Robust Dynamic Output Feedback Event-Triggering Synchronization for Complex Dynamical Networks

In this study, we investigate the co-design of dynamic output feedback synchronization controllers and event-triggering conditions for a class of complex dynamical networks (CDNs). The measurement output and control input are transmitted to the controller and actuators asynchronously over the individual communication channels of each network node. We address the sufficient conditions for the $\mathcal {L}_{2}$ stability of the error dynamics between the isolated and network nodes in the presence of external disturbances and noise in the measurement outputs and control inputs. Finally, we propose co-design conditions of the dynamic output feedback law and event-triggering conditions in terms of linear matrix inequalities (LMIs). Based on the proposed LMIs, we formulate an optimization problem to reduce the number of transmissions over each channel of the nodes. Numerical example results are provided to show the effectiveness of the proposed method.


I. INTRODUCTION
Complex dynamical networks (CDN) are typical large-scale dynamical systems with a large number of nodes, complex dynamic behaviors and topological structures and widely used in many natural and artificial systems. For example, the nervous system can be described as a network formed by many nerve cells interconnected through nerve fibers; a computer network can be seen as a network formed by autonomously working computers connected through communication media. Similarly, power, social relationship transportation networks, and dispatch networks are types of CDNs [1]- [3].
As one of the most important collective behaviors, the synchronization of CDNs has gained much attention in the past few decades and various types of control methods have been developed to deal with the synchronization problem of CDNs, such as continuous feedback control [4], adaptive control [5], intermittent control [6], pinning control [7], fuzzy The associate editor coordinating the review of this manuscript and approving it for publication was Feiqi Deng .
Lately, event-triggered control (ETC) has been considered an energy-efficient strategy to achieve the synchronization of CDNs [13], [14]. Under the framework of the ETC, sampled information is sent only at the instant when the error between the currently measured signal and the most recently transmitted signal exceeds a prescribed threshold. Therefore, this control strategy can effectively reduce the communication resources and bandwidth of networks by avoiding unnecessary signal transmission [15], [16]. Furthermore, a small transmission reduction for each node will incrementally be a significant reduction for practical applications in large CDNs. Many ETC schemes have been proposed owing to their strength and importance. For instance, [17] proposed a mechanism for event-triggering distributed sampling information to achieve synchronization in CDNs. In [18], exponential synchronization problems for Markovian jump delayed complex networks with partially unknown transition rates were investigated using a randomly occurring ETC strategy. In [19], the synchronization problem for a class of CDNs with uncertain VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ inner coupling was addressed by applying ETC methodology.
In [20], pinning event-triggered synchronization control for directed CDNs was studied. The periodic ETC approach of discrete-time CDNs was investigated and provided sufficient conditions for the ultimately bounded synchronization, see [21]. In [22], the synchronization problem for a class of discrete time-delay CDNs was studied by applying a discrete-time version of the dynamic event-triggering mechanism in terms of the absolute errors between control input updates. Please refer to [23]- [25], and references therein. Although the above-mentioned synchronization strategies have brought about significant advances, most studies have focused on full-state feedback controllers. However, full-state information is not always available in many realworld applications; it is often unbearably expensive to fully access all network states owing to the large scale of CDNs. Thus, it is essential to investigate the output-based ETC for CDNs. Few studies have presented observer-based event-triggering synchronization approaches. State estimation based event-triggered synchronization methods for a class of CDNs with mixed time delays [26], time-varying delays [27], and time-varying coupling weights [28] have been studied. In these studies, it was difficult to find the optimal solutions for the entire network system because the estimator and controller were designed separately. For example, the existing observer-based synchronization methods designed the observer-based controller. Based on the separately designed controller parameters, event-triggering conditions then designed in the sequel, meaning that the existing design methods does not provide the optimal controller. Recent studies on ETC systems have investigated the co-design of dynamic output feedback and event-triggering conditions. In [29], a co-design scheme to simultaneously synthesize the dynamic output feedback laws and event-triggering conditions for linear systems using linear matrix inequalities (LMIs) was proposed based on the analysis in [30]. Moreover, the results were used to formulate an optimization problem for designing an ETC that reduced the number of transmissions. [31] presented an improved co-design method for the output feedback law and event-triggering conditions for linear systems by applying additional constraints with free variables. Notably, the works on the co-design methods considered ideal linear systems, whereas the CDNs are nonlinear systems with complex interconnections.
In light of the above, we propose an effective strategy for the co-design of a robust dynamic output feedback law and triggering conditions to synchronize CDNs. We design individual controllers for each node to achieve synchronization. The measurement output and control input are transmitted to the controller and actuators asynchronously by an event-triggering mechanism. The output and control inputs are affected by the measurement disturbance and input noise, respectively. The event-triggering mechanism of the sensor-to-controller channel and that of the controller-toactuator operates asynchronously, reflecting more practical environment, while the event-triggering mechanisms in the existing literature operate synchronously [20]- [28]. We provide sufficient conditions for L 2 stability for the error dynamics between the isolated node and each node of the network. Furthermore, we propose sufficient conditions for the co-design of the dynamic output feedback law and eventtriggering conditions. The LMI-relaxation of the co-design conditions is achieved by introducing the zero equation and by using Young's inequality. By using the proposed LMI conditions, we can optimize the event-triggered dynamic output feedback synchronization controller to reduce the number of transmissions. Numerical examples demonstrated the effectiveness of the proposed method. The main contributions of this study are summarized as follows: 1) To the best of our knowledge, this study is the first to address the co-design of the dynamic output feedback law and event-triggering conditions for the synchronization of CDNs. 2) We provide sufficient conditions for the L 2 stability and the co-design of the dynamic output feedback law and event-triggering conditions. 3) A convex optimization problem is formulated based on the proposed LMI-based design, which optimizes the dynamic output feedback law and event-triggering conditions to reduce the number of transmissions. The remainder of this paper is organized as follows. Section II describes the preliminaries and model descriptions of the CDNs. In section III, we provide sufficient conditions for L 2 stability. Subsequently, we present the LMI-based design conditions of dynamic output feedback event-triggering synchronization controllers and an optimization problem. The numerical simulation results are presented in section IV. We denote the minimum and maximum eigenvalues of the symmetric matrix A as λ min (A) and λ max (A), respectively. We denote the transpose and inverse of the transpose of A as A T and A −T , respectively, and block-diagonal matrix with the entries A 1 , · · · , A N on the diagonal as diag(A 1 , · · · , A N ). The symbol represents the symmetric blocks. The Euclidean norm is denoted as · . We use (x, y) to represent the vector [x T , y T ] T for x ∈ R n and y ∈ R m . For a vector x ∈ R n , we denote |x| = √ x T x as its Euclidean norm. ⊗ denotes the Kroneker product which has the following properties:

