In this paper, new stability conditions are obtained by designing a Lyapunov function that contains polynomials of the system states and membership functions. An iterative algorithm is designed to solve the stabilization problem. Since more free variables are introduced by using the square root of the system states and square matrix representation, the proposed conditions are less conservative than existing ones. Furthermore, the time derivative of the membership functions is dealt with by applying polynomial switching mechanism, while the average dwell time approach is employed to address the switching signal. The effectiveness of the proposed method is demonstrated through three numerical examples, including a practical tank level control system, confirming its applicability to real-world scenarios. This paper shows that for positive fuzzy systems, better results can be achieved by utilizing the positivity of the system states and the information of the membership functions.

This article focuses on addressing the issue of asynchronous${H_\infty}$control for Takagi-Sugeno (T-S) fuzzy Markov jump systems with generally incomplete transition probabilities (TPs). The delay is assumed to vary periodically, resulting in one monotonically increasing interval and one monotonically decreasing interval during each period. Meanwhile, a new Lyapunov-Krasovskii functional (LKF) is devised, which depends on membership functions (MFs) and two looped functions formulated for the monotonic intervals. Since the modes and TPs of the original system are assumed to be unavailable, an asynchronous switching fuzzy controller on the basis of hidden Markov model is proposed to stabilize the fuzzy Markov jump systems (FMJSs) with generally incomplete TPs. Consequently, a stability criterion with improved practicality and reduced conservatism is derived, ensuring the stochastic stability and${H_\infty}$performance of the closed-loop system. Finally, this technique is employed to the tunnel diode circuit system, and a comparison example is given, which verifies the practicality and superiority of the method.Note to Practitioners—As a category of stochastic hybrid nonlinear systems driven by continuous time and discrete events, FMJSs have significant applications in practical engineering such as aircraft control systems and large-scale manufacturing systems. However, the real-time acquisition of system mode information and TPs is difficult due to technological constraints and limited resources. Moreover, the presence of periodically varying delay is prevalent in many industrial processes, resulting in degradation of dynamic system performance and instability. Therefore, it is necessary to study the FMJSs with generally incomplete TPs and periodically varying delays in asynchronous framework. Accordingly, unlike previous work, the designed LKF relies on system modes, MFs, and two looped functions for monotonic intervals. This new LKF exhibits a high degree of flexibility, fully leveraging the information of MFs and periodically varying delays, which can significantly reduce conservatism.

This article investigates the problem of event-based strictly dissipative filtering for stochastic fuzzy complex networks (SFCNs) subject to multiple cyberattacks, including deception attacks and denial-of-service (DoS) attacks. A resilient event-triggered mechanism (ETM) is proposed to reduce the communication burden while effectively countering these cyberattacks. The membership functions are assumed to be mismatched due to the impact of the network environment and sampling behavior. Furthermore, the system dynamics are modeled using Itô-type stochastic differential equations, which include deterministic fuzzy complex networks (CNs) as a special case. Unlike previous studies on stochastic systems, the auxiliary vector function method is employed to introduce more time-varying delay information into the piecewise Lyapunov–Krasovskii functional (LKF), thus reducing conservatism. Consequently, a series of delay-dependent sufficient conditions is derived to ensure the exponentially mean-square stability (EMSS) and strict dissipativity of the filtering error system. Finally, the effectiveness of the proposed method is demonstrated through an illustrative example.

This article addresses the problem of finite-time asynchronous switching control for fuzzy Markov jump systems (FMJSs) using polynomial membership functions. Firstly, a Lyapunov-Krasovskii functional (LKF) is designed, incorporating both system modes and polynomial membership functions. This LKF contains more information about the membership functions, closely aligning with the characteristics of FMJSs, and effectively reducing conservatism. Based on the polynomial matrices switching rule, a practical asynchronous switching controller is introduced. The objective is to ensure finite-time boundedness of the closed-loop FMJSs while satisfying a$H_{\infty } $performance index. Furthermore, the average dwell time method is applied to handle the switching signal, eliminating the predefined assumptions that switching numbers are finite or that system states are constrained to a specific region. Ultimately, the validity and practicability of the obtained results are verified through three examples.

