Distributed adaptive event-triggered control for general linear singular multi-agent systems

1.   Introduction

The consensus control is one of the most basic distributed coordination control problems in multi-agent networks. In the past decade, the study of consensus in multi-agent systems has attracted considerable attention from a great many scholars. It is mainly due to the potential applications of multi-agent systems in various fields, such as physics, biology, control theory, and engineering, etc. ^[1,2,3,4]. The leaderless consensus problem and the leader-following consensus problem are the research basis of consensus. Their purpose is to design a suitable controller so that a group of agents reach agreement on a quantity of particular interest ^[5]. In practical applications, the distributed control protocol can greatly reduce the cost of information transmission and centralized control because the controller designed for each agent only relates to available information of its neighbors. In recent decades, some remarkable achievements have been made based on distributed consensus control for different systems, such as first-order/second-order continuous multi-agent system ^[6,7,8], general linear continuous multi-agent system ^[9,10,11], discrete-time multi-agent system ^[12,13,14], heterogeneous linear multi-agent system ^[15,16], and so on. It is worth noting that there are few results of singular multi-agent systems in the existing literature.

In general, singular multi-agent systems can be decomposed into the exponential mode part (i.e., the normal dynamics part) and the impulsive mode part (i.e., the specific part of the descriptor dynamics) ^[17]. Singular multi-agent systems are more felicitous to describe the complicated dynamics phenomenon than normal systems and have been extensively used to describe power systems, robot systems, economic systems and so on. As more and more scholars study singular systems, many excellent conclusions are drawn in this respect ^{[18,19,20,21]}. For example, the authors in ^[18] investigated the containment control problem of singular heterogeneous multi-agent systems by state feedback and output feedback control protocols. The containment control of singular heterogeneous multi-agent systems is studied under directed interaction topology in ^[19]. The bipartite output regulation problem for singular heterogeneous multi-agent systems with signed graph is considered by state feedback and output feedback control protocols in ^[21]. For singular multi-agent systems, it is worth noting that the existing distributed control strategies need to use the global information of communication topology, that is, the eigenvalue of the Laplace matrix $L$. In practical applications, it may be difficult to obtain for large-scale networks. Therefore, there are still many challenges to solve the consensus problem of the general liner singular multi-agent systems.

Meanwhile, too many communication resources will be used to update controllers and exchange information between neighbors. Therefore, we introduce an event-based mechanism to achieve the consensus of the singular multi-agent systems. However, there is no research on event-based distributed control strategy for singular systems in the existing literature, and most of the research is for general liner non-singular multi-agent systems ^{[22,23,24,25,26]}. Two distributed filters are introduced to solve the problem of distributed target tracking under cyber attacks for targets with the nonlinear dynamics ^[22]. In ^[24], the authors discussed the event-triggered distributed optimization problem of second-order nonlinear multi-agent systems under undirected and connected communication topologies. The authors in ^[25] put forward a dynamic event-triggered control law and discussed its application in smart grids and intelligent transportation systems in detail. Based on the above, the distributed event-triggered control strategies in those literatures all need the global information of the communication topology and thereby are not fully distributed.

Therefore, inspired by the above papers, we are interested in the leader-following consensus of singular multi-agent systems based on fully distributed adaptive event-triggered control law which is a challenging problem and needs further research. The main contributions of this paper are highlighted as follows. First, a novel fully distributed adaptive event-triggered controller is proposed to solve the leaderless consensus problem of singular multi-agent systems, which reduce the number of information transmission and the load and cost of the communication. Meanwhile, the proposed controller avoids using global information by introducing an adaptive gain. Second, we further extend it to solve the problem of leader-following consensus for general liner singular multi-agent systems. Third, it is proved that the inter-event times are lower bounded by a positive constant to exclude the Zeno behavior.

The rest of this paper is organized as follows. Section 2 presents some preliminary information as well as the problem formulation. Then, we propose two novel fully distributed event-triggered strategies for the leaderless consensus and leader-following consensus and present the corresponding results in Sections 3 and Section 4, respectively. Section 5 includes simulations that validate the effectiveness and feasibility of the designed control law. Section 6 contains the conclusions and the further study.