A. PRELIMINARIES AND MODEL DESCRIPTION
Consider the following CDN with N identical coupled nodes:ẋ h ij x j + Bu i + Ew i , (1a) 51262 VOLUME 10, 2022 where x i ∈ R n x , u i ∈ R n u , and w i ∈ L 2 [0, ∞) are the system state, the control input, and the external disturbance of the ith node; i = 1, 2, · · · , N . y i ∈ R n y is the measured output, and d yi ∈ R n y is the measurement noise of the ith node, which is assumed that the noise signal d yi is absolutely continuous and its time-derivative exists for almost all points in time in L 2 .
A, R, B, E, and C a are the constant system matrices. c > 0 is the coupling strength. f (x i ) ∈ R n f is a nonlinear function of the plant. We assume that f (x i ) is β-Lipschitz with respect to x i , i.e., ∀x, y ∈ R n , f (x) − f (y) ≤ β x − y where β > 0 is called the Lipschitz constant of the function f . The network topology is indicated by the outer coupling matrix Suppose the following isolated nodė where s ∈ R n x and y s ∈ R n y are the solution and output, respectively, of the isolated node. Then, we define the following synchronization errors The objective of this study is to construct robust dynamic output feedback event-triggering synchronization controllers to guide CDNs (1) to synchronize with the isolated node (2). Namely, we design the controller to stabilize the errors e i for i = 1, · · · , N .
We use the following hybrid dynamical system framework to describe event-triggering control systems: where x ∈ R n x is the state vector, w ∈ R n w is the external disturbance, F and G represent the flow and jump maps, respectively. The system follows the flow map , if x is in the jump set J ), the system follows the jump map G(x, w). For more details on hybrid dynamical systems, refer to [32]. The definitions of L 2 norm of hybrid signals and L 2 stability for systems are as follows: 1 2 , provided that the right-hand side exists and is finite, where z ∈ L 2 .
Definition 2: System (4) is L 2 stable from the input (w) to the output z = h(x, w) with gain less than or equal to γ ≥ 0 if there exists κ ∈ K ∞ such that any solution pair (x, w) to (4) satisfies

