Edge event-triggered control and state-constraint impulsive consensus for nonlinear multi-agent systems

1.   Introduction

As the topic of multi-agent becomes hotter and hotter, for a long time in the past, there exist many researches put forward in the consensus problem in order to improve the theory. In ^[1,2,3], the model of integrator and first-order multi-agent systems which conclude linear and non-linear dynamics function had been studied. In ^[4,5,6,7,8], the consensus problem of second-order or high-order multi-agent systems were taken into consideration. In ^{[9,10,12,13,14]}, the authors thought about the consensus problem of the multi-agent system which contains a leader-followers consensus or leaderless consensus in which the system was controlled by impulsive protocols. However, there is still an issue about how to reduce unnecessary information interaction on multi-agent system or how to make multi-agent system converge to a consensus with less cost. In this paper, edge event-triggered is an effective control means of saving energy.

With the same time, the theoretical analysis of impulsive system had also aroused the interest of many scholars^{[15,16,17,18,19,20,21,22]}. In actual situation, there were many restrictions on actuators and inputs. In ^[18], the problem based on state-constraint impulsive protocol were taken into consideration. In ^[19], the issue of time-delay complex networks system based on the impulsive control were researched. On many occasions, impulsive control was regarded as a useful approach to handle some stability or synchronization problems. Then, it was evident that impulsive control is more beneficial in chaos from^[23,24] or complex networks ^{[25,26,27,28]} which compares with continuous control.

So far, there have been a great number of theorems and deductions about impulsive system. The sufficient conditions of the primary outcomes are obtained with high requirements. Nevertheless, there is no research on that state-constraint impulsive protocols are applied to nonlinear multi-agent systems via edge event-triggered control so far, then, we will discuss the problem in this paper.

The contributions of this article are twofold:

(1) In real life, it is a common phenomenon to have a limitation of the input or actuator. For the purpose of better closing to the actual situation, two kinds of impulsive control protocols which conclude input saturation and double actuator saturation are discussed in this paper, the sufficient conditions for system to reach consensus are obtained.

(2) Edge event-triggered strategy which is a novel control way of event-triggered can greatly reduce the energy in the process of exchanging information. In this paper, we combine impulse control with edge event trigger control and let the time triggered by the edge event be the impulse time to avoid the Zeno-behavior. When the state error of the agent is small at a certain impulse time, the information interaction can be eliminated. Then, compared to the single impulsive control^[11], the number of information exchange has been reduced and the energy consumption of the whole system is correspondingly reduced.

The framework of the paper is as follows. Section Ⅱ describes some preliminaries which conclude notations, graph theory, state constraints and edge event-triggered strategy are introduced. In section Ⅲ, the models of the nonlinear dynamics and two types of impulsive control rules are formulated. In section IV, some theorems and their proofs are offered. For the purpose of validating the feasibility of the proposed methods, some numerical simulation are offered in section Ⅴ. In section Ⅵ, conclusion of this paper are offered.

2.   Preliminaries

2.1.   Notations

${R}$ is defined as the set of real number and let ${N}$ be a set of positive real number. The matrix inequality$\ A > B$ stands for that every element of $A$ is bigger than B. Suppose that every eigenvalue of matrix $A$ is real. $\lambda_{min}{(C)}$ and $\lambda_{max}{(C)}$ are the smallest eigenvalue and largest eigenvalue of matrix $C$. Then, $diag[{a_1, a_2, \ldots, a_N}]$ represents a diagonal matrix with elements $a_i$ on the diagonal. $I$ is an identity matrix and $I_N$ denotes a $N$- dimensional identity matrix. $max(x_i)$ means the maximum of $x_i$ when $x_i\in R$ and $i = 1, 2, \ldots, N$. $co\{u^j:j = 1, 2, \ldots, N\}$ represents a convex hull. $I_N$ denotes an $N$- dimensional identity matrix.

2.2.   Graph theory

In the paper, the symbol $G$ = ($\mathit{v}$, $\varepsilon$) represents a graph in which $\mathit{v} = \{\nu_1, \nu_2, \ldots, \nu_N\}$ is a set of nodes and $\varepsilon$ $\subseteq$ $\nu \times \nu$ is a set of edge. Let $A = [a_{i j}]$ be a weighted adjacency matrix with nonnegative elements. For an undirect graph $G$, the element of adjacency matrix $A$ which is a symmetric matrix is $1$ if there exists an edge $(\nu_i, \nu_j)$ between node $\nu_i$ and node $\nu_j$, otherwise, the element is $0$. The out-degree of node $i$ is defined as ${deg}(i) = \sum_{j = 1}^{N}a_{ij}$ and let matrix $\widetilde D$ be the degree matrix which is a diagonal matrix with the out-degree of each node along the principal diagonal. Then, the Laplacian matrix $L = [l_{ij}]$ and the expression is:

where $N_{i}$ is a set which is made up of all the neighbors of node $i$.

