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

Authors: Yuting Li
Journal: Complexity
 (2020)

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## Abstract

This 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. Then, a sufficient condition is given in the format of linear matrix inequalities. Simulation example is given to validate the effectiveness of the approach.


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## Body

## 1. Introduction

Fractional order systems have attracted more and more attention due to its demonstrated application in many fields ([1–4]). 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 [5–7]. Control problem for fractional order multiagent systems were introduced in [8–10]. Dynamic properties, control, and synchronization of fractional order chaotic systems were discussed in [11–14].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 ([15–17]). 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]. Then, 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]. The problem of robust stabilization for positive fractional order interconnected systems with heterogeneous time-varying delays was proposed by [21]. The robust decentralized fault-tolerant resilient control for fractional order large-scale interconnected uncertain system was investigated in [22]. There 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 [23–25] and [26]. T-S fuzzy controllers for nonlinear multiple time-delay interconnected systems were studied in [27–29]. The 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 [32–34]. Stability and stabilization problem for T-S Interconnected Fuzzy Systems by using different Lyapunov functions with slack variables are considered in [23]. The 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.This paper is organized as follows. Some preliminaries and the problem formulation are introduced first. The main results on the sufficient conditions of stabilization of nonlinear fractional order interconnected system are derived in Section3. Section 4 interconnected fractional order chaotic systems to illustrate the effectiveness of the proposed results. Finally, the conclusions are drawn in Section 5.Notations. The transpose of a matrixA is denoted by AT. SymA is used to denote the expression AT+A and ∗ will be used in some expression to indicate a symmetric, i.e., XY∗Z=XYYTZ.


## 2. Preliminaries and Problem Formulation

Consider a fractional order nonlinear interconnected system composed ofJ fractional order subsystems Nj, j=1,…,J. The jth fractional order subsystem Nj is described as follows:(1)Dαxjt=fjxjt,ujt+Δfjxjt,ujt+∑n=1,n≠jJbnjxnt,where 0<α≤1 is the fractional commensurate order, fj⋅ is the nonlinear vector-valued function, Δfj⋅ is the system uncertainties, bnj is the nonlinear interconnection between the nth and jth subsystems, xjt is the state vector, and ujt is the input vector of the jth fractional order subsystem, respectively. The operator Dα denotes Dtαt0C.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 systemsN. The ith rule of the fuzzy model for the fractional order nonlinear interconnected subsystem Nj is proposed as follows:Plant Rulei:Ifz1jt is Mi1j and … and zpjt is MipjThenDαxjt=Aij+ΔAijxjt+∑n=1,n≠jJA^injxnt+Bij+ΔBijujtwherei=1,…,rj, rj is the number of IF-THEN rules, Mihjh=1,2,…,p are the fuzzy sets, and z1jt,…,zpjt are the premise variables. Aij, A^inj, and Bij are constant matrices with appropriate dimension, while ΔAij and ΔBij are real-valued function matrices representing the time-varying parameter uncertainties that have the following form:(2)ΔAijΔBij=DAijFAijtEAijDBijFBijtEBij,where DAij, DBij, EAij, EBij are known constant matrices, and FAijt, FBijt, are unknown matrices with Lebesgue measurable elements satisfying FAijTtFAijt≤I, FBijTtFBijt≤I. The final state of the fractional order fuzzy model is inferred as follows:(3)Dαxjt=∑i=1rjhijtAij+ΔAijxjt+∑n=1,n≠jJA^injxnt+Bij+ΔBijujt,where hijt=wijt/∑i=1rjwijt, wijt=∏q=1pMiqjzqjt, Miqjzqjt is the grade of the membership of zqjt in Miqj. Notice the facts wijt≥0 for i=1,2,…,rj and ∑i=1rjwijt>0 for all t. Therefore, hijt≥0 for i=1,2,…,rj and ∑i=1rjhijt=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 systemsN. The jth model-based fuzzy controller isControl rulei:IFz1jt is Mi1j and … and zpjt is Mipj,THENujt=Kijxjt.wherei=1,2,…,rj.Hence, the final output of the fuzzy controller has the form(4)ujt=∑i=1rjhijtKijxjt,i=1,2,…,rj.Substituting (4) into equation (3) yields the jth closed-loop subsystem as follows:(5)Dαxjt=∑i=1rj∑l=1rjhijthljtAij+ΔAij+Bij+ΔBijKljxjt+∑n=1,n≠jJA^injxnt.Lemma 1.
(see [37]). Let x=0 be an equilibrium point for the nonautonomous fractional order system Dαxt=ft,x. Let us assume that there exist a continuous Lyapunov function Vxt,t and a scalar class-K function γ1⋅ such that ∀x≠0(6)γ1xt≤Vxt,t,DαVxt,t≤0,withα∈0,1,then the origin of the system is Lyapunov stable.Lemma 2.
(see [37]). Let xt∈Rn be a vector of differentiable functions. Then, for any time instant t≥t0, the following relationship holds:(7)12DαxTtPxt≤xTtPDαxt,∀α∈0,1,∀t≥t0,where P∈Rn×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 XTY+YTX≤σXTX+σ−1YTY for any σ>0.Lemma 4.
(see [40]). Given matrices T, Π, Nt, and M of appropriate dimensions and with M symmetrical, then M+TNtΠ+ΠTNTtTT<0 holds for any Nt satisfying NTtNt≤I if and only if there exists ε>0, such that M+εTTT+ε−1ΠTΠ<0.Lemma 5.
(Schur Complement), see [41]). For a given matrix S=ST, the following assertions are equivalent:(8)1S=S11S12S21S22<0;2S11<0,S22−S12TS11−1S12<0;3S22<0,S11−S12S22−1S12T<0.


