Event-triggered scheduling for pinning networks of coupled dynamical systems under stochastically fast switching

This paper studies the stability of linearly coupled dynamical systems with feedback pinning algorithms. Here, both the coupling matrix and the set of pinned-nodes are time-varying, induced by stochastic processes. Event-triggered rules are employed in both diffusion coupling and feedback pinning terms, which can reduce the actuation and communication loads. Two event-triggered rules are proposed and it is proved that if the system with time-average couplings and pinning gains is stable and the switching of coupling matrices and pinned nodes is sufﬁciently fast, the proposed event-triggered strategies can stabilize the system. Moreover, Zeno behaviour can be excluded for all nodes. Numerical examples of networks of mobile agents are presented to illustrate the theoretical results.


INTRODUCTION
In recent years, the study of complex dynamical networks has attracted broad attentions due to its potential applications in various fields [1][2][3][4]. Among them, the synchronization problem is a hot research topic and has been extensively investigated [5,6]. In dynamical networks, synchronization means that all nodes approach to a uniform dynamical behaviour and are generally assured by the couplings among nodes and/or external distributed and cooperative control. Many control strategies are taken into account to steer the synchronization dynamics to a desired trajectory. Among them, pinning control is an effective scheme, which was introduced in [7] to study the coupled map lattice and was then extended to complex networks by applying some local feedback controllers only to a fraction of nodes. Pinning control strategies for different models have been proposed and investigated. For example, pinning controllability of complex networks was studied in [8], pinning synchronization of complex dynamical networks was investigated in [9,10], pinning complex dynamical networks via a single controller was considered in [11], and an adaptive pinning strategy for fuzzy coupled neural networks was proposed in [12].
In recent decades, a number of researchers have suggested that the event-based control algorithms can reduce communication and computation loads in networked systems while maintaining control performance [13][14][15][16][17][18][19][20][21]. Dimarogonas et al. [13] investigated centralized and distributed formulation of eventdriven strategies for consensus of multi-agent systems. Hu et al. [14] addressed output consensus of heterogeneous linear multiagent systems via event-triggered and self-triggered control. In [16,17], consensus problem of second-order multi-agent systems with event-triggered control was investigated. Eventtriggered control was also applied to pinning synchronization problem of complex networks [18][19][20][21]. Lu et al. [18] studied pinning synchronization of linearly coupled dynamical systems with time-varying coupling matrix and pinned node set induced by a Markov chain. Furthermore, pinning synchronization problem of Markovian switching complex networks with partly unknown transition rates was studied in [19]. Liu et al. [20] investigated cluster pinning synchronization problem for complex dynamical networks with switching signal characterized by average dwell-time constraint. In [21], the pinning quasi-synchronization problem of Markovian switching heterogeneous networks was studied. In [22], a novel memory sampled-data control scheme was proposed to ensure the synchronization of semi-Markov jumping complex networks. In these works, the network topology is assumed to be static or slowly switching.
However, in many biological and engineering networks, the on-off interactions among nodes lead to a stochastically fast switching system [23][24][25][26], and in networks of mobile agents that are equipped with spatial pinning controllers [27], when agents move randomly and rapidly, it will induce a stochastically fast switching of the network topologies and pinned node sets. Therefore, it is necessary to study the dynamic behaviours of fast switching complex networks. Wang et al. [28] investigated the state estimation issue for switched complex dynamical networks governed by the persistent dwell-time switching signal. Although the persistent dwell-time regulation could handle both slow and fast switching, the stability of every subsystem among switching is required. In our previous work [29], pinning synchronization problem of fast Markovian switching complex networks was investigated, and then the event-triggered control was employed to the fast Markovian switching system in [30]. However, the Markovian switching is restrictive because in practice switching rules can be non-Markovian, leading to a more general switching rule with boarder applicability. Moreover, the proposed event-triggered strategies in [30] cannot rule out Zeno behaviour for some cases. This motivates the present study.
