Self-triggered Consensus of Multi-agent Systems with Quantized Relative State Measurements

This paper addresses the consensus problem of first-order continuous-time multi-agent systems over undirected graphs. Each agent samples relative state measurements in a self-triggered fashion and transmits the sum of the measurements to its neighbors. Moreover, we use finite-level dynamic quantizers and apply the zooming-in technique. The proposed joint design method for quantization and self-triggered sampling achieves asymptotic consensus, and inter-event times are strictly positive. Sampling times are determined explicitly with iterative procedures including the computation of the Lambert $W$-function. A simulation example is provided to illustrate the effectiveness of the proposed method.


Introduction
With the recent development of information and communication technologies, multi-agent systems have received considerable attention.Cooperative control of multi-agent systems can be applied to various areas such as multi-vehicle formulation [1] and distributed sensor networks [2].A basic coordination problem of multi-agent systems is consensus, whose aim is to reach an agreement on the states of all agents.A theoretical framework for consensus problems has been introduced in the seminal work [3], and substantial progress has been made since then; see the survey papers [4,5] and the references therein.
In practice, digital devices are used in multi-agent systems.Conventional approaches to implementing digital platforms involve periodic sampling.However, periodic sampling can lead to unnecessary control updates and state measurements, which are undesirable for resource-constrained multi-agent systems.Event-triggered control [6][7][8] and self-triggered control [9][10][11] are promising alternatives to traditional periodic control.In both event-triggered and self-triggered control systems, data transmissions and control updates occur only when needed.Event-triggering mechanisms use current measurements and check triggering conditions continuously or periodically.On the other hand, selftriggering mechanisms avoid such frequent monitoring by calculating the next sampling time when data are obtained.Various methods have been developed for event-triggered consensus and selftriggered consensus; see, e.g., [12][13][14][15][16]. Comprehensive surveys on this topic are available in [17,18].Some specifically relevant studies are cited below.
The bandwidth of communication channels and the accuracy of sensors may be limited in multiagent systems.In such situations, only imperfect information is available to the agents.We also face the theoretical question of how much accuracy in information is necessary for consensus.From both practical and theoretical point of view, quantized consensus has been studied extensively.For continuous-time multi-agent systems, infinite-level static quantization is often considered under the situation where quantized measurements are obtained continuously; see, e.g., [19][20][21][22][23][24][25].Event-triggering mechanisms and self-triggering mechanisms have been proposed for continuous-time multi-agent systems with infinite-level static quantizers in [26][27][28][29][30][31][32][33][34][35].Self-triggered consensus with ternary controllers has been also studied in [36,37].For event-triggered consensus under unknown input delays, finitelevel dynamic quantizers have been developed in [38], where the quantization error goes to zero as the agent state converges to the origin.
In this paper, we consider first-order continuous-time multi-agent systems over undirected graphs.Our goal is to jointly design a finite-level dynamic quantizer and a self-triggering mechanism for asymptotic consensus.We focus on the situation where relative states, not absolute states, are sampled as, e.g., in [16,19,22,24,[29][30][31][32]34].We assume that each agent's sensor has a scaling parameter to adjust the maximum measurement range and the accuracy.For example, if indirect time-of-flight sensors [53] are installed in agents, then the modulation frequency of light signals determines the maximum range and the accuracy.In the case of cameras, they can be changed by adjusting the focal length; see Section 11.2 of [54] for a mathematical model of cameras.
In the proposed self-triggered framework, the agents send the sum of the relative state measurements to all their neighbors as in the self-triggered consensus algorithm presented in [14].In other words, each agent communicates with its neighbors only at the sampling times of itself and its neighbors.The sum is transmitted so that the neighbors compute the next sampling times, not the inputs.After receiving it, the neighbors update the next sampling times.Since the measurements are already quantized when they are sampled, the sum can be transmitted without error, even over channels with finite capacity.
The main contributions of this paper are summarized as follows: 1. We propose a joint algorithm for finite-level dynamic quantization and self-triggered sampling of the relative states.We also provide a sufficient condition for the consensus of the quantized selftriggered multi-agent system.This sufficient condition represents a quantitative trade-off between data accuracy and sampling frequency.Such a trade-off can be a useful guideline for sensing performance, power consumption, and channel capacity.
2. In the proposed method, the inter-event times, i.e., the sampling intervals of each agent, are strictly positive, and hence Zeno behavior does not occur.In addition, the agents can compute sampling times using an explicit formula with the Lambert W -function (see, e.g., [55] for the Lambert W -function). Consequently, the proposed self-triggering mechanism is simple and efficient in computation.
