# Mixedℋ∞ and Passivity Analysis of Delayed Fractional-Order Complex Dynamical Networks with Hybrid Coupling

Authors: Hamed Alsulami; M. Syed Ali; M. Hymavathi; Tareq Saeed; Bashir Ahmad; Ahmed Alsaedi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, global asymptotic stability analysis, and mixedℋ∞ and passive control for a class of control fractional-order systems is investigated. Based on the fractional-order Lyapunov stability theorem and some properties of fractional calculus, we propose sufficient conditions to ensure the mixed ℋ∞ and passivity performance. More relaxed conditions by employing the new type of augmented matrices by using Kronecker product terms can be handled, which can be introduced. The derived criteria are expressed in terms of linear matrix inequalities that which can be checked numerically using toolbox MATLAB. Finally, two numerical examples are provided to demonstrate the correctness of the proposed results.


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## Body

## 1. Introduction

Fractional calculus is a branch of applied mathematics that contain integrals and derivatives of any order, which may be developed as a superset of conventional integer order calculus. Many authors pointed out that the derivatives and integrals of non-integer-order are more appropriate for representing the properties of several real time materials and processes, e.g., polymers. That is, the generalization of the derivative order of a parameter enriches the system performance by expanding degree of freedom flexibility. The real-world applications of fractional-order differential equations are viscoelastic structure, system identification, quantitative finance and so on. Fractional order stability and synchronization results have been realized, see the reference therein [1–10].The dynamics of complex networks have been widely studied on the basis of complex network models, with a focus on the interaction between the complexity of the overall topology and the local dynamical characteristics of the connected nodes. Hence, fractional-order complex networks can better model and reveals some remarkable results have been obtained [11–14]. The behaviour of the network nodes is similar throughout the existing literature on complex dynamical networks that takes drive and response systems under consideration. Examples of such complex networks can be found everywhere in our daily life, from physical objects such as the Internet, multi agent systems, neural networks, living organisms, etc. Numerous collective phenomena, including self-organization, synchronization, and spatiotemporal chaos, occur in complex dynamical neural networks. Synchronization, as a very important phenomenon, has often appeared in papers [15–17].It is completely found out that the reason ofℋ∞ controllers/filters is to assure the closed-loop error systems are stable with an ℋ∞ norm bound restricted to disturbance attenuation [18, 19]. On the other hand, the Mixed ℋ∞ and passivity control has been proved [20, 21]. Recently, state feedback ℋ∞ control of commensurate fractional-order systems and ℋ∞ model reduction for positive fractional-order systems were investigated in [22, 23]. The main key of passivity idea is that the passive properties can preserve the system inside stable and explicit to the belongings of vitality application. For the other one, thinking mixed ℋ∞ and passive performance when designing controller is the ideal solution. It is really well worth bringing up that the ℋ∞ ideal performs an critical role with inside the layout and evaluation of linear and nonlinear systems, which has attracted significant interest over the past decades [24–28]. To the best of the author knowledge, no results have been reported on the mixed ℋ∞ and passive control for fractional-order system with hybrid couplings.Furthermore, time delay in practice for the coupling connections in the neural networks is inevitable because of the finiteness of signal transmission speed over the links. The delayed coupling could describe the decentralized nature of real-world couple systems. Studying the synchronization of fractional-order neural networks with hybrid coupling is thus still a difficult but interesting topic. In real-world situation, time delay is ubiquitous in many physical systems due to the finite switching speed of amplifiers, finite signal propagation time in networks and so on.The main contributions of this work are highlighted below:(1)
We studied the problem of mixedℋ∞ and passive analysis of delayed fractional-order complex dynamical networks with hybrid coupling.(2)
Novel Lyapunov functional consisting of Kronecker product is constructed to derive the main results.(3)
This paper is to discuss issue of the fractional-order complex networks under a new reliable protocol subject to coupling delay.(4)
The derivation process that leads in the feasible criteria applies a variety of inequality lemmas.(5)
By using the Lyapunov direct method, a new sufficient condition is proposed to ensure the system to be global asymptotically stable with mixedℋ∞ and passivity performance level.(6)
Then the criterion is adopted to design a state feedback controller in terms of linear matrix inequalities, which can be solved numerically by using standard computational LMI-based algorithms.(7)
All the derived conditions obtained here are expressed in terms of LMIs whose feasibility can be easily checked by using numerically efficient MATLAB LMI Control toolbox.


