Fuzzy Lyapunov Function Research Articles

Fuzzy H<SUB align="right">∞ filtering for nonlinear 2D systems in the Roesser model

This study focuses on the H∞ filtering problem for two-dimensional (2D) discrete Takagi-Sugeno (T-S) fuzzy systems in the Roesser model. The objective is to design a stable filter guaranteeing the asymptotic stability and a prescribed H∞ performance of the filtering error system. By using a new structure of the fuzzy Lyapunov function, and some analysis techniques, the stability and a prescribed H∞ performance index are guaranteed for the overall filtering-error system, such that the coupling between the Lyapunov matrix and the system matrices is removed. In addition, sufficient conditions for the existence of such a filter are established in term of linear matrix inequalities (LMIs). When these LMIs are feasible, the explicit expression of the desired filter can be characterised. An illustrative example is presented to demonstrate the effectiveness of the developed results.

A Novel Piecewise Affine Filtering Design for T-S Fuzzy Affine Systems Using Past Output Measurements.

This paper tackles the problem of piecewise affine memory filtering design for the discrete-time norm-bounded uncertain Takagi-Sugeno fuzzy affine systems. The objective is to design an admissible filter using past output measurements of the system, guaranteeing the asymptotic stability of the filtering error system with a given [Formula: see text] performance index. Based on the piecewise fuzzy Lyapunov functions and the projection lemma, a new sufficient condition for [Formula: see text] filtering performance analysis is first derived, and then the filter synthesis is carried out. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities. In addition, it is also shown that the filtering performance can be improved with the increasing number of past output measurements used in the filtering design. Finally, two examples are presented to show the advantages and effectiveness of the proposed approach.

선적분 퍼지 리아푸노프 함수를 이용한 연속시간 T-S 퍼지 시스템에 대한 정적 출력궤환 제어

The problem of static output feedback control design for continuous-time Takagi-Sugeno (T-S) fuzzy systems is addressed. The proposed stabilizing condition is based on a line-integral fuzzy Lyapunov function and a non-parallel distributed compensation (non-PDC). The existence condition of the stabilizing static output feedback control law is represented in terms of linear matrix inequalities (LMIs) without any transformation matrices and equality conditions. The proposed sufficient stabilizing condition is less conservative than previous results in the literature. An example is illustrated to verify this.

Formula omitted] and [formula omitted] fuzzy filters with memory for Takagi–Sugeno discrete-time systems

This paper is concerned with the problem of H2 and H∞ fuzzy filter design for Takagi–Sugeno (T–S) discrete-time systems. The novelty of the proposed method, in the context of T–S fuzzy systems, is the inclusion of an arbitrary number of past information (states and measurements) in the structure of the filter, producing a T–S fuzzy filter with memory. Linear matrix inequality relaxations based on multi-polynomial fuzzy Lyapunov functions are proposed for the filter design, providing filters that attain lower bounds for the H2 and H∞ performance criteria, when compared to other approaches available in the fuzzy literature that employ memoryless filters. The advantages of the proposed approach are illustrated by numerical examples.

Event-triggered fault detection filter design for nonlinear networked systems via fuzzy Lyapunov functions

This paper investigates the problem of event-triggered fault detection filter design for nonlinear networked control systems with both sensor faults and process faults. First, Takagi–Sugeno (T–S) fuzzy model is utilized to represent the nonlinear systems with faults and disturbances. Second, a discrete event-triggered communication scheme is proposed to reduce the utilization of limited network bandwidth between filter and original system. At the same time, considering network-induced delays and event-triggered scheme, a novel T–S fuzzy fault detection filter is constructed to generate a residual signal, which has nonsynchronous premise variables with the original T–S fuzzy system. Then, the fuzzy Lyapunov functional based approach and the reciprocally convex approach are developed such that the obtained sufficient conditions ensure that the fuzzy fault detection system is asymptotically stable with H∞ performance and is less conservative. All the conditions are given in terms of linear matrix inequalities (LMIs), which can be solved by LMI tools in MATLAB environment. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed results.

