A Fuzzy Control Strategy to Synchronize Fractional-Order Nonlinear Systems Including Input Saturation

One of the most important engineering problems, with numerous uses in the applied sciences, is the synchronization of chaos dynamical systems. Tis paper introduces a dynamic-free T-S fuzzy sliding mode control (TSFSMC) method for synchronizing the diferent chaotic fractional-order (FO) systems, when there is input saturation. Using a new defnition of fractional calculus and the fractional version of the Lyapunov stability theorem and linear matrix inequality concept, a Takagi–Sugeno fuzzy sliding mode controller is driven to suppress and synchronize the undesired behavior of the FO chaotic systems without any unpleasant chattering phenomenon. Finally, an example of synchronization of complex power grid systems is provided to illustrate the theoretical result of the paper in real-world applications.


Introductory Remarks.
With a history dating back more than 3 centuries, fractional calculus enables a more in-depth examination of natural phenomena. In FO diferential equations, the order of diferentiation [1] can be a noninteger. FO systems have garnered a great deal of attention in the past two decades from scientists and engineers who choose for modeling of numerous phenomena using FO Systems. In felds such as electrical systems [2], neural networks [3], quantum mechanics [4], aerospace sciences [5], economical systems [6], medical sciences [7], emotional sciences [8], psychological sciences [9], etc., the role of FO systems is extensively documented.
Tis is because of their oscillatory properties as well as their extreme sensitivity to the values they start with. In addition, according to several studies, the majority of FOSs display unpredictable behavior because of the oscillatory characteristics they possess and their extreme susceptibility to initial values. As a direct consequence of this, academicians have focused their eforts on the development of alternative methods for synchronizing and maintaining chaotic FO systems [10]. In this context, several diferent control strategies have been proposed, including the sliding mode controller [11,12], the adaptive controller [10,13], the fuzzy controller [14][15][16], the PID controller [17], the observer controller [18], and the optimal controller [19] to control and synchronize of chaotic FO Systems.
Te T-S fuzzy technique has recently become one of the most popular methods for coordinating and regulating nonlinear systems, both in theory and in reality. In fact, the T-S fuzzy method uses fuzzy weight and cost functions as tools to transform the nonlinear system into a linear equivalent. Te T-S fexible method has benefts like a good theory study, usability, and durability.
Te sliding mode control (SMC) methodology has quickly become one of the most well-liked control methodologies in recent times, both in terms of its theoretical foundations and its practical applications. Typically, the SMC divides into two parts described as follows: (1) Te development of an adequate and stable sliding surface (2) To generate control signals so that the chaotic trajectories of the FO systems are suppressed and remain on the specifed sliding surface