We now compare our results with previous studies.The finite-level dynamic quantizers developed in [39][40][41][42][43] and their aforementioned extensions require the absolute states.More specifically, they quantize the error between the absolute state and its estimate for communication over finite-capacity channels.In this framework, the agents have to estimate the states of all their neighbors for decoding.In contrast, we develop finite-level dynamic quantizers for relative state measurements.As in the existing studies above, we also employ the zooming-in technique introduced for single-loop systems in [56,57].However, due to the above-mentioned difference in what is quantized, the quantizer we study has several notable features.For example, the proposed algorithm can be applied to GPSdenied environments.Moreover, the estimation of neighbor states is not needed, which reduces the computational burden on the agents.
A finite-level quantizer may be saturated, i.e., it does not guarantee the accuracy of quantized data in general if the original data is outside of the quantization region.To achieve asymptotic consensus, we need to update the scaling parameter of the quantizer so that the relative state measurement is within the quantization region and the quantization error goes to zero asymptotically.In [29,32,34], infinite-level static quantizers have been used for quantized self-triggered consensus of first-order multi-agent systems.Hence the issue of quantizer saturation has not been addressed there.In [29], infinite-level uniform quantization has been considered, and consequently only consensus to a bounded region around the average of the agent states has been achieved.The quantized self-triggered control algorithm proposed in [32,34] achieves asymptotic consensus with the help of infinite-level logarithmic quantizers, but sampling times have to belong to the set {t = kh : k is a nonnegative integer} with some h > 0, which makes it easy to exclude Zeno behavior.Table 1 summarizes the comparison between this study and several relevant studies.
The difficulty of this study is that the following three conditions must be satisfied: • avoiding quantizer saturation; • decreasing the quantization error asymptotically; and • guaranteeing that the inter-event times are strictly positive.
To address this difficulty, we introduce a new semi-norm ||| • ||| ∞ for the analysis of multi-agent systems.
The semi-norm is constructed from the maximum norm and is suitable for handling errors of individual agents due to quantization and self-triggered sampling.Moreover, the Laplacian matrix L ∈ R N ×N of the multi-agent system has the following semi-contractivity property: There exists a constant γ > 0 such that for all v ∈ R N and t ≥ 0; see [58,59] for the semi-contraction theory.The semi-contractivity property facilitates the analysis of state trajectories under self-triggered sampling and consequently leads to a simple design of the scaling parameter for finite-level dynamic quantization.The rest of this paper is organized as follows.In Section 2, we introduce the system model.In Section 3, we provide some preliminaries on the semi-norm and sampling times.Section 4 contains the main result, which gives a sufficient condition for consensus.In Section 5, we explain how the agents compute sampling times in a self-triggered fashion.A simulation example is given in Section 6, and Section 7 concludes this paper.
Notation: We denote the set of nonnegative integers by N 0 .We define inf and the corresponding induced norm of A ∈ R M ×N with the (i, j)-th element A ij is given by we write λ 2 (P ) := λ 2 .We define and write ave(v) := 1v for v ∈ R N .The graph Laplacian of an undirected graph G is denoted by L(G).We denote the Lambert W -function by W (y) for y ≥ 0. In other words, W (y) is the solution x ≥ 0 of the transcendental equation xe x = y.Throughout this paper, we shall use the following fact frequently without comment: For a, ω > 0 and c ∈ R, the solution x = x * of the transcendental

System Model
2.1.Multi-agent system.Let N ∈ N be N ≥ 2, and consider a multi-agent system with N agents.Each agent has a label i ∈ N := {1, 2, . . ., N }.For every i ∈ N , the dynamics of agent i is given by ( 1) ẋi (t) = u i (t), t ≥ 0; where x i (t) ∈ R and u i (t) ∈ R are the state and the control input of agent i, respectively.The network topology of the multi-agent system is given by a fixed undirected graph G = (V, E) with vertex set V = {v 1 , v 2 , . . ., v N } and edge set If (v i , v j ) ∈ E, then agent j is called a neighbor of agent i, and these two agents can measure the relative states and communicate with each other.For i ∈ N , we denote by N i the set of all neighbors of agent i and by d i the degree of the node v i , that is, the cardinality of the set N i .Consider the ideal case without quantization or self-triggered sampling, and set (2) for t ≥ 0 and i ∈ N .It is well known that the multi-agent system to which the control input (2) is applied achieves average consensus under the following assumption.
Assumption 2.1.The undirected graph G is connected.
In this paper, we place Assumption 2.1.Moreover, we make two assumptions, which are used to avoid the saturation of quantization schemes.These assumptions are relative-state analogues of the assumptions in the previous studies on quantized consensus based on absolute state measurements (see, e.g., Assumptions 3 and 4 of [44]).Assumption 2.2.A bound E 0 > 0 satisfying is known by all agents.
