Asynchronous Distributed Consensus with Minimum Communication

A graph $G=(V,E)$ is a finite set of nodes $V$ connected by a set of edges $E\subseteq(V\times V)$. An edge between nodes $i$ and $j$ is represented by an ordered pair $(i,j)\in E$. A graph $G$ is said to be simple if $(i,i)\not\in E,~\forall i\in V$. A graph $G$ is said to be undirected if $(i,j)\in E$ implies $(j,i)\in E$. In an undirected graph $G$, if $(i,j)\in E$ (and equivalently $(j,i)\in E$), then the nodes $i$ and $j$ are said to be neighbours of each other. A path between nodes $i$ and $j$ in an undirected graph $G$ is a sequence of edges $(i,k_{1}),(k_{1},k_{2}),\dots,(k_{r-1},k_{r}),(k_{r},j)\in E$. An undirected graph $G$ is said to be connected if there exists a path between any two nodes in $G$. Let $n_{i}$ denote the number of neighbours of node $i$ and $|V|$ denote the cardinality of set $V$. Then, the Laplacian matrix $L\in{\mathbb{R}}^{|V|\times|V|}$ of a simple undirected graph $G=(V,E)$ is defined as

Consider a multi-agent system (MAS) of $n$ single-integrator agents, labeled as $a_{1}$, $a_{2},\dots,a_{n}$, with dynamics

where $x_{i}(t)\in\mathbb{R}$ and $u_{i}(t)\in\mathbb{R}$ are the state and the control input of agent $a_{i}$, respectively. Let $X:=[x_{1},x_{2},\dots,x_{n}]$ and ${\bf{u}}:=[u_{1},u_{2},\dots,u_{n}]$ be the augmented state and control vectors of MAS (1), respectively. Define the set $P:=\{1,2,\dots,n\}$.

Let $G$ be a time-invariant simple undirected graph, whose nodes represent agents in MAS (1) whereas the edges represent the communication links between agents, over which they exchange information with their neighbours. Let $S_{i}$ be the set of indices of neighbours of agent $a_{i}$. Note that $i\not\in S_{i}$ as $G$ is a simple graph. The cardinality of the set $S_{i}$ is denoted by $n_{i}$. Let $L$ denote the Laplacian matrix of $G$.

We make the following assumptions about MAS (1):

1. 1.

   The control input $u_{i}$ of each agent belongs to the set

   $$\mathcal{U}:=\{u\in\mathcal{M}~|~|u(t)|\leq\beta,~\forall t\geq 0\}$$

   where $\mathcal{M}$ denotes the set of measurable functions from $[0,\infty)$ to $\mathbb{R}$.

2. 2.

   The communication graph $G$ is connected.

3. 3.

   The communication delay is zero.

In this paper, we will be using two notions of consensus, which we define next.

Define $Z(t)=[z_{1}(t),\dots,z_{n}(t)]:=LX(t)$. Then, it follows from the definition of the Laplacian matrix that

As $z_{i}$ is the sum of differences of agent $a_{i}$’s state with its neighbours, we call it the local disagreement of agent $a_{i}$. It is well known [15] that for a MAS with connected, time-invariant communication graph, conventional consensus at instant $\widetilde{t}$ is equivalent to $z_{i}(t)=0,~\forall t\geq\widetilde{t},~\forall i\in P$. However, in many practical applications, it is not necessary that each $z_{i}$ becomes exactly zero. It is sufficient if each $|z_{i}|$ remains below a prespecified consensus bound. This motivates our next notion of consensus, namely $\alpha$-consensus.

In this paper, we consider a discrete communication model. Let $t^{k}_{i}$ denote the $k$th update instant (also referred to as the communication instant) of agent $a_{i}$, at which it accesses the state information of its neighbours and then, based on that information, updates its control. Our communication model is asynchronous, i.e., the update instants $t^{k}_{i}$’s of two different agents need not coincide.

As the communication model is discrete, the number of update instants is a good measure of communication effort. Hence, we define the communication cost of agent $a_{i}$, denoted by $C_{i}(t)$, as the number of update instants of agent $a_{i}$ in the time interval $[0,t]$. Then, we define the aggregate communication cost of MAS (1), denoted by $C_{MAS}(t)$, as

Let $X(0)$ be an initial condition of MAS (1), $\alpha$ be the prespecified consensus bound and $T$ be the prespecified consensus time. Our objective is to develop a protocol which minimizes the communication cost $C_{MAS}(T)$, under the constraint of achieving $\alpha$-consensus of MAS (1) within time $T$.

As discussed in Section I, due to information structure imposed by graph $G$, an individual agent in MAS (1) has access only to its own information and that of its neighbours. Therefore, the proposed protocol needs to be distributed, i.e., based only on the local information. In addition, an individual agent does not have global information such as the number of agents $n$ in MAS (1), the structure of the communication graph $G$, the complete initial condition $X(0)$ etc. To address these uncertainties, the proposed protocol must be able to achieve $\alpha$-consensus of MAS (1) within the prespecified time $T$, for any $n$, any connected $G$ and any $X(0)\in{\mathbb{R}}^{n}$.

