An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach

This paper addresses stability analysis and stabilization for Takagi-Sugeno (T-S) fuzzy systems with parametric uncertainties via a so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. First, the Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties is used as the model for the uncertain nonlinear system. Based on the fuzzy Lyapunov function approach and a parallel distributed compensation (PDC) scheme, we give stabilization conditions for closed-loop fuzzy systems with parametric uncertainties. Second, all the conditions are formulated in the format of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Finally, the T-S fuzzy model of the chaotic Lorenz system, which has complex nonlinearity, is developed as a test bed. A numerical example of the chaotic Lorenz system is given to illustrate the utility of the fuzzy Lyapunov function approach.

A novel linear matrix inequality (LMI) condition for the stability of the sampled-data fuzzy control system based on the Takagi-Sugeno fuzzy model is presented. Using the novel Lyapunov functional, the relaxed stability condition is presented for the sampled-data fuzzy control and represented in the LMI format. A simulation example is provided to verify the effectiveness of the proposed technique.

In this paper the fault tolerant synchronization of two chaotic systems based on fuzzy model and sample data is investigated. The problem of fault tolerant synchronization is formulated to study the global asymptotical stability of the error system with the fuzzy sampled-data controller which contains a state feedback controller and a fault compensator. The synchronization can be achieved no matter whether the fault occurs or not. To investigate the stability of the error system and facilitate the design of the fuzzy sampled-data controller, a Takagi–Sugeno (T–S) fuzzy model is employed to represent the chaotic system dynamics. To acquire good performance and produce a less conservative analysis result, a new parameter-dependent Lyapunov–Krasovksii functional and a relaxed stabilization technique are considered. The stability conditions based on linear matrix inequality are obtained to achieve the fault tolerant synchronization of the chaotic systems. Finally, a numerical simulation is shown to verify the results.

In this paper, a stability control approach of flexible joint robot (FJR) based T-S fuzzy model using fuzzy Lyapunov function is proposed. First, the Takagi and Sugeno (T-S) fuzzy model is employed to approximate the flexible joint robot, then a fuzzy controller is developed based on parallel distributed compensation principle (PDC). The non-quadratic stability conditions for the flexible joint robot control system are proposed by using fuzzy Lyapunov function combined with linear matrix inequality (LMI). Finally, the simulation example is given to illustrate the effectiveness of the proposed method.

Local stability analysis of continuous-time Takagi–Sugeno fuzzy systems: A fuzzy Lyapunov function approach

Stability analysis of discrete T-S fuzzy system with the common Lyapunov function approach and the fuzzy Lyapunov function approach is computationally attractive but conservative. Based on the definition of a discrete piecewise fuzzy Lyapunov function at each overlapping rules group, this paper introduces a new sufficient stability condition that only needs satisfying the stability condition of the fuzzy Lyapunov function approach in each maximal overlapped-rules group (MORG). A simulation example shows the condition is less conservative and easier to check than earlier approaches.

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

This paper investigates the sampled-data stabilization problem for a Takagi–Sugeno (T-S) fuzzy system with time delay. By taking the information of states within the intervals from $t_{k}$ to $t$ and $t$ to $t_{k+1}$ into account, a new two-side delay-dependent looped-functional is introduced which can not only relax the monotonic constraint of Lyapunov–Krasovskii functional (LKF), but also make better use of the actual sampling pattern. Furthermore the sampled-data fuzzy controller is designed to contain both the present and delayed state information, thereby enhancing the control performance and design flexibility. Based on the novel augmented LKF and improved bounding technique, less conservative stability criteria are derived in the form of linear matrix inequalities. The superiority of proposed results is shown by two simulation examples.

In this paper, the stabilization of Takagi-Sugeno (T-S) fuzzy control systems whose premise variables employ two-overlapped fuzzy partition is investigated. Based on the piecewise fuzzy Lyapunov function and the nonparallel-distributed-compensation (non-PDC) control laws, sufficient conditions for stabilization of T-S fuzzy systems with parametric uncertainties are derived in the form of linear matrix inequalities (LMIs). The piecewise fuzzy Lyapunov function approach avoid the disadvantages of the fuzzy Lyapunov function approach who is valid only when the time derivatives of the membership functions exist in the whole fuzzy region. Moreover, compared to the PDC fuzzy controller, the non-PDC control laws shows greater potential to relax the stability conditions. An illustrative example is given to validate the effectiveness of the proposed approach.

