Fuzzy Lyapunov Function Research Articles

Fuzzy Event-Triggered Control for Back-to-Back Converter Involved PMSG-Based Wind Turbine Systems

The main objective of this study is to focus on dynamical analysis of full-scale direct-driven permanent-magnet-synchronous-generator-based wind energy conversion systems (WECSs) configured with back-to-back voltage source converters. The mathematical methods are employed to discuss the properties WECSs in a theoretical manner, which are proven to be cost effective. It can be seen that the considered model contains the nonlinearities in terms of currents and rotational speed of the shaft that results in nonlinearities. The Takagi–Sugeno fuzzy approach is considered to be an effective tool to approximate the nonlinear model into local linear submodels without compromising the system characteristics. In general, WECSs are networked to each other, the event-triggered-based control (ETC) is considered to be an appropriate control regarding the reduction of packet loss and the ability to ensure the stable performance of the WECSs in an effective manner. The main advantage of utilizing the ETC is that it will activate the controllers with a user-designed event-triggering condition that helps to restrict the unnecessary network transmissions and reduce the leakages. By the virtue of the fuzzy Lyapunov function, intensive attention is focused on deriving the theoretical-based sufficient conditions in terms of solvable linear matrix inequalities, which ensure the global stability of the closed-loop model based on the Lyapunov stability theory. Detailed numerical simulations are performed with an experimental range of system parameters that illustrate the effectiveness of the proposed ETC scheme.

A Fuzzy Lyapunov Function Approach to Positive Ll Observer Design for Positive Fuzzy Semi-Markovian Switching Systems With Its Application

This paper concerns a positive G1 observer for positive nonlinear semi-Markovian switching systems (MSSs) via the expansion of Taylor formula and the fuzzy Lyapunov function approach, in which semi-Markovian switching parameters, positivity, Takagi-Sugeno (T-S) fuzzy, and external disturbance are all considered in a unified framework. A fuzzy Lyapunov function approach with less conservativeness is introduced into the research of positive systems. In the system under consideration, positive S-MSSs with the semi-Markovian process can describe more complex systems in a practical control process. The main motivation of this paper is that the practical system subject to positivity and abrupt changes can be described by positive nonlinear S-MSSs, which always needs to consider the external disturbance. First, by using the normalized membership function approach, positive nonlinear S-MSSs can be represented by local positive T-S fuzzy S-MSSs. Second, by constructing the fuzzy Lyapunov function, some sufficient conditions are proposed for stochastic stability and G1-gain performance analysis, respectively. Then, a positive G1 observer in a novel standard linearprogramming condition is designed to guarantee the resulting closed-loop augmented system is positive and stochastically stable with a required G1-gain performance. Finally, a practical example about the epidemiological model is introduced to show the effectiveness of the main theory.

Relaxed Stabilization Conditions for Interconnected Nonlinear Systems

This paper deals with the conservatism reduction of stability and stabilization conditions for nonlinear continuous-time interconnected systems. Based on Takagi–Sugeno modeling, the interconnected system is described by a convex combination of interconnected linear systems. Line integral fuzzy Lyapunov functions are considered to develop stability and stabilizability criteria for the considered class of interconnected nonlinear systems. All analysis and design conditions are formulated in LMI terms. The effect of the coupling due to interconnection terms is also investigated and taken into account to obtain less conservative conditions. Examples are given to illustrate the merit of the obtained results.

Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions

This paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequalities (LMI). The FLF is given by $V(x(t)) = x(t)^{T}P(\alpha (x(t)))x(t)$ , where $x(t)$ is the plant state vector, $P(\alpha (x(t))) = \alpha _{1}(x(t))P_{1} + \alpha _{2}(x(t))P_{2} + \cdots + \alpha _{r}(x(t))P_{r}$ , $P_{i}=P_{i}^{T} > 0$ and $\alpha _{i}(x(t))$ is the weight related to the local model $i$ in the representation of the plant by TS fuzzy models, for $i=1,2,\cdots,r$ . When one calculates the time derivative of this $V(x(t))$ , it appears the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , that is usually handled using conservative upper bounds, supposing that the bounds of the time derivative of $\alpha _{i}(x(t))$ , $i=1,2,\cdots,r$ , are available. The main result of this paper is a procedure to obtain optimal upper bounds for the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , such that they contemplate the maximum value and are always smaller than or equal to the maximum value. It is a relevant result on this subject, because these optimal upper bounds do not add any constraint. With these optimal upper bounds, a relaxed design method for stabilization of TS fuzzy models is proposed. Two numerical examples illustrate the effectiveness of this procedure.