Survey of the Relevant Literature.
Te T-S fuzzy method has been adopted by many academics as a means for stabilization and synchronization of the FO nonlinear systems.
For instance, in [20], a model free state feedback strategy is designed based on a cost guaranteed function, in the T-S fuzzy space topology. According to [21], a group of T-S fuzzy delayed nonlinear complex networks can be synchronized via a fuzzy memory management method. In [22,23], fuzzy observer-based state-feedback methods are designed to control uncertain nonlinear MIMO nonlinear systems. In [24], to gain the control of the heart rate of runners, a discrete-time observer-based state feedback technique is proposed. Wu et al. [25] suggest a fexible T-S fuzzy control algorithm to be a solution to the synchronization challenge facing FO fuzzy nonlinear unpredictable systems with delay. A fnite-time fuzzy control method to synchronize a collection of T-S fuzzy riotous delayed artifcial neural networks has been developed by the authors of [26]. In addition, new data-based fuzzy techniques have been developed in [27,28] to synchronize and regulate Markovian jump systems as well as complicated distributed systems. In the paper referred to as [29], a novel controller for synchronizing the hyperchaos networks with application in encrypted communication for nonideal networks is suggested. Tis controller relies on the T-S fuzzy model and the LMI technique. An adaptive T-S fuzzy technique has been given, and the problem of synchronization failure in chaos gyrostat is taken into consideration and discussed in [30]. Te method makes use of a wavelet-based model. A switching control strategy is presented as a solution to synchronize problems in the paper referred to as [31]. Tis approach is employed to synchronize a collection of T-S fuzzy artifcial complex networks which include coupled-delayed. Te FO integral fuzzy SMC strategy is suggested in [32] to stabilize uncertain FO nonlinear systems exposed to uncertainties and external. A strong FOSMC is created in [33] using the frequency distribution model to error-control the chaotic FO Jafari-Sprott systems. In [34], a fnite-time FO recursive SMC method is developed for unknown linear/ nonlinear systems. In [35], homogeneous controllers are proposed to synchronize FO chaotic networks, based on utilizing the properties of the Presnov decomposition. Te authors of [36] have addressed adaptive feedback controllers for complete and fnite-time synchronization for a class of FO fuzzy neural networks. Te challenge of synchronization of FO chaotic systems is considered in [37] using a nonsingular fuzzy-based SMC technique. In [38], a reliable SMC approach based on neural networks is recommended for the fnite-time synchronization of uncertain hyperchaotic FO systems. A controller for the synchronization of both transmitter and receiver systems is created in [39]. Tis control is based on a four-dimensional nonlinear FO hyperchaotic system with an external disturbance. In [40,41], adaptive terminal SMC methods are designed for the synchronization of FO uncertain chaotic systems in a fnite time. In [42], an adaptive neural controller has been presented to stabilize unknown FO delayed systems in the presence of input saturation as nonlinearity. Te authors of [43] have considered an intermittent control FT synchronization problem for FO complicated dynamical neural networks.
But generally, these mentioned research works have at least one of the following drawbacks: (1) Te Caputo fractional derivative, which has been used in almost all of these works, has been contested by a number of researchers who fnd it not adequate for pseudo-state-space developments because it does not permit taking into account the physical behavior of the system (with a fractional-order system, all the system past must be taken into account to defne its future, even at time t � 0). (2) Te majority of these works have contemplated the synchronization of two identical FOSs; this is despite the fact that in practice, it is extremely uncommon for anyone to have access to two identical FOSs. Terefore, the abovementioned disadvantages motivate us to combine the FO edition of Lyapunov-stability-theorem (LST) with the LMI and T-S fuzzy theory for designing a chattering-free T-S fuzzy SMC synchronization technique, which is both dynamic-free and robust against uncertainties.

Contribution and Motivation.
According to the aforementioned talks, it is important to create and propose a nochatter T-S fuzzy SMC technique (TSFSMC) to synchronize complicated FO chaotic systems in the existence of system uncertainties, extrinsic disturbances, and input saturation. In addition, the no-chatter TSFSMC technique has not been sufciently studied to synchronize FO chaotic systems until this point, so the primary goal of this research is to take this into consideration. Furthermore, input saturation is another crucial nonlinearity in the construction of practical controllers. Because of input saturation, the control energy does not increase beyond a certain point. Terefore, it signifcantly lowers energy waste in the control system. Also, utilizing a new defnition of the noninteger derivative ensures us about the results.
As a result, this work suggests that constructing a nochatter TSFSMC approach is the best way to address the synchronization problem of the FO chaos systems in the face 2 International Journal of Intelligent Systems of the system's uncertainties, outside disturbances, and input saturation. An easy-to-design and simple-to-use sliding surface is initially ofered based on the FO integration concept. Ten, the LST's FO version is used to create a suitable dynamic-free T-S fuzzy control strategy to ensure the sliding motion. It should be made clear that neither linear nor nonlinear phrases of dynamical components of the system were utilized in the construction of this FTSMC technique. To demonstrate the usefulness and efectiveness of the recommended dynamic-free FTSMC approach, certain simulations are also provided as examples.
In conclusion, the following is a summary of the most signifcant motivations and contributions made by this study: (1) Developing a no-chatter TSFSMC approach with the goal of synchronizing a large area of complicated and disorganized FO chaotic systems based on a new defnition of fractional calculus with the use of an efcient continuous function. (2) Te suggested TSFSMC is resilient and can mitigate the impacts of system uncertainties as well as external disturbances or input saturation. (3) To obtain trustworthy fndings regarding the global and asymptotical stability of the coordinated closedloop chaotic FO systems, the linear matrix inequality (LMI) and FO type of the LST were used. (4) When applied to real-world issues, the suggested TSFSMC technique works better than other methodologies.