Notations: $\mathbb{R}^{m}$ and $\mathbb{R}^{n \times m}$ denote the set of $m$-dimensional real column vectors and $n \times m$ real matrices, respectively. Any matrix $A > 0$ means A is positive definite. $\otimes$ denotes the Kronecker product. $\Vert \cdot \Vert$ denote Euclidean vector norm or induced 2-norm for matrix. For any matrix $B$, $\lambda_{min}(B)$ and $\lambda_{2}(B)$ refer to the minimum and minimum nonzero eigenvalues, respectively.

2.   Problem formulation and preliminaries

2.1.   Graph theory

The communication graph $\mathcal{G}$ among the N agents is presented by a fixed undirected graph $\mathcal{G} = (\mathcal{V}, \mathcal{\varepsilon}, \mathcal{A})$ with a finite set of N nodes $\mathcal{V} = \{v_1, v_2, ..., v_N \}$, a set of edges $\mathcal{\varepsilon} \subset \mathcal{V} \times \mathcal{V}$, and the adjacency matrix $\mathcal{A} = [a_{ij}] \in \mathbb{R}^{N \times N}$. For $i, j \in \mathcal{V}$, where $a_{ij} > 0$, if $(i, j) \in \mathcal{\varepsilon}$; otherwise, $a_{ij} = 0$. Define the in-degree matrix as $\mathcal{D} = diag (d_i) \in \mathbb{R}^{N \times N }$ with $d_i = \sum_{j = 1}^N a_{ij}$. The Laplacian matrix is defined as $\mathcal{L} = \mathcal{D} - \mathcal{A}$.

2.2.   Problem formulation

In the subsection, a group of $N$ identical agents with continuous-time general linear singular dynamics are considered. The dynamics of the $i$th agent is given by

where $x_i(t) \in \mathbb{R}^{n}$ and $u_i(t)\in\mathbb{R}^{m}$ are the state information and the control input of the $i$th agent, respectively. $E, A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ are constant matrices with appropriate dimensions. The systems (2.1) is said to be singular, if $rank(E) < n$.

The main purpose of this article is to solve the adaptive event-triggered consensus problem for the singular agents in the systems (2.1). The critical of the adaptive event-triggered consensus problem is to design a fully distributed event-based consensus protocol that consist of the event-based control laws and the triggering functions for each agent. Specifically, the goal is to solve the adaptive event-triggered leaderless consensus problem for the singular agents in the systems (2.1) by ensuring that $\lim\limits_{t \rightarrow +\infty} \Vert x_i(t) - x_j(t) \Vert = 0$, where $i, j \in \{1, 2, ..., N \}$ and excluding Zeno behavior, i.e., the event-triggered number in a limited time is finite.

3.   Fully distributed adaptive event-triggered control law for leaderless consensus

In the section, we will design fully distributed adaptive event-triggered laws for leaderless consensus of the singular multi-agent systems. The following assumptions and lemmas are needed.

Assumption 3.1. $(E, A)$ is regular and $(E, A, B)$ is stabilizable.

Remark 3.1. Assumption 3.1 ensures that the adaptive event-triggered consensus problem is well posed. For the regularity of $(E, A)$, it makes sure the existence and uniqueness of the solution of the system equations (2.1) of the agents. For the stabilizable of $(E, A, B)$, it guarantees the existence of the distributed control law.

Assumption 3.2. The undirected graph $\mathcal{G}$ is connected, i.e., there exists a root agent that has directed paths to all other agents.

Remark 3.2. Similar to Assumption 3.1, Assumption 3.2 also guarantees that the adaptive event-triggered consensus problem is well posed. Without this assumption, at least one isolated agent can't get any information from other agents.