## 3. Main Results

In this section, the stability of the fractional order nonlinear interconnected systemN is studied. A sufficient condition is established for system (5). Then the following theorem presents the main result.Theorem 1.
The closed-loop fractional order nonlinear interconnected system (5) is asymptotically stable if there are symmetric positive definite matrices Qjj=1,…,J, matrices Wiji=1,2,…,rj, and real scalar constants εij, ηij, δij, μlj, ρlj and μ¯ such that(9)ΛijQjEAijTWijTEBijTQj∗−εijI00∗∗−ηijI0∗∗∗−μ¯J−1−1I<0,j=1,…,J,i=1,…,rj,where(10)Λij=SymAijQj+BijWij+εijDAijDAijT+ηijDBijDBijT+∑n=1Jμ¯A^injA^injT,(11)ΛiljQjEAijTWljTEBijTQjEAljTWijTEBljTQj∗−γijI0000∗∗−δijI000∗∗∗−μljI00∗∗∗∗−ρljI0∗∗∗∗∗−μ¯J−1−1I<0,1≤i<l≤rj,where(12)Λilj=12SymAijQj+BijWlj+AljQj+BljWij+γijDAijDAijT+δijDBijDBijT+μljDAljDAljT+ρljDBljDBljT+∑n=1Jμ¯A^injA^injT.
The asymptotically stabilizing state feedback gain matrix isKij=WijQj−1.Proof.
Let the Lyapunov function for the fractional order interconnected systemN be defined as(13)Vt=∑j=1Jvjt=∑j=1J2xjTtPjxjt.
P
j is real symmetric positive definite matrix. It follows from Lemma 1, the closed-loop fractional order nonlinear interconnected system (5) is asymptotically stable if DαVxt,t≤0. Note that(14)DαVt=∑j=1JDαvjt.
Applying Lemma2 to Dαvjt, it can be obtained that(15)DαVt≤∑j=1JDαxjtTPjxjt+xjTtPjDαxjt.
Right side of inequality (15) can be represented by(16)∑j=1JxjTtPj∑i=1rj∑l=1rjhijthljtAij+ΔAij+Bij+ΔBijKljxjt+∑n=1,n≠jJA^injxnt+∑i=1rj∑l=1rjhijthljtAij+ΔAij+Bij+ΔBijKljxjt+∑n=1,n≠jJA^injxntTPjxj=∑j=1J∑i=1rj∑l=1rjhijthljtxjTtPjAij+ΔAij+Bij+ΔBijKlj+Aij+ΔAij+Bij+ΔBijKljTPjxjt+∑j=1J∑i=1rj∑l=1rj∑n=1,n≠jJhijthljtxjTtPjA^injxn+xnTA^injTPjxj.
By applying Lemma3, it can be obtained(17)DαVt≤∑j=1J∑i=1rj∑l=1rjhijthljtxjTtSymPjAij+ΔAij+Bij+ΔBijKljxjt+∑j=1J∑i=1rj∑n=1JhijtμJ−1JxjTtxjt+μ−1xjTtPjA^injA^injTPjxj−1JxjTPjA^ijjA^ijjTPjxj.
In view of the matrixA^ijj is equal to zero and ∑l=1rjhlj=1, we have(18)∑j=1J∑i=1rj∑l=1rjhijthljtxjTtSymPjAij+ΔAij+Bij+ΔBijKlj+∑n=1Jμ−1PjA^injA^injTPj+μJ−1Ixjt=∑j=1J∑i=1rjhij2txjTtSymPjAij+ΔAij+Bij+ΔBijKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1Ixjt+2∑j=1J∑i<lrjhijthljtxjTt12SymPjAij+ΔAij+Bij+ΔBijKlj+Alj+ΔAlj+Blj+ΔBljKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1Ixjt<0.
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:(19)SymPjAij+ΔAij+Bij+ΔBijKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1I<0.
Equation (19) can be represented by(20)SymPjAij+BijKij+SymPjDAijFAijtEAij+SymPjDBijFBijtEBijKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1I<0.
By applying Lemma3 to (20), one obtains (20) holds if and only if there exist εij and ηij such that(21)SymPjAij+BijKij+εijPjDAijDAijTPj+εij−1EAijTEAij+ηijPjDBijDBijTPj+ηij−1EBijKijTEBijKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1I<0.
