# Sliding Mode Matrix-Projective Synchronization for Fractional-Order Neural Networks

Authors: Jinman He; Tengfei Lei; Limin Jiang
Journal: Journal of Mathematics
 (2021)

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## Abstract

This work generalizes the projection scaling factor to a general constant matrix and proposes the matrix-projection synchronization (MPS) for fractional-order neural networks (FNNs) based on sliding mode control firstly. This kind of scaling factor is far more complex than the constant scaling factor, and it is highly variable and difficult to predict in the process of realizing the synchronization for the driving and response systems, which can ensure high security and strong confidentiality. Then, the fractional-order integral sliding surface and sliding mode controller for FNNs are designed. Furthermore, the criterion for realizing MPS is proved, and the reachability and stability of the synchronization error system are analyzed, so that the global MPS is realized for FNNs. Finally, a numerical application is given to demonstrate the feasibility of theory analysis. MPS is more general, so it is reduced to antisynchronization, complete synchronization, projective synchronization (PS), and modified PS when selecting different projective matrices. This work will enrich the synchronization theory of FNNs and provide a feasible method to study the MPS of other fractional-order dynamical models.


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## Body

## 1. Introduction

Neural network is an important part of artificial intelligence, which is composed of a large number of highly connected neurons. It is a mathematical model based on the preliminary understanding of the physiological structure and activity mechanism of the brain. The neural network has the unique knowledge representation structure and can process information, learn, and adapt to the unknown system efficiently and quickly, which provide new research ideas for control problems and intelligent information processing. Integer order differential equations cannot describe the memory properties of neurons and the dependence on past history, but the fractional-order calculus [1–4], which has strong memory and hereditary characteristic, contains all of the information from the start point to the current moment and can describe the memory properties and dynamical behaviors of neurons more accurately. Therefore, FNNs can improve the computational ability of neurons, speed up the information transmission of neurons, and solve the problem of parameter identification effectively. With the development of fractional-order calculus, FNNs are becoming more and more popular and their dynamical behaviors have been widely investigated, such as stability [5–8], bifurcations [9], chaos and hyperchaos [10], and synchronization [11–14].Two systems are called PS when the drive and response systems are synchronized to a scaling factor. Recently, scholars have researched the PS of FNNs and achieved a lot of valuable results. In [15–17], by using different control methods, the authors have researched the PS for FNNs. In [18], Zhang et al. have used the adaptive control method to achieve the PS of FNNs in quaternion field. In [19–21], researchers have explored the PS for fractional-order memristive neural networks with different characters. In [22–24], the authors extend the FNNs to the complex domain and study their PS and quasiprojective synchronization by using different control strategies. In [25], Ding and Shen constructed the fractional-order integral sliding mode surface, designed fractional-order sliding mode controller, and realized PS for two FNNs with different structures. In [26, 27], by means of the fractional Lyapunov-like method, the authors realized PS in finite-time and mixed H∞/passive projective synchronization for nonidentical FNNs via a new sliding mode controller. In [28], by using sliding mode control, Wu et al. realized the finite-time interlayer PS of fractional-order two-layer networks based on Caputo derivates.To the best of our knowledge, the scaling factor of PS in most research studies is a diagonal matrix or a fixed constant, but in fact this kind of scaling factor may not ensure high security of communication. Our work will generalize the proportion factor to a general constant matrix, which is far more complex than the constant scaling factor. Also, it is highly variable and difficult to predict in the process of realizing the synchronization for the driving and response systems, which can ensure the high security and strong confidentiality. Based on the characteristics of matrix scaling factor, our work presents a new kind of MPS for FNNs, whose complexity and unpredictability can effectively increase the difficulty for hackers to track the right path, improve the antiattack capability of the system, and enhance the confidentiality of secure communication.As everyone knows, synchronization of FNNs can be reached via various control methods. Especially, sliding mode control strategy has many important and special advantages, which include low sensitivity to the parameter perturbation and external disturbance, implementation simplicity, and fast response. In our work, because of the complexity of the MPS and nonidentical FNNs, it greatly increases the difficulty of control in actual operation. Hence, it is extremely necessary to use fractional-order sliding mode control strategy to research the global MPS.According to the above discussion and main research content, our work is divided into the following chapters. Section2 introduces some lemmas and establishes the FNNs. In Section 3, by using the fractional-order sliding mode control method, MPS is defined and sufficient criterion is proved. As applications, a numerical application is given to demonstrate theory analysis in Section 4. Section 5 concludes the whole work.