Te
Outline of the Paper's Structure. Te structure of this paper can be observed in the following statements: Section 2 provides preliminary and fundamental concepts concerning the FO calculus and the FO systems. Te problem description of synchronization of the FO chaotic system is presented in Section 3. Ten, a dynamic-free TSFSMC technique is designed to solve the synchronization problem in Section 4. After that, in Section 5, applied examples are illustrated to show the efectiveness and efciency of the desired TSFSMC method in practice. Finally, the obtained results, consequences, and future plans are covered in detail in Section 6.

Basic Concepts
Defnition 1 (See [44]). Let H(t) be a continuous function in R. Ten, the FO integral defned by Riemann-Liouville for G(t) is represented as follows: where t 0 shows the beginning time, Γ(.) is the Gamma function, and β is the order of derivative.
Feature 1 [45]: If w ∈ R is a genuinely constant number, then one gets D β w � 0. Feature 2 [45]: At any point in t > 0, let 0 < β ≤ 1 be true and f and g be diferentiable. Ten, Defnition 3 (See [46]). Suppose that W ∈ R n×n and I be the identity matrix. Ten, the matrix sign function has the following defnition: where in l is the order of the approximation and Also, for any sliding surface like S(t) ∈ R m sgn (l) (S(t)) � (I + S(t)) l − (I − S(t)) l · (I + S(t)) l +(I − S(t)) l − 1 .

(5)
Lemma 4 (See [47]). Te following inequality is true for constant matrices G and E and a symmetric constant matrix S of acceptable dimensions: If and only if for any ε > 0 where in G fulflls G T G ≤ R.
Lemma 5 (See [48]). Given any three matrices X, Y, and Z of proper dimensions such that Z > 0, we get the following: Theorem 6 (See [49]). Let β ∈ (0,1), and assuming that the FO system D β c(t) � g(c, t) satisfes the Lipschitz requirement, including c � 0 for an equilibrium-point. Also, suppose that there are class-K functions k 1 , k 2, and k 3 with a Lyapunov-function V(t, c(t)) satisfying International Journal of Intelligent Systems where 0 < m < 1. Following that, the equilibrium point of the system D β c(t) � g(c, t) will reach a state of asymptotically stable behavior.

Notation.
Te symbol * will be utilized for symmetric elements of a symmetric-matrix. Moreover, OT means orthogonal transformation.

The Description of the Problem and the T-S Fuzzy Forming
Here, frst, an explanation of the issue will be provided, and then T-S fuzzy description of the problem is going to be taken into consideration.

Description of the Problem.
Tat which follows is the outcome of two n-dimensional nonlinear structures with indeterminate drives and unpredictable FO response structure: Drive system: Response system: where β ∈ (0, 1), x(t) and y(t) are in R n×1 and are the vectors of states of the system, and A and B determine the matrixes of the system structure. Furthermore, ∆A and ∆B show the uncertainties and external disturbances. Moreover, u(t) denotes the controller and ϕ(u(t)) � [ϕ 1 (u 1 (t)), . . . , ϕ n (u n (t))] T is the actuator saturation, which for i � 1, . . . , n are introduced as where where u + , u + ∈ R + , and u − , u − ∈ R − represent the ranges of the measured-saturation function, while θ ∈ R + represents the measured-saturation slope. Now, error system's structure will be introduced as follows: In accordance with this, the error structure to synchronize scenario is carried out as follows: Te main objective of this paper is to propose a T-S fuzzy method that is plausible and modifable in such a way that Constructing equation (14) is essential in order to solve the synchronization problem that occurs between equations (10) and (11).