Improved results on stability analysis and controller synthesis for T-S fuzzy systems

ABSTRACTThis paper is concerned with the stability analysis and synthesis issues for T-S fuzzy systems. By fully using the properties of fuzzy weighting functions and matrix inequalities, two improved sufficient stability conditions are derived based on the common Lyapunov function and fuzzy Lyapunov function, respectively. It is not necessary to require every fuzzy subsystem to be stable in these conditions. Following the analysis, both parallel and non-parallel distributed compensation controllers are obtained by solving linear matrix inequalities. Finally, four examples are given to show the effectiveness of our proposed approach and the advantages of the approach over existing methods.

Observer-based robust H∞ control for uncertain Markovian jump systems via fuzzy Lyapunov function

This paper investigates the problem of observer-based robust [Formula: see text] control for a class of continuous-time nonlinear Markovian jump systems (MJSs) with uncertainties, external disturbance and unavailable states that can be represented by Takagi-Sugeno (T-S) fuzzy models. Based on a mode-dependent fuzzy Lyapunov function and by introducing slack matrix variables, a sufficient condition for the existence of the state observer and observer-based robust [Formula: see text] controller for such MJSs are derived by constructing an augmented fuzzy system. Further, by means of congruent transformation in matrix and linear matrix inequality (LMI) method, the results are given in the form of LMIs that can be easily solved by using the convex optimization techniques. Moreover, we also give the result obtained via common stochastic Lyapunov function to compare with the proposed approach. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.

Fault-tolerant control for T–S fuzzy systems with sensor faults: Application to a ship propulsion system

This paper is concerned with fault-tolerant control issues for T–S fuzzy systems with sensor faults. Firstly, a design procedure for fuzzy descriptor observer is proposed to estimate the states of the fuzzy system and unknown sensor faults. Subsequently, a fault-tolerant controller is constructed, which is informed by the fuzzy descriptor observer. Based on the properties of fuzzy weighting functions and fuzzy Lyapunov function, relaxed sufficient conditions for the existence of the fuzzy descriptor observer and the fault-tolerant controller are derived in terms of linear matrix inequalities (LMIs). Finally, one numerical example and the ship electric propulsion system are given to show the effectiveness of our proposed approach and the advantages of the approach over existing methods.

Diagnostic Observer Design for T–S Fuzzy Systems: Application to Real-Time-Weighted Fault-Detection Approach

This paper deals with a real-time-weighted observer-based fault-detection (FD) scheme for Takagi–Sugeno (T–S) fuzzy systems. The essential idea is to develop a weighted diagnostic observer-based FD system to optimize the worst case robustness and fault sensitivity simultaneously by using the information provided by each local system. To achieve an early detection of potential fault, the robustness issue is investigated in the $\mathcal {L}_{\infty }/\mathcal {L}_2$ observer-based FD context. Meanwhile, the $\mathcal {L}_-$ fault sensitivity condition is addressed to optimize the fault detectability. Using fuzzy Lyapunov functions, sufficient conditions on the FD system design are studied. Two examples are given in the end to show the efficiency of the proposed results.

A Fuzzy Lyapunov–Krasovskii Functional Approach to Sampled-Data Output-Feedback Stabilization of Polynomial Fuzzy Systems

This paper presents an output-feedback exponential stabilization condition of sampled-data polynomial fuzzy control systems under variable sampling rates. Compared with previous work, the proposed method is less conservative because of the newly developed time-dependent fuzzy Lyapunov–Krasovskii functional that is based on the conventional fuzzy Lyapunov function. Moreover, the controller is allowed to contain polynomial gain matrices, thereby improving the control performance and design flexibility. This is realized by assuming the difference between the continuous- and discrete-time state vectors as time-varying norm-bounded uncertainties, which are manipulated using a robust control technique. A new sufficient condition is introduced to cast the stability condition containing the integral term as the sum-of-square conditions. Finally, the effectiveness of the proposed method is validated by simulations.

Observer‐based non‐PDC controller design for T–S fuzzy systems with the fractional‐order

This study focuses on observer-based non-parallel-distributed-compensation (non-PDC) controller design for T–S fuzzy systems with fractional-order 0 < α < 1 . A rule-dependent fuzzy Lyapunov function is used to analyse the (non-PDC) controller. For a dynamical output feedback design, the immeasurable premise variables of the fuzzy observer and fuzzy controller are considered. Under the assumption that the membership derivatives for a given compact set can be represented by a kind of weighted sum, the difficulty in knowing a priori the appropriate bounds of the membership derivatives to satisfy linear matrix inequalities (LMIs) is removed. Using the matrix singular value decomposition method, new less-conservative strict LMI conditions are derived for the locally asymptotical stabilisation of the fractional-order T–S fuzzy system with immeasurable state variables. A fractional-order Lorenz chaotic system is provided to show the effectiveness of the proposed method.