Assumption 2.3.A bound d ∈ N satisfying is known by all agents.
We make an assumption on the number R of quantization levels.
In this paper, we study the following notion of consensus of multi-agent systems under Assumption 2.2.
Definition 2.5.The multi-agent system achieves consensus exponentially with decay rate ω > 0 under Assumption 2.2 if there exists a constant Ω > 0, independent of E 0 , such that for all t ≥ 0 and i, j ∈ N .
2.2.Quantization scheme.Let E > 0 be a quantization range and let R ∈ N be the number of quantization levels satisfying Assumption 2.4.We assume that E and R are shared among all agents.We apply uniform quantization to the interval [−E, E].Namely, a quantization function Q E,R is defined by The agents use a fixed R but change E in order to achieve consensus asymptotically.In other words, E is the scaling parameter of the quantization scheme.Let {t i k } k∈N 0 be a strictly increasing sequence with t i 0 := 0, and t i k is the k-th sampling time of agent i.To describe the quantized data used at time t = t i k for k ∈ N 0 , we assume for the moment that a certain function E : [0, ∞) → (0, ∞) satisfies the unsaturation condition (4) Agent i measures the relative state x i (t i k )−x j (t i k ) for each neighbor j ∈ N i and obtains its quantized value ]. Then agent i sends to each neighbor j ∈ N i the sum The neighbors use the sum q i (t i k ) to calculate the next sampling time, not the input.This data transmission implies that the agents use information not only about direct neighbors but also about two-hop neighbors as in the self-triggering mechanism developed in [14].
The sum q i (t i k ) consists of the quantized values, and therefore agent i can transmit q i (t i k ) without errors even through finite-capacity channels.In fact, since R is an odd number under Assumption 2.4, the sum q i (t i k ) belongs to the finite set where d ∈ N is as in Assumption 2.3.The encoder of agent i assigns an index to each value 2pE/R and transmits the index corresponding to the sum q i (t i k ) to the decoder of each neighbor j ∈ N i .Since the agents share E, R, and d, the decoder can generate the sum q i (t i k ) from the received index.

2.3.
Triggering mechanism.Let a strictly increasing sequence {t i k } k∈N 0 with t i 0 := 0 be the sampling times of agent i ∈ N as in Section 2.2, and let k ∈ N 0 .As in the ideal case (2), the control input u i (t) of agent i is given by the sum of the quantized relative state, (5) for t i k ≤ t < t i k+1 when the unsaturation condition ( 4) is satisfied.Then the dynamics of agent i can be written as ẋi where f i (t) and g i (t) are, respectively, the errors due to sampling and quantization defined by We make a triggering condition on the error f i due to sampling.From the dynamics (1) and the input (5) of each agent, we have that for all t i k ≤ t < t i k+1 , Substituting ( 9) and ( 10) into (7) motivates us to consider the following function obtained only from the inputs: . Using the quantization range E(t), we define the (k + 1)-th sampling time t i k+1 of agent i ∈ N by ( 12) , where δ i > 0 is a threshold and τ i max , τ i min > 0 are upper and lower bounds of inter-event times, respectively, i.e., τ i min ≤ τ i k ≤ τ i max .The behaviors of the errors f i (t) and g i (t) can be roughly described as follows.Under the triggering mechanism (12), the error |f i (t)| due to sampling is upper-bounded by δ i E(t).The error |g i (t)| due to quantization is also bounded from above by a constant multiple of E(t i k ) for t i k ≤ t < t i k+1 when the quantizer is not saturated.Hence, if E(t) decreases to zero as t → ∞, then both errors f i (t) and g i (t) also go to zero.
After some preliminaries in Section 3, Section 4 is devoted to finding a quantization range E(t), a threshold δ i , and upper and lower bounds τ i max , τ i min of inter-event times such that consensus (3) as well as the unsaturation condition (4) are satisfied.In Section 5, we present a method for agent i to compute the sampling times {t i k } k∈N 0 in a self-triggered fashion.We conclude this section by making two remarks on the triggering mechanism (12).First, the constraint τ i k ≥ τ i min is made solely to simplify the consensus analysis, and agent i can compute the sampling times {t i k } k∈N 0 without using the lower bound τ i min .Second, continuous communication with the neighbors is not required to compute the sampling times, although the inputs of the neighbors are used in the triggering mechanism (12).It is enough for agent i to communicate with the neighbor j ∈ N i at their sampling times {t i k } k∈N 0 and {t j k } k∈N 0 .In fact, the inputs are piecewise-constant functions, and agent i can know the input u j of the neighbor j from the received data q j (t j k ).Based on the updated information on q j (t j k ), agent i recalculates the next sampling time.We will discuss these issues in detail in Section 5.

