#### Abstract

In this article, global asymptotic stability analysis, and mixed 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 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.

#### 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 of controllers/filters is to assure the closed-loop error systems are stable with an norm bound restricted to disturbance attenuation [18, 19]. On the other hand, the Mixed and passivity control has been proved [20, 21]. Recently, state feedback control of commensurate fractional-order systems and 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 and passive performance when designing controller is the ideal solution. It is really well worth bringing up that the 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 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 mixed 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 mixed 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 order .

*Definition 1 (see [29]). *The Caputo fractional derivative of order for a function is defined bywhere *n* is a positive integer. In particular, when , we have

Lemma 1 (see [30]). *Let be a differentiable vector-valued function. Then for , we havewhere is a symmetric positive definite matrix.*

Lemma 2 (see [31]). *If and , then**In particular, when , we have*

Lemma 3 (see [32]). *For a given matrix**With , , then the following conditions are equivalent:*

Lemma 4 (see [33]). *Let denote the notation of Kronecker product. Then, the following relationships hold*

Lemma 5 (see [34]). *Let , , and , , , and , with , then*

Lemma 6 (see [35]). *Let H, S be any real matrix, , , , , and , with , and and are functions and defined in system (1). Then for any vectors p and q with appropriate dimensions, the following matrix inequality holds:*

*Property 1 (see [9]). *If , then the following property

*Property 2 (see [9]). *For any constants , and two functions , , we have

*Assumption 1 (see [36]). *Bounded functions satisfying and for all Now the activation function , and , some constants are , , , , , and satisfyWe denote

#### 3. Main Result

Consider the fractional-order complex dynamical network model consisting of coupled nodes of the formwhere , be the state vector of the th node at time ; is Bernoulli distributed stochastic variable such that , is a coupling delay and assumed to satisfying and , where , are known constants. *c* is coupling strength; , , denote the connection weight matrix and the delayed connection weight matrix, respectively. and represent constants inner coupling matrix at time *t* and the time ; is real constant matrices with appropriate dimension. and are the coupled configuration matrix and its elements satisfies the following conditions: if there is a connection between the nodes *k* and *j*, then ; otherwise for and the diagonal elements = .

Combining with the sign of Kronecker product, model (1) can be rewritten as

*Definition 2 (see [35]). *The system (1) is said to be globally asymptotically stable with mixed and passivity performance , if the following requirements are satisfied simultaneously;(i)The system (1) is globally asymptotically stable whenever ;(ii)Under zero initial condition, there exists a scalar such that the following condition is satisfiedFor all and any non-zero , where denotes a weighting parameter that represents the trade-off between the weighted performance index and the passivity performance index.

Theorem 1. *For given , the system (1) is mixed and passive performance level , if there exist positive definite matrices , and positive diagonal matrices , , , , , such that the following LMIs holds*

*Proof. *Choose the Lyapunov functional candidate:It follows from Lemma 1 that we obtain the -order Caputo derivative of as follows:From Assumption, for any positive diagonal matrices , , , , one hasTo show the mixed and passivity performance of of system (1). Then we have the following estimate:where,From (4), we getIntegrating (8) with respect to *t* from 0 to , we getBy using Property 1, Property 2 and Lemma 2, we haveOn the other hand, we haveUnder zero initial condition, we obtainHence with zero initial condition. Therefore, we haveBy Definition 2, system (1) is mixed and passivity performance . The proof of theorem is completed.

Theorem 2. *The System (1) with is globally asymptotically stable if there exist positive definite matrix , and positive diagonal matrices , , , , , such that the following LMI holds*

*Proof. *Choose the Lyapunov functional candidate:It follows from Lemma 1 that we obtain the -order Caputo derivative of as follows:,From Assumption, for any positive diagonal matrices , , , , one hasCombining (32)–(34),where,From (10), . Therefore, the system (1) with is globally asymptotically stable. This completes the proof of the theorem.

*Remark 1. *The System (1) can be rewritten asCombining with the sign of Kronecker product, model (15) can be rewritten as

Theorem 3. *For given , the system (16) is mixed and passive performance level , if there exist positive definite matrices , and positive diagonal matrices , , , , , such that the following LMIs holds*

*Proof. *Choose the Lyapunov functional candidate:It follows from Lemma 1 that we obtain the -order Caputo derivative of as follows:From Assumption, for any positive diagonal matrices , , , , one hasTo show the mixed and passivity performance of of system (16). Then we have the following estimate:where,From (17), we getIntegrating (22) with respect to *t* from 0 to , we getBy using Property 1, Property 2 and Lemma 2, we haveOn the other hand, we haveUnder zero initial condition, we obtainHence with zero initial condition. Therefore, we haveBy Definition 2, system (16) is globally asymptotically stable with mixed and passivity performance . The proof of theorem is completed.

#### 4. State Feedback Control

The state feedback controller is designed as