A Novel Fuzzy Observer-Based Steering Control Approach for Path Tracking in Autonomous Vehicles

In this paper, the problem of steering control is investigated for vehicle path tracking in the presence of parametric uncertainties and nonlinearities. In practice, the vehicle mass varies due to the number of passengers or amount of payload, while the vehicle velocity also changes during normal cruising, which significantly influences vehicle dynamics. Moreover, the vehicle dynamics are strongly nonlinear caused by the tire/road forces under different road surface conditions. With fuzzy modeling method, the original nonlinear path tracking system with parameter variations is first formulated as a T–S fuzzy model with additive norm-bounded uncertainties, and then an approach to the fuzzy observer-based output feedback steering control for vehicle dynamics is proposed under a fuzzy Lyapunov function framework. By employing matrix inequality convexifying techniques, a sufficient condition is developed in the form of linear matrix inequalities such that the closed-loop path tracking error system is asymptotically stable with a guaranteed $\mathcal {H}_{\infty }$ level. Finally, the effectiveness of the proposed fuzzy observer-based output feedback controller is demonstrated in Carsim/Matlab joint simulation environment, via which the advantage of a T–S fuzzy observer-based output controller over the closed-loop driver model embedded in Carsim is also shown with parametric uncertainties and nonlinearities.

Adaptive Synchronization of Unknown Complex Dynamical Networks with Derivative and Distributed Time-Varying Delay Couplings

This paper investigates the synchronization problem of unknown non-identical complex dynamical networks with derivative and distributed time-delay couplings. To deal with the unknown topological structure, a new general class of T–S fuzzy model is proposed in this paper. Then, the unknown network is presented by a fuzzy system. In order to achieve synchronization of the above fuzzy network, this paper designs a fuzzy adaptive feedback controller which is only connected with the dynamical behavior of directly related nodes. Meanwhile, a new class of fuzzy Lyapunov function is constructed in this paper. The sufficient asymptotic synchronization condition is obtained in terms of linear matrix inequality. Numerical example is given to demonstrate the effectiveness of the proposed theoretical results.

Dynamic Output Feedback Control for Continuous-Time T–S Fuzzy Systems Using Fuzzy Lyapunov Functions

A novel relaxation approach is proposed for the analysis and dynamic output feedback control of continuous-time Takagi–Sugeno (T–S) fuzzy systems using fuzzy Lyapunov functions. Previous relaxation methods for T–S fuzzy systems have some drawbacks in relaxing quadratic functions depending on normalized fuzzy weighting functions. They often introduce conservatism or lead to computational difficulty in analysis and control synthesis. Different from previous works, the proposed approach employs linear fractional transformation mechanism and full-block $S$ -procedure to reduce the conservatism in analysis. Furthermore, the relaxation technique proposed in this paper can be used in solving the controller synthesis problem effectively. As a result, a design procedure of a nonparallel distributed compensation output feedback controller, which ensures asymptotic stability and optimizes ${\mathcal L}_2$ gain performance of the closed-loop systems, is provided. Several examples have been used to illustrate the advantages and efficiency of the proposed method extensively.

Fuzzy Fault Detection Filter Design for T–S Fuzzy Systems in the Finite-Frequency Domain

This paper deals with the fault detection filter design for a nonlinear discrete-time system in the Takagi–Sugeno fuzzy form with faults and unknown inputs. Both unknown input and fault frequencies are assumed to be known and to reside in low-/middle-/high-frequency ranges. A filter is proposed in the finite-frequency domain to reduce the conservatism generated by those designed in the entire-frequency domain. In order to guarantee the best robustness to disturbances and sensitivity to faults, the developed filter combines the $H_{-}$ / $H_{\infty }$ performances. The asymptotic stability of the filtering error dynamics is ensured by using a fuzzy Lyapunov function and a linear matrix inequality approach. Finally, two examples are presented to validate the proposed new design techniques.

