊容错控制方法不但使得模糊控制系统渐近稳定,而且能够取得 H∞性能指标.计算机仿真结果进一步验证了所提出方法的正确性.     

基于模糊双线性模型的非线性系统的跟踪控制

研究一种基于T-S模糊双线性系统的跟踪控制器设计及稳定性分析.使用分布并行补偿法(PDC)设计了模糊控制器,得到模糊双线性系统跟踪控制渐近稳定的充分条件,仿真结果验证了该方法改进了闭环系统的性能.     

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.     

A two-stage approach to self-learning direct fuzzy controllers

In this paper, a novel approach is presented to fine tune a direct fuzzy controller based on very limited information on the nonlinear plant to be controlled. Without any off-line pretraining, the algorithm achieves very high control performance through a two-stage algorithm. In the first stage, coarse tuning of the fuzzy rules (both rule consequents and membership functions of the premises) is accomplished using the sign of the dependency of the plant output with respect to the control signal and an overall analysis of the main operating regions. In stage two, fine tuning of the fuzzy rules is achieved based on the controller output error using a gradient-based method. The enhanced features of the proposed algorithm are demonstrated by various simulation examples.     

Trajectory stabilization of a model car via fuzzy control

This paper deals with trajectory stabilization of a computer simulated model car via fuzzy control. Stability conditions of fuzzy systems are given in accordance with the definition of stability in the sense of Lyapunov. First, we approximate a computer simulated model car, whose dynamics is nonlinear, by T-S (Takagi and Sugeno) fuzzy model. Fuzzy control rules, which guarantee stability of the control system under a condition, are derived from the approximated fuzzy model. The simulation results show that the fuzzy control rules effectively realize trajectory stabilization of the model car along a given reference trajectory from all initial positions under a condition and the dynamics of the approximated fuzzy model agrees well with that of the model car.