## 2. Model Description and Preliminaries

There are four common definitions of fractional calculus, namely Riemann Liouville, Hadamard and Caputo definitions. Among them, Caputo fractional-order derivative is well understood in physical situations and more applicable to real-world problems, because of the same initial conditions as in integer-order derivatives. Thus, only Caputo fractional-order derivative is used in this paper. We firstly give some useful definitions and lemmas fractional derivative of orderβ>0.Definition 1 (see [29]).
The Caputo fractional derivative of orderβ for a function ht is defined by(1)Dtβ0Cht=1Γn−β∫0tt−sn−β−1hnsds,t≥0,n−1<β≤n,where n is a positive integer. In particular, when 0<β<1, we have(2)Dtβ0Cht=1Γ1−β∫0tt−s−βh′sds,t≥0.Lemma 1 (see [30]).
Letpt∈ℛn be a differentiable vector-valued function. Then for t≥0, we have(3)Dtβ0CpTtΥpt≤2pTtΥDtβ0Cpt,β∈0,1,where Υ∈ℛn×n is a symmetric positive definite matrix.Lemma 2 (see [31]).
Ifpt∈Cn0,+∞,R and n−1<β<n, then(4)ℐtβ0Dtβ0Cpt=pt−∑k=0n−1tkk!pk0.
In particular, when0<β<1, we have(5)ℐtβ0Dtβ0Cpt=pt−p0.Lemma 3 (see [32]).
For a given matrix(6)Θ=Θ11Θ12Θ12TΘ22,
WithΘ11=Θ11T, Θ22=Θ22T, then the following conditions are equivalent:(7)1Θ<0,2Θ22<0,Θ11−Θ12Θ22−1Θ12T<0.Lemma 4 (see [33]).
Let⊗ denote the notation of Kronecker product. Then, the following relationships hold(8)1.αA⊗B=A⊗αB,2.A+B⊗C=A⊗C+B⊕C,3.A⊗BC⊗D=AC⊗BD.Lemma 5 (see [34]).
Lete=1,1,…,1T, EN=eeT, and U=NIN−EN, T∈Rn×n, p=p1T,p2T,….,pNTT, and q=q1T,q2T,….,qNTT , with pk,qk∈Rn,k=1,2,…,N, then(9)pTU⊗Tq=∑1≤k<j≤NNpk−pjTTqk−qj.Lemma 6 (see [35]).
LetH, S be n×n any real matrix, e=1,1,…,1T, EN=eeT, U=NIN−EN, p=p1T,p2T,….,pNTT, and q=q1T,q2T,….,qNTT, with pk,qk∈Rn,k=1,2,…,N, and f⋅ and F⋅ are functions and defined in system (1). Then for any vectors p and q with appropriate dimensions, the following matrix inequality holds:(10)pTU⊗TFℐN⊗Sq=∑1≤k<j≤NNpk−pjTTTfSqk−fSqj.Property 1 (see [9]).
Ifα,β>0, then the following property(11)DαD−βht=Dα−βht.Property 2 (see [9]).
For any constantsk1,k2∈0,1, and two functions ft, gt, we have(12)Dtβ0Ck1ft+k2gt=k1Dtβ0Cft+k2Dtβ0Cgt.Assumption 1 (see [36]).
Bounded functions satisfyingfk0=gk0 and for all Now the activation function f⋅, g⋅ and h⋅, some constants are uk−, uk+, vk−, vk+, wk−, wk+ and satisfy(13)uk−≤fkp1−fkp2p1−p2≤uk+,vk−≤gkp1−gkp2p1−p2≤vk+.
We denote(14)Δ1=diagu1+u1−,…,un+un−,Δ2=diagu1++u1−2,…,un++un−2,Δ3=diagv1+v1−,…,vn+vn−,Δ4=diagv1++v1−2,…,vn++vn−2.