## 2. Preliminaries

Definition 1.
(see [1]). Fractional integral for function f:t0,+∞⟶R is(1)Itαt0ft=1Γα∫t0tt−ξα−1fξdξ,α>0,t≥t0,where Γα=∫0+∞e−ttα−1dt is gamma function.Definition 2.
(see [1]). Caputo derivative for a function ft∈Cnt0,+∞,R is defined as(2)Dtαt0Cft=1Γn−α∫t0tfnξt−ξα−n+1dξ,α>0,t≥t0,where n is the positive integer satisfying n−1<α<n, fn is the n-th-order derivative of fx, and Cnt0,+∞,R is the space which is composed of n order continuous differentiable functions from t0,+∞ to R. In particular, when 0<α<1, Dtαt0Cft=1/Γ1−α∫t0tf′ξ/t−ξαdξ; when α=1, the Caputo derivative operation Dtαt0Cft is identified with integer order ones dft/dt.Lemma 1.
If Caputo derivativeDtαt0Cft is integrable, then(3)Itαt0CDtαt0Cft=ft−∑k=0n−1fkt0k!t−t0k.
Especially, for0<α<1, one can obtain Itαt0CDtαt0Cft=ft−ft0.Lemma 2.
For Caputo derivativeDtαt0Cft, the following equality holds:(4)Dt1−αt0CDtαt0Cft=Dt1t0Cft=dftdt.Lemma 3.
If the constantC≠0, then Dtαt0CC=0.Lemma 4.
If functionft∈L∞t0,t, for α>0 and β>0, then(5)Dtαt0CItβt0Cft=Dtα−βt0Cft,when α=β,(6)Dtαt0CItαt0Cft=ft.Lemma 5.
(see [2]). For any time instant t≥0, if xt∈R is continuously differentiable, then the inequality 1/2DtαaCx2t≤xtDtαaCxt0<α<1 holds; if xt∈Rn, the inequality 1/2DtαaCxTtxt≤xTtDtαaCxt also holds.Lemma 6.
(see [6]). If ft∈C10,+∞,R is a continuously differentiable function, the inequality(7)Dtα0Cft+≤a.e.sgnftDtα0Cft,holds almost everywhere, where 0<α<1 and ft+=Δlimτ⟶t+fτ.Consider two nonidentical FNNs as the drive and response system:(8)Dtα0Cxt=−Cxt+Afxt,zt=Pxt,(9)Dtα0Cyt=−Dyt+Bgyt+ut,z^t=Pyt,0<α<1,where xt=x1t,x2t,…,xntT∈Rn and yt=y1t,y2t,…,yntT∈Rn denote the state variables and zt and z^t are the outputs. fxt=f1x1t,f2x2t,…,fnxntT∈Rn and gyt=g1y1t,g2y2t,…,gnyntT∈Rn are neuron activation functions at time t. n is the number of units. C=diagC1,C2,…,Cn and D=diagD1,D2,…,Dn are self-regulating parameters of neurons. A=Aijn×n and B=Bijn×n denote synaptic connection weight matrices. ut is the controller to be designed later.Assumption 1.
The continuous neuron activation functionsfandg meet Lipschitz condition(10)fϕ−fφ<Fϕ−φ,gϕ−gφ<Gϕ−φ,ϕ,φ∈R,where F,G>0 are Lipschitz constants.


