#### Abstract

This article examines mixed -infinity and passivity synchronization of Markovian jumping neutral-type complex dynamical network (MJNTCDN) models with randomly occurring coupling delays and actuator faults. The randomly occurring coupling delays are considered to design the complex dynamical networks in practice. These delays complied with certain Bernoulli distributed white noise sequences. The relevant data including limits of actuator faults, bounds of the nonlinear terms, and external disturbances are available for designing the controller structure. Novel Lyapunov–Krasovskii functional (LKF) is constructed to verify the stability of the error model and performance level. Jensen’s inequality and a new integral inequality are applied to derive the outcomes. Sufficient conditions for the synchronization error system (SES) are given in terms of linear matrix inequalities (LMIs), which can be analyzed easily by utilizing general numerical programming. Numerical illustrations are given to exhibit the usefulness of the obtained results.

#### 1. Introduction

Complex dynamical network (CDN) models are firstly investigated by Watts and Strogatz [1]. These models are sets of large-scale coupled nodes of interconnected systems, e.g., chemical substrates, microprocessors, and computers [2, 3]. Generally, CDN models are epitomized in practice, e.g., the World Wide Web, food webs, cellular and metabolic networks, neural systems, aviation systems, transportation systems, and electricity distribution systems [4, 5]. A number of these systems exhibit complexities in their topological and dynamical properties. The application of these networks can be found in streamlining the structure as well as reducing the control costs [6].

Particularly, synchronization remains normal along with the significant group of powerful performance as concerns CDNs, which has turned into an expanding concern furthermore, close inquire about the theme with its expressive ongoing years [7]. Synchronization is a regular and significant aggregate conduct of a complex network. As a key look into zone broad, endeavors are given to the comprehension of the synchronization system of such organization. As a central wonder in moving practices about systems, synchronization issue as concerns CDNs has appealed to an impressive consideration against different range, for example, environment, aerology, building, control matrices as well as banking sector [8, 9], because of its productive applications. Since inadequate data frequently show up in some practical systems, for example, just of excitatory neurotransmitters can be received between two associated cortical regions of brain systems [10], not all the diverted data of clients in wireless systems are known to the transmitter [11]. Immediately, the synchronization issue of complex networks with partial coupling is tended to in [12]. For the model, a time-varying complex dynamical networks model and its synchronization criteria are investigated in [13]. Synchronization of CDNs with a weighted time-varying adjacency matrix was explored by means of distributed adaptive control in [14]. The coupled network models are regularly utilized as scientific instruments to demonstrate in break down systems. For asynchronous complex networks, it is specifically noteworthy to locate a lot of controllers to accomplish synchronization of dynamical practices of all nodes in systems.

Markovian jump networks (MJNs) are a class of stochastic hybrid models. They are dynamic networks whose structures are subjected to random parameter that changes abrupt condition disturbances, changing subsystem interconnections that occur nonlinearly [15–21]. When CDN models get unexpected changes in their structure, we can represent them using Markovian jumping complex networks, which have been studied by many researchers [22,23].

In practical systems, time delays are regularly experienced. Consequently, time delays in couplings and in dynamical nodes have received considerable attention [24, 25]. On the other hand, time delays are unavoidable as a result of constrained transmission speeds and traffic congestion. Many researchers have investigated the effects of time delays on stability and synchronization [26]. Indeed, the existence of time delays can be found in numerous systems, such as atomic reactors, population dynamic models, aircraft stabilization, natural frameworks, chemical engineering systems, and ship stabilization circuit theory [27, 28].

It is natural that actuator faults occur frequently, which can lead to unstable system operation [29]. It is imperative to design controllers with the capability of ensuring the stability of closed-loop systems in the event of faults. In this regard, various control methodologies are available to realize the control objectives subject to faults. Fault-tolerant control (FTC) systems can be divided into two types: passive and active approaches. Passive FTC controllers usually have basic structures with restricted fault-tolerant capabilities. In contrast, active FTC controllers have structures with better self-organization. Fault identification or estimation generally requires dynamic fault-tolerant controllers that can be changed in accordance with data limitations (see [30, 31]).

Motivated by the above discussion, we study mixed -infinity and passivity-based synchronization of Markovian jumping neutral-type complex dynamical network (MJNTCDN) models with time-varying distributed coupling delays and actuator faults in this paper that are as follows:(1)We analyze the mixed -infinity and passivity synchronization of MJNTCDN models with distributed random coupling time-varying delays and actuator faults(2)The randomly occurring coupling delays satisfy the Bernoulli random binary procedure(3)Delay-dependent conditions are derived to guarantee the MJNTCDN models is mixed -infinity and passive performance at level (4)Improved Jensen’s inequalities and integral inequalities are utilized to infer the sufficient conditions in terms of LMIs(5)Numerical results are provided to exhibit the effectiveness of the proposed method

*Notations*. The following notations are used throughout this paper. and denote the -dimensional Euclidean space and the set of all real matrices, respectively, while means that matrix is real symmetric and positive definite. Superscript “” stands for the transpose; stands for a block diagonal matrix; denotes the matrix inverse of ; stands for the mathematical expectation operator with respect to the given probability measure; stands for the space of square-integrable vector functions over interval ; denotes the Euclidean norm of a vector and its induced norm of a matrix. The symbol “” is used to represent the term of a symmetric matrix which can be inferred by symmetry; symbol “” stands for the Kronecker product. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. Given a complete probability space with a natural filtration fulfilling the typical conditions (i.e., the filtration contains all -invalid sets and is right continuous), where is the sample space, is the algebra of events, and is the probability measure defined on . The process is a continuous homogeneous Markovian process with right continuous trajectories and takes values in the finite space . In particular, is related to the change likelihood network given by the transition rates:where is the visit time and . Here, is the change rate from mode at time to mode at time if and .