Preliminaries
In this section, we introduce a semi-norm on R N and basic properties of sampling times.The reader eager to pursue the consensus analysis of multi-agent systems might skip detailed proofs in this section and return to them when needed.
3.1.Semi-norm for consensus analysis.Inspired by the norm used in the theory of operator semigroups (see, e.g., the proof of Theorem 5.2 in Chapter 1 of [60]), we introduce a new semi-norm on R N , which will lead to the semi-contractivity property [58,59] of the matrix exponential of the negative Laplacian matrix.) is satisfied for some γ > 0, then e −Lt is a semicontraction with respect to the semi-norm ||| • ||| for all t > 0. In Lemma 9 of [58], a more general method is presented for constructing such semi-norms.The tuning parameter of this method is a matrix whose kernel coincides with the span {α1 : α ∈ R}.Since the constants Γ and γ in ( 13) are easier to tune for the joint design of a quantizer and a self-triggering mechanism, we will use Lemma 3.1 in the consensus analysis.
3.2.Basic properties of sampling times.Let {t i k } k∈N 0 be a strictly increasing sequence of real numbers with t i 0 := 0 for i ∈ N = {1, 2, . . ., N }.Set t 0 := 0 and k i (0) := 0 for i ∈ N .Define . ., t ℓ+1 } for ℓ ∈ N 0 and i ∈ N .Roughly speaking, in the context of the multi-agent system, {t ℓ } ℓ∈N 0 are all sampling times of the agents without duplication, and k i (ℓ) is the number of times agent i has measured the relative states on the interval (0, t ℓ ].Hence t i k i (ℓ) is the latest sampling time of agent i at time t = t ℓ .Define I(0) := N and In our multi-agent setting, I(ℓ) represents the set of agents measuring the relative states at t = t ℓ .Proposition 3.3.Let {t i k } k∈N 0 be a strictly increasing sequence of real numbers with t i 0 := 0 for i ∈ N = {1, 2, . . ., N }.The sequences {t ℓ } ℓ∈N 0 and {k i (ℓ)} ℓ∈N 0 defined as above have the following properties for all ℓ ∈ N 0 and i ∈ N : a) . In this case, holds for ℓ = 0. Suppose that the inequality ( 14) holds for some ℓ ∈ N 0 .Then . Therefore, the inequality ( 14) holds for all ℓ ∈ N 0 by induction.b) Assume that k i (ℓ + 1) = k i (ℓ) + 1.By the definition of t ℓ+1 , we obtain t ℓ+1 ≤ t i k i (ℓ)+1 .On the other hand, t i k i (ℓ+1) ∈ {t 0 , . . ., t ℓ+1 } by the definition of k i (ℓ + 1).Since {t ℓ } ℓ∈N 0 is a nondecreasing sequence by a), it follows that ℓ+1) .Conversely, assume that t ℓ+1 = t i k i (ℓ)+1 .Then t i k i (ℓ)+1 ∈ {t 0 , t 1 , . . ., t ℓ+1 }.By the definition of k i (ℓ + 1), we obtain c) The definition of k i (ℓ) directly yields It remains to show that t ℓ < t ℓ+1 .By construction, I(ℓ) ̸ = ∅ holds for all ℓ ∈ N 0 .First, we consider the case Since I(0) = N by definition, we obtain ℓ ≥ 1.Let i ∈ I(ℓ + 1).Then t ℓ+1 = t i k i (ℓ)+1 .On the other hand, i ̸ ∈ I(ℓ) and hence If ℓ ≥ 1, then we have from i ∈ I(ℓ) and b) that , it follows that t ℓ < t ℓ+1 .d) We have from c) that This and the assumption t i k ≤ t ℓ 1 yield t i k < t i k i (ℓ 1 )+1 , and therefore k ≤ k i (ℓ 1 ).Let If ℓ 0 = 0, then we obtain This and b) yield by the definition of t ℓ+1 , it follows that f) For all ℓ ∈ N 0 , there exists i ∈ N such that t ℓ ∈ {t i k } k∈N 0 .We have from c) that t ℓ ̸ = t ℓ+1 .Since N is a set with finite elements, there exist i ∈ N and a subsequence {t ℓ(p) } p∈N 0 of {t ℓ } ℓ∈N 0 such that Assume, to get a contradiction, that sup ℓ∈N 0 t ℓ < ∞.Take 0 < ε < τ i min .
There exists p 0 ∈ N 0 such that for all p ≥ p 0 .Choose p ≥ p 0 arbitrarily.We obtain By assumption, (15).□

Consensus Analysis
In this section, first we define a semi-norm based on the maximum norm.Next, we obtain a bound of the state with respect to the semi-norm for the design of the quantization range.After these preparations, we give a sufficient condition for consensus in the main theorem.Finally, we find bounds of the constant Γ in (13) corresponding to our multi-agent setting.