This paper works on the event-triggered scheduling for complex dynamical networks under stochastically fast switching. Sufficient conditions are given to guarantee the pinning synchronization of coupled dynamical systems under fast switching couplings and event-triggered communications. The main contributions and novelties can be illustrated as follows.
First, the concerned complex dynamical networks under stochastically fast switching can be employed to describe a broader class of practical stochastic systems. Second, two novel event-triggered rules are proposed to stabilize the switched complex dynamical networks. Under the proposed eventtriggered rules, the interevent interval will be greater than some positive number and Zeno behaviour will be excluded. Third, criteria for synchronization of coupled dynamical systems under fast switching couplings and event-triggered communications are derived. In comparison with the existing literature, we do not suppose the stability of every subsystem, that is, subsystems among switching can be unstable.
This paper is organized as follows. In Section 2, the underlying problem is formulated. In Section 3, we propose the eventtriggered schemes of diffusion configuration and pinning terms to pin the coupled systems to a homogenous pre-assigned trajectory of the uncoupled node system. Theoretical analysis is given in this section. Simulations are given in Section 4 to verify the theoretical results. In Section 5, we discuss future research directions and conclude the paper.
Notations: Denote the identity matrix of m dimensions by I m . A i j denotes the (i, j )th element of matrix A and A ⊤ the transpose of A. For a square matrix A, A sym = (A + A ⊤ )∕2 denotes its symmetry part; A > 0(≥ 0) denotes that A is positive (semi-) definite and A < 0(≤ 0) denotes A is negative (semi-) definite; (A) and (A) are the largest and smallest eigenvalues in module, respectively. For a symmetric matrix B, denote its i-th largest eigenvalue by i (B). ‖A‖ denotes the matrix norm of A induced by the vector norm ‖ ⋅ ‖. For a matrix A, In particular, without special notes, L 2 -vector norm is used in this paper and denote it by The symbol ⊗ represents the Kronecker product.

PROBLEM FORMULATION
Consider linearly coupled ordinary differential equations (LCODEs) as follows: where x i (t ) ∈ ℝ n denotes the state vector of node i; the continuous map f (⋅, ⋅) ∶ ℝ n × ℝ + → ℝ n denotes the node dynamics if there are no couplings; c is the uniform coupling strength at each node; L i j (t ) ≥ 0 for i, j = 1, … , m, i ≠ j , denote the coupling coefficients, and L ii (t ) = − ∑ j ≠i L i j (t ); Γ = [Γ kl ] n k,l =1 ∈ ℝ n,n is the inner configuration matrix with Γ kl ≠ 0 if two nodes are connected by the k-th and l -th state components, respectively.
The desired trajectory s(t ) satisfies: where s(t ) can be an equilibrium, a periodic orbit or even a chaotic orbit. The pinning controlled network is described as follow:ẋ where D i (t ) = 1 if note i is pinned at time t , otherwise D i (t ) = 0; is the pinning strength gain over the coupling strength.
Denote D(t ) = diag{D 1 (t ), … , D m (t )}. This paper supposes the network topology and pinned node set are switching with respect to time and the switching rate is finite in any time period [0, t ], namely, that only finite switches occur in any finite time interval. Hence, the solution of (1) exists for the interval [0, +∞) and is unique. Moreover, we assume L(t ) and D(t ) satisfy the following assumption. Assumption 1. For matrices L(t ) and D(t ), there exist two average matricesL andD such that where the limits hold uniformly with respect to t .
Here the convergence of limits in Assumption 1 is uniform in the sense that there are two continuous and strictly decreasing [31][32][33]. There are many switching rules of time-varying coupling topologies and pinned node sets satisfy Assumption 1, including periodic switching, Markovian switching and the switching subject to the Cox process [34].
Under Assumption 1, it was proven in [29] that the switching system (1) is stable if the time-average system is stable and the network topology and pinned node sets switch sufficiently fast. Motivated by the setup in [31,33,35] to describe the fast switching network, we consider the coupling matrix and the pinned node matrix of the form L(t ∕ ) and D(t ∕ ), where parameter > 0 determines the switching speed of the network topology and pinned node sets.
