T-S Fuzzy Control of Uncertain Fractional-Order Systems with Time Delay

In this article, the adaptive control of uncertain fractional-order time-delay systems (FOTDSs) with external disturbances is discussed. A Takagi-Sugenu (T-S) fuzzy model with if-then rules is adopted to characterize the dynamic equation of the FOTDS. Besides, a fuzzy adaptive method is proposed to stabilize the model. By utilizing the Lyapunov functions, a robust controller is constructed to stabilize the FOTDS. Due to the uncertainty of system parameters, some fractional-order adaptation laws are designed to update these parameters. At the same time, some if-then rules with linear structure based on the fuzzy T-S adaption concept are established. The designed method not only guarantees that the state of closed-loop system asymptotically converges to origin but also keeps the signal in the FOTDS bounded. Finally, the applicability of the control method is proved by simulation examples.


Introduction
In the last century, fractional-order calculus has been widely used in many electrical systems, robot systems, engineering systems, and thermodynamic systems, which has attracted extensive attention [1][2][3][4][5]. e fractional-order system (FOS) has more advantages than classical integer-order system [6][7][8]. Firstly, FOSs focus on the characteristics of the whole time domain. Secondly, FOSs show the properties of the whole space. Based on the above two characteristics, FOSs have played a significant role in the control field. Actually, the description of FOS is more consistent with the real dynamic system. It can be used to characterize more complex systems, and it also provides a guarantee for the genetic and memory properties of different substances. So, the application and theoretical research of FOSs will become more and more popular [9,10]. e traditional integer-order theory is not suitable for FOSs because of its unique definition, so the researchers adopt two new methods to stabilize the FOS. e first solution is Lyapunov function, and the other is Laplace transformation to stabilize FOS [11][12][13][14][15]. In control analysis, it is necessary to make the state trajectory of the system follow the desired command. In this study, many control methods are proposed to achieve good performance for noninteger-order systems. In addition, the researchers are also working to develop various control methods, such as sliding mode control [16][17][18], pinning control [19,20], adaptive fuzzy control [21,22], PID control [23,24], and backstepping control [25,26], to achieve the effective control. For example, an adaptive sliding mode control for FOS stabilization was studied in [27]. In [28], according to Lyapunov theory, the stability of the adaptive control method for the synchronous error system was analyzed. In [29], a class of global Mittag-Leffler stability problems with coupled FOSs was proposed. However, in modeling dynamics, the existence of nonlinearity makes it more difficult to deal with stability and other performance. To resolve the difficulties, many methods have been adopted to deal with these obstacles, among which the Takagi-Sugeno (T-S) model method is the most effective tool to describe the dynamic equations [30][31][32]. e most obvious advantage of this method is that it uses less fuzzy rules, and then the linear subsystems are fuzzy combined to get the overall model of the system. e results of control analysis and stability analysis with the T-S control method are also reported in the literature [33][34][35]. In [33], under the condition of quantization, a dynamic output feedback controller was used to control the fuzzy T-S fractional-order neural network. A static feedback controller with fuzzy T-S structure was designed in [34]. e regularization and control of singular rectangular FOSs were considered in [35]. Although there are many reports about the stability of T-S systems, few studies on time-delay systems were published.
In fact, FOSs often contain uncertainty and time delays, which are two common phenomena [36][37][38]. eir existence will increase the instability of the model and make it more complex. erefore, it will be more important to solve such problems in theoretical application. Even so, researchers have developed lots of methods to solve the uncertainties and time delay. Moreover, many studies focus on the stability of fractional-order time-delay systems (FOTDSs) with uncertainties, such as [37,39,40]. e recently developed immersion boundary method was used to model the complex solid boundary in [39]. A single input and output delay model was given in [37]. In [40], the control problem of the first-order dynamic discrete system was discussed. Among these methods, some of them are easier to control by using their unique characteristics [41,42]. At the same time, the external disturbance in the time-delay system is inevitable; in other words, such system will become more complex. However, for our known knowledge, there are still some studies on the control of FOSs with disturbances [43][44][45][46][47][48].
Based on the inspiration of the above literature, this article focuses on T-S fuzzy control for uncertain FOTDSs. Aiming at uncertain FOSs, a fuzzy adaptive control strategy is established. At the same time, a fuzzy T-S system is constructed to characterize the system equation. Next, according to the adaptive control theory and Lyapunov function, simple linear control rules are designed to guarantee the asymptotic stability of the unknown system. Some adaptive laws for estimating unknown parameters of the system are given. Lastly, simulation results show the superiority of the T-S method. e purpose of this paper is to obtain the stability conditions by fractional Lyapunov functions, and the main contributions are as follows.
(1) A suitable Lyapunov function is constructed, and the Lyapunov stability theory is used to ensure that the model is stable. (2) e controller obtained by using the linear structure can improve the stability of the FOTDS. (3) e adaptive laws are given to update the uncertainty of the model. ese can enhance the control effect of the system and cause lower complexity.
Other structures of the article are as follows. e modeling of the T-S fuzzy system and some related concepts of fractional calculus are introduced in Section 2. Section 3 gives the stability analysis results of FOTDSs. Numerical simulations are shown in Section 4. Section 5 summarizes this full text.

