Nonlinear control design based on generalized Takagi–Sugeno fuzzy systems

This paper is concerned with control design for a generalized Takagi–Sugeno fuzzy system. The Takagi–Sugeno fuzzy system generally describes nonlinear systems by employing local linear system representations, while a generalized fuzzy system to be considered in this paper describes even a wider class of nonlinear systems by representing locally nonlinear systems. For such a generalized system, a stabilizing controller design method is proposed by introducing a new class of non-PDC controllers. A non-PDC controller is a generalized controller of PDC one, which is a traditional fuzzy controller. Stabilizing controller design conditions are given in terms of a set of linear matrix inequalities (LMIs), which are easily numerically solvable. A relaxation method is used to reduce the conservatism of design conditions. Finally, numerical examples are given to illustrate our nonlinear control design and to show the effectiveness over other existing results.     

On delay-dependent stability conditions for Takagi–Sugeno fuzzy systems

This paper presents new less conservative stability analysis conditions for Takagi–Sugeno fuzzy systems subject to interval time-varying delay. The methodology is based on the direct Lyapunov method allied with an appropriate Lyapunov–Krasovskii functional choice and the use of the integral inequalities, Finsler lemma, Newton–Leibniz formula manipulations and convex combination properties. Particularly, the main result differs from previous ones since the positiveness of the Lyapunov–Krasovskii functional is guaranteed by new relaxed conditions. Two examples illustrate the effectiveness of the proposed methodology.     

Networked H∞ filtering for Takagi–Sugeno fuzzy systems under multi-output multi-rate sampling

This paper studies networked H_∞ filtering for Takagi–Sugeno fuzzy systems with multi-output multi-sensor asynchronous sampling. Different output variables in a dynamic system are sampled by multiple sensors with different sampling rates. To estimate the signals of such a system, a continuous multi-rate sampled-data fusion method is proposed to design a novel networked filter. By considering a class of decentralized event-triggered transmission schemes, multi-channel network-induced delays, and the updating modes of the MOMR sampled-data, a networked jumping fuzzy filter is proposed to estimate system signals based on the transmitted multi-rate sampled-data of fuzzy system and the multi-rate sampled states of filter, and the jumping among filter modes is governed by a Markov process which depends on the arrival times of sampled output sub-vectors. To deal with asynchronous membership functions, the networked fuzzy filtering system is modeled as an uncertain fuzzy stochastic system with membership function deviation bounds. Based on stability and H_∞ performance analysis, several membership-function-dependent conditions are presented to co-design the event-triggered transmission schemes and the fuzzy filter such that the filtering error system is robustly mean-square exponentially stable with a prescribed H_∞ attenuation level. Finally, the improvement in estimation performance and comparison with the existing filtering methods are discussed through simulation examples.     

Stabilization of positive Takagi–Sugeno fuzzy discrete-time systems with multiple delays and bounded controls

This paper deals with the problem of stabilization by state feedback control of Takagi–Sugeno (T–S) fuzzy discrete-time systems with multiple fixed delays while imposing positivity in closed-loop. The obtained results are presented under linear programming (LP) form. In particular, the synthesis of state feedback controllers is first solved in terms of Linear programming for the unbounded controls case. This result is then extended to the stabilization problem by nonnegative controls, and stabilization by bounded controls. The stabilization conditions are derived using the single Lyapunov–Krasovskii functional (LKF). An example of a real plant is studied to show the advantages of the design procedure. A comparison between linear programming and LMI approaches is presented.     

Robust delay-dependent H∞ filtering for uncertain Takagi–Sugeno fuzzy neutral stochastic time-delay systems

This paper addresses the robust H_∞ filter design problem for a class of uncertain fuzzy neutral stochastic system with time-delay through Takagi–Sugeno (T–S) fuzzy model. By constructing an augmented Lyapunov–Krasovskii functional, some novel delay-dependent stability criteria for uncertain fuzzy neutral stochastic system with time varying delay are obtained in terms of linear matrix inequalities. By using the integral inequality in the neutral stochastic setting combined with delay decomposition approach, the H_∞ fuzzy filter is designed to guarantee the corresponding filtering error systems robustly asymptotically stable with a specified H_∞ performance index. At last, two numerical examples are presented to show the less conservatism than the previous results.     

On stability and stabilization of singular uncertain Takagi–Sugeno fuzzy systems

This paper deals with the problem of robust stability and robust stabilization for a class of continuous-time singular Takagi–Sugeno fuzzy systems. Sufficient conditions on stability and stabilization are proposed in terms of strict LMI (Linear Matrix Inequality) for uncertain T–S fuzzy models. In order to reduce the conservatism of results developed using quadratic method, an approach based on non-quadratic Lyapunov functions and S-procedure is proposed. Illustrative examples are given to show the effectiveness of the given results.     