## 3. Main Results

In this part, based on the fractional sliding mode control method, the sliding mode controllers will be designed to research the MPS between systems (8) and (9).The error function of MPS is(11)et=yt−Λxt,where e=e1,e2,…,enT and Λ=Λijn×n is a general constant matrix.Definition 3.
If any two solutionsxt and yt with initial values x0 and y0 meet(12)limt⟶+∞et1=limt⟶+∞yt−Λxt=0,systems (8) and (9) are said to be MPS, where 1=∑i=1n denotes the Euclidean norm.
Taking Caputo derivative of both sides of error functionet=yt−Λxt and substituting into (8) and (9), the error system can be obtained as(13)Dtα0Cet=Dtα0Cyt−ΛDtα0Cxt=−Dyt+Bgyt+ut−Λ−Cxt+Afxt=−Dyt−Λxt−DΛxt+Bgyt−gΛxt+BgΛxt+ut−Λ−Cxt+Afxt=−Det+Bgyt−gΛxt−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut,(14)z¯t=Pet.


### 3.1. Sliding Mode Controller Design

The design principle of sliding mode control is usually to design a suitable sliding surface as required and then construct a controller to force the system to move on the sliding surface and stay on it forever. First, the fractional integral sliding surface is designed as(15)St=et+Itα0CC+KPet−Bgyt−gΛxt,where K∈Rn×n is the gain matrix. Integrate system (13) from 0 to t, and using Lemma 1, we have(16)Itα0CDtα0Cet=et−e0=Itα0C−Det+Bgyt−gΛxt−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut,then(17)et=e0+Itα0C−Det+Bgyt−gΛxt−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut.Next, from equations (15) and (17), we have(18)St=e0+Itα0C−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut,and sliding surface (18) and (15) are equivalent. Based on the sliding mode control method, when error system (13) moves on sliding surface, the formulas St=0 and S˙t=0 have to be satisfied. Then, one can obtain S˙t=Dt1−α0CDtα0CSt=0 by using Lemma 2, which means Dtα0CSt=0. Using (18) and Lemmas 3 and 4, we obtain(19)Dtα0CSt=Dtα0Ce0+Dtα0CI0tαC−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut=C−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut=0.From equation (19), the equivalent sliding mode controller is designed as(20)ueqt=−C−D+KPet+DΛxt−BgΛxt+Λ−Cxt+Afxt.Substituting controller (20) into (13), the sliding mode error system is described as(21)Dtα0Cet=−C+KPet+Bgyt−gΛxt.Obviously,et=0 is an equilibrium point of (21). In order to make the trajectories of the system reach the sliding surface more efficiently and faster, the reaching law is selected as(22)urt=−k∗sgnSt,where the switching gain is k∗>0 and sgnSt=−1St<00St=01St>0. Then, the sliding mode control ut is denoted by(23)ut=ueqt+urt=−C−D+KPet+DΛxt−BgΛxt+Λ−Cxt+Afxt−k∗sgnSt.Theorem 1.
According to sliding switching surface (15) and sliding mode controller (23), synchronization error trajectories (13) asymptotically reach sliding surface St=0.Proof.
Construct a Lyapunov function as(24)Vt=12STtSt,taking Caputo derivative of (24) with respect to time t and based on Lemma 5, we obtain(25)Dtα0CVt≤STtDtα0CSt=STtC−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut=STt−k∗sgnSt=−k∗St.
Ask∗>0, system trajectories asymptotically converge to St=0, which means synchronization error trajectories (21) reach the predetermined sliding surface globally and stay on it forever.