## 3. Main Result

Consider the fractional-order complex dynamical network model consisting ofN coupled nodes of the form(15)Dtα0Cpkt=ℛ1pkt+ℛ2pkt−λt+ζ¯tc∑j=1NfkjQ1pjt,+1−ζ¯tc∑j=1NgkjQ2pjt−λt+ℛwktpkt=ϕkt,k=1,2,3……,N,where 0<α≤1, pkt=pk1t……pknt∈ℛn be the state vector of the k th node at time t; ζ¯t is Bernoulli distributed stochastic variable such that Eζ¯t=ζ0, λt is a coupling delay and assumed to satisfying 0<λt≤λ and λ.t≤μ<1, where λ, μ are known constants. c is coupling strength; ℛ1, ℛ2, denote the connection weight matrix and the delayed connection weight matrix, respectively. Q1 and Q2 represent constants inner coupling matrix at time t and the time t−λt; ℛ is real constant matrices with appropriate dimension. ℱ=fkjN×N and G=gkjN×N are the coupled configuration matrix and its elements satisfies the following conditions: if there is a connection between the nodes k and j, then gkj>0; otherwise gkj=0 for k≠j and the diagonal elements gkk = −∑k=1,k≠jNgkj.Combining with the sign⊗ of Kronecker product, model (1) can be rewritten as(16)Dtα0Cpt=ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+cζ¯tℱ⊗Q1pt,+c1−ζ¯tG⊗Q2pt−λt+ℐN⊗ℛwt,Definition 2 (see [35]).
The system (1) is said to be globally asymptotically stable with mixed ℋ∞ and passivity performance γ, if the following requirements are satisfied simultaneously;(i)
The system (1) is globally asymptotically stable wheneverwt=0;(ii)
Under zero initial condition, there exists a scalarγ>0 such that the following condition is satisfied(17)∫tf0−γ−1θpTtpt+21−θpTtwtdt≥−γ∫0tfwTtwtdt,
For alltf>0 and any non-zero wt∈L20,∞, where θ∈0,1 denotes a weighting parameter that represents the trade-off between the weighted ℋ∞ performance index and the passivity performance index.Theorem 1.
For givenθ∈0,1, the system (1) is mixed ℋ∞ and passive performance level γ, if there exist positive definite matrices T1>0, and positive diagonal matrices J1, J2, ℒ1, ℒ2, Δ=diagδ1,δ2,…δn, such that the following LMIs holds(18)Φ=Φ11Φ12Φ130Φ150Φ17∗Φ220Φ240Φ260∗∗Φ330000∗∗∗Φ44000∗∗∗∗Φ5500∗∗∗∗∗Φ660∗∗∗∗∗∗Φ77<0,Φ11=2T1ℛ1−ζ¯cNfkjT1Q1−Q1TJ1Δ1Q1−Q2TJ2Δ3Q2−γ−1θ,Φ12=T1ℛ2−1−ζ¯cNgkjT1Q2,Φ13=Q1TJ1Δ2,Φ15=Q2TJ2Δ4,Φ17=T1ℛ−1−θ,Φ22=−Q1Tℒ1Δ1Q1−Q2Tℒ2Δ3Q2,Φ24=Q1Tℒ1Δ2,Φ26=Q2Tℒ2Δ4,Φ33=−J1,Φ44=−ℒ1,Φ55=−J2,Φ66=−ℒ2,Φ77=−γℐ.Proof.