Throughout this and the next sections, we consider the quantized self-triggered multi-agent system presented in Section 2. Let {t i k } k∈N 0 with t i 0 := 0 be the sampling times of agent i ∈ N , which are given in (12).Define {t ℓ } ℓ∈N 0 and {k i (ℓ)} ℓ∈N 0 as in Section 3.2.We let L := L(G), where G is the undirected graph of the multi-agent system.Define ⊤ for t ≥ 0. Then we have from the dynamics (6) of individual agents that 4.1.Semi-norm based on the maximum norm.We start by showing the following simple result.Then Γ < ∞ and the inequalities hold for all v ∈ R N and t ≥ 0.
Next we study |g i (t)| for t ℓ ≤ t < t ℓ+1 , where g i is defined as in (8) and is the error due to quantization.Since the unsaturation condition ( 4) is satisfied until t = t ℓ , we have that ) R for all j ∈ N i .Proposition 3.3.e)shows that t ℓ+1 − t i k i (ℓ) ≤ τ i max , which gives for all t ∈ [t ℓ , t ℓ+1 ).Hence From the inequalities ( 34) and (38), we obtain for all t ∈ [t ℓ , t ℓ+1 ) and i ∈ N .This and Lemma 3.1.a)with F = I give for all t ∈ [t ℓ , t ℓ+1 ).Therefore, we have from Lemma 3.1.c)and d) that Combining the inequalities ( 39) and ( 40), we obtain 3 is in implicit form with respect to the decay parameter ω.We rewrite this condition in explicit form by using the Lambert W -function.To this end, we define (41) ω := min where Lemma 4.4.Assume that the threshold δ i > 0 and the number R ∈ N of quantization levels satisfy the inequality (42) for all i ∈ N .Then ω defined by (41) satisfies ω > 0.Moreover, the decay parameter ω satisfies the condition (32) if and only if 0 < ω ≤ ω.
Proof.Let i ∈ N .The inequality ( 42) is equivalent to Since it follows that for all sufficiently small ω > 0, the inequality holds.The inequality (43) is equivalent to Therefore, using the Lambert W -function, one can write the inequality (43) as Since (43) holds for all sufficiently small ω > 0, we obtain ω > 0.
By definition, γ − 2Γ ∞ κ(ω) = min{η i − ξ i e ωτ i max : i ∈ N }.From this, it follows that ω ≤ γ − 2Γ ∞ κ(ω) if and only if (43) holds for all i ∈ N .We have shown that (43) holds for all i ∈ N if and only if ω ≤ ω.Thus, the condition ( 32) is equivalent to 0 < ω ≤ ω. □ 4.3.Main result.Before stating the main result of this section, we summarize the assumption on the parameters of the quantization scheme and the triggering mechanism.Assumption 4.5.Let upper bounds τ i max > 0 be given for all i ∈ N .The following three conditions are satisfied: a) The threshold δ i > 0 and the number R ∈ N of quantization levels satisfy the inequality (42) for all i ∈ N .b) For all i ∈ N , the lower bound τ i min satisfies 0 < τ i min ≤ min{τ i min , τ i max }, where τ i min is as in (31).c) The decay parameter ω of the quantization range E(t) defined by (29) where ω is as in (41).
Theorem 4.6.Suppose that Assumptions 2.1-2.4 and 4.5 hold.Then the unsaturation condition (4) is satisfied for all k ∈ N 0 and i ∈ N .Moreover, Σ MAS achieves consensus exponentially with decay rate ω.
Proof.Since 0 < ω ≤ ω, Lemma 4.4 shows that the condition (32) on ω is satisfied.By Lemmas 4.2 and 4.3, we obtain for all t ≥ 0 and i, j ∈ N .Therefore, the unsaturation condition ( 4) is satisfied for all k ∈ N 0 and i ∈ N .The inequality (44) and the definition ( 29) of E(t) give for all t ≥ 0 and i, j ∈ N .Thus, Σ MAS achieves consensus exponentially with decay rate ω.□ Recall that the maximum decay parameter ω is the minimum of which is the solution of the equation ω = η i − ξ i e ωτ i max ; see the proof of Lemma 4.4.Moreover, ξ i becomes smaller as d i /R decreases, and η i becomes larger as δ i decreases.Therefore, ω becomes larger as d i , δ i , and τ i max decreases and as R increases.This also means that if agent i has a large d i , i.e., many neighbors, then we need to use small δ i and τ i max in order to achieve fast consensus of the multi-agent system.