### 3.2. Stability Analysis

Theorem 2.
If Assumption1 and the inequality(26)Ci>∑j=1n∑l=1nkjlPli+∑j=1nbjimi,hold, then the drive system (8) and response system (9) can realize MPS based on controller (23).Proof.
First, error system (21) is converted as(27)Dtα0Ceit=−Cieit−∑j=1n∑l=1nkilpljejt+∑j=1nbijgjyjt−gj∑i=1nΛjixit.
Next, design Lyapunov functionVt=et1=∑i=1neit, according to Assumption 1 and Lemma 6, taking Caputo derivative of Lyapunov function Vt along error trajectories (21), then(28)Dtα0CVt=Dtα0C∑i=1neit=∑i=1nDtα0Ceit≤∑i=1nsgneitDtα0Ceit=∑i=1nsgneit−Cieit−∑j=1n∑l=1nkilpljejt+∑j=1nbijgjyjt−gj∑i=1nΛjixit=∑i=1nsgneit−Cieit−∑i=1nsgneit∑j=1n∑l=1nkilpljejt+∑i=1nsgneit∑j=1nbijgjyjt−gj∑i=1nΛjixit≤∑i=1n−Cieit+∑j=1n∑l=1nkilpljejt+∑j=1nbijGjejt=∑i=1n−Cieit+∑j=1n∑l=1nkjlplieit+∑j=1nbjiGieit=−∑i=1nCi−∑j=1n∑l=1nkjlpli−∑j=1nbjiGieit.
Letλi=Ci−∑j=1n∑l=1nkjlpli−∑j=1nbjiGi, and we can obtain(29)λ=min1≤i≤nλi=min1≤i≤nCi−∑j=1n∑l=1nkjlpli−∑j=1nbjiGi>0,then(30)Dtα0CVt≤−λVt=−λet1≤0,t≥0,and Vt is monotonously nonincreasing. Hence, Vt≤V0t≥0 and eit is bounded on t≥0 from the definition of Vt. There exists a positive constant satisfying DCαet1≤M; then, we claim that limt⟶∞et1=0 whose proof is similar to Theorem 1 in [16]. Therefore, error system (21) is globally asymptotically stable and MPS between systems (8) and (9) is realized based on controller (23).Remark 1.
When activating controller (23), response system (9) converts to(31)Dtα0Cyt=−C+KPyt+CΛ−ΛCxt+kpΛxt+Bgyt−BgΛxt+ΛAfxt,and then based on drive system (8), controlled response system (31), and error system (27), the MPS behaviors between systems (8) and (9) can be analyzed furthermore.Remark 2.
Theorems1 and 2 are still true and new for α=1.
In particular, the MPS is a kind of more general synchronization. Selecting different projection matrices and controllers, it degenerates to some special synchronization types as Remark3.Remark 3.
(1)
IfΛ=I, the sliding mode controller becomes(32)ut=ueqt+urt=−C−D+KPyt+KPxt−Bgxt+Afxt−k∗sgnSt,where sliding surfaceSt is given by (15), and then systems (8) and (9) can achieve complete synchronization.(2)
IfΛ=−I, based on sliding surface (15), the sliding mode controller becomes(33)ut=ueqt+urt=−C−D+KPyt−Bg−xt−KPxt−Afxt−k∗sgnSt,and then systems (8) and (9) can realize antisynchronization.(3)
IfΛ=cIc=const andc≠±1, the sliding mode controller is similar to (23) and sliding surface St is given by (15), and then systems (8) and (9) can achieve the PS.(4)
IfΛ=diagc1,c2,…,cnci=const,i=1,2,…,n and at least two ci of them are unequal, the sliding mode controller is similar to (23) and sliding surface St is given by (15), and then systems (8) and (9) can achieve the modified PS.