Choose the Lyapunov functional candidate:(19)Vpt=pTtU⊗T1pt,
It follows from Lemma1 that we obtain the α-order Caputo derivative of Vpt as follows:(20)Dtα0CVpt≤2pTtU⊗T1Dtα0Cpt,≤2pTtU⊗T1ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+ζ¯cℱ⊗Q1pt+1−ζ¯cG⊗Q2pt−λt+ℐN⊗ℛwt,≤2pTtU⊗T1ℐN⊗ℛ1pt+2pTtU⊗T1ℐN⊗ℛ2pt−λt+2pTtU⊗T1ζ¯cℱ⊗Q1pt+2pTtU⊗T11−ζ¯cG⊗Q2pt−λt+2pTtU⊗T1ℐN⊗ℛwt=∑1≤k<j≤Npkt−pjtTT1ℛ1+ℛ1TT1pkt−pjt,+2pkt−pjtTT1ℛ2pkt−λt−pjt−λt−2pkt−pjtTζ¯cNfkjT1Q1pkt−pjt−2c1−ζ¯pkt−pjtTNgkjT1Q2pkt−λt−pkt−λt+2pkt−pjtTT1ℛwkt−wjt.
From Assumption, for anyn×n positive diagonal matrices J1, ℒ1, J2, ℒ2, Δ one has(21)pkt−pjtfpkt−fpjtT−Q1TJ1Δ1Q1Q1TJ1Δ2∗−J1pkt−pjtfpkt−fpjt,+pkt−λt−pjt−λtfpkt−λt−fpjt−λtT−Q1Tℒ1Δ1Q1Q1Tℒ1Δ2∗−ℒ1,×pkt−λt−pjt−λtfpkt−λt−fpjt−λt≥0,pkt−pjtgpkt−gpjtT−Q2TJ2Δ3Q2Q2TJ2Δ4∗−J2pkt−pjtgpkt−gpjt+pkt−λt−pjt−λtgpkt−λt−gpjt−λtT−Q2Tℒ2Δ3Q2Q2Tℒ2Δ4∗−ℒ2×pkt−λt−pjt−λtgpkt−λt−gpjt−λt≥0.
To show the mixedℋ∞ and passivity performance of γ of system (1). Then we have the following estimate:(22)Dtα0CVpt+γ−1θpTtpt−21−θpTtwt−γwTtwt≤∑1≤k<j≤Nξ^kjTtΦ7×7ξkjt,where,(23)ξ^kjt=pkt−pjtTpkt−λt−pjt−λtTfpkt−fpjtTfpkt−λt−fpjt−λtTgpkt−gpjtTgpkt−λt−gpjt−λtTwks−wjsTT.
From (4), we get(24)Dtα0CVpt+γ−1θpTtpt−21−θpTtwt−γwTtwt≤0,∀t≥0.
Integrating (8) with respect to t from 0 to tf, we get(25)Itf10Dtfα0CVpt+∫0tfγ−1θpTtpt−21−θpTtwt−γwTtwtdt≤0.
By using Property1, Property 2 and Lemma 2, we have(26)Itf10Dtfα0CVpt=Itf1−α0Itfα0Dtfα0CVpt=Itf1−α0Itfα0Dtfα0CVpt=Itf1−α0Vxt−Vx0=Itf1−α0Vxt−Itf1−α0Vx0.
On the other hand, we have(27)Itf1−α0Vpt=1Γ1−α∫0tftf−s−αpTsU⊗T1psds≥0,∀tf≥0.
Under zero initial condition, we obtain(28)Itf1−α0Vp0=1Γ1−α∫0tftf−s−μpT0U⊗T1p0ds=0,∀tf≥0.
HenceItf10Dtfα0CVpt≥0,∀tf≥0 with zero initial condition. Therefore, we have(29)∫0tf−γ−1θpTtpt+21−θpTtwtdt≥−γ∫0tfwTtwtdt,∀tf≥0.
By Definition2, system (1) is mixed ℋ∞ and passivity performance γ. The proof of theorem is completed.Theorem 2.