Suppose there are $\varrho$ edges in graph $G$ and label them be $e_1, e_2, \ldots, e_{\varrho}$, then, it's obvious that each edge $e_g = (\nu_i, \nu_j)$ where $g\in[1, \varrho]$. Denote $D = [d_{ij}]$ be a incidence matrix and the elements of $D$ is that

So, there exists a relationship between the Laplacian matrix $L$ and the incidence matrix $D$ is that $L = DWD^T$. Then, $W = [w_{ii}]$ is a diagonal matrix and define $w_{ii}$ be weight value of $i_{th}$ edge.

From now on, we assume that the topologies of multi-agent systems are all undirected graphs which are all connected.

2.3.   State-constraint strategy

In practical industrial applications, the parameters will be limited by various physical conditions. So it is a common phenomenon to limit the actuator or input. For example, the cost of electronic devices during power transmission makes the input limited. In addition, almost all actual physical systems are subject to state constrains which conclude actuator saturation constraints and input saturation constraints. Then, define the saturation function be

where $y \in R$.

2.4.   Edge event-triggered strategy

For a multi-agent system, we should first formulate some edge event-triggered rules. Then, at each time of sampling, whether the state of each agent and its neighbors are updated depend on the edge event is triggered or not. Namely, if agent $j$ is a neighbor of agent $i$ and the edge event is triggered by their communication link, the relative state of the two agents are all sampled and their controllers are also updated respectively. Else, the edge event is not triggered, their relative state don't renew and is the same as the one at the most recent sampling moment. For convenience, let $t_{g}^{(t)}$ be the event-triggered moment at the $g_{th}$ time and $t_{g^{ij}{(t)}}$ be the time that is the last sampling time before $t$.

In this paper, we assume that the moment of event-trigger is the impulse constant.

3.   Problem description

The following nonlinear system is considered in the paper and the dynamics of each agent in the system is expressed as follows:

in which ${x_i}\left(t\right) \in R$ is the desired state of the $i$th agent. $f(t, x_i)$ is a nonlinear functions. The nominal control input of $i$th agent is represented by ${b_i}\left(t\right)$.

In the paper, the following two types of control protocols are formulated:

1: Input Saturation

2: Double Actuator Saturation

where $r_i$ is the strength of impulsive. The constant $\{t_k\}$ satisfies the inequality $0 < t_0 < t_1 < \cdots < t_k < t_{k+1} < \cdots$. We let $\lim\limits_{k\to\infty }t_k = +\infty$ and denote $\delta(t-t_k)$ be a Dirac function. Then we assume that $\vartriangle x_i(t_k) = x_i(t_k^+)-x_i(t_k^-)$, $x_i(t_k) = x_i(t_k^-)$ and $x_i(t_k^-) = \lim\limits_{t \to {t_k^-}}x_i(t)$, $x_i(t_k^+) = \lim\limits_{t \to {t_k^+}}x_i(t)$.

Suppose that there exist $m$ edges of graph $G$ and label the $m$ edges be $e_1, e_2, \ldots, e_m$. Then, for any edge $e_g = (\nu_i, \nu_j)$ in which $1\leqslant g\leqslant m$, let $z_g = x_i(t)-x_j(t)$, $\tilde z_g(t_k^-) = x_i(t_{ij}(t_k^-))-x_j(t_{ij}(t_k^-))$ and define $x(t) = [x_1(t), x_2(t), \ldots, x_N(t)]^T$, $z(t) = [z_1(t), z_2(t), \ldots, z_m(t)]^T$, $\tilde z(t_k^-) = [\tilde z_1(t_k^-), \tilde z_2(t_k^-), \ldots, \tilde z_m(t_k^-)]^T$. So, it's clear that $z(t) = D^Tx(t)$.

Here, we bring forward the rule of edge event-triggered. For the system, the edge event $e_g$ will be turned on at the time $t_k$ unless the following inequality is invalid.

where $\mu_g$ and $\sigma_g$ are all system parameters to be designed.