Since the input set $\mathcal{U}$ is magnitude bounded, for a fixed $T$, it will not be possible to achieve $\alpha$-consensus within time $T$, for every $X(0)\in{\mathbb{R}}^{n}$. Thus, the consensus time must be specified as a function of $X(0)$. To highlight this dependence on $X(0)$, we modify the notation of the prespecified consensus time from $T$ to $T\big{(}X(0)\big{)}$. Similarly, the communication costs $C_{i}$ and $C_{MAS}$ depend on $X(0)$. Thus, we modify their notations from $C_{i}(t)$ and $C_{MAS}(t)$ to $C_{i}\big{(}t,X(0)\big{)}$ and $C_{MAS}\big{(}t,X(0)\big{)}$, respectively. Now, we formalize our objective as follows:

In Problem 3, the time $T\big{(}X(0)\big{)}$ can be specified as any function of $X(0)$. However, in practical applications, it is desirable to set $T\big{(}X(0)\big{)}$ to the minimum feasible value.

In [21], the time-optimal control rule is proposed which achieves conventional consensus of MAS (1) in minimum time. This control rule and the corresponding consensus time, denoted by $u^{\star}_{i}$ and $T^{\ast}\big{(}X(0)\big{)}$, respectively, are presented below. Let $X(0)=[x_{1}(0),\dots,x_{n}(0)]$ be an initial condition of MAS (1). Define $x^{min}(0):=\min\big{\{}x_{1}(0),\dots,x_{n}(0)\big{\}}$ and $x^{max}(0):=\max\big{\{}x_{1}(0),\dots,x_{n}(0)\big{\}}$. Recall that $z_{i}$ denote the local disagreement of agent $a_{i}$. Let $sign$ denote the standard signum function. Then, the time-optimal consensus rule from [21] is

with the corresponding consensus time

Notice that the control rule (5) requires instantaneous access to the local disagreement $z_{i}$, and as a result, demands continuous communication. Then, it follows from the time optimality of $T^{\ast}\big{(}X(0)\big{)}$ that a discrete communication based protocol cannot achieve $\alpha$-consensus of MAS (1) within time $T^{\ast}\big{(}X(0)\big{)}$. This implies that Problem 3 is infeasible for $T\big{(}X(0)\big{)}\leq T^{\ast}\big{(}X(0)\big{)}$ and we should assume $T\big{(}X(0)\big{)}>T^{\ast}\big{(}X(0)\big{)}$. We further assume that $T\big{(}X(0)\big{)}\geq 2T^{\ast}\big{(}X(0)\big{)}$. This assumption makes Problem 3 tractable and results in a particularly simple closed form solution for the control inputs. We solve Problem 3 for $T\big{(}X(0)\big{)}=2T^{\ast}\big{(}X(0)\big{)}$ in Sections III-V and later extend it for $T\big{(}X(0)\big{)}>2T^{\ast}\big{(}X(0)\big{)}$ in Section VI.

A discrete communication based distributed protocol is said to be a feasible solution of Problem 3 if it satisfies constraint (4), i.e., achieves $\alpha$-consensus of MAS (1) within time $T\big{(}X(0)\big{)}$, for any number of agents $n$, any connected graph $G$ and any initial condition $X(0)\in{\mathbb{R}}^{n}$. Recall that $\Psi(n)$ denotes the set of connected graphs with $n$ nodes. Then, the following lemma gives a necessary condition on the feasible solutions of Problem 3.

The following lemma shows that Protocol $\Romannum{1}$ satisfies the necessary condition (11). The proof of this lemma relies on the derivation of control rules (7)-(10) in Protocol $\Romannum{1}$, which is deferred to Section V for better structure of the paper. Hence, we defer the proof of the said lemma to Section V-C.

The following theorem shows that Protocol $\Romannum{1}$ achieves $\alpha$-consensus of MAS (1) within time $2T^{\ast}\big{(}X(0)\big{)}$.

According to Theorem 8, under Protocol $\Romannum{1}$, the duration $2T^{\ast}\big{(}X(0)\big{)}$ is an upper bound on the time required to achieve $\alpha$-consensus of MAS (1). Next, we show with the following example that there exists a MAS of the form (1) for which the $\alpha$-consensus time under Protocol $\Romannum{1}$ is arbitrarily close to $2T^{\ast}\big{(}X(0)\big{)}$.

The following theorem shows that under Protocol $\Romannum{1}$, the time required to achieve $\alpha$-consensus of the MAS in Example 9 is arbitrarily close to $2T^{\ast}\big{(}X(0)\big{)}$.