Parallel and Nonparallel Distributed Compensation Controller Design for T-S Fuzzy Discrete Singular Systems With Distinct Difference Item Matrices

The issues of admissibility and controller design for a class of extended T-S fuzzy discrete singular systems (FDSSs) with distinct difference item matrices are discussed in this article. A new augmented system which is the equivalent of the original system is introduced to convert these distinct difference item matrices into the easy treatment form. On this basis, using the fuzzy Lyapunov function (FLF), relaxed sufficient condition is given to ensure the admissibility of unforced systems. This condition is described via strict linear matrix inequalities (LMIs), which facilitates to analyze the admissibility. Meanwhile, the design methods of parallel and nonparallel distributed compensation (PDC and Non-PDC) controllers are also proposed. Finally, the advantages of the developed admissibility and controller design approach are illustrated by three examples.

Robust Set-Invariance Based Fuzzy Output Tracking Control for Vehicle Autonomous Driving Under Uncertain Lateral Forces and Steering Constraints

This paper is concerned with a new control method for path tracking of autonomous ground vehicles. We exploit the fuzzy model-based control framework to deal with the time-varying feature of the vehicle speed and the highly uncertain behaviors of the tire-road forces involved in the nonlinear vehicle dynamics. To avoid using costly vehicle sensors for feedback control while favoring the simplest control structure for real-time implementation, a new fuzzy static output feedback (SOF) scheme is proposed. In particular, though the robust set-invariance property and Lyapunov-based arguments, the physical constraints on the steering input saturation and the vehicle state can be taken into account in the control design to improve the driving safety and comfort. The theoretical development relies on the use of fuzzy Lyapunov functions and the non-parallel distributed compensation control concept to reduce the design conservatism. Exploiting some specific convexification techniques, the control design is reformulated as an optimization problem under linear matrix inequalities with a single line search, which are efficiently solved via semidefinite programming techniques. The proposed fuzzy path tracking controller is evaluated through various dynamic driving tests conducted with high-fidelity CarSim/Matlab co-simulations. Moreover, to emphasize the advantages of the new fuzzy SOF controller, a performance comparison with the CarSim driver model is also performed.

Event-based fuzzy control for T-S fuzzy networked systems with various data missing

This paper is concerned with the design of fuzzy controller for networked control systems (NCSs). T-S fuzzy system approach is adopted to study the problem. In account of NCSs, various data missing on both sides of the controller caused by event-triggered scheme, data losses and event-driven controller are involved. For handling this issue, an auxiliary random series method is presented to describe the data transferring in the network. Based on this method, the closed-loop NCSs is constructed and then analyzed via Lyapunov stability theory. Additionally, the probability of random data losses at each instant is supposed to be different which is more practical. For the purpose of obtaining more relaxed stabilization conditions, the fuzzy Lyapunov function obtained on the basis of both plant and controller fuzzy rules is employed. Furthermore, variations of membership functions are taken into account, and consequently, slack matrices are introduced to obtain less conservative result. At last, by considering a cart-damper-spring system, the effectiveness of the proposed method is illustrated.

Membership Function-Dependent Local Controller Design for T–S Fuzzy Systems

For the conservatism issue of state-feedback control for Takagi–Sugeno (T–S) fuzzy systems, a novel local controller design method and stability analysis approach are proposed in this article. First, based on a membership function piecewise linearization technique, a local linear controller is proposed to form a closed-loop system. Then, a series of system stability conditions are derived using quadratic and piecewise function approaches. Further, with a designed fuzzy local controller and fuzzy Lyapunov function, less conservative stability conditions are obtained. Finally, a simulation example is adopted to make comparisons with existing and proposed methods. The results clearly show the conservatism reduction and excellent performance of the proposed approach.

Stability of Fuzzy Dynamical Systems via Lyapunov Functions

The purpose of this paper is to introduce the concept of fuzzy Lyapunov functions to study the notion of stability of equilibrium points for fuzzy dynamical systems associated with fuzzy initial value problems, through the principle of Zadeh. Our contribution consists in a qualitative characterization of stability by a study of the trajectories of fuzzy dynamical systems, using auxiliary functions, and they will be called fuzzy Lyapunov functions. And, among the main results that have been proven is that the existence of fuzzy Lyapunov functions is a necessary and sufficient condition for stability. Some examples are given to illustrate the obtained results.