## 3.1. Sliding Mode Controller Design

The design principle of sliding mode control is usually to design a suitable sliding surface as required and then construct a controller to force the system to move on the sliding surface and stay on it forever. First, the fractional integral sliding surface is designed as(15)St=et+Itα0CC+KPet−Bgyt−gΛxt,where K∈Rn×n is the gain matrix. Integrate system (13) from 0 to t, and using Lemma 1, we have(16)Itα0CDtα0Cet=et−e0=Itα0C−Det+Bgyt−gΛxt−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut,then(17)et=e0+Itα0C−Det+Bgyt−gΛxt−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut.Next, from equations (15) and (17), we have(18)St=e0+Itα0C−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut,and sliding surface (18) and (15) are equivalent. Based on the sliding mode control method, when error system (13) moves on sliding surface, the formulas St=0 and S˙t=0 have to be satisfied. Then, one can obtain S˙t=Dt1−α0CDtα0CSt=0 by using Lemma 2, which means Dtα0CSt=0. Using (18) and Lemmas 3 and 4, we obtain(19)Dtα0CSt=Dtα0Ce0+Dtα0CI0tαC−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut=C−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut=0.From equation (19), the equivalent sliding mode controller is designed as(20)ueqt=−C−D+KPet+DΛxt−BgΛxt+Λ−Cxt+Afxt.Substituting controller (20) into (13), the sliding mode error system is described as(21)Dtα0Cet=−C+KPet+Bgyt−gΛxt.Obviously,et=0 is an equilibrium point of (21). In order to make the trajectories of the system reach the sliding surface more efficiently and faster, the reaching law is selected as(22)urt=−k∗sgnSt,where the switching gain is k∗>0 and sgnSt=−1St<00St=01St>0. Then, the sliding mode control ut is denoted by(23)ut=ueqt+urt=−C−D+KPet+DΛxt−BgΛxt+Λ−Cxt+Afxt−k∗sgnSt.Theorem 1.
According to sliding switching surface (15) and sliding mode controller (23), synchronization error trajectories (13) asymptotically reach sliding surface St=0.Proof.
Construct a Lyapunov function as(24)Vt=12STtSt,taking Caputo derivative of (24) with respect to time t and based on Lemma 5, we obtain(25)Dtα0CVt≤STtDtα0CSt=STtC−D+KPet−DΛxt+BgΛxt−Λ−Cxt+Afxt+ut=STt−k∗sgnSt=−k∗St.
Ask∗>0, system trajectories asymptotically converge to St=0, which means synchronization error trajectories (21) reach the predetermined sliding surface globally and stay on it forever.


## 3.2. Stability Analysis

Theorem 2.
If Assumption1 and the inequality(26)Ci>∑j=1n∑l=1nkjlPli+∑j=1nbjimi,hold, then the drive system (8) and response system (9) can realize MPS based on controller (23).Proof.
First, error system (21) is converted as(27)Dtα0Ceit=−Cieit−∑j=1n∑l=1nkilpljejt+∑j=1nbijgjyjt−gj∑i=1nΛjixit.
Next, design Lyapunov functionVt=et1=∑i=1neit, according to Assumption 1 and Lemma 6, taking Caputo derivative of Lyapunov function Vt along error trajectories (21), then(28)Dtα0CVt=Dtα0C∑i=1neit=∑i=1nDtα0Ceit≤∑i=1nsgneitDtα0Ceit=∑i=1nsgneit−Cieit−∑j=1n∑l=1nkilpljejt+∑j=1nbijgjyjt−gj∑i=1nΛjixit=∑i=1nsgneit−Cieit−∑i=1nsgneit∑j=1n∑l=1nkilpljejt+∑i=1nsgneit∑j=1nbijgjyjt−gj∑i=1nΛjixit≤∑i=1n−Cieit+∑j=1n∑l=1nkilpljejt+∑j=1nbijGjejt=∑i=1n−Cieit+∑j=1n∑l=1nkjlplieit+∑j=1nbjiGieit=−∑i=1nCi−∑j=1n∑l=1nkjlpli−∑j=1nbjiGieit.
Letλi=Ci−∑j=1n∑l=1nkjlpli−∑j=1nbjiGi, and we can obtain(29)λ=min1≤i≤nλi=min1≤i≤nCi−∑j=1n∑l=1nkjlpli−∑j=1nbjiGi>0,then(30)Dtα0CVt≤−λVt=−λet1≤0,t≥0,and Vt is monotonously nonincreasing. Hence, Vt≤V0t≥0 and eit is bounded on t≥0 from the definition of Vt. There exists a positive constant satisfying DCαet1≤M; then, we claim that limt⟶∞et1=0 whose proof is similar to Theorem 1 in [16]. Therefore, error system (21) is globally asymptotically stable and MPS between systems (8) and (9) is realized based on controller (23).Remark 1.
When activating controller (23), response system (9) converts to(31)Dtα0Cyt=−C+KPyt+CΛ−ΛCxt+kpΛxt+Bgyt−BgΛxt+ΛAfxt,and then based on drive system (8), controlled response system (31), and error system (27), the MPS behaviors between systems (8) and (9) can be analyzed furthermore.Remark 2.
Theorems1 and 2 are still true and new for α=1.
In particular, the MPS is a kind of more general synchronization. Selecting different projection matrices and controllers, it degenerates to some special synchronization types as Remark3.Remark 3.
(1)
IfΛ=I, the sliding mode controller becomes(32)ut=ueqt+urt=−C−D+KPyt+KPxt−Bgxt+Afxt−k∗sgnSt,where sliding surfaceSt is given by (15), and then systems (8) and (9) can achieve complete synchronization.(2)
IfΛ=−I, based on sliding surface (15), the sliding mode controller becomes(33)ut=ueqt+urt=−C−D+KPyt−Bg−xt−KPxt−Afxt−k∗sgnSt,and then systems (8) and (9) can realize antisynchronization.(3)
IfΛ=cIc=const andc≠±1, the sliding mode controller is similar to (23) and sliding surface St is given by (15), and then systems (8) and (9) can achieve the PS.(4)
IfΛ=diagc1,c2,…,cnci=const,i=1,2,…,n and at least two ci of them are unequal, the sliding mode controller is similar to (23) and sliding surface St is given by (15), and then systems (8) and (9) can achieve the modified PS.


