Robust Stabilization of Nonlinear Fractional Order Interconnected Systems Based on T-S Fuzzy Model

*is paper concerns robust stabilization of nonlinear fractional order interconnected systems. Based on uncertain fractional order Takagi–Sugeno fuzzy model and the fractional order extension of lyapunov direct method, a parallel distributed compensate controller is designed to asymptotically stabilize the fractional order interconnected systems. *en, a sufficient condition is given in the format of linear matrix inequalities. Simulation example is given to validate the effectiveness of the approach.


Introduction
Fractional order systems have attracted more and more attention due to its demonstrated application in many fields ( [1][2][3][4]). Recently, a considerable literature has grown up around the theme of fractional order systems. For example, the problem of state estimation and synchronization for fractional order neural networks was discussed in [5][6][7]. Control problem for fractional order multiagent systems were introduced in [8][9][10]. Dynamic properties, control, and synchronization of fractional order chaotic systems were discussed in [11][12][13][14].
Large-scale interconnected system consists of a number of independent subsystems connected by some interconnections. Because interconnected systems are efficiently applied to practical systems such as economic systems, computer communication networks, and transportation systems, a considerable amount of literature has been published ( [15][16][17]). Much of the previous research has focused on integer order. In 2013, a class of fractional order linear interconnected systems' stabilization problem was considered [18].
en, the problem of robust resilient controllers synthesis for uncertain fractional order linear interconnected system was studied, and the state feedback nonfragile controller was designed under the additive and multiplicative gain perturbations [19]. Positive reduced-order functional observers for positive fractional order interconnected time-delay systems were designed by [20]. e problem of robust stabilization for positive fractional order interconnected systems with heterogeneous timevarying delays was proposed by [21]. e robust decentralized fault-tolerant resilient control for fractional order large-scale interconnected uncertain system was investigated in [22]. ere are relatively few historical studies in the area of fractional order nonlinear interconnected systems.
Takagi-Sugeno (T-S) fuzzy model is one of the most common effective methods for approximating complex nonlinear systems. Over the past several years, there has been rapid development of interconnected systems by using T-S fuzzy model. For example, fuzzy large-scale interconnected systems were discussed in [23][24][25] and [26]. T-S fuzzy controllers for nonlinear multiple time-delay interconnected systems were studied in [27][28][29]. e finite-time stabilization problem for type-2 T-S interconnected nonlinear systems was investigated in [30,31]. H ∞ control design for fuzzy discrete-time interconnected systems based on T-S fuzzy model was studied in [32][33][34]. Stability and stabilization problem for T-S Interconnected Fuzzy Systems by using different Lyapunov functions with slack variables are considered in [23]. e LMI stability conditions of fractional order uncertain T-S system were introduced in [35,36]. In this paper, we study the stability problem of nonlinear fractional order interconnected systems based on T-S fuzzy model and the fractional order extension of Lyapunov direct method. is paper is organized as follows. Some preliminaries and the problem formulation are introduced first. e main results on the sufficient conditions of stabilization of nonlinear fractional order interconnected system are derived in Section 3. Section 4 interconnected fractional order chaotic systems to illustrate the effectiveness of the proposed results. Finally, the conclusions are drawn in Section 5.
Notations. e transpose of a matrix A is denoted by A T . Sym A { } is used to denote the expression A T + A and * will be used in some expression to indicate a symmetric, i.e.,

Preliminaries and Problem Formulation
Consider a fractional order nonlinear interconnected system composed of J fractional order subsystems N j , j � 1, . . . , J. e jth fractional order subsystem N j is described as follows: where 0 < α ≤ 1 is the fractional commensurate order, f j (·) is the nonlinear vector-valued function, Δf j (·) is the system uncertainties, b nj is the nonlinear interconnection between the nth and jth subsystems, x j (t) is the state vector, and u j (t) is the input vector of the jth fractional order subsystem, respectively. e operator D α denotes C t 0 D α t . A set of fractional order T-S fuzzy model is employed here to deal with the control design problem of the fractional order nonlinear interconnected systems N. e ith rule of the fuzzy model for the fractional order nonlinear interconnected subsystem N j is proposed as follows: are the fuzzy sets, and z 1j (t), . . . , z pj (t) are the premise variables. A ij , A inj , and B ij are constant matrices with appropriate dimension, while ΔA ij and ΔB ij are real-valued function matrices representing the time-varying parameter uncertainties that have the following form: where D Aij , D Bij , E Aij , E Bij are known constant matrices, and F Aij (t), F Bij (t), are unknown matrices with Lebesgue measurable elements satisfying F T Aij (t)F Aij (t) ≤ I, F T Bij (t)F Bij (t) ≤ I. e final state of the fractional order fuzzy model is inferred as follows: where is the grade of the membership of z qj (t) in M iqj . Notice the facts w ij (t) ≥ 0 for i � 1, 2, . . . , r j and erefore, h ij (t) ≥ 0 for i � 1, 2, . . . , r j and r j i�1 h ij (t) � 1. According to the decentralized fuzzy control scheme, a set of fuzzy controllers is synthesized via the parallel distributed compensation (PDC) to deal with the stabilization control for the fractional order nonlinear interconnected systems N. e j th model-based fuzzy controller is Control rule i: where i � 1, 2, . . . , r j . Hence, the final output of the fuzzy controller has the form Substituting (4) into equation (3) yields the jth closedloop subsystem as follows: Lemma 1 (see [37]). Let x � 0 be an equilibrium point for the nonautonomous fractional order system

x). Let us assume that there exist a continuous Lyapunov function V(x(t), t) and a scalar class-K function
then the origin of the system is Lyapunov stable.
Lemma 2 (see [37]). Let x(t) ∈ R n be a vector of differentiable functions. en, for any time instant t ≥ t 0 , the following relationship holds:

Complexity
where P ∈ R n×n is a constant, square, symmetric, and positive matrix.
Lemma 3 (see [38,39]). For any matrices X and Y with appropriate dimensions, we have Lemma 4 (see [40]). Given matrices T, Π, N(t), and M of appropriate dimensions and with M symmetrical, then M + Lemma 5 (Schur Complement), see [41]). For a given matrix S � S T , the following assertions are equivalent:

Main Results
In this section, the stability of the fractional order nonlinear interconnected system N is studied. A sufficient condition is established for system (5). en the following theorem presents the main result.