Lemma 3.1. ^[18] For $E, A \in \mathbb{R}^{n \times n}$, and $B \in \mathbb{R}^{n \times m}$, suppose that $(E, A)$ is regular and impulse free, and $(E, A, B)$ is R-controllable. Then, for any given positive definite matrices $Q > 0$, there exists a positive define matrix $P > 0$ which is the solution of the following equality:

Lemma 3.2. (Cauchys covergence criterion)^[27] The sequence $V(t_i)$, $i = 0, 1, 2, ...$ converges to something if and only if this holds: For any $\phi > 0$, there exists a positive number $\mathcal{W}_\phi$ such that $\forall s > \mathcal{W}_\phi$,

Compared with the traditional controller, we consider the following fully distributed adaptive event-triggered controller for the $i$th agent:

where $\alpha \in (0, 1)$, the gain matrix $K$ is to be designed; $\tilde{x}_i(t) \in \mathbb{R}^{n}$, which are described by (3.4), is the state estimate value of $x_i(t)$.

where the sampling instant $t_{k}^i$ means the $k$th triggering time instant of the $i$th agent.

Remark 3.3. The distributed control means that the $i$th agent only communicates with its neighbor agents, instead of communicating with all agents in (3.3). The event-triggered control laws of ^{[28,29,30,31]} require absolute real state information, namely $x_i(t_{k}^i)$ and $x_j(t_{k'}^j)$. Especially, the adaptive gains $c_{ij}(t)$ in ^[28] are required to satisfy $c_{ij}(t) = c_{ji}(t)$ for $\forall t \geq 0$. It undoubtedly complicates the triggering mechanism, raises the computing cost, and increases the communication load, implying that it is not feasible in some practical applications. However, the estimate value $\tilde{x}_i(t)$ of $x_i(t)$ only requires real state information at discrete event-triggered time instants, which greatly makes up these shortcomings. Meanwhile, in ^{[32,33,34,35]}, the adaptive event-triggered consensus protocols require the global eigenvalue information of the communication graph $\mathcal{G}$. In this paper, the gain $c_i(t)$ in (3.3) is adaptively adjusted which is based on sampling relative estimate value information to avoid using the global information of the Laplace matrix.

We further propose the following triggering condition to determine the sampling instants:

where $\mu > 0$, $\nu > 0$, $e_i(t) = \tilde{x}_i(t) - x_i(t)$ the measurement error and the error $z_i(t) = x_i(t) - \frac{1}{N}\sum_{k = 1}^Nx_k(t)$ between the state of each agent and the average state of all states, respectively. Obviously, $\tilde{x}_i(t) - \tilde{x}_j(t) = e_i(t) - e_j(t)+ (z_i(t) - z_j(t))$. Then, the time derivative of $e_i(t)$ can be written as

Therefore, (3.7) can be rewritten as

where $e(t) = col(e_1(t), e_2(t), ..., e_N(t))$, $c(t) = diag \{c_1(t), c_2(t), ..., c_N(t)\}$ and $z(t) = col(z_1(t), z_2(t), ..., z_N(t))$. Let $\varepsilon_i(t) = E x_i(t)$.

Then, the time derivative of $\varepsilon(t)$ can be written as

Definition 3.1. To simplify the calculation, define $\acute{e}_i(t) = \sum_{j = 1}^Na_{ij}(e_i(t) - e_j(t))$, $\acute{\tilde{x}}_i(t) = \sum_{j = 1}^Na_{ij}(\tilde{x}_i(t) - \tilde{x}_j(t))$, $\acute{z}_i(t) = \sum_{j = 1}^Na_{ij}(z_i(t) - z_j(t))$, $M = I_N - \frac{1}{N} \bf{11^T}$.

Therefore,

We are now ready to present the main results of this section.

Theorem 3.1. Given Assumptions 3.1, 3.2 and Lemma 3.1, the leader-following multi-agent system (2.1) under the control protocol (3.3) with the dynamic event-triggered strategy i.e. described as (3.5) and (3.6), will achieve leaderless consensus if $\alpha \in (0, 1)$ and $K = -B^TPE$.

Consider the Lyapunov-like function candidate

where $c_0$ is a positive constant satisfying $c_0 \geq \frac{2(1+\alpha)(1+\kappa)}{\lambda_{2}(L)\kappa(1-\alpha)(1-\alpha-\kappa)}$ with $(0 < \alpha < 1, 0 < \kappa < \frac{1}{\alpha} -1)$.