Let $z_{i}\big{(}t^{k}_{i}\big{)}$ be the local disagreement of agent $a_{i}$ at its update instant $t^{k}_{i}$ such that $\big{|}z_{i}\big{(}t^{k}_{i}\big{)}\big{|}\leq\alpha$. The goal of agent $a_{i}$ is to maximize the inter-update duration $\big{(}t^{k+1}_{i}-t^{k}_{i}\big{)}$ by delaying its next update instant $t^{k+1}_{i}$. However, while doing so, agent $a_{i}$ must satisfy the necessary condition (11). For that purpose, agent $a_{i}$ needs to ensure that $|z_{i}(t)|\leq\alpha$ for all $t\in[t^{k}_{i},t^{k+1}_{i}]$. Recall the definition of $z_{i}$ in (2). Then, the evolution of $z_{i}$ in the interval $[t^{k}_{i},t^{k+1}_{i}]$ is given as

This evolution depends on control inputs $u_{i}$ and $u_{j},~\forall j\in S_{i}$, in the interval $[t^{k}_{i},t^{k+1}_{i}]$. Note that any instant $t\in[t^{k}_{i},t^{k+1}_{i}]$ can be an update instant of a neighbouring agent $a_{j},~j\in S_{i}$, at which $a_{j}$ updates its control. This updated value is not known to agent $a_{i}$ in advance, at instant $t^{k}_{i}$. Thus, while maximizing the inter-update duration $\big{(}t^{k+1}_{i}-t^{k}_{i}\big{)}$, agent $a_{i}$ needs to consider the worst-case realizations of the neighbouring inputs $u_{j},~\forall j\in S_{i}$, which result in the minimum value of the inter-update duration. This leads to the following max-min optimization:

The following lemma shows that control law (7)-(8) is the solution of optimization (14)-(16).

Let $z_{i}\big{(}t^{k}_{i}\big{)}$ be the local disagreement of agent $a_{i}$ at its update instant $t^{k}_{i}$ such that $\big{|}z_{i}\big{(}t^{k}_{i}\big{)}\big{|}>\alpha$. Then, in order to achieve $\alpha$-consensus, it is necessary to steer $\big{|}z_{i}\big{(}t^{k}_{i}\big{)}\big{|}$ below the consensus bound $\alpha$. If that could be done in time-optimal manner, it is an added advantage. Recall that in Section II-F, we discussed the time-optimal consensus rule (5) which achieves conventional consensus of MAS (1) in minimum time. Motivated from this rule, we choose our control rule as

As mentioned in Section V-A, the goal of agent $a_{i}$ is to maximize the inter-update duration $\big{(}t^{k+1}_{i}-t^{k}_{i}\big{)}$ by delaying its next update instant $t^{k+1}_{i}$. However, while doing so, agent $a_{i}$ must satisfy the necessary condition (11).

Recall that $\big{|}z_{i}\big{(}t^{k}_{i}\big{)}\big{|}>\alpha$. Without loss of generality, assume that $z_{i}\big{(}t^{k}_{i}\big{)}<-\alpha$. Then, it follows from (17) that $u^{\ast}_{i}(t)=\beta,~\forall t\in[t^{k}_{i},t^{k+1}_{i})$. Let $\hat{t}\in[t^{k}_{i},t^{k+1}_{i})$ be the first time instant at which $z_{i}(\hat{t}~\!)=-\alpha$. Then, in order to satisfy (11), agent $a_{i}$ needs to ensure that

which is equivalent to

As discussed in Section V-A, at instant $t^{k}_{i}$, agent $a_{i}$ does not know the future values of the neighbouring inputs in the interval $[t^{k}_{i},t^{k+1}_{i})$. Thus, while maximizing the inter-update duration $\big{(}t^{k+1}_{i}-t^{k}_{i}\big{)}$, agent $a_{i}$ needs to consider the worst-case realizations of the neighbouring inputs $u_{j},~\forall j\in S_{i}$, which result in the minimum value of the inter-update duration. This leads to the following max-min optimization:

The following lemma shows that control law (9)-(10) is the solution of optimization (18)-(19).

In this section, we present the proof of Lemma 7 which claims that Protocol $\Romannum{1}$ satisfies the necessary condition (11).

By using Lemma 7, it is already shown in Theorem 8 that Protocol $\Romannum{1}$ satisfies constraint (4) in Problem 3 for time $2T^{\ast}\big{(}X(0)\big{)}$, i.e., achieves $\alpha$ consensus of MAS (1) within time $2T^{\ast}\big{(}X(0)\big{)}$, for every $n$, every connected $G$ and every $X(0)\in{\mathbb{R}}^{n}$. In the next section, we prove the optimality of Protocol $\Romannum{1}$.

Consider MAS (1) with initial condition $X(0)\in{\mathbb{R}}^{n}$. Let $Q$ be any protocol which is a feasible solution of Problem 3 for $T\big{(}X(0)\big{)}=2T^{\ast}\big{(}X(0)\big{)}$, i.e., a discrete communication based distributed protocol which achieves $\alpha$-consensus of MAS (1) within time $2T^{\ast}\big{(}X(0)\big{)}$, for every $n$, every connected $G$ and every $X(0)\in{\mathbb{R}}^{n}$. Let $C^{Q}_{MAS}\Big{(}2T^{\ast}\big{(}X(0)\big{)},X(0)\Big{)}$ and $C^{\ast}_{MAS}\Big{(}2T^{\ast}\big{(}X(0)\big{)},X(0)\Big{)}$ denote the value of the aggregate communication cost $C_{MAS}$ defined in (3), under Protocol $Q$ and Protocol $\Romannum{1}$, respectively.