Admissibility Analysis and Robust ${H_\infty }$ Control for T–S Fuzzy Descriptor Systems With Structured Parametric Uncertainties

This article considers admissibility analysis and robust ${H_\infty }$ control for continuous-time Takagi–Sugeno fuzzy descriptor systems with norm-bounded uncertainties in all parametric matrices. The system under consideration contains singular derivative matrices and different membership functions, thus it generalizes other related forms. First, uncertainties in derivative matrices are divided into two cases, i.e., one is expressed by a constant matrix left multiplied by an invertible uncertain matrix and the other is produced by its dual form. Then, admissible conditions and ${H_\infty }$ performance for the system with the first case of uncertainties are derived based on a new augmented system. As for the second case, the admissibility analysis is converted into the first case by an equivalent companion system then solved as well. All conditions are cast into strict linear matrix inequalities. Besides, due to the introduction of a new nonquadratic fuzzy Lyapunov function and slack decision variables, the proposed methods are less conservative than related ones. Finally, simulation examples are provided to illustrate improvements and effectiveness of the main results.

Stability analysis of Takagi–Sugeno systems using a switched fuzzy Lyapunov function

In this paper, a switched fuzzy Lyapunov function approach is proposed to analyze the stability of continuous-time Takagi–Sugeno fuzzy systems. The results are established by exploring properties of the membership functions. The key point is that the time derivatives of the membership functions are represented as a finite polytope and less conservative linear matrix inequalities are obtained. Numerical examples illustrate the efficiency of the new stabilizing conditions.

Reliable H∞ guaranteed cost control for uncertain switched fuzzy stochastic systems with multiple time-varying delays and intermittent actuator and sensor faults

In this paper, the issue of delay-dependent passive fault-tolerant control and optimal guaranteed cost control is considered for multiple time-varying delayed switched Takagi–Sugeno (T–S) fuzzy stochastic systems against intermittent actuator and sensor faults, model uncertainties, external disturbances and measurement noise. This is the first attempt to implement H∞ guaranteed cost control for switched T–S fuzzy stochastic systems with intermittent actuator and sensor faults which allows more kinds of faults. Thus, the method has a wider application range. A dynamic output feedback controller is designed. Then, a closed-loop system is constructed. Stability results of reliable H∞ guaranteed cost control are given via piecewise fuzzy Lyapunov function in terms of linear matrix inequalities. Slack matrices are not required at all. The conservatism of this paper is relatively low. At last, the effectiveness of the addressed approach is presented by explanatory examples.

Robust H∞ Takagi–Sugeno fuzzy output-feedback control for differential speed steering vehicles

This paper studied the modelling and control of four-wheel independently driven electric vehicles using differential speed steering. The Takagi–Sugeno fuzzy modelling approach represents the nonlinearities of the four-wheel independently driven electric vehicle state variables in several system models. The proposed controller design is a robust Takagi–Sugeno fuzzy output-feedback control based on a fuzzy Lyapunov function approach. More precisely, the Lyapunov function is chosen to be dependent on the membership functions. Sufficient conditions for the existence of the robust Takagi–Sugeno fuzzy controller are given in terms of linear matrix inequality constraints. The designed parameters are tested by simulating the four-wheel independently driven electric vehicles under varying operating conditions. The simulation results underscore the robustness and disturbance rejection importance of the proposed controller, which is then contrasted to better highlight the improved performance of the proposed approach over a fixed robust controller design.

Observer and fault‐tolerant controller design for discrete‐time multiple state‐delayed T–S fuzzy systems

This study is concerned with observer-based fault estimation (FE) and fault-tolerant controller design for a class of discrete-time Takagi–Sugeno (T–S) fuzzy systems. There exist multiple time-varying state delays, sensor and actuator faults, local non-linear dynamics and exogenous disturbances in the systems. In comparison with the existing results, the approach suggested in this study is more flexible and feasible. By means of the n-step induction FE, a novel fuzzy adaptive descriptor observer is developed to obtain the n-step error functions. Then, an active dynamic output feedback fault-tolerant controller is designed to stabilise the closed-loop fuzzy system. Furthermore, a set of delay-dependent sufficient conditions are provided by the fuzzy Lyapunov function which utilises the form of linear matrix inequalities. The stability results from the observer and the controller in this study have less conservatism compared with the ones from the existence of observers and fault-tolerant controllers. At last, a simulation example is presented to demonstrate the advantages and effectiveness of the approach proposed in the study.