The Lipschitz and(or) the QUAD condition are widely used in the synchronization literature [5,6,[8][9][10][11] to assure the nonincreasing of some proposed Lyapunov function. Throughout this paper, we assume the node dynamics f (x, t ) satisfies the following assumption.
In fact, we do not need the uniformly Lipschitz condition hold for all x, y ∈ ℝ n but for a region Λ ⊂ ℝ n which contains the global attractors of the coupling systems.
if there exist a positive definite matrix G ∈ ℝ n,n and constants ∈ ℝ, > 0, such that holds for all x, y ∈ ℝ n .
The QUAD condition implies that linear state feedback with suitable control gains can stabilize the system. Consider the system: , then a sufficiently large can guarantee that lim t →∞ ‖x(t ) − s(t )‖ = 0. It should be noted that the QUAD condition can be satisfied by many well-known systems, such as the Lorenz system, the Chen system, the Chua's circuit and so on. . ( For more discussions on the relations among QUAD, Lipschitz and contracting conditions, we refer readers to the work [36]. This study employs an event-triggered strategy to pin the network to the desired trajectory. Each node updates the diffusion coupling term and pinning control term (if pinned) at its latest event-triggered time point based on some criteria, which giveṡ Here, t i k , k ∈ ℕ are the event-triggered time points for node i. At triggering time t i k , node i collects the state information of its neighbours and the target (if pinned), then updates its coupling and pinning control (if pinned) terms accordingly. This kind of updating rule is called the pull-based one [37] and adopted in many works [14,18,38,39]. The triggered event is defined based on the neighbours', the target trajectory's and its own states with some prescribed rule to be defined later.
Here, the diffusion coupling term of a node will remain unchanged until its next triggering time, even if one of its neighbours is triggered in this period. This could reduce computation loads of the network. However, every node needs to acquire its neighbours' states and the target state (if pinned) at every triggering time. To realize this, one can equip each node with an embedded microprocessor, which is in charge of information collection, computation and controller actuation [13]. The microprocessor monitors the triggering condition. It will update the controller and actuate the latest controller to the node once an event is triggered.
In the following, we study the stability of trajectory s(t ) of system (4). (4) is said to be stable at s(t ) in the mean square sense, if

Stability analysis
Theorem 1. Suppose that there exist a diagonal matrix P > 0, a matrix G > 0 and positive constants , , c, such that: Then there exist * > 0 such that for any 0 < ≤ * , system (4) is stable at s(t ) in the mean square sense, under either of the following two updating rules: (1) set t i k+1 as the next triggering time point by the rule (2) set t i k+1 as the next triggering time point by the rulê where ′ < , T ≥ T 1 can be any positive constants and Moreover, Zeno behaviour can be excluded for each node.
Proof of Theorem 1 is deferred to Appendix A.1.
Remark 1. Zeno behaviour refers to the phenomenon that an infinite number of events happens in finite time period. It can be excluded if the interevent interval is greater than some positive number. Inspired by [14,38], two positive constants T 1 and T 2 are incorporated in rule (8) and (10), respectively, such that the interevent interval is lower bounded by T 1 or T 2 . Specifically, for event-triggered rule (7), (8), t i k+1 − t i k ≥ T 1 holds for ∀i, ∀k, while for event-triggered rule (9),(10), t i k+1 − t i k ≥ T 2 holds for ∀i, ∀k.
For the average coupling matrixL, define its underlying graph by (L) in such a way that there exists a link from the i-th vertex to the j -th one if and only ifL i j > 0. For the average pinned matrixD, we denote the pinned node set by  (D) and define that j ∈  (D) if and only ifD ii > 0. Then Condition 3 in the above Theorem can be obtained by the following remark.