Analysis and design for continuous-time string-connected Takagi–Sugeno systems

Distributed systems consist of interconnected, lower-dimensional subsystems. For such systems, distributed analysis and design present several advantages, such as modularity, easier analysis and design, and reduced computational complexity. A special case of distributed systems is when the subsystems are connected in a string. Applications include distributed process control, traffic and communication networks, irrigation systems, hydropower valleys, etc. By exploiting such a structure, in this paper, we propose conditions for the distributed stability analysis of Takagi–Sugeno fuzzy systems connected in a string. These conditions are also extended to observer and controller design and illustrated on numerical examples.     

Active fault tolerant control of wind turbine systems based on DFIG with actuator fault and disturbance using Takagi–Sugeno fuzzy model

This paper presents an active fault tolerant control (FTC) for doubly fed induction generator (DFIG) with actuator fault and disturbance using Takagi–Sugeno (TS) fuzzy model. The control structure has two parts: fault and disturbance estimation part and FTC part. First, a TS fuzzy model is used to describe the DFIG system. Using a special linear transformation, the original system is decoupled into three independent subsystems: state subsystem without fault and disturbance, disturbance subsystem without fault, and fault subsystem without disturbance. By solving linear matrix inequalities (LMIs), a TS fuzzy observer is designed for the state subsystem without fault and disturbance. Second, estimations of faults and disturbance are obtained using the other subsystem models. Third, an active FTC scheme is developed to reduce the effect of disturbance and actuator faults. Finally, the performance of the proposed FTC is tested for a wind turbine system based on DFIG with actuator faults and disturbance. The simulation results demonstrate that the new FTC scheme makes possible to obtain an efficient fault and disturbance estimation and to reduce the peak current in the transient process.     

Robust non-fragile proportional plus derivative state feedback control for a class of uncertain Takagi–Sugeno fuzzy singular systems

This paper proposes a novel robust non-fragile proportional plus derivative state feedback (PDSF) control scheme for a class of uncertain nonlinear singular systems. The Takagi–Sugeno (T–S) fuzzy model is employed to represent the nonlinear singular system with parameter uncertainties appearing not only in distinct state matrices, but also in distinct derivative matrices. By using the free-weighting matrix technique, some sufficient conditions, which guarantee the resulting closed-loop system to be normal and stable (NS), are presented. With these conditions, the problems of non-fragile PDSF controllers design with additive and multiplicative uncertainties are respectively solved in terms of linear matrix inequalities (LMIs), which can be conveniently solved via the convex optimization technique. Finally, two examples are provided to illustrate the validity of the presented results.     

Sampled-data H∞ filtering of Takagi–Sugeno fuzzy systems with interval time-varying delays

The sampled-data _H∞$H_{\infty }$ filtering for a continuous-time Takagi–Sugeno fuzzy system with an interval time-varying state delay is investigated, where the measurement outputs from the plant to the filter are assumed to be sampled at discrete instants with a variable period. Firstly, by means of a newly proposed inequality bounding technique and a new Lyapunov–Krasovskii functional, the fuzzy sampled-data _H∞$H_{\infty }$ filtering performance analysis is carried out such that the resultant filter error system is asymptotically stable with a prescribed _H∞$H_{\infty }$ attenuation performance index. Secondly, sufficient conditions on the existence of fuzzy sampled-data _H∞$H_{\infty }$ filters are derived in the simultaneous presence of the time-varying state delay and the variable sampling period. The proposed bounding inequality lies in its more tightness and alleviates the enlargement of some inverse “coefficients” resulting from the utilization of the well-known Jensen integral inequality  . Compared with some existing Lyapunov–Krasovskii functionals, more information about the relationship among the current state and its delayed state is considered. The upper bound of the derivative of the time-varying state delay is not required to be less than one. Different from some existing results in the literature, by applying the proposed results, each different value of such an upper bound (greater than one) leads to a different _H∞$H_{\infty }$ disturbance attenuation level. Finally, a numerical example and a modified continuous stirred tank reactor system are given to show the effectiveness of the proposed results.     

H∞ reduced-order observer-based controller synthesis approach for T-S fuzzy systems

For continuous-time nonlinear systems represented by Takagi–Sugeno fuzzy models, a new H_∞ reduced-order-observer based controller synthesis structure is investigated in this paper. By the fuzzy reduced-order observer and fuzzy controller, an augmented error system composed of the estimation and control errors is obtained. The fuzzy modeling residual terms are seen as part of the external disturbance, and an extra design matrix is added to facilitate the design process. The robustness and stability conditions are given based on Lyapunov function approach, then the conditions are transformed into convex form to facilitate the numerical solving process. Finally, by the comparison with existing methods in simulation section, the control performance and conservativeness reduction effects of the proposed methods are verified.     

Hybrid state observer‐based event‐triggered control for switched linear systems with quantized input

The paper investigates the design of hybrid state observer-based event-triggered controller for switched linear systems subject to quantized input and unknown but bounded additional disturbance and measurement noise. Firstly, by introducing a hybrid state observer and constructing a mode-dependent event-triggered mechanism, we design event-triggered controller for the considered switched linear systems. Then, by modeling the closed-loop system as an augmented asynchronous switched time-delay system, we deal with the asynchronous control problem caused by the switching between two consecutive trigger instants for the switched linear system. Thirdly, based on merging signal technique and multiple Lyapunov functional method, we obtain the sufficient criteria to guarantee the stability of the switched system when the switching signal meets an average dwell time condition, and further establish the hybrid observer-based event-triggered controller gains. Finally, a simulation example illustrates the validity of the results.     