Unknown input observer synthesis for discrete-time T–S fuzzy singular systems with application to actuator fault estimation

This work studies an unknown input observer (UIO) design dedicated to the Takagi–Sugeno fuzzy-approximation-based nonlinear singular systems with unknown inputs in the discrete-time domain. Not only the input equation, but also the output equation are assumed to be affected by unknown inputs. By restoring to a fuzzy Lyapunov function together with the well-developed linear matrix inequality (LMI) technique, some new relaxed sufficient criteria for the convergence of the designed UIO are exactly given and accurately proved. Significantly, consider that the actuator fault may occur frequently, which may result in a disastrous consequence for modern control systems. Therefore, an extended UIO called unknown input proportional integral observer is further investigated to reconstruct the actuator fault. All the design procedures are formulated in terms of LMIs. Finally, simulation studies on a numerical example as well as a practical one are provided to test the performance of the developed two procedures.

Non-weighted [formula omitted] gain and asynchronous [formula omitted] control for continuous-time switched T–S fuzzy systems

This article considers the non-weighted L2 gain and asynchronous H∞ control issues for continuous-time switched nonlinear systems. Firstly, Takagi–Sugeno (T–S) fuzzy models are employed to describe the nonlinear subsystems. By taking full advantage of the asynchronous features, a novel time-varying fuzzy Lyapunov function (TVFLF) and the corresponding controllers are all constructed separately in synchronous and asynchronous intervals. Based on the TVFLF approach, sufficient conditions of asynchronous H∞ controller are established. Finally, two simulation examples are discussed to certify the effectiveness of the proposed results.

Switching fuzzy modelling and control scheme using T-S fuzzy systems with nonlinear consequent parts

This paper extends the idea of switching T-S fuzzy systems with linear consequent parts to nonlinear ones. Each nonlinear subsystem is exactly represented by a T-S fuzzy system with Lure’ type consequent parts, which allows to model and control wider classes of switching systems and also reduce the computation burden of control synthesis. With the use of a switching fuzzy Lyapunov function, the LMI conditions for asymptotic stability of the system with maximum decay-rate and disturbance attenuation properties under arbitrary switching law are derived. Moreover, several numerical examples are provided to demonstrate the effectiveness of the proposed approaches to reduce the computational burden of control synthesis and improving the closed-loop performance of the system.

“선적분 퍼지 리아푸노프 함수를 이용한 연속시간 T-S 퍼지 시스템에 대한 정적 출력궤환 제어”에 대한 정정

“선적분 퍼지 리아푸노프 함수를 이용한 연속시간 T-S 퍼지 시스템에 대한 정적 출력궤환 제어”에 대한 정정

선적분 퍼지 Lyapunov 함수를 이용한 T-S 퍼지 시스템에 대한 완화된 안정성

선적분 퍼지 Lyapunov 함수를 이용한 T-S 퍼지 시스템에 대한 완화된 안정성

Multiple intermittent fault estimation and tolerant control for switched T-S fuzzy stochastic systems with multiple time-varying delays

This paper focuses on the observer-based fault estimation (FE) and active dynamic output feedback controller design for switched Takagi–Sugeno (T-S) fuzzy Itô stochastic systems against multiple time-varying delays as well as intermittent actuator faults and sensor faults. External disturbances are also considered. First, a novel descriptor sliding mode observer (SMO) is investigated to establish the error dynamics. Compared with the existing results, the proposed observer could be utilized to more cases and applied to a wider scope. Next, in light of the online FE information, a fault tolerant controller is suggested to make the closed-loop system mean-square exponentially stable. Furthermore, a novel piecewise fuzzy Lyapunov function is depicted and then the delay dependent sufficient conditions can be got by linear matrix inequalities (LMIs). The designed SMO has less conservatism than the existing ones. The reachability of the sliding mode surface in the estimation error space is assured under the adopted method. And the finite-time boundedness (FTB) issue is discussed. At last, a simulation example is acquired to validate the novelty and validity of the method proposed in the paper.