-gain for positive impulsive Takagi-Sugeno (T-S) fuzzy systems are further studied in this article. Different from the Lyapunov function in the literature, where the Lyapunov matrices are time-invariant or only linearly dependent on the impulse interval, in this article, a novel polynomial impulse-dependent (ID) copositive Lyapunov function (CLF) is constructed by using the polynomial impulse time function. In addition, the binomial coefficients are applied to derive new finite linear programming conditions. Less conservative results are obtained since the polynomial ID CLF contains more impulse interval information. Three examples demonstrate the influence of the polynomial degree on the results and the effectiveness of the developed new results.

Till now, most of the results on Takagi–Sugeno (T-S) fuzzy Markov jump systems are synchronous and independent of the analysis of membership functions (MFs), which leads to conservatism. Therefore, it is significant to study the MFs-dependent Lyapunov–Krasovskii functional (LKF) while considering asynchrony. This article investigates the${H_\infty }$control issue of T-S fuzzy Markov jump systems with time-varying delay. First, delay-product-type terms and MFs are introduced to design a new LKF, which fully utilizes the information of time-varying delay and MFs simultaneously. Moreover, based on a hidden Markov model, the controller mode is assumed to operate asynchronously with the system mode, and an asynchronous switching fuzzy controller is proposed by employing the switching approach. Thus, a more practical and less conservative stability criterion is obtained to ensure that the closed-loop system is stochastically stable with${H_\infty }$performance. Finally, three examples are provided to illustrate the superiority and practicability of the presented method.

This article addresses the issue of adaptive event-triggered${H_{\infty} }$control for Markov jump systems based on the Takagi–Sugeno (T–S) fuzzy model. First, a new double asynchronous switching controller is presented to deal with the problem of the mismatch of premise variables and modes between the controller and the plant, which is widespread in real network environment. To further reduce the power consumption of communication, a switching adaptive event-triggered mechanism is adopted to relieve the network transmission pressure while ensuring the control effect. In addition, a new Lyapunov–Krasovskii functional (LKF) is constructed to reduce conservatism by introducing the membership functions (MFs) and time-varying delays information. Meanwhile, the invariant set is estimated to ensure the stability of the system. And the disturbance rejection ability is measured by the optimal${H_{\infty} }$performance index. Finally, two examples are presented to demonstrate the effectiveness of the proposed approach.

Recently, a switching method is applied to deal with the membership function-dependent Lyapunov-Krasovskii functional (LKF) for fuzzy systems with time delay; however, the Lyapunov matrices are only linear dependent on the grades of membership which leads to linear switching (Wang and Lam, 2019). In this article, the linear dependence on the grades of membership is extended to homogenous polynomially membership function dependent (HPMFD) and the linear switching is extended to polynomial matrix switching, based on which the obtained result contains the previous one as a special case. Furthermore, in order to fully use the introduced variables without speial structure, an iteration algorithm is designed to construct the switching controller and the initial condition of the algorithm is also discussed. The final simulation demonstrates the effectiveness of the developed new results.

It is observed that if the premise variables are differentiable, the time derivatives of membership functions are positive or nonpositive. Based on the above fact, the switching observer gains and controller gains are designed to guarantee that the fuzzy system is globally exponentially asymptotically stable by applying the average dwell-time (ADT) technique. In addition, two methods are proposed to find the controller gains and observer gains by dividing the matrices into$3\times 1$blocks or$2\times 2$blocks. Finally, two examples are given to verify the conclusion.

In the recently published paper, a switching method has been proposed to deal with the time derivative of the membership functions and less conservative results can be obtained due to this method; however, this method is based on the assumption that the switching times are finite. In this article, this method is further studied and the average dwell-time (ADT) switching technique is applied to ensure the stability if there is no such assumption. In addition, an algorithm is proposed to find the switching controller gains. The final simulation demonstrates the effectiveness of the developed new results.

In this article, the homogenous polynomially membership functions dependent (HPMFD) matrices are used to study the interval type-2 Takagi-Sugeno fuzzy systems. First, some necessary notations of the HPMFD matrices are introduced. Next, based on these notations, the time derivative of the HPMFD matrices is discussed and a switching method is proposed to ensure that the time derivative of the HPMFD matrices is negative. Then, a HPMFD controller is designed and new stabilization conditions are obtained by using the HPMFD Lyapunov function. In the end, the simulations show that the method in this article is less conservative than the existing ones in the literatures.