Remark 2. If the pinned node set  (D) can access all other nodes in the graph (L), there exist positive constants , c, and a positive diagonal matrix P such that {P ( I m + cL − cD)} sym ≤ 0, see [29] for the detailed proof.
In [10], it was proved that if the pinned node set can access all other nodes in the graph and the QUAD condition is satisfied, then the dynamics of nodes can be steered to the desired trajectory. Therefore, Conditions 1-3 in Theorem 1 implies that the system with the average coupling matrix and the average pinning gains is stable at s(t ). In other words, Theorem 1 indicates that if the system with the average coupling matrix and average pinning gains is stable and the switching is sufficiently fast, switched system could be stabilized under the event-triggered schemes.
Remark 3. In Theorem 1, triggering rule (7), (8) is based on error e i (t ), i = 1, … , m in (5), which depends on the coupling coefficients and pinned node sets in average, while triggering rule (9), (10) is based on errorê i (t ), i = 1, … , m in (6), depending on the instantaneous coupling coefficients and pinned node sets. Different from triggering rules in [30], positive constants T 1 and T 2 are incorporated into the triggering rules.

3.2
The switching rule follows an ergodic Cox process The Cox process t is a combination of two processes: { n , n ∈ ℕ} as a discrete adapted process with respect to filtration (Ω,  n ), n = 1, 2, …, embedded in (Ω,  ), and the switching point process {t n , n ∈ ℕ} [34]. In detail, let N [t 1 ,t 2 ) be the counting number of the switches in [t 1 , t 2 ) and N (t ) = N [0,t ) with N (0) = 1. All switches occur independently of each other and the switching rate (t ) follows a stochastic process depending on the adapted process { n , n ∈ ℕ}: (t ) = ( N (t ) ), where (⋅) is a positive measurable function with respect to (Ω,  N (t ) ) with upper bound 0 , that is, ( ) ≤ 0 for all ∈ Ω. Namely, for Thus, the point process N (t ) is also an adapted process, measurable on some filtration, denoted by  t . The process t is right-continuous and defined as Thus ,{ t , t ∈ ℝ + } is a well-defined adapted process with respect to a joint filtration (Ω,  t ×  N (t ) ). Denote the state space of the adapted process { n , n ∈ ℕ} by . An ergodic Cox process { t , t ∈ ℝ + } satisfies that: holds for some probability distribution (i ), i ∈ and any t with probability 1. One can check that (2) hold for L( t ) and Here, we suppose the switching of coupling topologies and pinned node sets follow an ergodic Cox process t ∕ , where is the parameter that affects the switching rate of coupling topologies and pinned node sets. Then we have the following corollary.
Take a constant ′ ∈ (0, ) with given in (3) and consider system (4) with switching process t ∕ , we have that (2) under updating rule (9),(10), the system is stable at s(t ) in the mean square sense if where T ≥ T 1 is any positive constant and Proof of corollary 1 is deferred to Appendix A.2.
Remark 4. In the above updating rules, continuous-time monitoring is needed to calculate the next triggering time. To get rid of continuous monitoring, the discrete-time monitoring strategy can be derived analog to the procedures given in our previous work [18,39]. However, it should be pointed out that as a payoff for the small cost of discrete monitoring, triggering events will happen more frequently than those in continuous-time monitoring.