Development of the T-S Fuzzy Algorithm to FO Systems.
Tis section will focus on the formulation of the synchronizing problem for the T-S fuzzy uncertain error FO system (15). Suppose that the following rule is a model that can be used for the complex FO error system (15) with saturation input nonlinearities.

Main Results
Here, the sliding surface will be introduced based on the TS fuzzy method and LMI inequality approach. Ten, the controller methodology is going to design, and the analytical result will be presented. We consider that the submatrix K i is a full rank m, and we suppose that there is an OT matrix H in such a way that Also, K∈ R m×m shows a nonsingular matrix, and let K be decomposed into the singular value subsets: , then following relation can be derived from (19): (21) and (22), the sliding surface is designed as It is clear that the requirement s(t) � 0 will be satisfed whenever there is sliding motion; consequently, So, in (21), one gets.
Theorem 7. Te sliding surface (23) ensures that the slidingmode dynamic motion described in equation (25) is asymptotically stable if for any constant p > 0, there exist positive symmetric matrices M and N that fulfll the following LMI: Tis is such that ξ j � ξ 2 j and N � SM.

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Ten, where in I is the case of D α V(t), and for any σ 1 (t) ≠ 0, μ < 0 must be fulflled.
Here, by multiplying M − 1 to both sides of μ and symbolizing According to Lemma 4, the LMI (32) satisfes for all L j (t) holding L T j (t)L j (t) ≤ I if there exists a positive number ξ − 1 j so that Ten, the condition of Lemma 4 is satisfed and when the Schur complement is applied to (33), the result is (26). Tis concludes the proof.
After constructing the sliding surface such that the sliding dynamic systems have the proper response, the second phase in the SMC design methodology is to develop a switched control rule that ensures the ease of access of the defned sliding surface (22). In this case, the conditions are written as follows: Now, the FSMC rule is presented as where 6 International Journal of Intelligent Systems where, F > 0 and q > 0 and In which, φ min and φ max are the minimum and maximum eigenvalues of F i , respectively. In addition, Figure 1 demonstrates the structure of the suggested TSFSMC control mechanism in a block diagram as follows. □ Theorem . We consider the TS fuzzy chaotic error system (19). We suppose that the LMIs (26) are practicable and the sliding surface (23) be provided, where S may be determined by (26). In that case, all the error states in the closed-loop T-S fuzzy chaotic error system with the control rule (37) will be asymptotically stable.
Proof. Based on S T (t)D α S(t) and the sliding surface (23), one can get the following results: In which where F > 0 and q > 0. Also, we have the following conditions: Tey lead to Using Lemmas 4 and 5, it is clear that ℧ 2 ≤ ℧ 4 and one obtains Terefore, when the control law (37) is operated, the FO dynamic systems (21) and (22) will be converging to the sliding surface (23). Now, according to the abovementioned results and Teorem 7, the FO dynamic systems (21) and (22) are asymptotically stable. Terefore, since σ(t) � He(t), if σ(t) ⟶ 0, then e(t) will converge to zero asymptotically. Terfore, the proof is completed. □ Remark 9. It is difcult to establish a combination of SMC and TS fuzzy techniques to develop a dynamic-free approach that takes saturation nonlinearities into account. Despite the fact that the suggested method may appear to be traditional, such a combination does not exist very often. In addition, making use of the defnition of the sign function (formulas (4) and (5)) for one of the other advances that this approach possesses is the design of the method that is being sought after.