B. EVENT-TRIGGERED CONTROL SYSTEM REPRESENTATION
In this subsection, we present the event-triggering mechanisms ETM yi and ETM ui , which manage the measurement output and control input data transmissions, respectively, for each node (Fig 1). For example, when t = t y i,k , k ∈ Z ≥0 , ETM yi allows measurement output y ei to be sent over the network. Subsequently, a zero-order hold element is used to keep the most recently transmitted data constant on the receiver's side. Therefore, we represent the network node and its output with ETM yi as follows: whereû i ∈ R n u is the sampled value of the control input u ∈ R n u ,ŷ ei ∈ R n y is the last transmitted value of the output measurement y ei ∈ R n y , and τ yi is the elapsed time since the most recent transmission instant of the measured output. d yi ∈ R n y is the measurement noise. The transmission instant is determined by the triggering laws, the design method of which is presented below. ETM ui operates analogously to ETM yi for control input data transmission. ETM ui allows u i to be sent over the network at t = t u i,k , k ∈ Z ≥0 , otherwise, transmission is restricted. ETM ui operates asynchronously with ETM yi . Therefore, the dynamic output feedback synchronization controller with ETM ui shown in Fig. 1 is given as follows: where φ i ∈ R n φ is the controller state, τ ui is the elapsed time since the most recent transmission instant of the control input, and d ui ∈ R n u is the noise that corrupts the control input. Noise d ui can be quantization errors, model computational glitches, or any disturbance that can affect the control input. We assume that the noise signals d ui are absolutely continuous, that is, its time derivative exists for almost all points in time in L 2 .

A. STABILITY ANALYSIS
In this subsection, the L 2 stability analysis of system (7) is presented. We develop the following lemmas to prove our main theorem.
Remark 1: Lemma 2 extends the result in [30] to the stability of the CDN synchronization problem. Furthermore, Lemma 2 separates the disturbances into two termsw andd when selecting the function α s in Lemma 2.
The following theorem provides the sufficient conditions for L 2 -stability of system (7) with the flow and jump sets defined in (8) in terms of LMIs.
Remark 2: In [30], the LMI-based L 2 stabilization conditions (Proposition 1) were addressed for a linear system. In this study, we derive the LMI-based sufficient conditions for the L 2 stability of the nonlinear error system of CDNs (7).
We use (14) to obtain sufficient conditions for the co-design of the dynamic output feedback laws and eventtriggering conditions. However, there are two challenges in achieving this: 1) the nonlinear terms A T 2 A 2 and A T 3 A 3 make the LMI relaxation of (14) difficult when the dynamic control law, A c , B c , and C c are set as decision variables [29], [31]. VOLUME 10, 2022 To overcome this problem, [29], [31] added three additional LMI constraints to deal with the above-mentioned issue but the results are only available for ideal single linear systems; 2) Furthermore, the coupling matrix H makes the relaxation more difficult. To handle this challenge,H 3 is defined in Theorem 1.H 3 which is a zero equation ofC uL1 is expressed with the interconnected termH 1 , making it easier to obtain LMI conditions for the controller design. In the next subsection, we describe an ETC design for the synchronization of CDNs.

B. CONTROLLER DESIGN
In this subsection, a co-design method of the output feedback law and triggering condition is proposed for the synchronization of CDNs. The following theorem proposes the LMI-based design conditions and procedures for obtaining the controller parameters.

C. OPTIMIZATION PROBLEM
In this subsection, the following optimization problem is presented for the optimal controller design to reduce the number of transmissions of y ei and u i , which is transmitted if |ẽ yi | and |ẽ ui | violate the thresholds η y |y ei | and η u |u i |, respectively, as follows: where δ 1 , δ 2 , δ 3 , and δ 4 ≥ 0 are weight values. The formulated problem (20) indirectly enables the optimal maximum η yi and η ui in (8) to be obtained while guaranteeing the stability of system (7). That is, we can achieve the objective by minimizing µ y , µ u , σ y , and σ u of the relationships η y = µ y σ y − 1 2 , η u = (µ u σ u ) − 1 2 in Theorem 2 for the given positive values α 1 , α 2 , α 3 ,γd , and scalars λ y , λ u .