Then, the next definition and lemmas are shown for supporting derivation better:

Definition 1. The system achieves consensus with the control protocols while

where $i, j = 1, 2, \cdots, N$.

Lemma 1. ^[18] Let $w^1, w^2, \ldots, w^U \in R^{n1}$, $m^1, m^2, \ldots, m^V\in R^{n2}$, $w = (w_1, w_2, \cdots, w_n)^{ \rm T}$, $m = (m_1, m_2, \cdots, m_n)^{ \rm T}$ and $U, V, n1, n2$ are positive integers. If $w\in co\{w^u:u = 1, 2, \ldots, U\}$ and $m\in co\{m^v:v = 1, 2, \ldots, V\}$, then

Lemma 2. ^[18] Let $w$, $m \in {R}^n$, $w = (w_1, w_2, \cdots, w_n)^{ \rm T}$, $m = (m_1, m_2, \cdots, m_n)^{ \rm T}$, $n \in N^+$. $E$ is a set of $n\times n$ diagonal matrices and its diagonal elements are either $1$ or $0$. Assume that $E_i$ stands for every element of $E$. $i = 1, 2, \ldots 2^n$. Then, $E = \{E_i:i \in \{1, 2, \cdots, 2^n \}\}$. Denote $E_i^- = I-E_i$. It is obvious that $E_i^-$ is also one element of $E$ when $E_i\in E$. When $|m_i| \leqslant 1$, $sat(w) \in co\{E_i w+E_i^{-}m:i \in \{1, 2, \cdots, 2^n \}\}$.

For example, if $n = 2$, then

If $x \in R^n$ and $T, H \in R^{n\times n}$ are two matrices, when $\|Hx\|_{\infty} \leqslant 1$, we can get $sat(Tx) \in co\{E_iTx+E_i^{-}Hx:i \in \{1, 2, \cdots, 2^n \}\}$. There exists $0 \leqslant \varrho_i \leqslant 1$ and satisfies $\sum_{i = 1}^{2^n} \varrho_i = 1$. Then $sat(Tx) = \sum_{i = 1}^{2^n} \varrho_i(E_iT+E_i^{-}H)x$.

Lemma 3. Let $x = [x_1, x_2, \ldots, x_n]^T$, $SAT(x) = [sat(x_1), sat(x_2), \ldots, sat(x_n)]^T$ and $H = diag[h_1, h_2, \ldots, h_n]$. Let $V = diag[v_1, v_2, \ldots, v_n]$, $W = diag[w_1, w_2, \ldots, w_n]$ and $0\leqslant v_i < 1$. Denote $W = I-V$, it is clearly that $0 < w_i\leqslant 1$. Then, for $SAT(x)$, there exists a matrix $H$ such that $SAT(x) = Vx+WHx$ when $\|Hx\|_{\infty} \leqslant 1$.

Proof. Let $V = diag[v_1, v_2, \ldots, v_n]$, $W = diag[w_1, w_2, \ldots, w_n]$, $H = diag[h_1, h_2, \ldots, h_n]$. We assume $0\leqslant v_i < 1$and $v_i+w_i = 1$, then $0 < w_i\leqslant1$. So there exists a constant $h_i$ such that $sat(x_i) = v_ix_i+w_ih_ix_i$. For example, we choose $v_i = a$, $w_i = 1-a$, then $sat(x_i) = ax_i+(1-a)h_ix_i$. So we let $h_i = 1$ if $-1\leqslant x_i\leqslant 1$ and let $h_i = \frac{sign(x_i)-ax_i}{(1-a)x_i}$ when $|x_i| > 1$. Then, we can easily obtain that $SAT(x) = Vx+WHx$ when $\|Hx\|_{\infty} \leqslant 1$. The proof is completed.

Lemma 4. ^[3] If there exist two matrices $P$ and $B$ belong to $R^{n\times n}$, which are all positive definite and symmetric, then for any $x\in R^n$, the following inequation holds

Remark 1. For Lemma 4, If matrix $P$ is a identity matrix $I_n$, then the following inequation holds

Assumption 1. Nonlinear function $f(\cdot)$ satisfies a condition that there exists a positive constant $\tau$ such that:

4.   Main results

4.1.   Input saturation

Consider a consensus problem of system (3.1) based on protocol 1 (shown as (3.2)). Then the system can be considered as follows:

According to Lemma 2, we can get:

denote $R = diag[r_1, r_2, \dots, r_N]$. Assume $S = \sum_{i = 1}^{2^n} \varrho_i(E_iR+E_i^{-}H)$ and if the matrix $H$ is selected for a diagonal matrix, $S$ is also a diagonal matrix.