#### 2. Problem Statement and Preliminaries

We recognize a class of MJNTCDN models composed of indistinguishable coupled nodes, in which each node has -dimensional randomly occurring distributed coupling delays and actuator faults and position of the elements of the th hub is spoken to through the accompanying nonlinear dynamical subsystem:where stands for the state vector th node with respect to the model at time , is the measured output corresponding to the th node; is the continuous initial function of the th node; denotes the fault control input vector of the th hub; represents the constant external input vector which belongs to ; is continuously differentiable smooth nonlinear vector-valued field, which satisfies sector-bounded conditions which will be characterized later; are the outer-coupling configuration matrices representing the topological structure of the model, in which is defined as follows: if there is a connection from node to node , then the coupling matrix and ; otherwise, and . The diagonal elements are defined by are constant diagonal inner-coupling matrices. In addition, , , , , , , and are constant matrices with appropriate dimensions. For the time delay signals, , , and denote the retarded delays, distributed delays, and neutral time-varying delays, respectively, and are assumed to satisfy the following inequalities:where , , , , , , , and are real constant scalars.

The randomly occurring coupling delay denotes a stochastic variable which is in the form of a Bernoulli distributed sequence defined by

The probability occurrence of stochastic variable is given bywhere is a known constant.

Then, we can obtain

*Assumption 1. *For nonlinear function , there exist constant matrices and such thatfor any .

To synchronize all the indistinguishable nodes in model (2) to a common value, let us characterize the synchronization error vector be the synchronization error, where represents the state vector of the unforced isolate node and is chosen aswhere is the unforced isolate output vector. We define the synchronization error dynamics aswhere , , and .

The actuator fault model is used as [32, 33]where denotes the th agent and denotes the th actuator, is the input signal of the actuator, is the output signal from the actuator, is an unknown and piecewise continuous bounded actuator failure factor which shows the degree of effectiveness of the actuator, and is a known constant which represents the upper limits of , . Note that, when , there is no deficiency of the actuator, i.e., the th actuator of the th agent is sound or typical; when , the th actuator is subject to fault.

We indicateIn this case, a uniform actuator fault model is represented asTaking the actuator fault (13) into account, model (10) can be represented as follows:where .

The following state feedback controller law is employed for the synchronization error model:where is a real constant matrix representing the gain matrix of the feedback controller to be determined.

Therefore, the control input is substituted into the error model (14) leading toFor convenience, every conceivable estimation of is meant by , in the sequel. As such, we havewhere , , , , , , and for any are known constant matrices of appropriate dimensions.

Then, error dynamical model (16) in virtue of the Kronecker product can be written in the following compact form:whereIn the following, we present the associated definitions and lemmas, which are required to derive the main results of this paper.

*Definition 1 (see [34]). *Model (18) is said to have mixed -infinity and passive performance , if there exists a constant such thatfor any and any nonzero , where represents the parameter that defines the trade-off between -infinity and passive performance.

*Definition 2. *The mixed -infinity and passive synchronization of model (18) can achieve the mixed -infinity and passive performance at level .

*Definition 3 (see [35]). *For model (18), if there exists a scalar such that the derivative of the Lyapunov function with respect to time satisfiesthen model (18) with is said to be quadratically stable.

*Remark 1. *This article considered the complex dynamic network system with the linear coupling strength on the mode at time , while the rest of the coefficient is constant matrix, in the process of jump. This makes the complex dynamical network system more general.

*Remark 2. *Note that model (20) incorporates mixed -infinity and passivity performance level, which is a unique instance of dissipativity. For instance, if is estimated as 1, model (20) decreases to -infinity performance level; if is estimated as 0, model (20) decreases to the passivity performance; when takes , model (20) represents the mixed -infinity and passivity performance level.

Lemma 1 (see [36]). *For any matrix and , and given scalar , the vector function is such that the following relation holds:*

Lemma 2 (see [36]). *For any matrix and , and given a scalar function , the vector-valued function is such that the following relation holds:*

Lemma 3 (see [37]). *For a given matrices , , , and with appropriate dimensions, holds if and only if , .*

Lemma 4 (see [38]). *The Kronecker product has the following properties:*(1)*(2)**(3)**(4)*

#### 3. Main Results

In this section, we establish the LMI to determine a fault-tolerant controller which is mixed -infinity and passive with synchronization error model (18).

Theorem 1. *Given some constants such as , , , , , , , , , , , , and diagonal matrices , , , , and , are known constant matrices, error model (18) satisfies a mixed -infinity and passivity performance in the sense of Definitions 1 and 2, if there exist symmetric positive definite matrices , , , , and and are of appropriate dimension matrices such that the following successive LMIs hold:whereand the remaining parameters are zero. Then, the feedback controller gain is given by which stabilizes the error dynamics.*

*Proof. *We consider the following LKF with integral terms for the synchronization error model (18)whereWe use to denote the weak infinitesimal operator of [39], which is defined asThe derivative and mathematical expectation of (26) along the trajectories of the error model (18) isAccording to Lemmas 1 and 2, the integral terms in (29) can be rewritten asApplying Lemma 1 in (29), it follows thatBesides, as per Assumption 1, we can obtain the following inequality:Besides that, for any matrix , with appropriate dimension and scalars , , and taking the mathematical expectation, the following condition holds:For the performance index, we can obtainSubstituting (32) and (33) into , and from (29)–(31), it yields thatwhere