Consensus analysis of systems with time-varying interactions : An event-triggered approach

TL;DR

This paper analyzes how multi-agent systems with time-varying interactions can achieve consensus using event-triggered control, demonstrating exponential convergence with dynamic triggers and extending results to switching topologies.

Contribution

It introduces a novel event-triggered control approach for systems with time-varying graphs, including switching topologies, and proves exponential consensus under dynamic triggers.

Findings

Dynamic trigger functions ensure exponential convergence.

Static triggers lead to convergence within a bounded region.

Results are validated through simulations.

Abstract

We present consensus analysis of systems with single integrator dynamics interacting via time-varying graphs under the event-triggered control paradigm. Event-triggered control sparsifies the control applied, thus reducing the control effort expended. Initially, we consider a multi-agent system with persistently exciting interactions and study the behaviour under the application of event-triggered control with two types of trigger functions- static and dynamic trigger.We show that while in the case of static trigger, the edge-states converge to a ball around the origin, the dynamic trigger function forces the states to reach consensus exponentially. Finally, we extend these results to a more general setting where we consider switching topologies. We show that similar results can be obtained for agents interacting via switching topologies and validate our results by means of simulations.

Equations

$$L\left(\mathcal{G}\right)=D\left(\mathcal{G}\right)D{\left(\mathcal{G}\right)}^{\top }$$

$$L\left(\tilde{\mathcal{G}}\left(t\right)\right)=D\left(\mathcal{G}\right)W\left(t\right)D{\left(\mathcal{G}\right)}^{\top }.$$

$${\mu }_2{I}_n\ge {\int }_t^{t+T}g(\tau )g(\tau {)}^{T}d\tau \ge {\mu }_1{I}_n\forall t\ge t_0$$

$${\dot{x}}_i=$$

$$u_i=$$

$$\dot{x}(t)=$$

$$=$$

$$x_e=$$

$${\dot{x}}_e=$$

$$x_e=$$

$$L_e\left(\mathcal{G}\right)=$$

$$=$$

$$\dot{{\rm Y}}=$$

$${\alpha }_v\ge \frac{1}{2T}\ln \frac{1}{\left[1-\frac{2k{\lambda }_{\min }(\Lambda ){\mu }_1}{{\left(1+k\sqrt{p}\parallel \Lambda \parallel {\mu }_2\right)}^2}\right]}$$

$$\Vert {\rm Y}(t)\Vert \le$$

$$\Vert x_e\Vert =$$

$$\le$$

$$\Vert x_e\Vert \le$$

$$u_i=$$

$$f_i(e_i(t))=$$

$$f_i(t,e_i(t))=$$

$$\dot{x}(t)=$$

$$\dot{{\rm Y}}=$$

$$\Vert e_i\Vert \le$$

$${\bar{e}}_i=e_j-e_k$$

$$\Vert {\bar{e}}_i\Vert$$

$$\le \Vert e_j\Vert +\Vert e_k\Vert$$

$$\le 2c{e}^{-\beta t}.$$

$$\tilde{e}$$

$$={\Gamma }^{\top }\left[I_p{0}_{m-p}\right]\bar{e}$$

$$={\Gamma }^{\top }{\left[{\bar{e}}_1{\bar{e}}_2\cdots {\bar{e}}_p\right]}^{\top }.$$