The time derivative of $V_1(t)$ along the trajectory of $z_i(t)$ in (3.1) with $K = -B^TPE$ is given by

Obviously, $LM = L$. Then,

Moreover, let $\Gamma = E^TPBB^TPE$,

Then, by triggering condition (3.6), we can further imply

Substituting (3.14) and (3.15) into (3.13) yields

It can be verified that

where $0 < \kappa < 1/\alpha-1$. According to (3.17) and the triggering condition (3.5), one has

Thus, it further follows that

where $\varrho = \frac{\kappa(1-\alpha)(1-\alpha-\alpha\kappa)}{2(1+\alpha)(1+\kappa)} > 0$ and $\theta = \frac{2N\mu}{1+\alpha} + \frac{c_0 (1-\alpha) \kappa\mu}{2(1+\alpha) \min c_i(0)} > 0$.

Under Assumption 3.2, the Laplacian matrix $L$ is symmetric and a $M$-matrix. Thus, there exists an orthogonal matrix $U$ such that $U^TLU = \Lambda = diag \{ \lambda_1, \lambda_2, ..., \lambda_N \}$, with $0 = \lambda_1 < \lambda_2\leq\lambda_3...\leq\lambda_N$. Let $\bar{z}(t) = (U^T \otimes I_N)z(t) = col(\bar{z}_1(t), \bar{z}_2(t), ..., \bar{z}_N(t), )$ and $\hat{z}_i(t) = E \bar{z}_i(t)$. Then, one has

where $Q$ satisfies the equality (3.1) and $\lambda_{min}(Q)$ is the minimal eigenvalue of $Q$. Consider the following function:

which is continuous on the interval $[0, +\infty)$. It follows from (3.20) that

Therefore, it can be seen from (3.22) that $W_1(t)$ is non-increasing on the interval $[0, +\infty)$. Because of $W_1(t) \geq 0$, $W_1(t)$ is bounded. It implies that $W_1(t)$ is bounded, i.e., $\lim\limits_{t \rightarrow +\infty}W_1(t)$ exists. Thus, from the definition of $W_1(t)$, it can be seen that $z_i(t)$ and $c_i(t)$ are bounded over $[0, +\infty)$, for $i, j = 1, 2, ..., N$.

Since $\lim\limits_{t \rightarrow +\infty}W_1(t)$ exists, by the Cauchy convergence criterion in Lemma 3.2, there exists a positive number $T_0$, such that for any $T_2 > T_1 > T_0$, $V_1(T_1) - V_1(T_2) < \varepsilon$. Suppose that the triggering time instants for all agents over $(T_1, T_2)$ are given by $t_l^1 < t_l^2 < t_l^3 < ... < t_l^r$. Then, it follows from (3.22) that

By the Cauchy criterion, we can conclude that $\lim\limits_{t \rightarrow +\infty} \lambda_{2}(L) \lambda_{min}(Q) \int_{0}^{t} \sum_{i = 1}^N \hat{z}_i^T(s) \hat{z}_i(s)\, ds$ exists, i.e., $\lim\limits_{t \rightarrow +\infty} \int_{0}^{t} \hat{z}^T(s) \hat{z}(s)\, ds$ exists. Because of (3.9) and (3.10), $\int_{0}^{t} \hat{z}^T(s) \hat{z}(s)\, ds$ is twice differentiable on each interval $[t_k^i, t_{k+1}^i)$. Note that the boundedness of $\hat{z}(t)$ and $\dot{\hat{z}}(t)$ are bounded over the interval $[0, +\infty)$. Therefore, there exists a positive constant H, such that

Then, by Lemma 1 in ^[36], i.e., the general Barbalat's lemma, $\lim\limits_{t \rightarrow +\infty} \hat{z}^T(t) \hat{z}(t) = 0$, which means that $\lim\limits_{t \rightarrow +\infty} z^T(t) z(t) = 0$.

The proof is thus completed.