NUMERICAL SIMULATIONS
We employ theoretical results to the "random waypoint" (RWP) model [40], which is widely used in the performance evaluation of protocols of ad hoc networks. The RWP model contains (among some other technical assumptions) the following conditions: realistic movements of agents, a sensible transmission range, and limited buffering storage spaces. In detail, we consider m agents moving in the planar space Λ ∈ ℝ 2 according to the RWP model. The agent moves towards a randomly selected target with a random velocity. After approaching the target, the agent waits for a time interval of random length and then continues the process. The motion of each agent is stochastically independent of the others and of time. Two agents are considered to be coupled at time t with adjacent coefficient 1 if the distance between them is less than a given interaction radius r > 0.  [41]. For networks with time-varying links, implementing a selective pinning is difficult, since the controller must move with the pinned agents. Inspired by [27], the reference model is equipped in a fixed region Λ c ⊂ Λ such that once an agent enters Λ c , it can gather the information of the reference, in other words, it is pinned. Therefore, the pinned node set is assigned by the spatial distribution of the agents at each time. For a pinned subset  ⊂ {1, … , m}, we define a pinning control diagonal matrix D in such a way: if i ∈ , then D ii = 1, otherwise D ii = 0. Denote 2 = {D(1), … , D(N 2 )} all possible pinning control matrices and D( t ) the pinning control matrix at time t , here t ∶ ℝ + → {1, … , N 2 } Similarly, D( t ) is also a homogeneous continuous Markov chain with a finite state space 2 . Hence, ( t , t ) is a higher dimensional homogeneous Markov chain. For convenience, we suppose t = t in the following.
The node distribution of RWP model was studied in [42], which proved that the node distribution is ergodic and the stationary distribution has strictly positive probability everywhere in the region. That is, every pair of agents has a positive probability to be coupled and every agent has a positive probability to be pinned, with the stationary distribution. Therefore, the expected coupling matrixL corresponds to a complete graph with all non-diagonal elements being identical, and the diagonals ofD are all positive and identical.
As shown in Figure 2, the events of updating rule based on the instantaneous couplings and pinning gains are more than the rule based on average ones. As a trade-off, the performance of the updating rule (9),(10) in terms of convergence rate of V (t ) is higher than the updating rule (7), (8), as shown in Figure 1.

DISCUSSIONS AND CONCLUSIONS
This study employs the event-triggered configurations and pinning control to stabilize linearly coupled dynamical systems with fast switching in both coupling matrix and pinned node set, towards reducing communication and computation loads. The event-triggered rules were proved to perform well and can exclude Zeno behaviours. Simulations were given to verify these theoretical results for networks of mobile agents. The present study has several limitations. First, we assume the state information of the complex network is available. This assumption has been widely made in the literature. This assumption is reasonable because obtaining the state information is a research problem that is orthogonal to the present study.
Recently, various state estimation methods for complex dynamical networks under different constraints have been developed. For example, delay-compensation-based state estimation for networks with communication delays, fading observations and dynamical bias disturbances was considered in [44], event-triggered state estimation was investigated for networks with bounded distributed delay [45] and networks with randomly switching topologies and multiple missing measurements [46]. It is an interesting future work to study the event-triggered state estimation for complex dynamical networks under stochastically fast switching.
Second, we apply the pinning control strategy with local feedback controllers. Additionally, we assume the feedback gains of the controllers and the coupling strength are constant. Future work will focus on developing effective control strategies for fast switching complex dynamical networks, including sliding mode control and adaptive pinning control strategy.
Third, time-delays and non-linearity are inevitable for complex dynamical networks. Therefore, it is a significant future work to explore event-triggered strategies for switched complex dynamical networks with constraints, such as quantization or transmission delays.  (4), the measurements e i (t ) andê i (t ) defined in (5) and (6) satisfy that: (11) and (12); (2) ‖e i (t )‖ ≤ ‖x(t )‖ holds for any i and t ∈ [t i k , t i k + T 1 ] with , T 1 given in (11) and (12).
Proof. The dynamics ofx i (t ) satisfies:x Then, applying the Lipschitz assumption of f and the assumption ‖ê i (t )‖ ≤ ‖x(t )‖ on system (A1), we have that Noting that L(t ∕ ) and D(t ∕ ) are right continuous, the righthand Dini derivative ofê i (t ) is By estimations (A2) and (A3), we have whereĉ 2 is given in (13). Then it follows that ) .

A.1
Proof of Theorem 1 Proof. We only prove that system (4) is stable in mean square sense under the triggering rule (7) and (8). The same procedure can be adapted to obtain the stability of system under the triggering rule (9) and (10).