The System (1) with wt=0 is globally asymptotically stable if there exist positive definite matrix T1, T2 and positive diagonal matrices J1, J2, ℒ1, ℒ2, Δ=diagδ1,δ2,…δn, such that the following LMI holds(30)Θ=Θ11Θ12Θ130Θ150∗Θ220Θ240Θ26∗∗Θ33000∗∗∗Θ4400∗∗∗∗Θ550∗∗∗∗∗Θ66<0,Θ11=2T1ℛ1+T2−ζ¯cNfkjT1Q1−Q1TJ1Δ1Q1−Q2TJ2Δ3Q2,Θ12=T1ℛ2−1−ζ¯cNgkjT1Q2,Θ13=Q1TJ1Δ2,Θ15=Q2TJ2Δ4,Θ22=−1−μT2−Q1Tℒ1Δ1Q1−Q2Tℒ2Δ3Q2,Θ24=Q1Tℒ1Δ2,Θ26=Q2Tℒ2Δ4,Θ33=−J1,Θ44=−ℒ1,Θ55=−J2,Θ66=−ℒ2.Proof.
Choose the Lyapunov functional candidate:(31)Vpt=pTtU⊗T1pt+0CDt1−α∫t−λttpTsU⊗T2psds,
It follows from Lemma1 that we obtain the α-order Caputo derivative of Vpt as follows:,(32)Dtα0CVpt≤2pTtU⊗T10CDtαpt+pTtU⊗T2pt−1−μpTt−λtU⊗T2pt−λt,≤2pTtU⊗T1ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+ζ¯c(ℱ⊗Q1)pt+1−ζ¯cG⊗Q2pt−λt+ℐN⊗ℛwt+pTtU⊗T2pt−1−μpTt−λtU⊗T2pt−λt,≤2pTtU⊗T1ℐN⊗ℛ1pt+2pTtU⊗T1ℐN⊗ℛ2pt−λt+2pTtU⊗T1ζ¯cℱ⊗Q1pt+2pTtU⊗T11−ζ¯cG⊗Q2pt−λt+pTtU⊗T2pt−1−μpTt−λtU⊗T2pt−λt=∑1≤k<j≤Npkt−pjtTT1ℛ1+ℛ1TT1pkt−pjt+2pkt−pjtTT1ℛ2pkt−λt−pjt−λt−2pkt−pjtTζ¯cNfkjT1Q1pkt−pjt−2c1−ζ¯pkt−pjtTNgkjT1Q2pkt−λt−pkt−λt+pkt−pjtTT2pkt−pjt−pkt−λt−pjt−λtTT2pkt−λt−pjt−λt1−μ
From Assumption, for anyn×n positive diagonal matrices J1, ℒ1, J2, ℒ2, Δ one has(33)pkt−pjtfpkt−fpjtT−Q1TJ1Δ1Q1Q1TJ1Δ2∗−J1pkt−pjtfpkt−fpjt+pkt−λt−pjt−λtfpkt−λt−fpjt−λtT−Q1Tℒ1Δ1Q1Q1Tℒ1Δ2∗−ℒ1×pkt−λt−pjt−λtfpkt−λt−fpjt−λt≥0,(34)pkt−pjtgpkt−gpjtT−Q2TJ2Δ3Q2Q2TJ2Δ4∗−J2pkt−pjtgpkt−gpjt+pkt−λt−pjt−λtgpkt−λt−gpjt−λtT−Q2Tℒ2Δ3Q2Q2Tℒ2Δ4∗−ℒ2×pkt−λt−pjt−λtgpkt−λt−gpjt−λt≥0.
Combining (32)–(34),(35)Dtα0CVpt≤∑1≤k<j≤Nπ^kjTtΘ6×6πkjt,where,(36)π^kjt=pkt−pjtTpkt−λt−pjt−λtTfpkt−fpjtTfpkt−λt−fpjt−λtTgpkt−gpjtTgpkt−λt−gpjt−λtTT.
From (10), Θ<0. Therefore, the system (1) with wt=0 is globally asymptotically stable. This completes the proof of the theorem.Remark 1.
The System (1) can be rewritten as(37)Dtα0CVpt=ℛ1pkt+ℛ2pkt−λt+ζ¯tfQ1pkt+1−ζ¯tgQ2pkt−λt+ℛwkt,pkt=ϕkt,k=1,2,3……,N
Combining with the sign⊗ of Kronecker product, model (15) can be rewritten as(38)Dtα0CVpt=ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+ζ¯0tℱℐN⊗Q1pt+1−ζ¯0tGℐN⊗Q2pt+ζ¯t−ζ¯0ℱℐN⊗Q1pt−GℐN⊗Q2pt+ℐN⊗ℛwtTheorem 3.
For givenθ∈0,1, the system (16) is mixed ℋ∞ and passive performance level γ, if there exist positive definite matrices T1>0, and positive diagonal matrices J1, J2, ℒ1, ℒ2, Δ=diagδ1,δ2,…δn, such that the following LMIs holds(39)φ=φ11φ12φ13φ1400φ17∗φ2200φ25φ260∗∗φ330000∗∗∗φ44000∗∗∗∗φ5500∗∗∗∗∗φ660∗∗∗∗∗∗φ77<0,φ11=2T1ℛ1−Q1TJ1Δ1Q1−Q2TJ2Δ3Q2−γ1−1θ,φ12=T1ℛ2,φ13=Q1TJ1Δ2+T1ζ¯0+T1ζt¯−ζ¯0,φ14=Q2TJ2Δ4+T11−ζ¯0t−T1ζt¯−ζ¯0,φ17=T1ℛ−1−θ,φ22=−Q1Tℒ1Δ1Q1−Q2Tℒ2Δ3Q2,φ25=Q1Tℒ1Δ2,φ26=Q2Tℒ2Δ4,φ33=−J1,φ44=−J2,φ55=−ℒ1,φ66=−ℒ2,φ77=−γℐ.Proof.