This chapter investigates the problem of stabilization of nonlinear discrete-time networked control systems (NCS) with event-triggering communication scheme in the presence of signal transmission delay. A Takagi–Sugeno (T–S) fuzzy model and parallel-distributed compensation (PDC) scheme are first employed to design a nonlinear fuzzy event-triggered controller for the stabilization of nonlinear discrete-time NCS. The idea of the event-triggering communication scheme (a soft computation algorithm) under consideration is that the current sensor data is transmitted only when the current sensor data and the previously transmitted one satisfy a certain state-dependent trigger condition. By taking the signal transmission delay into consideration and using delay system approach, a T–S fuzzy delay system model is established to describe the nonlinear discrete-time NCSs with event-triggering communication scheme. Attention is focused on the design of fuzzy event-triggered controller which ensures asymptotic stability of the closed-loop fuzzy systems. Linear matrix inequality- (LMI-) based conditions are formulated for the existence of admissible fuzzy event-triggered controller. If these conditions are feasible, a desired fuzzy event-triggered controller can be readily constructed. A nonlinear mass-spring-damper mechanical system is presented to demonstrate the effectiveness of the proposed method.

The purpose of this paper is to use the sampled-data fuzzy control (SDFC) method with exponential time-varying gains (ETVGs) for solving the control problem with regard to nonlinear parabolic PDE systems. Firstly, a model is established via the T-S fuzzy approach towards the PDE system. Next, a SD fuzzy controller with ETVGs is designed for the system. Via a Lyapunov functional, the design condition of the SD fuzzy controller with ETVGs expressed by the linear matrix inequalities is obtained, which is used to make the system exponentially stable (ES). Lastly, by carrying out simulation verification through a concrete example, we test and verify the validity of the SDFC approach.

This study investigates the stability of fuzzy‐model‐based (FMB) control system, which is formed by a Takagi–Sugeno (T–S) fuzzy model representing a non‐linear plant and a fuzzy controller connected in a closed loop. The proposed fuzzy controller considers a higher design flexibility that the premise membership functions and the number of fuzzy rules are allowed to be not the same as those of the T–S fuzzy model. By employing simple membership functions and a smaller number of fuzzy rules, the complexity of the fuzzy controller can be reduced to facilitate its implementation. Based on the Lyapunov stability theory, a membership‐function‐dependent (MFD) stability analysis is proposed to develop stability conditions in the form of linear matrix inequalities (LMIs) to determine the system stability and facilitate the control synthesis. The information of the membership functions is brought to the MFD LMI‐based stability conditions, which are dedicated to the FMB control system under consideration. It is thus the proposed MFD stability analysis result is more relaxed compared with the existing membership‐function‐independent stability analysis results. A simulation example is given to illustrate the effectiveness of the proposed approach.

The issue of non-fragile observer-based adaptive integral sliding mode control for a class of Takagi–Sugeno (T-S) fuzzy descriptor systems with uncertainties and unmeasurable premise variables is investigated. General nonlinear systems are represented by nonlinear T-S fuzzy descriptor models, because premise variables depend on unmeasurable system states and fuzzy models have different derivative matrices. By introducing the system state derivative as an auxiliary state vector, original fuzzy descriptor systems are transformed into augmented systems for which system properties remain the same. First, a novel integral sliding surface, which includes estimated states of the sliding mode observer and controller gain matrices, is designed to obtain estimation error dynamics and sliding mode dynamics. Then, some sufficient linear matrix inequality (LMI) conditions for designing the observer and the controller gains are derived using the appropriate fuzzy Lyapunov functions and Lyapunov theory. This approach yields asymptotically stable sliding mode dynamics. Corresponding auxiliary variables are introduced, and the Finsler's lemma is employed to eliminate coupling of controller gain matrices, observer gain matrices, Lyapunov function matrices, and/or observer gain perturbations. An observer-based integral sliding mode control strategy is obtained to assure that reachability conditions are satisfied. Moreover, a non-fragile observer and a non-fragile adaptive controller are developed to compensate for system uncertainties and perturbations in both the observer and the controller. Finally, example results are presented to illustrate the effectiveness and merits of the proposed method.

An optimal dual-rate digital redesign method is proposed that can be applied to continuous-time and fast-rate discrete-time control systems. The paper presents a technique, which relies on the solution of a dual-rate H2 discrete-time control problem, to convert such systems to either a slow-rate or a dual-rate sampled-data control system while guaranteeing closed-loop stability and performance in the discrete-time H2 sense. The proposed technique results in sampled-data control systems providing satisfactory closed-loop performances over an extended range of sampling rates, as compared with other widely used methods of digital redesign. Furthermore, the proposed digital redesign technique is useful in practice since it allows the designer to constrain the complexity of the digital controllers, thus resulting in low-order digital controllers easily implementable in software code and free of causality and real-time problems. Numerical examples provide a comparison of the proposed digital redesign technique with well-known approaches.

The stabilization analysis of networked control systems (NCSs) with transmitted data dropout is presented based on a stochastic fuzzy system approach. In this paper, the plant in the NCS is a stochastic time-delay nonlinear system that is represented by a Takagi-Sugeno (T-S) stochastic time-delay fuzzy system. A sufficient condition for the stabilization of the NCS is obtained by using a fuzzy Lyapunov function, which is defined by fuzzily blending quadratic functions. The stabilization controller can be designed by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software. A simulation example is provided to demonstrate the effectiveness of the proposed methods.