Theorem 1.
e closed-loop fractional order nonlinear interconnected system (5) is asymptotically stable if there are symmetric positive definite matrices Q j (j � 1, . . . , J), matrices W ij (i � 1, 2, . . . , r j ), and real scalar constants ε ij , η ij , δ ij , μ lj , ρ lj and μ such that where where The asymptotically stabilizing state feedback gain matrix Proof. Let the Lyapunov function for the fractional order interconnected system N be defined as P j is real symmetric positive definite matrix. It follows from Lemma 1, the closed-loop fractional order nonlinear interconnected system (5) Applying Lemma 2 to D α v j (t), it can be obtained that Complexity 3 Right side of inequality (15) can be represented by By applying Lemma 3, it can be obtained In view of the matrix A ijj is equal to zero and

Complexity
If it is possible to assume each sum of (18) to be negative definite, respectively, then the fractional order nonlinear interconnected system is asymptotically stable.
First, assume that the first sum of the last equation in (18) is negative definite: Equation (19) can be represented by By applying Lemma 3 to (20), one obtains (20) holds if and only if there exist ε ij and η ij such that By the Schur complement, we can get where Define transformation matrix as diag P − 1 j I I I and take a congruence transformation to (22); this yields Denoting Q j � P − 1 j , W ij � K ij P − 1 j , and μ � μ − 1 , we have where e second LMI (11) can be established through a similar procedure. Assume that the second sum of the last equation in (18) is negative definite: Equation (27) can be represented by By applying Lemma 3 to (28), one obtains (28) holds if and only if there exist c ij , δ ij , ρ ij and μ ij , such that Complexity 5 By the Schur complement, we can get where Ω ilj � 1 2 Sym P j A ij + B ij K lj + A lj + B lj K ij + c ij P j D Aij D T Aij P j + δ ij P j D Bij D T Bij P j + μ lj P j D Alj D T Alj P j + ρ lj P j D Blj D T Blj P j Define transformation matrix as diag P − 1 j I I I I I and take a congruence transformation to (30); this yields 6 Complexity we have is completes the proof.

□
When ΔA ij and ΔB ij are 0, it is easy to get the following Corollary.

Corollary 1.
e closed-loop fractional order nonlinear interconnected system (5) is asymptotically stable if there are symmetric positive definite matrices Q j (j � 1, . . . , J), matrices W ij (i � 1, 2, . . . , r j ), and a real scalar constant μ such that with with The asymptotically stabilizing state feedback gain matrix is K ij � W ij Q − 1 j .

Remark 1.
Since T-S fuzzy system can effectively approximate complex systems with nonlinearity, our model can be applied to a broad class of nonlinear fractional order interconnected systems. Most of all, stabilization of system can be developed by solving a set of LMIs. Moreover, when the number of the rules r 1 , . . . , r j is one, our model can be applied to solving fractional order linear interconnected system with uncertainties.

Numerical Examples
In this part, in order to show the effectiveness of the proposed method, a numerical example on interconnected fractional order chaotic systems will be provided. Consider the asymptotical stability of nonlinear fractional order interconnected systems, and each subsystem is fractional order uncertain Lorenz chaotic system: And when a � 10, △a � sint, b � 8/3, c � 28, △c � 0.14, and α � 0.993, chaotic behaviors of the fractional order uncertain Lorenz chaotic system are shown in Figure 1.
Let us consider two interconnected fractional order uncertain Lorenz chaotic system as follows:     e state curves Nonlinear Fractional Order Interconnected Systems that without control, i.e., u 1 ≡ 0, u 2 ≡ 0 are shown in Figures 2 and 3.
Step 1. To stabilize the above fractional order interconnected system, we firstly establish fractional order T-S fuzzy model for each nonlinear fractional order subsystem. Assume that x 1j (t) ∈ [− d, d] and d > 0, d � 30, then we have e fuzzy model of Subsystem j: , l ≠ j and j � 1, 2.
Here ΔA 1j and ΔA 2j can be represented by − 0.1 0 0 0 0.005 0 Fuzzy controller of Subsystem j: e final output of the fuzzy controller is u j (t) � 2 i�1 h ij (t)K ij x j (t).
Step 2. By applying eorem 1 and using packages YALMIP in Matlab, we find the LMI (9) and (11) in eorem 1 is feasible, a feasible solution is as follows: erefore, the fractional order nonlinear fractional order interconnected system under fuzzy control law is determined to be asymptotically stable. e simulation results of each subsystem states under control are illustrated in   10 Complexity Figures 4 and 5 shows that it is asymptotically stable, and the control curve of system is shown in Figures 6 and 7.

Conclusion
is paper focuses on the stability of the nonlinear fractional order interconnected systems. A useful stabilization approach has been given. e basis of this approach is to apply fractional order uncertain T-S fuzzy model to nonlinear fractional order interconnected systems. e PDC control design is carried out based on the fractional order T-S fuzzy model and the fractional order extension of Lyapunov direct method, a sufficient condition was given in terms of LMI. Finally, nonlinear fractional order interconnected systems was given to illustrate the effectiveness of the proposed theoretical results.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

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