Remark 10.
Recently, in [50,51], the problems of synchronization/control of discrete nonhomogeneous Markovian switching systems are considered via H ∞ performance and fnite-time control methods. However, the use and combination of the T-S fuzzy method with the described controllers in these papers show that the combination of the concept of T-S fuzzy and nonlinear controllers can be very reliable and popular.

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Remark 11. Research on synchronization and stabilized switched systems in the FO nonlinear chaotic system is an emerging feld. Although it has obtained very acceptable results, there are still many problems and developments, such as providing some conservativeness analysis of the proposed stability methods (Refs. [52,53]), which should be considered as a future research issue in this feld of study.

Remark 12.
Research on the proposed TSFSMC approach for synchronization of an FO chaotic system is a growing feld with many limits; although it has yielded extremely satisfactory results, there are still many restrictions that we hope will be eliminated in the future. For instance, we have (I) the restriction on the order of derivation between 0 and 1. (II) Complicated system modeling using the T-S fuzzy approach. (III) Choosing the controller settings by trial and error.

Numerical Simulations
Here, to demonstrate the applicability and efectiveness of the designed TSFSMC approaches, two examples are provided. In the frst scenario, the real situation involving the synchronization of two distinct chaotic FO power grid systems is studied. Furthermore, in the second scenario, the synchronization of PMSM and BLDC motors is taken into account. In addition, simulations are programmed using a revolutionary numerical technique in [54]. In addition, it should be mentioned that the controller starts operating after t � 3 seconds.

Scenario 1.
Here, the synchronization of the FO chaotic systems Lu and Chen, which exhibit chaotic behaviors for = 0.94, demonstrates the efciency of the recommended TSFSMC (37). Te chaotic Lü system is one of the dynamical systems employed most often in complicated power grid systems [55]. Figure 2 is a real-world example of a complex power grid system. Furthermore, one of the intriguing aspects of the fractional-order Lü system is that it exhibits chaotic behavior for orders lower than three when we know that is not the case for orders lower than three in comparable integer-order situations [56]. Te following introduces FO chaotic Lu system's dynamics as a driving system [56].

Saturation Nonlinearity
Response System  Step-up Transformer

Uncertainties and External Distributions
Step-down Transformer

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Also, Λ 1 , Λ 2 , Λ 3 , and Λ 4 are the system's unknown parameters; also, for i � 1, . . . , 4, ∆Λ i � P i L i (t)Q i shows uncertainty terms of the system, defning them as the following matrixes.
In order to demonstrate the practicability of the suggested TSFSMC strategy and the correctness of the proposed stabilization condition, the system parameters are arbitrarily 8. In addition, it should be mentioned that the controller starts operating after t � 3 seconds. Figure 3 is displayed in order to demonstrate how T-S fuzzy FO nonlinear systems Lu and Chen are synchronized in equations (44) and (45). In addition, the error structure among T-S fuzzy FO chaotic systems (44) and (45), which has approximated the value of zero, is depicted in Figure 4. Te synchronization between two distinct chaotic FO systems (44) and (45) is clearly visible, as can be seen from the pictures in the article. In addition, Figure 5 illustrates the saturated control signals (37) and Figure 6 shows the sliding surface (23), respectively. Based on Figure 5, there are no indications of the chattering occurrence in the controller impulses at all.
In addition, as shown in Figure 5, the saturation condition suppresses the control laws as they near the saturation limits, resulting in the leaping occurrence. Terefore, transitioning and leaping states are easily applied, particularly when switches and predetermined saturation conditions are employed. From Figure 6, one can observe that the sliding surface (23) is approached to its origin. Tis means that the proposed TSFSMC can successfully synchronize the T-S fuzzy FO chaotic systems, Lu and Chen, in equations (44) and (45). Now, in order to compare the TSFSMC technique, the FO chaotic systems Lu and Chen are synchronized with β � 0.92 by utilizing an adaptive controller suggested in [58]. Research by Luo et al. has resulted in the development of an adaptive control strategy, which has recently been used to the synchronizing of FO systems. In this context, the operating of the control technique in [58] may be seen as follows: After operating the designed adaptive control approach (49a) and (49b), Figure 7 displays the state trajectories of the error between equations (44) and (45), synchronized via both of the controllers (37) and (49a) and (49b). As can be observed, even though the error states converges to zero, but the suggested TSFSMC (37) has a better convergence in comparison with the adaptive method (49a) and (49b).