The following theorem excludes the Zeno behavior.

Theorem 3.2. Under the conditions in Theorem 3.1, the network does not exhibit the Zeno behavior and the interval between two consecutive triggering instants for any agent is strictly positive.

Proof. For the $i$th agent at time interval $t \in [t_k^i, t_{k+1}^i)$, it follows from (3.8) that

From Theorem 3.1, we know that $z_i(t)$, $e_i(t)$ and $c_i(t)$ are bounded over $[0, +\infty)$, for $i = 1, 2, ..., N$. Therefore, $\frac{d \Vert K \sum_{j = 1}^Na_{ij}(e_i(t) - e_j(t)) \Vert ^2 }{dt}$ is an upper bound, which is assumed to be $\varpi_i$. Note that $\Vert K \sum_{j = 1}^Na_{ij}(e_i(t_k^i) - e_j(t_k^i)) \Vert ^2 = 0$. Using these two properties, one has

When the triggering function (3.6) is at $t = t_{k+1}^i$, there is

Combining (3.26) and (3.27), one has

According to the above inequality, it is obvious that $\tau_k^i = t_{k+1}^i - t_k^i > 0$ for any finite horizon. Thus, no agent exhibits Zeno behavior for each agent in the system for any limited time. The proof is over.

4.   Fully distributed adaptive event-triggered control law for leader-follower consensus

In this section, we consider the event-triggered consensus problem in the presence of a leader. Without loss of generality, assume that the agent indexed by $x_1(t)$ is the leader whose control input $u_1(t) = 0$. The communication graph $\overline{\mathcal{G}}$ among the agents is assumed to satisfy the following assumption.

Assumption 4.1. The pair $(A, B)$ in (2.1) is stabilizable. The subgraph associated with the followers is undirected and the graph $\overline{\mathcal{G}}$ contains a directed spanning tree with the leader as the root.

Because the leader has no neighbors, the Laplacian matrix $L$ can be partitioned as $L = \begin{bmatrix} 0 & 0_{1 \times (N-1)}\\ L_2&L_1 \end{bmatrix}$, where $L_1 \in \mathbb{R}^{(N-1) \times (N-1)}$ is symmetric and $L_2 \in \mathbb{R}^{(N-1) \times 1}$. In light of Lemma 1 in ^[28], $L_1$ is positive definite.

Definition 4.1. To simplify the calculation, define $\grave{e}_i(t) = \sum_{j = 2}^Na_{ij}(e_i(t) - e_j(t)) + a_{i1}e_i(t)$; $\grave{\tilde{x}}_i(t) = \sum_{j = 2}^Na_{ij}(\tilde{x}_i(t) - \tilde{x}_j(t)) + a_{i1}(\tilde{x}_i(t) - x_1(t))$; $\grave{z}_i(t) = \sum_{j = 2}^Na_{ij}(z_i(t) - z_j(t)) + a_{i1}z_i(t)$; $\tilde{z}_i(t) = \tilde{x}_i(t) - x_1(t)$.

For each follower, we propose the following adaptive event-based control law:

where $i = 2, 3, ..., N$.

We further propose the following triggering condition for determining the sampling instants:

where

For the $i$th follower, we define the measurement error $e_i(t) = \tilde{x}_i(t) - x_i(t)$ and the error $z_i(t) = x_i(t) - x_1(t)$, respectively. Then, the time derivative of $e_i(t)$ can be written as

where $H = L_1 + \Delta$ and $\Delta = diag \{a_{11}, a_{21}, ..., a_{N-1 1} \}$. Then, (4.4) can be rewritten as

Therefore, the time derivative of $z_i(t)$ can be written as

Then, (4.6) can be rewritten in a compact form as

Theorem 4.1. Given Assumption 4.1 and Lemma 3.1, the leader-following multi-agent system (2.1) under the control protocol (4.1) with the dynamic event-triggered strategy (4.2), will achieve consensus if $\alpha \in (0, 1)$ and $K = -B^TPE$.