Then, obviously

Furthermore, the system can be described as follow

define $F(t, x(t)) = (f(t, x_1(t)), f(t, x_2(t)), \dots, f(t, x_N(t)))^T$.

Theorem 1. Suppose that there exists a matrix ${H}$ such that $\|HLx\|_{\infty} \leqslant 1$ under the impulsive control protocol 1, if Assumption 1 holds and there exist two constants $\hat\xi$, $\hat\gamma$ that make the following inequities hold:

$(i)$ There exists a constant $\hat\varpi$ to make the expression $0 < t_{k+1}-t_{k}\leqslant \hat\varpi, k\in N^+$ holds.

$(ii)\; 0 < 1-\hat\gamma\leqslant \hat\xi$;

$(iii)\; \ln(\hat\xi)+2\tau\hat\varpi \leqslant 0$;

The system (3.1) will be aligned under protocol 1.

Proof. Consider the following Lyapunov function:

When $t \neq t_k$, we can easily obtain that the derivative of $V$ is:

When $t = t_k$, we can get that

where

For the convenience of the next, the case that the edge events labeled by 1 to $g$ are triggered at $t = t_k$ is assumed. Also, we can get $L = DWD^T$, $z(t) = D^Tx(t)$ and denote $z = z(t)$, $\tilde z = \tilde z(t_k^-)$. Then, according to Lemma 4 and Remark 1, it's easy to obtain that

denote

and $Q$ is a diagonal matrix in which $g$ elements are $2\hat\alpha-\hat\beta$ and $m-g$ elements are $\frac{2\hat\alpha\mu_{p}-\hat\beta}{\sigma_{p}^2}$, where $p = {m-g, m-g+1, \ldots, m}$.

Based on (4.7) and (4.8), it's clearly that

From (4.6) and (4.9), one obtains

When $t\in [t_0, t_1)$, in view of inequalities (4.10), we can get

According to mathematical deduction, when $t\in [t_k, t_{k+1})$, it implies that

From (4.11) and (4.12), one gets

Notice that $\ln(\hat\xi)+2\tau\hat\varpi \leqslant 0$ from $(iii)$ in Theorem 1, one obtains

It is obvious that the system (4.1) which controlled by protocol 1 can reach consensus. The proof is completed.

4.2.   Double actuator saturation

Discuss a consensus problem of the system (3.1) under control protocol 2 whose expression is (3.3). For the sake of convenience, suppose this situation that the edge events labeled by 1 to $g$ are triggered at the moment $t = t_k$. Then multi-agent system can be rewritten as follows:

According to Lemma 3, we can get the impulsive instant expression of agent is

where $SAT(x(t)) = (sat(x_1(t)), sat(x_2(t)), \dots, sat(x_N(t)))^T$ and $\bar H = diag(h_1, h_2, \ldots, h_N)^T$. We let $O = V+W\bar H$, it's clearly that $O$ is a diagonal matrix. Then

Then the system can be reformulated as:

Theorem 2. Assume that there exists a matrix ${\bar H}$ such that $\|\bar Hx\|_{\infty} \leqslant 1$ under the impulsive control protocol 2, if Assumption 1 holds and there exist two constants $\bar\xi$, $\bar\gamma$ that make the following inequities hold

$(i)$ There exists a constant $\bar\varpi$ to make the inequation $0 < t_{k+1}-t_{k}\leqslant \bar\varpi, k\in N^+$ holds.

$(ii)\; 0 < 1-\bar\gamma\leqslant \bar\xi$;

$(iii)\; \ln(\bar\xi)+2\tau\bar\varpi \leqslant 0$;

Then, system (3.1) can achieve consensus based on the action of protocol 2.

Proof. Taking the following Lyapunov function into account:

When $t \neq t_k$, the proof is the same as Theorem 1, so we can easily obtain:

When $t = t_k$, one can gather

where

Then, according to the above, it's easy to obtain that

where

and $\bar Q$ is a diagonal matrix in which $g$ elements are $2\zeta\bar\alpha-\delta^2\bar\beta$ and $m-g$ elements are $\frac{2\zeta\bar\alpha\mu_{p}-\delta^2\bar\beta}{\sigma_{p}^2}$ in which $p = {m-g, m-g+1, \ldots, m}$.