By the Schur complement, we can get(22)ΩijEAijTEBijKijTI∗−εijI00∗∗−ηijI0∗∗∗−μ−1J−1−1I<0,j=1,…,J,i=1,…,rj,where(23)Ωij=SymPjAij+BijKij+εijPjDAijDAijTPj+ηijPjDBijDBijTPj+∑n=1Jμ−1PjA^injA^injTPj.
Define transformation matrix asdiagPj−1III and take a congruence transformation to (22); this yields(24)Pj−1ΩijPj−1Pj−1EAijTPj−1EBijKijTPj−1I∗−εijI00∗∗−ηijI0∗∗∗−μ−1J−1−1I<0.
DenotingQj=Pj−1, Wij=KijPj−1, and μ¯=μ−1, we have(25)ΛijQjEAijTWijTEBijTQj∗−εijI00∗∗−ηijI0∗∗∗−μ¯J−1−1I<0,j=1,…,J,i=1,…,rj,where(26)Λij=SymAijQj+BijWij+εijDAijDAijT+ηijDBijDBijT+∑n=1Jμ¯A^injA^injT.
The second LMI (11) can be established through a similar procedure. Assume that the second sum of the last equation in (18) is negative definite:(27)12SymPjAij+ΔAij+Bij+ΔBijKlj+Alj+ΔAlj+Blj+ΔBljKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1I<0.
Equation (27) can be represented by(28)12SymPjAij+BijKlj+Alj+BljKij+SymPjDAijFAijt2EAij+SymPjDBijFBijt2EBijKlj+SymPjDAljFAljt2EAlj+SymPjDBljFBljt2EBljKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1I<0.
By applying Lemma3 to (28), one obtains (28) holds if and only if there exist γij, δij, ρij and μij, such that(29)12SymPjAij+BijKlj+Alj+BljKij+γijPjDAijDAijTPj+γij−1EAijTEAij+δijPjDBijDBijTPj+δij−1EBijKljTEBijKlj+μljPjDAljDAljTPj+μlj−1EAljTEAlj+ρljPjDBljDBljTPj+ρlj−1EBljKijTEBljKij+∑n=1Jμ−1PjA^injA^injTPj+μJ−1I<0.
By the Schur complement, we can get(30)ΩiljEAijTEBijKljTEAljTEBljKijTI∗−γijI0000∗∗−δijI000∗∗∗−μljI00∗∗∗∗−ρljI0∗∗∗∗∗−μ−1J−1−1I<0,where(31)Ωilj=12SymPjAij+BijKlj+Alj+BljKij+γijPjDAijDAijTPj+δijPjDBijDBijTPj+μljPjDAljDAljTPj+ρljPjDBljDBljTPj+∑n=1Jμ−1PjA^injA^injTPj.
Define transformation matrix asdiagPj−1IIIII and take a congruence transformation to (30); this yields(32)Pj−1ΩiljPj−1Pj−1EAijTPj−1EBijKljTPj−1EAljTPj−1EBljKijTPj−1I∗−γijI0000∗∗−δijI000∗∗∗−μljI00∗∗∗∗−ρljI0∗∗∗∗∗−μ¯J−1−1I<0,we have(33)ΛiljQjEAijTWljTEBijTQjEAljTWijTEBljTQj∗−γijI0000∗∗−δijI000∗∗∗−μljI00∗∗∗∗−ρljI0∗∗∗∗∗−μ¯J−1−1I<0,1≤i<l≤rj,where(34)Λilj=12SymAijQj+BijWlj+AljQj+BljWij+γijDAijDAijT+δijDBijDBijT+μljDAljDAljT+ρljDBljDBljT+∑n=1Jμ¯A^injA^injT.
This completes the proof.WhenΔAij and ΔBij are 0, it is easy to get the following Corollary.Corollary 1.
The closed-loop fractional order nonlinear interconnected system (5) is asymptotically stable if there are symmetric positive definite matrices Qjj=1,…,J, matrices Wiji=1,2,…,rj, and a real scalar constant μ¯ such that(35)ΛijQj∗−μ¯J−1−1I<0,i=1,…,rj,j=1,…,J,with(36)Λij=SymAijQj+BijWij+∑n=1Jμ¯A^injA^injT,ΛiljQj∗−μ¯J−1−1I<0,1≤i<l≤rj,with(37)Λilj=12SymAijQj+BijWlj+AljQj+BljWij+∑n=1Jμ¯A^injA^injT.
The asymptotically stabilizing state feedback gain matrix isKij=WijQj−1.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 rulesr1,…,rj is one, our model can be applied to solving fractional order linear interconnected system with uncertainties.