Proof. Consider the Lyapunov-like function candidate

where $c_0 \geq \frac{2(1+\alpha)(1+\kappa)}{\lambda_{2}(L_1)\kappa(1-\alpha)(1-\alpha-\alpha\kappa)}$ with $0 < \alpha < 1, \; 0 < \kappa < \frac{1}{\alpha} -1$.

The time derivative of $V_2(t)$ along the trajectory of $Ez_i(t)$ in (4.7) with $K = -B^TPE$ is given by

Then, one has

Moreover,

The triggering condition (4.3) implies

Substituting (4.11) and (4.12) into (4.10), one gets

It can be verified that

where $\kappa$ is a positive constant which is determined later. By using of the triggering condition (4.3), one has

Thus, it further follows that

where $\varrho = \frac{\kappa(1-\alpha)(1-\alpha-\alpha\kappa)}{2(1+\alpha)(1+\kappa)} > 0$ and $\theta = \frac{2(N-1)\mu}{1+\alpha} + \frac{c_0 (1-\alpha) \kappa \mu}{2(1+\alpha) \min c_i(0)} > 0$.

Under Assumption 3.3, the Laplacian matrix $H$ is symmetric. Thus, there exists an orthogonal matrix $U$ such that $U^THU = \Lambda = diag \{ \lambda_1, \lambda_2, ..., \lambda_{N-1} \}$, with $0 = \lambda_1 < \lambda_2\leq...\leq \lambda_{N-1}$. Let $\bar{z}(t) = (U^T \otimes I_N)z(t) = col(\bar{z}_1(t), \bar{z}_2(t), ..., \bar{z}_{N-1}(t), )$. Then, it follows from condition in Theorem 3.1, $\hat{z}_i(t) = E \bar{z}_i(t)$ and (3.1) that

Then, the rest of the proof is similar with in Theorem 3.1.

Theorem 4.2. Under the conditions of Theorem 4.1, the Zeno behavior can be excluded in the leader-following system.

The proof of Theorem 4.2 is similar with Theorem 3.2 and it is omitted the details of the proof for brevity.

5.   Illustrative examples

In this section, an example is provided to demonstrate the effectiveness of the results obtained. Consider the singular multi-agent systems with six agents, Its communication topology is $\mathcal{G}$, which is described in Figure 1. The dynamic model of the $i$th agent is based on (2.1), where

In the trigger function (3.15), the parameter is selected as $\beta = 0.5$, $\gamma = 0.95$, $\mu = 8$ and $\nu = 3$. Later, to satisfy (3.1), we use the tool of the LMI toolbox to get

Then, for the initial conditions, $c_i(0) = 0$ and $x_i(0)$, $\tilde{x}_i(0)$ can be selected randomly, $i = 1, 2, ..., 6$. Finally, various initial conditions are simulated. The results corresponding to an initial condition are shown in the Figures 2–4. Adaptive parameters $c_i(t)$ is described in Figure 2, which ultimately converge to some positive values. Triggered instants under adaptive event-triggered control law in (3.6) is presented in Figure 3, which shows that the Zeno behavior can be excluded. Figure 4 shows that the errors of all agents converges to zero, which proves the effectiveness and authenticity of the proposed fully distributed adaptive event-triggered control strategy.

6.   Conclusions and future work

In the paper, we propose the fully distributed adaptive event-based control laws to solve the leaderless consensus problem and the leader-following consensus problem of the general linear singular multi-agent systems. The triggering conditions and the adaptive gains depend on the relative state estimate information between adjacent agents for each agent. Therefore, the proposed event-triggered protocol does not require continuous communication between neighbors, which greatly alleviate the negative impact of communication. Based on the designed adaptive gain, the protocol does not need to know the Laplace of the communication graph which can be designed in a fully distributed. It has been shown that Zeno behavior is avoided during triggering process. In addition, numerical examples show the effectiveness and reliability of the method. Extending the results to heterogeneous singular multi-agent systems is a promising future direction.

Acknowledgments

This research was supported by the Sichuan Petroleum and Natural Gas Development Research Center under Grant No. SKB19-06.

Conflict of interest

The authors declare that there are no conflicts of interest.