Choose the Lyapunov functional candidate:(40)Vpt=pTtU⊗T1pt.
It follows from Lemma1 that we obtain the α-order Caputo derivative of Vpt as follows:(41)Dtα0CVpt≤2pTtU⊗T10CDtαpt,≤2pTtU⊗T1⋅ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+ζ¯0ℱℐN⊗Q1pt+1−ζ¯0GℐN⊗Q2pt+ζ¯t−ζ¯0ℱℐN⊗Q1pt−GℐN⊗Q2pt+ℐN⊗ℛwt,=∑k=1N−1∑j=k+1Npkt−pjtTT1ℛ1+ℛ1TT1pkt−pjt+2pkt−pjtTT1ℛ2pkt−λt−pjt−λt+2ζ¯0pkt−pjtTT1fpkt−fpjt+21−ζ¯0pkt−pjtTT1gQ2pkt−gQ2pjt+2ζ¯t−ζ¯0pkt−pjtTT1fQ1pkt−fQ1pjt−2ζ¯t−ζ¯0pkt−pjtTT1gQ2pkt−gQ2pjt+2pkt−pjtTT1ℛwkt−wjt.
From Assumption, for anyn×n positive diagonal matrices J1, ℒ1, J2, ℒ2, Δ one has(42)pkt−pjtfQ1pkt−fQ1pjtT−Q1TJ1Δ1Q1Q1TJ1Δ2∗−J1pkt−pjtfQ1pkt−fQ1pjt+pkt−λt−pjt−λtfQ1pkt−λt−fQ1pjt−λtT−Q1Tℒ1Δ1Q1Q1Tℒ1Δ2∗−ℒ1×pkt−λt−pjt−λtfQ1pkt−λt−fQ1pjt−λt≥0,pkt−pjtgQ2pkt−gQ2pjtT−Q2TJ2Δ3Q2Q2TJ2Δ4∗−J2pkt−pjtgQ2pkt−gQ2pjt+pkt−λt−pjt−λtgQ2pkt−λt−gQ2pjt−λtT−Q2Tℒ2Δ3Q2Q2Tℒ2Δ4∗−ℒ2×pkt−λt−pjt−λtgQ2pkt−λt−gQ2pjt−λt≥0.
To show the mixedℋ∞ and passivity performance of γ of system (16). Then we have the following estimate:(43)Dtα0CVt,pt+γ−1θpTtpt−21−θpTtwt−γwTtwt≤∑1≤k<j<NΨ^kjTtφ7×7Ψkjt,where,(44)Ψ^kjt=pkt−pjtTpkt−λt−pjt−λtTf(Q1pkt−fQ1pjtTgQ2pkt−gQ2pjtT,fQ1pkt−λt−fQ1pjt−λtTgQ2pkt−λt−gQ2pjt−λtTwks−wjsTT.
From (17), we get(45)Dtα0CVt,pt+γ−1θpTtpt−21−θpTtwt−γwTtwt≤0,∀t≥0.
Integrating (22) with respect to t from 0 to tf, we get(46)Itf10Dtfα0CVpt+∫0tfγ−1θpTtpt−21−θpTtwt−γwTtwtdt≤0.
By using Property1, Property 2 and Lemma 2, we have(47)Itf10Dtfα0CVpt=Itf1−α0Itfα0Dtfα0CVt,pt,=Itf1−α0Itfα0Dtfα0CVt,pt,=Itf1−α0Vxt−Vx0,=Itf1−α0Vxt−Itf1−α0Vx0.
On the other hand, we have(48)Itf1−α0Vpt=1Γ1−α∫0tftf−s−αpTsU⊗T1psds≥0,∀tf≥0.
Under zero initial condition, we obtain(49)Itf1−α0Vp0=1Γ1−α∫0tftf−s−μpT0U⊗T1p0ds=0,∀tf≥0.
HenceItf10Dtfα0CVpt≥0,∀tf≥0 with zero initial condition. Therefore, we have(50)∫0tf−γ−1θpTtpt+21−θpTtwtdt≥−γ∫0tfwTtwtdt,∀tf≥0.
By Definition2, system (16) is globally asymptotically stable with mixed ℋ∞ and passivity performance γ. The proof of theorem is completed.