Fault detection observer design for fuzzy systems with local nonlinear models via fuzzy Lyapunov function

This paper concerns with the problem of designing fault detection (FD) observer for Takagi-Sugeno (T-S) fuzzy systems subject to local nonlinear models. Different from the quadratic Lyapunov function approaches, fuzzy Lyapunov function approach is applied to design the fault detection observer, and corresponding sufficient conditions are given in terms of linear matrix inequalities. In addition, combined with nonlinear local models scheme and slack variable technique, the T-S fuzzy systems have fewer fuzzy rules to reduce the computational burden. Compared with the existing results, less conservative results on existence of fuzzy fault detection observers are derived. Moreover, the designed fuzzy fault detection observer not only guarantees the residual system stability but also improves the H_ index estimation. It is noted that the H_ index is employed to measure the worst-cast fault sensitivity performance. Finally, a numerical simulation is given to illustrate the effectiveness of the designed fault detection observer.

Fuzzy Observer-Based Fault Detection Design Approach for Nonlinear Processes

This paper is concerned with the analysis and integrated design of a type of observer-based fault detection (FD) system for general nonlinear processes. To this end, the existence and design condition for this type of nonlinear observer-based FD systems is first introduced. In this context, the integrated design scheme is investigated by dealing with the design condition with the aid of Takagi–Sugeno (T–S) fuzzy dynamic modeling technique. To be specific, the universal T–S fuzzy observer-based residual generator is developed via fuzzy Lyapunov functions. Subsequently, an integrated observer-based FD scheme is proposed with an embedded dynamic threshold, which is generated to meet the real-time FD requirements from industrial processes. In the end, a numerical example and case simulation study on a continuous stirred tank heater process are performed to show the application of the proposed method.

Admissibility Analysis and Control Synthesis for T–S Fuzzy Descriptor Systems

The problem of admissibility analysis and control synthesis for Takagi–Sugeno fuzzy descriptor systems is investigated. First, based on Nonquadratic fuzzy Lyapunov function and fully using the information of fuzzy membership functions, a new relaxed sufficient condition ensuring a fuzzy descriptor system to be admissible (regular, impulse-free, and stable) is proposed, in which it is not necessary to require every fuzzy subsystem to be stable. Second, the other sufficient condition for the admissibility is obtained without the information of time derivatives of fuzzy membership functions. Following the analysis, both parallel and nonparallel distributed compensation controllers are designed, linear matrix inequalities conditions are given to construct the controllers. Finally, some examples are provided to illustrate the main results in this paper less conservative than some earlier related results.

Dissipativity Analysis and Synthesis for a Class of T–S Fuzzy Descriptor Systems

This paper is concerned with the problems of dissipativity analysis and synthesis for a class of Takagi–Sugeno (T–S) fuzzy descriptor systems with different derivative matrices. First, a new augmented system which is equivalent to the original system is given to deal with these different derivative matrices. By employing a fuzzy Lyapunov function and slack matrices, a set of relaxed sufficient conditions are developed to guarantee that the unforced systems are admissible (regular, impulse free, and stable), which includes the existing related results as special cases. The developed conditions are expressed in terms of strict linear matrix inequalities (LMIs), which is convenient for checking the admissibility. Then, a novel method is proposed to design an admissible fuzzy controller. Second, by applying the dissipativity theorem and Lyapunov stability theorem, a set of sufficient conditions are derived to ensure that the resultant closed loop systems are admissible and dissipative, and the fuzzy controllers are designed using LMIs techniques. Finally, some examples are provided to illustrate that the main results in this paper are feasible and less conservative than the earlier related ones.

$H_{\infty }$ Filtering for Continuous-Time T–S Fuzzy Systems With Partly Immeasurable Premise Variables

This paper is concerned with the $H_{\infty }$ filtering problem for T–S fuzzy systems with partly immeasurable premise variables. By using measurable premise variables of fuzzy models as the premise variables of fuzzy filters, a new fuzzy filter scheme is constructed. Further based on the new filter scheme and a class of new line integral fuzzy Lyapunov functions, a convex condition for designing $H_{\infty }$ filters is proposed. In contrast to the existing approaches, the new condition can take full use of measurable premise variables for less conservative design. A numerical example is given to illustrate the effectiveness of the proposed method.