Focus on (4.21) and (4.22), it's clearly that

From (4.20) and (4.23), we can observe

When $t \in [t_0, t_1)$, from the inequality of (4.24), we can get

Thus, we can easily obtain that

According to (4.26), by mathematical deduction, it holds that

From $(iii)$ in Theorem 2, we can receive $\ln(\bar\xi)+2\tau \bar\varpi \leqslant 0$. Then,

It's evident that the system (4.15) which controlled by protocol 2 reach consensus. The proof is completed.

5.   Numerical simulations

Example 1. We consider the multi-agent system (4.1) under the control of protocol 1.

For the system (4.1), the undirect graph is chosen as the Figure 1.

From the graph of Figure 1, we can clearly acquire the incidence matrix

And we select that nonlinear functions are as follows: $f(t, x_i(t)) = cos^2(x_i(t))-|sin(x_i(t))|$. And choose $x_1(0) = 3, x_2(0) = -0.4, x_3(0) = -2.7$, $x_4(0) = 1.3$ and $\tau = 1.2$.

With regard to Theorem 1, let's define the step size to be 0.001, $R = diag[0.5, 0.7, 0.6, 0.4]$, $\mu_1 = \mu_2 = \mu_3 = \mu_4 = 0.8$, $\sigma_1 = \sigma_2 = \sigma_3 = \sigma_4 = 1.2$, $\varrho_1 = 0.2, \varrho_2 = 0.1, \varrho_3 = 0.3, \varrho_4 = 0.1, \varrho_5 = 0.3$, the rest of $\varrho_i$ are all zero and $\hat\xi = 0.86$, $t_k-t_{k-1} = 0.2$, $k\geqslant 1$. According to calculation, the conditions of $(ii), (iii)$ in Theorem 1 are all hold. Then, Figure 2 displays the state value of $i$th agent under the control of impulsive and event-triggered, Figure 3 indicates the error between every two agents, Figure 4 reveals the event-triggered time of each edge.

We can obviously see the system is consensus under the control of protocol 1 from Figure 2 and Figure 3.

Example 2. We take the multi-agent system (4.15) into consideration which is controlled by protocol 2.

As the system (4.15), we let the graph and the direction of every edge as follows in Figure 5:

So, the incidence matrix $D$ is

Then we assume that the nonlinear function of the dynamics of each agent are $f(t, x(t))\cos^2(x(t))-|0.4\cos(x(t))-1|$ and choose $x(0) = [3, -0.4, -2.28, 0.3], \tau = 1.6$.

About Theorem 2, suppose step size be 0.001, $R = diag[0.4, 0.6, 0.6, 0.4]$, $\mu_1 = \mu_2 = \mu_3 = \mu_4 = 0.72$, $\sigma_1 = \sigma_2 = \sigma_3 = \sigma_4 = 1.46$ $\bar\xi = 0.79$, $\varrho_1 = 0.3, \varrho_3 = 0.2, \varrho_5 = 0.2, \varrho_7 = 0.1, \varrho_9 = 0.2$, the rest of $\varrho_i$ are all zero and $t_k-t_{k-1} = 0.2$, $k\geqslant 1$. By calculation, we can verify that the conditions of Theorem 2 are established. Then, the state of $i$th agent with the impulsive control is manifested in Figure 6 and the error between every two agents is evident in Figure 7. The event-triggered moment of each edge is displayed in Figure 8.

It is clear that the multi-agent system reach consensus under the control of protocol 2 from Figure 6 and Figure 7.

6.   Conclusions

We discuss the nonlinear multi-agent consensus issue under the control state-constrain impulsive and edge event-triggered tactics in the above. Impulsive protocol based on the relative information between the agents and their neighbors has been adopted to deal with the problem of consensus. Edge event-triggered strategy can reduce cost by reducing the number of information exchange. According to theoretical analysis, we can gather sufficient conditions to ensure the consensus of the system in this paper. Numerical simulations have verified that the consensus problem are solved by the control protocols. The asynchronous event-based problem of multi-agent systems will be taken into consideration.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants 61873213 and 61633011, and in part by National Key Research and Development Project under Grant 2018AAA0100101.

Conflict of interest

All authors declare no conflicts of interest in this paper.