## 4. State Feedback Control

The state feedback controller is designed as(51)ukt=Kkpkt,Dtα0Cpt=ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+ζ¯0tℱℐN⊗Q1pt,+1−ζ¯0tGℐN⊗Q2pt+ζ¯t−ζ¯0ℱℐN⊗Q1pt,−GℐN⊗Q2pt+ℐN⊗ℛwt+ℐN⊗Kpt.Theorem 4.
For givenθ∈0,1, the system (26) is mixed ℋ∞ and passive performance level γ, if there exist positive definite matrices T1>0, and positive diagonal matrices J1, J2, ℒ1, ℒ2, Δ=diagδ1,δ2,…δn, such that the following LMIs holds:(52)Υ=Υ11Υ12Υ13Υ1400Υ17∗Υ2200Υ25Υ260∗∗Υ330000∗∗∗Υ44000∗∗∗∗Υ5500∗∗∗∗∗Υ660∗∗∗∗∗∗Υ77<0.
Then system (26) is stabilizable by the control ut=Kpt=K¯T1−1¯pt with disturbance attenuation γ.Proof.
Through derivation similar to that for Theorem3,(53)Ξ=Ξ11Ξ12Ξ13Ξ1400Ξ17∗Ξ2200Ξ25Ξ260∗∗Ξ330000∗∗∗Ξ44000∗∗∗∗Ξ5500∗∗∗∗∗Ξ660∗∗∗∗∗∗Ξ77<0,Ξ11=2T1ℛ1+T1K−Q1TJ1Δ1Q1−Q2TJ2Δ3Q2−γ1−1θ,Ξ12=T1ℛ2,Ξ13=Q1TJ1Δ2+T1ζ¯0+T1ζt¯−ζ¯0t,Ξ14=Q2TJ2Δ4+T11−ζ¯0t−T1ζt¯−ζ¯0,Ξ17=T1ℛ−1−θ,Ξ22=−Q1Tℒ1Δ1Q1−Q2Tℒ2Δ3Q2,Ξ25=Q1Tℒ1Δ2,Ξ26=Q2Tℒ2Δ4,Ξ33=−J1,Ξ44=−J2,Ξ55=−ℒ1,Ξ66=−ℒ2,Ξ77=−γℐ.
Pre and post-multiply matrixdiagT1−1,ℐ,ℐ,ℐ,ℐ,ℐ,ℐ and applying the change of variable, T1−1=T¯1(54)Λ=Λ11Λ12Λ13Λ1400Λ17∗Λ2200Λ25Λ260∗∗Λ330000∗∗∗Λ44000∗∗∗∗Λ5500∗∗∗∗∗Λ660∗∗∗∗∗∗Λ77,where,(55)Λ11=ℛ1T¯1+T¯1ℛ1T+KT¯1−T¯1Q1TJ1Δ1Q1T¯1−T¯1Q2TJ2Δ3Q2T¯1−T¯1γ1−1θT¯1,Λ12=ℛ2,Λ13=T¯1Q1TJ1Δ2+ζ¯0+ζt¯−ζ¯0,Λ14=T¯1Q2TJ2Δ4+1−ζ¯0t−ζt¯−ζ¯0,Λ17=ℛ−T¯11−θ,Λ22=−Q1Tℒ1Δ1Q1−Q2Tℒ2Δ3Q2,Λ25=Q1Tℒ1Δ2,Λ26=Q2Tℒ2Δ4,Λ33=−J1,Λ44=−J2,Λ55=−ℒ1,Λ66=−ℒ2,Λ77=−γℐ.
Applying the change of variable,KT¯1=K¯(56)Υ=Υ11Υ12Υ13Υ1400Υ17∗Υ2200Υ25Υ260∗∗Υ330000∗∗∗Υ44000∗∗∗∗Υ5500∗∗∗∗∗Υ660∗∗∗∗∗∗Υ77,where,(57)Υ11=ℛ1T¯1+T¯1ℛ1T+K¯−Q¯1TJ1Δ1Q¯1−Q¯2TJ2Δ3Q¯2−γ1−1θ,Υ12=ℛ2,Υ13=T¯1Q1TJ1Δ2+ζ¯0+ζt¯−ζ¯0,Υ14=T¯1Q2TJ2Δ4+1−ζ¯0t−ζt¯−ζ¯0,Υ17=ℛ−T¯11−θ,Υ22=−Q1Tℒ1Δ1Q1−Q2Tℒ2Δ3Q2,Υ25=Q1Tℒ1Δ2,Υ26=Q2Tℒ2Δ4,Υ33=−J1,Υ44=−J2,Υ55=−ℒ1,Υ66=−ℒ2,Υ77=−γℐ.Through derivation similar to that for Theorem3, the proof of Theorem 4 can be completed.Remark 2.
If we consider the orderα=1 in system (1), then the fractional-order neural networks with a time-varying delay and external disturbance will degenerate to an integer-order one. Correspondingly, the ℋ∞ control result in Theorem 1 is still valid for the integer-order delayed neural networks model.Remark 3.
Some recent works investigated the asymptotic stability of the zero solution of fractional-order nonlinear systems. By using Lyapunov directed method, Theorem2 derives a sufficient condition for the globally asymptotically stable of the fractional-order systems (1) with wt = 0. The result in Theorem 2 is quite general since many factors global asymptotic stability, are considered. Therefore, the obtained result in Theorem 1 generalizes those given in the previous literature.Remark 4.
Neural networks with time-varying delays have found widespread applications in many different fields such as signal and image processing, pattern recognition, optimization, and learning algorithm. To apply the neural networks to these applications, the stability must be guaranteed, so the stabilization problem has become a hot topic. On the other hand, external disturbances are usually unavoidable in neural networks systems because many reasons such as linear approximation, measurement errors, external noises and modeling inaccuracies. The external disturbances destroy stability, and hinder the performance of the system. One solution can beℋ∞ control method which reduces the effects of external disturbances on the system to a certain acceptable value namely ℋ∞ performance. ℋ∞ control method have adopted on the stabilization of neural networks with external disturbances.Remark 5.
This paper deals with the problem ofℋ∞ control for delayed Fractional order neural networks in the sense of Caputo fractional derivative. How to extend the results in this paper to fractional-order neural networks in the sense of generalized fractional derivative are interesting problems. These require further investigation in future works.Remark 6.
It should be mentioned here that stability theory and Lyapunov stability theory are independent concepts, which neither implies nor exclude each other. Thus, the problem ofℋ∞ control for delayed fractional-order neural networks in sense of Lyapunov stability theory is open issues. In this paper, we have investigated the problem of ℋ∞ control for fractional-order neural networks with a time-varying delay and external disturbance for the first time. This is the main difference between the strategy developed in this paper and previous works.Remark 7.
Many results on fractional-order neural networks have been published in the recent years. For examples, the problem of stability analysis for fractional-order neural networks with or without time-varying delays was studied in previous decades. Some interesting results on the control problem for some kind of fractional order neural networks with time-varying delays have been proposed.Remark 8.
In recent years, many researchers have investigated synchronization of the isolated fractional-order neural networks with or without delay. They provide many approaches to study synchronization, but they seldom consider delay coupling for fractional-order neural networks. The general fractional-order model includes fractional-order delayed neural networks, fractional-order delayed cellular neural networks, and some famous chaotic systems, such as fractional-order Chuas system, fractional-order Chen system, fractional-order Rossler system, and so forth. We propose a complex networks with hybrid coupling.