Scenario 2.
In this part, the synchronization of two FO electrical motors including the permanent magnet synchronous motor (PMSM) and brushless DC motor (BLDCM) systems, which exhibit chaotic behaviors for β � 0.99, illustrates the efectiveness of the designed TSFSMC (37). (20).
Te following introduces FO chaotic PMSM system's dynamics as a driving system [59].
Let e 3 (t) ∈ (− d, d) and d � 20, the following equations are used to determine the membership functions in fuzzy structure in order to synchronize the FO PMSM and FO BLDCM: (52) Ten, the fuzzy relations will be achieved as follows: 10 International Journal of Intelligent Systems  (44) and (45). International Journal of Intelligent Systems where Also, Λ 1 , Λ 2 , Λ 3 , and Λ 4 elucidate the system's unidentifed parameters and characteristics. Furthermore, for i � 1, . . . , 4, ∆Λ i � P i L i (t)Q i indicates uncertainty terms of the systems, introduced as follows:  (44) and (45).  (44) and (45).
In order to demonstrate the practicability of the suggested TSFSMC strategy and the correctness of the proposed stabilization condition, the system parameters are arbitrarily  Figure 8 is shown to demonstrate the synchronization in the T-S fuzzy FO chaotic systems proposed in equations (50) and (51). Ten, the error function in the T-S fuzzy FO chaotic systems developed by PMSM and BLDCM (50) and (51), which is shown in Figure 9, has also almost reached zero. In addition, Figures 10 and 11, respectively, display the saturated control signals (37) and sliding surface (23), respectively. Figure 10 shows that the control signals show no Proposed TSFSMC Existed method in [58] Proposed TSFSMC Existed method in [58] Proposed TSFSMC Existed method in [58] Figure 7: Comparison of the FO error systems (44) and (45), controlled with designed TSFSMC (37) and active control method (49a) and (49b) in [58].  (50) and (51). evidence of the chattering phenomena. In addition, the saturation condition suppresses the control laws as they approach the saturation boundaries, resulting in the leaping phenomena. Terefore, switching and jumping states are easily applicable, especially when relays and predetermined saturation conditions are employed. In Figure 11, the sliding surface (23), which is approaching its origin, can be seen. Tis suggests that the T-S fuzzy FO chaotic systems PMSM and BLDCM in equations (50) and (51) can be successfully synchronized using the proposed TSFSMC (37).

Conclusion
In this research, a dynamic-free T-S fuzzy sliding mode control approach is proposed for the purpose of synchronizing the various chaotic fractional-order systems in the presence of input saturation. A Takagi-Sugeno fuzzy sliding mode controller is driven to suppress and synchronize the undesirable behavior of FO chaotic systems without any unpleasant chattering phenomenon by utilizing a new defnition of fractional calculus, the fractional version of the Lyapunov stability theorem, and the linear matrix inequality concept. Te design of this method is completely independent of the system's dynamics, and it can synchronize a variety of FO chaotic systems despite the presence of model errors, external disturbances, and input saturation. An example of the synchronization of two diferent realworld application chaotic systems is illustrated to show the applicability of the theoretical result presented in the paper. Also, for future work, the problem of synchronizing FO  (50) and (51). delayed chaotic systems in a fnite time can be considered. Also, for fnding the best parameters of the controller, the machine/deep learning method can be utilized to select optimal control parameters. Moreover, work on the unlimited nature of β is an essential issue. Finally, experimental results will be added to increase the quality and impact of the studies.

Data Availability
No data were used to support the fndings of this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.