Remark 4.7.The condition on the lower bound τ i min in Assumption 4.5.b) is not used when each agent computes the next sampling time; see Section 5 for details.Therefore, Theorem 4.6 essentially shows that asymptotic consensus is achieved if (42) holds for each i ∈ N and if 0 < ω ≤ ω for given upper bounds τ 1 max , . . ., τ N max of inter-event times.
Remark 4.8.To check the conditions obtained in Theorem 4.6, the global network parameters, λ 2 (L) and Γ ∞ , are needed.In addition, the quantization range E(t) is common to all agents as the scaling parameter of finite-level dynamic quantizers studied, e.g., in the previous works [40,41].These are drawbacks of the proposed method.
Remark 4.9.Although the proposed method is inspired by the self-triggered consensus algorithm presented in [14], the approach to consensus analysis differs.In [14], a Lyapunov function and LaSalle's invariance principle have been employed.In contrast, we develop a trajectory-based approach, where the semi-contractivity property of e −Lt plays a key role.Moreover, we discuss the convergence speed of consensus, by using the global parameters mentioned in Remark 4.8 above.The utilization of the global parameters also enables us to investigate the minimum inter-event time in a way different from that of [14].
4.4.Bounds of Γ ∞ .We use the constant Γ ∞ in the definition (29) of E(t) and the conditions for consensus given in Assumption 4.5.To apply the proposed method, we have to compute Γ ∞ numerically by (25) or replace Γ ∞ with an available upper bound of Γ ∞ .In the next proposition, we provide bounds of Γ ∞ by using the network size.The proof can be found in Appendix A.
Proposition 4.10.Let N ∈ N satisfy N ≥ 2 and let G be a connected undirected graph with N vertices.Define L := L(G).Then the following statements hold for Γ ∞ (γ) defined as in (25): We conclude this section by using Proposition 4.10.b) to examine the relationship between the network size of complete graphs and the design parameters for quantization and self-triggered sampling.For real-valued functions Φ, Ψ on N, we write We see from the condition (42) that if the number R of the quantization levels satisfies then the quantized self-triggered multi-agent system achieves consensus exponentially for some threshold δ i .Hence, the required sensing accuracy for asymptotic consensus is Θ(1) as N → ∞.

Number of indices for data transmission:
Recall that the agents send the sum of relative state measurements to all neighbors for the computation of sampling times.The number of indices used for this communication is 2 dR 0 + 1, where R 0 ∈ N 0 and d ∈ N satisfy R = 2R 0 + 1 and for all i ∈ N , respectively.Hence, the required number of indices for asymptotic consensus is Θ(N ) as N → ∞.Threshold for sampling: We see from the condition (42) that the threshold δ i of the triggering mechanism (12) of agent i has to satisfy Combining this inequality with (45), we have that the required threshold for asymptotic consensus is Θ(N ) as N → ∞.

Computation of Sampling Times
In this section, we describe how the agents compute sampling times in a self-triggered fashion.We discuss an initial candidate of the next sampling time and then the first update of the candidate, followed by the p-th update.Finally, we present a joint algorithm for quantization and self-triggering sampling.
Let i ∈ N and k ∈ N 0 .Define τ i min by (30).By Proposition 3.3.d)and f), there exists ℓ 0 ∈ N 0 such that t i k = t ℓ 0 .5.1.Initial candidate of the next sampling time.First, agent i updates q i at time t = t ℓ 0 .If the neighbor j also updates q j at time t = t ℓ 0 , then agent i receives q j .Next, agent i computes a candidate of the inter-event time, τ i k,0 := min{τ i k,0 , τ i max }, where By ( 36) and ( 37), τ i k,0 ≥ τ i min .Agent i takes t i k + τ i k,0 as an initial candidate of the next sampling time.If agent i does not receive an updated q j from any neighbors j on the interval (t i k , t i k + τ i k,0 ), then t i k + τ i k,0 is the next sampling time, that is, agent i updates q i at t = t i k + τ i k,0 .Using the Lambert W -function, one can write τ i k,0 more explicitly.To see this, we first note that the solution τ = τ * of the equation Define the function ϕ 0 by for a, c ∈ R and b > 0. We also set 5.2.First update.If agent i receives an updated q j from some neighbor j by t = t i k + τ i k,0 , then agent i must recalculate a candidate of the next sampling time as in the self-triggered method proposed in [14].We will now consider this scenario, i.e., the case where I(ℓ) is defined as in Section 3.2.Let t ℓ 1 ∈ (t i k , t i k + τ i k,0 ) be the first instant at which agent i receives updated data after t = t i k .Since t i k = t ℓ 0 , one can write ℓ 1 as ℓ 1 = min{ℓ ∈ N : ℓ > ℓ 0 and I(ℓ) ∩ N i ̸ = ∅}.