## 5. Illustrative Examples

In this section, some numerical results are provided to demonstrate the main results.Example 1.
Consider the following fractional order complex networks (1),(58)Dtα0Cpkt=ℛ1pkt+ℛ2pkt−λt+ζ¯tc∑j=1NfkjQ1pjt,+1−ζ¯tc∑j=1NgkjQ2pjt−λt+ℛwkt,where α∈0,1, λt=0.05, pt is the state vector, wt∈R is the unknown disturbance input vector. The system parameters are characterize as:(59)ℛ1=−4.20.80.8−2,ℛ2=−2.50.70.75−1.6,Q1=−1.30.50.4−0.6,Q2=−1.30.40.4−1.3,c=−1.50.50.8−1.3,ℛ=0.110.22.
We see that the activation functions(60)fpkt=0.5pk1−tanh0.1pk1t+0.4pk2t0.98pk2t−tanh0.76pk2t,gpkt=0.2pk1−tanh0.2pk1t0.98pk2t,
We consider the problem of mixedℋ∞ and passivity performance for system (1). Then it is easy verify that α=0.98, μ=0.67, ζ¯0=0.34 and ζ¯t=0.14(61)ℐ=1001,Δ1=−3003,Δ2=1001,Δ3=−4004,Δ4=−2002.
By using the Matlab solve the LMIs (4) in Theorem 1, we get the feasible solutions:(62)T1=−0.00200.00060.0006−0.0001,J1=4.2349004.2349,J2=0.0514000.0514,ℒ1=0.0028000.0028,ℒ2=0.0172000.0172,γ=13.987.
The above results exhibits that the conditions expressed in Theorem1 have satisfied and hence system (31) with the above given parameters are Mixed ℋ∞ passive.Example 2.
Consider the following fractional order complex networks (26),(63)Dtα0Cpt=ℐN⊗ℛ1pt+ℐN⊗ℛ2pt−λt+ζ¯0ℱℐN⊗Q1pt+1−ζ¯0GℐN⊗Q2pt+ζ¯t−ζ¯0ℱℐN⊗Q1pt−GℐN⊗Q2pt+ℐN⊗ℛwt)+ℐN⊗Kut.
The system parameters are characterize as:(64)ℛ1=−3.10.60.6−2,ℛ2=−2.80.50.5−0.6,Q1=−1.80.80.3−0.2,Q2=−1.60.20.2−1.6,ℛ=0.10.2.
Let the function and noise intensity function vector be given by(65)fpkt=0.6pk1−tanh0.2pk1t+0.3pk2t0.8pk2t−tanh0.6pk2t,gpkt=0.1pk1−tanh0.1pk1t0.87pk2t.
Then it is easy verify thatα=0.55, λt=0.07, μ=0.69, ζ¯0=0.76 and ζ¯t=0.98(66)ℐ=1001,Δ1=−9009,Δ2=110011,Δ3=−4004,Δ4=−2002.
By using the Matlab solve the LMIs (27) in Theorem 4, we get the feasible solutions:(67)T1=−0.00900.00020.0002−0.0008,J1=−2.134900−2.1349,J2=0.0316000.0316,ℒ1=0.0055000.0055,ℒ2=0.0552000.0552,γ=11.987.
According to Theorem4, system (32) is asymptotically stable with mixed ℋ∞ and passivity performance γ=11.987 under state feedback controller is given by(68)K=−1.7877−1.1336.The above results exhibits that the conditions expressed in Theorem4 have satisfied and hence system (32) with the above given parameters are Mixed ℋ∞ passive. Thus the results show that the designed feedback controller is suitable for stabilizing the system (32).


## 6. Conclusions

Here, we investigate the Mixedℋ∞ and passive analysis of delayed fractional order complex dynamical networks with hybrid coupling. By using the most updated technique on stability analysis of fractional order neural networks, we derive delay-dependent and order-dependent stabilization criteria for fractional order neural networks. Then, the problem of ℋ∞ control for fractional order neural networks with a time-varying delay and external disturbance is solved based on the proposed stabilization criteria and some auxiliary properties of fractional calculus. By utilizing a appropriate Lyapunov functional, we have shown that the Mixed ℋ∞ passive performance of complex dynamical networks is solvable if a set of linear matrix inequalities (LMIs) are feasible. Finally, numerical examples has been presented to demonstrate the effectiveness of the proposed method. In the future, considering the applications of fractional order complex valued neural networks and stochastic type complex valued neural networks are an interesting topic.



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