Basic Knowledge and System Introduction
In the control process, some definitions of fractional derivative are stated. Since the solutions of differential equations are related to the initial conditions given by traditional methods, Caputo derivative is useful in practical dynamics. In the article, we utilize Caputo's fractional derivative.
Lemma 1 (see [50]). Let 5(t) ∈ R n×n be a differentiable matrix. So, for any t ≥ t 0 , one obtains Lemma 2 (see [50]). Let x(t) ∈ R n be a derivative vector. en, for any t ≥ t 0 , where B ∈ R n×n is a positive definite symmetric constant matrix.
Defining an uncertain FOS as follows: where c ∈ (0, 1), t is the time, x is state vector, u represents control input, and ∇ denotes uncertain variable.

Generalized T-S Fuzzy Model.
Based on the expansion of the traditional T-S system, the generalized T-S system of FOS is derived. en, the system dynamics can be expressed by the local fractional linear model. Finally, the linear composition is fuzzy mixed to get the whole model of the system. Consider the following generalized T-S fuzzy structure.
where R ıj , j � 1, 2, . . . , k, denotes the fuzzy set, r represents the number of fuzzy rules, A ı ∈ R n×n is a constant matrix, and T represent the state vector and premise variable, respectively.
e system output is inferred as where where R ıj (κ j (t)) is a fuzzy membership function, satisfying

Main Results
In the part, the stability of T-S fuzzy FOTDS with uncertainty is studied. And a fuzzy control method is given to control uncertain FOTDS.

System
Description. Suppose that model (7) with unknown terms and time delay is designed as follows: where x(t) ∈ R n , κ(t) � [κ 1 (t), κ 2 (t), . . . , κ k (t)] T , and R ıj are the same as the above definitions. x(t − ι) ∈ R n denotes the time-delay term, Q ı and J ı are some constant matrices, Q sı and J sı are unknown parameters, and η(t) ∈ R m and u(t) ∈ R n are disturbance and control input, respectively. So, the final output is To achieve the asymptotic stability of system (15), a fuzzy adaptive controller is given as follows.
where Q sı and J sı are estimations of Q sı and J sı , which can be obtained by some adaptive laws, E ı and F ı are feedback gain matrices that can be given.
From equations (15) and (16), the fuzzy controller is obtained as e adaptive laws are where ϱ sı and ϱ tı are positive constant adaption gains.

Stability Analysis.
To verify the stability of system (21), a useful theorem is proposed. (17)- (19), then the system will converge to origin asymptotically.

Theorem 2. Consider the uncertain FOTDS (15). If model (15) is controlled by control laws
Proof. Consider the following Lyapunov function candidate where tr(C) represents the trace of a matrix and B is a symmetric positive definite matrix.
Using Lemmas 1 and 2, one obtains By utilizing equation (21) and D c Q sı � −D c Q sı and D c J sı � −D c J sı , one has According to the equations For the vectors x ∈ R n×1 and y ∈ R n×1 , (26) is rewritten as Substituting equations (18) and (19) into equation (27), one obtains

Journal of Mathematics
So, D c V ≤ 0 is equivalent to To sum up, if there exists E ı and F ı and a symmetric positive matrix B such that inequality (29) holds, which shows that D c V ≤ 0 and the uncertain system (15) with time delay converges to zero asymptotically. e proof is over. □ Remark 1. In a word, the fuzzy T-S system with Gauss membership function or other membership function can approach any a continuous function. It is worth emphasizing that compared with other literatures, such as [7,27,28,52], in comparison with other techniques, this method has the advantage of the designed T-S fuzzy controller has a simple linear structure. In the simulation of this paper, a special membership function is adopted to control the FOTDS, it should be noted that due to the limitations of existing modeling methods, only this type of membership function is utilized for research. (17), the proportional relationship between u(t) and the parameters E ı and F ı can be obtained. In other words, the smaller the values of E ı and F ı , the smaller the control effect caused, and the reverse is also correct. In the adaptive laws (18) and (19), the sizes of ϱ sı and ϱ tı will also affect the update parameters E ı and F ı accordingly, and furthermore, it will also affect this control input u(t). erefore, the greater the values of ϱ sı and ϱ tı , the greater the control impact caused. On the contrary, one can choose the appropriate ϱ sı and ϱ tı to make the control effect better. And according to inequality (29), E ı and F ı should be considered to achieve the asymptotic convergence of system (15).

Simulation Results
In the part, an example is provided to demonstrate the availability of the designed controller.
Consider the following uncertain T-S fuzzy FOS with time delay: e fuzzy rules is as follows.

Conclusions
is article studies the stability and control of uncertain FOTDSs by utilizing the T-S fuzzy structure. At the beginning, a model with if-then rules is developed to depict the differential equations of the system. Next, the fuzzy adaptive method is adopted to update the uncertainty and unknown parameters. en, according to the Lyapunov function and T-S fuzzy control method, some control rules are adopted such that the system states tend to zero. Finally, the designed controller is used to stabilize FOTDSs. Simulation examples are proposed to verify the rationality of the devised adaptive control method. Our experimental results can not only enrich the control theory of FOSs but also can be implemented to other FOSs. Future research work will be devoted to adaptive control of incommensurate nonlinear FOSs.

Data Availability
All datasets generated for this study are included in the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.