This article further studies the problem of observer design for the continuous-time Takagi-Sugeno (T-S) fuzzy system with unmeasurable premise variables. A membership function-dependent Lyapunov function is designed to obtain the observer-based controller. Different from the existing results, a switching method is proposed to deal with the time derivative of membership functions. Several problems such as, too many parameters, and the small local stabilization region in the existing papers are solved by applying the switching method. In addition, two algorithms are designed to obtain the controller gains and observer gains. In the end, two examples are provided to demonstrate the effectiveness of the proposed approach.

In this paper, the stability and positivity of positive polynomial fuzzy model based (PPFMB) control system are investigated, in which the positive polynomial fuzzy model and positive polynomial fuzzy controller are allowed to have different premise membership functions from each other. These mismatched premise membership functions can increase the flexibility of controller design; however, it will lead to the conservative results when the stability is analyzed based on the Lyapunov stability theory. To relax the positivity/stability conditions, the improved Taylor-series-membership-functions-dependent (ITSMFD) method is introduced by introducing the sample points information of Taylor-series approximate membership functions, local error information and boundary information of substate space of premise variables into the stability/positivity conditions. Meanwhile, the ITSMFD method is extended to the PPFMB control system with input saturation to relax the estimation of domain of attraction. Finally, simulation examples are presented to verify the feasibility of this method.

In this technical paper, a new Lyapunov-Krasovskii functional (LKF) is designed to study the stability of continuous-time Takagi-Sugeno fuzzy systems with time-varying delay. The integrand of the LKF depends on integral variable and time t which can help to reduce the number of linear matrix inequalities (LMIs). Then, a new stability criterion is derived by analyzing the sign of the time derivatives of membership functions. Compared with the existing results, larger delay bounds can be obtained by applying the new criterion. In the end, two examples show the effectiveness of the conclusions.

This brief paper investigates the local stabilization for continues-time Takagi-Sugeno fuzzy systems with constant time delay. In order to deal with the time delay, we design a Lyapunov-Krasovskii functional that is dependent on the membership function. Based on the Lyapunov-Krasovskii functional and the analysis of the time derivative of the membership function, less conservative results can be obtained; however, the Lyapunov-Krasovskii functional is designed so complicated that the Lyapunov level set is hard to be measured directly. Alternatively, two sets are obtained to estimate the local stabilization. One set is for the time-varying initial conditions and the other is for the time-invariant initial conditions. The relationship between the two sets are also discussed. In the end, two examples are given to illustrate the effectiveness of the proposed approach.

Till now, there are lots of stability and stabilization results about Takagi-Sugeno (T-S) fuzzy systems with time delay, but most of them are independent of the analysis of membership functions. Since the membership functions are an essential component to make a fuzzy system different from others, the conditions without its information are conservative. In this brief paper, a new Lyapunov-Krasovskii functional is designed to investigate the stability and stabilization of continuous-time T-S fuzzy systems with time delay. Different from the existing results in the literature, the integrand of the Lyapunov-Krasovskii functional in this paper depends not only on the integral variable, but also on the membership functions, and thus, the information of the time derivative of membership functions can also be used to reduce the conservativeness of finding the maximum delay bounds. Utilizing the information of the time derivative of membership, a number of controllers are designed according to their sign, and then, a switching idea is applied to stabilize the fuzzy system. In the end, two examples are given to illustrate the feasibility and validity of the design and analysis.

This paper deals with the problem of observer design for continuous-time Takagi-Sugeno fuzzy models with unmeasurable premise variables. First, in order to improve the existing results of observer design, a new method is proposed to bound the time derivatives of the membership function. Then, by applying the nonquadratic Lyapunov function and the matrix decoupling technique, the controller gains and observer gains are designed to guarantee that the error system is asymptotically stale. Furthermore, better H ∞ performance can be obtained by solving an optimization problem. All of the results are presented as linear matrices inequalities and three examples are provided to demonstrate the merits of the proposed approach.