IV. NUMERICAL EXAMPLES
In this section, a numerical example and simulation results are presented to illustrate the feasibility and effectiveness of the proposed conditions. We consider the synchronization of a CDN with three coupled identical nodes which are Chua's chaotic circuit [34] as follows: . According to (7) and (10), we can describe the CDNs and controlled output as follows:   We set λ u = λ y = 0.1, and δ 1 = δ 3 = 1, δ 2 = δ 4 = 30 and used a Nelder-Mead simplex algorithm, available as fminsearch in the optimization toolbox of MATLAB [35] to select the parameters α k for k ∈ {1, 2, 3}. Given γ = 50, the obtained values were α 1 = 0.0006, α 2 = 0.2159, α 3 = 0.8324, ρ = 1.9946, and ε H = 0.0149. Here, we set γw = γd = γ 2 . Using these parameters, we solved the optimization problem (20), and the control law and triggering conditions were obtained as follows:  x 2 (0) = (−0.5, 0.9, 0.5), x 3 (0) = (0.6, 0.5, −0.5), and φ i (0) = (1.5, −1.5, 1.5) for i = 1, 2, 3. The unknown exogenous disturbance for each node is w i ∈ R 3 is a random variable, which is uniformly generated within the range [−0.05, 0.05] for any t ≥ 0; the noise on the output and control input is d yi (t) = 0.005 sin (50t) and d ui (t) = 0.005 sin (60t), respectively for any t ≥ 0. Figs. 2-5 show the time-domain simulation results. The minimum inter-event times T y and T u are selected as 0.0043 and 0.0068 respectively. Fig. 2 shows the chaotic behavior of Chua's circuit and the trajectories of the synchronization errors e i of CDNs. One can observe that synchronization of the CDNs is successfully achieved by Theorem 2. In Figs.3-5, we can observe the trajectories and release intervals of the measurement output y ei (t) and control input u i (t) for the nodes, respectively. The transmission interval varies and depends on the error between the lastly transmitted signal and current signal. When the error was small, the release interval was large because the triggering condition was not satisfied. We can see that the synchronization was achieved despite of the intermittent transmission of the input and outputs.

V. CONCLUSION
This study has proposed a co-design of the dynamic output feedback law and event-triggering conditions for a class of CDNs. The error dynamics between the isolated and network nodes with event-triggering conditions have been described using a hybrid dynamical system framework. The ETC systems have measurement outputs and dynamic output controller that transmits the measured output and control input using their sampling rules. First, we have presented sufficient conditions for the L 2 -stability of error systems for CDNs with ETC. We then have proposed LMI-based design conditions for the dynamic output feedback law and event-triggering conditions for CDNs. The LMI-based conditions have been obtained by applying the modified Young's inequality to deal with the interaction terms among the nodes of CDNs, and the number of transmissions have been minimized using the proposed LMIs. The simulation results have demonstrated that the proposed method effectively designed a controller to synchronize CDNs. In future research, the authors will be devoted to the robust output feedback event-triggering synchronization schemes for a class of the CDNs having the uncertainties in the node dynamics and controller. Furthermore, the co-design of static output feedback controller and event-triggering condition for CDNs is still an open problem and would be an interesting research direction.

APPENDIX
Before proceeding the proof of Theorem 2, we introduce the following lemmas. Lemma 3 [36]: For given matrices M and N with appropriate dimensions, we have for any invertible matrix S and scalar ε > 0, Lemma 4 [31]: For given matrix R ∈ R n×n , we have for any matrix T > 0 and scalar α, Proof of Theorem 2: Let us define the following matrices such that S 1i = P i S 2i and M i N T i = I − X i Y i with a positive definite matrix P i ∈ R nẽ i . Then, we can obtain where G i = λ 2 y C a 0 0 λ 2 u C ci . By applying Schur complement on them, one can obtain the following inequalities: After pre-and post-multiplying the inequalities in (25) by P −1 i , we have by Lemma 4: where P = diag(P 1 , · · · , P N ),Ḡ = diag(G 1 , · · · , G N ), T r = diag(T r1 , · · · ,T rN ) for r = 1, 2, 3. Therefore, (27), as shown at the bottom of the previous page, holds from (24) and (26). Applying the Schur complement on the first diagonal element of the second term of (27), condition (14) in Theorem 1 can be obtained withḠ 1 (7) with (8) is L 2 stable from (w,d) toz with an L 2 gain less than or equal to γ = max{γw, γd }.