Adaptive T–S fuzzy controller using reinforcement learning based on Lyapunov stability

In this paper, an adaptive Takagi–Sugeno (T–S) fuzzy controller based on reinforcement learning for controlling the nonlinear dynamical systems is proposed. The parameters of the T–S fuzzy system are learned using the reinforcement learning based on the actor-critic method. This on-line learning algorithm improves the controller performance over the time, which it learns from its own faults through the reinforcement signal from the external environment and tries to reinforce the T–S fuzzy system parameters to converge. The updating parameters are developed using the Lyapunov stability criterion. The proposed controller is faster in learning than the T–S fuzzy that parameters learned using the gradient descent method under the same conditions. Moreover, it is able to handle the load changes and the system uncertainties. The test is carried out based on two mathematical models. In addition, the proposed controller is applied practically for controlling a direct current (DC) shunt machine. The results indicate that the response of the proposed controller has a good performance compared with other controllers.     

Formation-containment control for multi-robot systems with two-layer leaders via hierarchical controller–estimator algorithms

This paper studies the formation-containment control for multi-robot systems with two-layer leaders in the presence of parametric uncertainties, input disturbances and directed interaction topologies. To cope with the aforementioned issues, we establish a novel formation-containment control framework, where the analysis of the systems is carried out step by step. A hierarchical controller–estimator (HCE) algorithm, containing distributed sliding-mode estimators in each sub-algorithm, is proposed for the two-layer leaders system. Moreover, by invoking finite-time stable and input-to-state stable theories, the sufficient conditions for convergence of the proposed HCE algorithm are presented. Finally, numerous simulations are performed to demonstrate the validity of the theoretical results.     

Sum-of-squares-based robust H∞ controller design for discrete-time polynomial fuzzy systems

This paper investigates a robust H_∞ controller design for discrete-time polynomial fuzzy systems based on the sum-of-squares (SOS) approach when model uncertainties and external disturbances are simultaneously considered. At the beginning of the controller design procedure, a general discrete-time polynomial fuzzy control system proposed in this paper is used to represent a nonlinear system containing model uncertainties and external disturbances. Subsequently, through use of a nonquadratic Lyapunov function and the H_∞ performance index, the novel SOS-based robust H_∞ stability conditions are derived to guarantee the stability of the entire control system. By solving those stability conditions, control gains of the robust H_∞ polynomial fuzzy controller are obtained. Because the model uncertainties and external disturbances are considered simultaneously in the controller design procedure, the closed-loop control system achieves greater robustness and H_∞ performance against model uncertainties and external disturbances. Moreover, the novel operating-domain-based robust H_∞ stability conditions are derived by considering the operating domain constraint to relax the conservativeness of solving the stability conditions. Finally, simulation results demonstrated the availability and effectiveness of the proposed stability conditions, which are more general than those used in existing approaches.     

T–S fuzzy control for dithered nonlinear singularly perturbed systems with multiple time delays

This paper is concerned with the stability problem of nonlinear multiple time-delay singularly perturbed (NDSP) systems. To overcome the effect of modeling error between the reduced-order model of the NDSP plant and Takagi–Sugeno (T–S) fuzzy models, a robustness design of model-based fuzzy control is proposed in this study. A stability criterion in terms of Lyapunov’s direct method is derived to guarantee the asymptotic stability of NDSP systems. According to this criterion, a model-based fuzzy controller is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the NDSP system. If the designed fuzzy controller cannot stabilize the NDSP system, a high-frequency signal, commonly referred to as dither, is simultaneously introduced to stabilize it. Based on the relaxed method, the NDSP system can be stabilized by regulating appropriately the parameters of dither. If the dither’s frequency is high enough, the output of the dithered reduced system and that of its corresponding mathematical model – the relaxed reduced system – can be made as close as desired. This makes it possible to obtain a rigorous prediction of the stability of the dithered reduced system based on the one of the relaxed reduced system.     

Robust H∞ control for a class of uncertain nonlinear systems with mixed time-delays

This paper is concerned with the robust H_∞ control problem for a general class of uncertain nonlinear systems with mixed time-delays. The mixed time-delays consist of both discrete and distributed delays. We aim to design a memoryless state feedback controller such that the closed-loop system is robustly stable for all admissible uncertainties with guaranteed H_∞ disturbance rejection attenuation level. By introducing a new Lyapunov–Krasovskii functional that reflects the mixed delays, sufficient conditions are established for the closed-loop system ensuring the robust stability as well as the H_∞ performance requirement. The controller design is facilitated in terms of the solvability of a Hamilton–Jacobi inequality. Two numerical examples are utilized to demonstrate the effectiveness of the proposed methods.