## 4. 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:(38)Dαx1t=−a+△ax1t+a+△ax2t,Dαx2t=c+△cx1t−x2t−x1tx3t,Dαx3t=x1tx2t−bx3t.And whena=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.Figure 1
Chaotic behaviors of the fractional order uncertain Lorenz system.Let us consider two interconnected fractional order uncertain Lorenz chaotic system as follows:Subsystem 1:(39)Dαx11t=−a+△ax11t+a+△ax21t+x12t+x22t+u1t,Dαx21t=c+△cx11t−x21t−x11tx31t+x12t+x32t,Dαx31t=x11tx21t−bx31t.Subsystem 2:(40)Dαx12t=−a+△ax12t+a+△ax22t+x11t+x21t+u2t,Dαx22t=c+△cx12t−x22t−x12tx32t+x11t+x31t,Dαx32t=x12tx22t−bx32t.The state curves Nonlinear Fractional Order Interconnected Systems that without control, i.e.,u1≡0,u2≡0 are shown in Figures 2 and 3.Figure 2
The state of subsystem 1 in nonlinear fractional order interconnected systems.Figure 3
The state of subsystem 2 in nonlinear fractional order interconnected systems.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 thatx1jt∈−d,d and d>0, d=30, then we have
The fuzzy model of Subsystemj:Rule 1: IFx1jt is M1j, THENDαxjt=A1j+△A1jxjt+A^1ljxlt+B1jujt,Rule 2: IFx1jt is M2j, THENDαxjt=A2j+ΔA2jxjt+A^2ljxlt+B2jujt.wherexjTt=x1jtx2jtx3jt, l≠j and j=1,2.(41)A1j=−aa0c−1−d0d−b,A2j=−aa0c−1d0−d−b,A^1lj=A^2lj=110101000,ΔA1j=ΔA2j=−ΔaΔa0△c00000,B1j=B2j=100.
HereΔA1j and ΔA2j can be represented by(42)−0.10000.0050000sint0001000110−1002800000,then we have DA1j=DA2j=−0.10000.0050000, EA1j=EA2j=10−1002800000, and DB1j=DB2j=000000000, EB1j=EB2j=000 and the membership functions for Rule 1 and Rule 2 are: M1jx1jt=1/21+x1jt/d, M2jx1jt=1/21−x1jt/d.
Fuzzy controller of Subsystemj:Rule 1: IFx1jt is M1j, THEN ujt=K1jxjt,Rule 2: IFx1jt is M2j, THEN ujt=K2jxjt.The final output of the fuzzy controller is ujt=∑i=12hijtKijxjt.Step 2.
By applying Theorem1 and using packages YALMIP in Matlab, we find the LMI (9) and (11) in Theorem 1 is feasible, a feasible solution is as follows:(43)K11=−7.4134−17.37623.7393,K21=−9.6733−20.59940.8517,K12=−10.7060−21.34315.6562,K22=−13.8411−25.51201.4921.
Therefore, the fractional order nonlinear fractional order interconnected system under fuzzy control law is determined to be asymptotically stable. The simulation results of each subsystem states under control are illustrated in Figures4 and 5 shows that it is asymptotically stable, and the control curve of system is shown in Figures 6 and 7.Figure 4
Control results of the fractional order nonlinear interconnected system.Figure 5
Control results of the fractional order nonlinear interconnected system.Figure 6
Control curveu1t of the fractional order interconnected system.Figure 7
Control curveu2t of the fractional order interconnected system.


## 5. Conclusion

This paper focuses on the stability of the nonlinear fractional order interconnected systems. A useful stabilization approach has been given. The basis of this approach is to apply fractional order uncertain T-S fuzzy model to nonlinear fractional order interconnected systems. The 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.



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