Note that agent i may receive updated data from several neighbors at time t = t ℓ 1 .
By using the new data, agent i computes the following inter-event time at time t = t ℓ 1 : .
is a new candidate of the next sampling time.By (36) and (37), we obtain As in the initial case, if agent i does not receive an updated q j from any neighbors j on the interval (t ℓ 1 , t i k + τ i k,1 ), then t i k + τ i k,1 is the next sampling time.Otherwise, agent i computes the next sampling time again in the same way.
One can rewrite τ i k,1 by using the Lambert W -function.To see this, we define e −ωτ can be written as ).Next we consider the case a i k,1 c i k,1 < 0. In this case, the condition ( 48) is equivalent to For the latter inequality, we have that ).It may also occur that To see this, we first observe that Let W −1 be the secondary branch of the Lambert W -function, i.e., W −1 (y) is the solution x ≤ −1 of the equation xe x = y for y ∈ [−e −1 , 0).We obtain the infimum of the set in (49) from the following proposition, whose proof is given in Appendix B.
To apply Proposition 5.1, note that From Proposition 5.1, we conclude that 5.3.p-th update.Let p ∈ N and let We consider the case where agent i receives new data from its neighbors at times t = t ℓ 1 , . . ., t ℓp before the next candidate sampling times.
At time t = t ℓp , agent i computes where Combining this with the inequality (52), we obtain , where τ i k is defined as in (12).Thus, we obtain the desired result (53).□ 5.4.Algorithm for quantization and self-triggered sampling.We are now ready to present a joint algorithm for finite-level dynamic quantization and self-triggered sampling.Under this algorithm, the unsaturation condition (4) is satisfied for all k ∈ N 0 and i ∈ N , and the multi-agent system achieves consensus exponentially with decay rate ω; see Theorems 4.6 and 5.2.Moreover, the inter-event times t i k+1 − t i k are bounded from below by the constant τ i min > 0 for all k ∈ N 0 and i ∈ N .
Algorithm 5.3 (Action of agent i on the sampling interval t i k ≤ t < t i k+1 ).
Step 0. Choose the threshold δ i > 0 and the number R = 2R 0 + 1, R 0 ∈ N 0 , of quantization levels such that the inequality (42) holds for all i ∈ N .Choose the upper bounds τ 1 max , . . ., τ N max > 0 of inter-event times and the decay parameter ω of the quantization range E(t) such that 0 < ω ≤ ω, where ω is defined as in (41).
Step 1.At time t = t i k =: t ℓ 0 , agent i performs the following actions i)-v).i) Measure the quantized relative state q ij (t i k ) for all j ∈ N i and deactivate the sensor.ii) Encode the sum q i (t i k ) of the quantized measurements to an index in a finite set with cardinality 2 dR 0 + 1 and transmit the index to each neighbor j ∈ N i .iii) If an index is received from a neighbor at time t = t ℓ 0 , then decode the index and update the sum of the relative state measurements of the neighbor.iv) Compute τ i k,0 by (47), where a i k,0 , b i k,0 , and c i k,0 are defined as in (46).v) Set p = 0.
Step 2. Agent i plans to activate the sensor at time t = t i k + τ i k,p .
Step 3-a.If agent i receives an index from some neighbor on the interval (t ℓp , t i k + τ i k,p ), then agent i performs the following actions i)-iii).Then go back to Step 2.
i) Set p to p + 1 and store the time t ℓp at which the index is received.ii) Decode the index and update the sum of the relative state measurements of the neighbor.If several indices are received at time t = t ℓp , then this action is applied to all indices.iii) Compute τ i k,p by (50), where a i k,p , b i k,p , and c i k,p are defined as in (51).
Step 3-b.If agent i does not receive any indices on the interval (t ℓp , t i k + τ i k,p ), then agent i sets t i k+1 := t i k + τ i k,p .Step 4. Agent i sets k to k + 1.Then go back to Step 1.
Remark 5.4.The proposed method takes advantage of the simplicity of the first-order dynamics in the following way.Assume that the dynamics of agent i is given by ẋi (t) = Ax i (t) + Bu i (t), where A ∈ R n×n and B ∈ R n×m .Then the error x i (t i k + τ ) − x i (t i k ) due to sampling is written as for τ ≥ 0. Since e Aτ − I ̸ = 0 in general, the absolute state x i (t i k ) is required to describe the error x i (t i k + τ ) − x i (t i k ).However, one has e Aτ − I = 0 in the first-order case A = 0, and hence the absolute state x i (t i k ) needs not be measured in the proposed algorithm.Moreover, since the input u i is constant on the sampling interval, the integral term is a linear function with respect to τ in the first-order case A = 0.This enables us to use the Lambert W -function for the computation of sampling times.