## 4. Numerical Application

Two nonidentical drive and response systems are considered as(34)Dtα0Cxt=−Cxt+Afxt,zt=Pxt,(35)Dtα0Cyt=−Dyt+Bgyt+ut,z^t=Pyt,where C=D=100010001, A=2−1.201.81.711.15−4.7501.1, B=2.5−1.5021.71.2−4.701, P=100010001, fxt=tanhxt, gyt=tanhyt, and ut is the controller. In order to analyze chaotic synchronization for systems (34) and (35), the value of derivative order is chosen as α=0.98, which causes the two systems to exhibit chaotic behaviors described in Figures 1(a)–1(d) with initial conditions x10,x20,x30=y10,y20,y30=0.5,−0.6,0.4.Figure 1
Chaotic behaviors of systems (34) and (35) at α=0.98. (a) x1−x2−x3 phase portrait of system (34). (b) x1−x2 phase portrait of system (34). (c) y2−y1−y3 phase portrait of system (35). (d) y2−y3 phase portrait of system (35).
(a)(b)(c)(d)Projection matrixΛ is chosen as(36)Λ=11−4−0.5−21−3−0.5−1,and then error functions are calculated as(37)e1=y1−∑j=13Λ1jxj=y1−x1+x2−4x3,e2=y2−∑j=13Λ2jxj=y2−−0.5x1−2x2+x3,e3=y3−∑j=13Λ3jxj=y3−−3x1−0.5x2−x3.Next, choosingk1=5,k2=5,k3=5,andk∗=20 and according to Theorems 1 and 2, the MPS between systems (34) and (35) is realized.Figures2(a)–2(c) depict the time trajectories of variables x1∼y1, x2∼y2, and x3∼y3. Simultaneously, it is clearly seen that variables x1+x2−4x3∼y1, −0.5x1−2x2+x3∼y2, and −3x1−0.5x2−x3∼y3 realize the complete synchronization as depicted in Figures 2(d)–2(f). Figure 3 describes that the synchronization error trajectories converge to zero asymptotically for drive system (34) and response system (35). These numerical simulations and figures clearly indicate the applicability and effectiveness of the sliding controller for MPS.Figure 2
MPS behaviors between systems (34) and (35). Time history of (a) x1,y1; (b) x2,y2; (c) x3,y3; (d) x1+x2−4x3,y1; (e) −0.5x1−2x2+x3,y2; and (f) −3x1−0.5x2−x3,y3.
(a)(b)(c)(d)(e)(f)Figure 3
Trajectories of synchronization errorse1,e2, and e3.


## 5. Conclusions

For general FNNs, the paper comes up with the global MPS, constructs the fractional-order sliding surface, designs a sliding mode controller, establishes and proves the sufficient condition, and then realizes the global MPS. Our theory analysis provides important theoretical basis and technical support for enhancing signal security by using MPS of FNNs and contributes to the development of artificial intelligence. In future works, we will extend the MPS to the fractional-order memristive neural networks and consider applying them into secret communication.



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