Numerical simulation
In this section, we consider the connected network shown in Figure 1, where the number N of agents is N = 6.For each i ∈ N = {1, 2, . . ., 6}, the initial state x i0 is given by x i0 = sin(i).Since and then numerically compute Γ ∞ = 5/3, where Γ ∞ is defined by (25).
The threshold δ i and the upper bound τ i max of inter-event times for the triggering mechanism (12) are given by δ i = 0.04 if i = 1, 6 0.09 otherwise, τ i max = 1 if i = 1, 6 1.5 otherwise, respectively.The reason why agents 1 and 6 have smaller thresholds and upper bounds of interevent times is that these agents have more neighbors than others.For these thresholds, the minimum odd number R satisfying the condition (42) for all i ∈ N is 13.By Theorem 4.6, if the number R of quantization levels is odd and satisfies R ≥ 13, then the multi-agent system achieves consensus exponentially for a suitable decay parameter ω of the quantization range E(t).We use R = 19 for the simulation below.Then R 0 ∈ N 0 with R = 2R 0 + 1 is given by R 0 = 9.When each agent knows d = 3 as a bound of the number of neighbors, as stated in Assumption 2.3, the number of quantization levels for the transmission of the sum of the relative states is 2 dR 0 + 1 = 55, which can be represented by 6 bits.Under this setting of the parameters γ, δ i , τ i max , and R, the maximum decay parameter ω, which is defined as in (41), is given by ω = 0.2145.
In the simulation, we set ω = ω.
Note, however, that these lower bounds are not used for the real-time computation of inter-event times, because all candidates of the inter-event times computed by the agents are greater than or equal to these lower bounds as shown in Section 5.
The state trajectory and the corresponding sampling times of each agent are shown in Figures 2 and  3, respectively, where the simulation time is 16 and the time step is 10 −4 .From Figure 2, we see that the deviation of each state from the average state converges to zero. Figure 3 shows that sampling occurs frequently on the interval [0, 1] but less frequently on the interval [1,16].Agent 3 measures relative states more frequently on the interval [4,7] than on other intervals.This is because the state of agent 3 oscillates due to coarse quantization.Such oscillations can be observed also for other agents, e.g., agent 1 on the interval [3,4].Moreover, we find in Figure 2 that the states of agents 2 and 5 do not change on the intervals [2,4] and [2,7], respectively.This is also caused by coarse quantization.In fact, the quantized values of their relative state measurements are zero on these intervals.However, the proposed algorithm ensures that the quantization errors exponentially converge to zero, and hence the multi-agent system achieves asymptotic consensus.

Conclusion
We have proposed a joint design method of a finite-level dynamic quantizer and a self-triggering mechanism for asymptotic consensus by relative state information.The inter-event times are bounded from below by a strictly positive constant, and the sampling times can be computed efficiently by using the Lambert W -function.The quantizer has been designed so that saturation is avoided and quantization errors exponentially converge to zero.The new semi-norm introduced for the consensus analysis is constructed based on the maximum norm, and the matrix exponential of the negative Laplacian matrix has the semi-contractivity property with respect to the semi-norm.Future work will focus on extending the proposed method to the case of directed graphs and agents with high-order dynamics.

Acknowledgments
This work was supported by JSPS KAKENHI Grant Number JP20K14362.

Lemma 4 . 1 .
Let N ∈ N satisfy N ≥ 2 and let L be the Laplacian matrix of a connected undirected graph with N vertices.Let ∥ • ∥ be an arbitrary norm on R N and the corresponding induced norm on R N ×N .Fix γ ≤ λ 2 (L), and define Γ := sup t≥0 ∥e γt (e −Lt − 1 1)∥.

Table 1 .
Comparison between this study and several relevant studies.
number of data transmissions from neighbors occur until the next sampling time.The next theorem shows that when the neighbors do not update the measurements on the interval (t ℓp , t i Theorem 5.2.Let i ∈ N and k, p ∈ N 0 .Let t ℓ 0 , ..., t ℓp and τ i k,p be as above, and assume that agent i does not receive any measurements from its neighbors on the interval (t ℓp , t i k + τ i k,p ).Then By the definition of t ℓp , we obtain|f i k (τ )| < δ i E(t ℓ 0 + τ ) for all τ ∈ [0, t ℓp − t ℓ 0 ].Moreover, the arguments given in Sections 5.1 and 5.2 show that ϕ (12)τ i k,p ), the candidate t i k + τ i k,p of the next sampling times constructed as above coincides with the next sampling time t i k+1 computed from the triggering mechanism(12).