$$\Vert \tilde{e}\Vert$$

$$\le \sqrt{p}\Vert \Gamma \Vert \Vert {\bar{e}}_i\Vert$$

$$\le 2c\sqrt{p}\Vert \Gamma \Vert e^{-\beta t}.$$

$$\Vert \tilde{e}(t)\Vert \le C{e}^{-\beta t}.$$

Full text

1 Introduction

A flock of birds, a swarm of bees, a school of fishes, a colony of ants -all display a wonderful coordination and complex patterns which have caught our attention since time immemorial. On closer observation we realize that these seemingly complex patterns emerge out of simple yet powerful, local rules on each agent of the group based on interactions with only the neighbours. When we attempt to algorithmize such a 'decentralized' behaviour, two types of questions can be posed- what would be the result of a particular decentralized control law or what decentralized control law would result in a desired global pattern or formation. Consensus is one of the most common and powerful 'global' behaviours usually studied, primarily because it can be easily extended to other problems like formation, rendezvous, flocking etc. Vicsek et al. (1995) proposed an average-based decentralized control law for a multi-agent system based on only local information and observed that the agents attain consensus while Jadbabaie et al. (2003) gave a proof of the same assuming the agents interact via connected graphs that can switch at different time instants. Various results have been proposed by Ren and Beard (2005), Ren and Atkins (2005), Olfati-Saber and Shamma (2005), and others on consensus of a multi-agent system under directed or undirected graphs, with switching topologies and also with time delays.

Studying consensus behaviour for systems with switching or time-varying graphs is naturally of interest as in real-life scenarios it is not possible to assume that each agent has the same set of neighbours at all times. Time-varying graphs are useful when the information from each neighbour is assigned a weight proportional to the reliability of the information or the distance between them. Martin and Girard (2013) proved consensus under the assumptions of persistent-connectivity and cut-balance interactions. Chowdhury et al. (2016) obtained bounds on the rate of convergence for single-integrator and double integrator dynamics under persistent interactions.

With control laws being implemented on digital computers, developing discrete-time counterparts to continuous-time control laws is an eventuality. Among discrete control laws event-triggered control is preferred over time-triggered control as it comes into play only when an 'event' is triggered, thus sparsifying control. Event-triggered control of multi-agent systems has been studied by many. Dimarogonas et al. (2012), Seyboth et al. (2011) prove consensus under a connected time-invariant graph for single integrator dynamics, while Yu et al. (2015), Zhu et al. (2014) have extended the results for agents with general linear systems dynamics. Chen and Dai (2016) has studied the consensus of time varying systems with non-linear dynamics under event-triggered control but under a constant interaction topology. Our work considers the broader case of time-varying graphs that can have different spanning-trees and we prove consensus of single-integrator systems under event-triggered control structure.

The paper is organized as follows. We brush up on graph theory and persistent excitation in section 2. The system dynamics for single integrator systems are introduced in section 3. In section 4, we introduce the notion of event-triggered control, define the trigger conditions and evaluate convergence under event-triggered control in section 5. We extend the results to switching graphs in section 5.1 and show simulation results in section 6. The results that we obtained are summarized in section 7.

2 Preliminaries

2.1 Notions of graph theory

In this work we consider agent interactions represented by undirected graphs $\mathcal{G}=\left(V,\,E\right)$, where $V=\left\{v_{1},v_{2},\cdots v_{n}\right\}$ denotes a non-empty set of nodes and $\left[V\right]^{2}\supseteq E=\left\{e_{1},e_{2},\cdots e_{m}\right\}$ is the edge set, where $\left[V\right]^{2}$ is the set of all subsets of $V$ containing two elements. Each node represents an agent of the system and each edge $\left({v_{i},v_{j}}\right)$ signifies that the agents occupying nodes $v_{i}$ and $v_{j}$ can exchange information with each other. We define $D\left({\mathcal{G}}\right)\left({=[d_{ij}]}\right)\in\mathbb{R}^{n\times m}$ to be the incidence matrix associated with the graph $\mathcal{G}$ by arbitrarily assigning orientation to each edge $e_{j}\in E$. Then $[d_{ij}]=-1$ if $v_{i}$ is the tail of $e_{j}$ ,$[d_{ij}]=1$ if $v_{i}$ is the head of $e_{j}$, $[d_{ij}]=0$ otherwise. The Laplacian of $\mathcal{G}$,

[TABLE]

is a symmetric square matrix that captures the inter-connections between each pairs of nodes. We model the time-varying interactions between the agents by assuming a constant, underlying graph $\mathcal{G}$ with edge weights that can be time-varying. The Laplacian can be tweaked to reflect the resultant time-varying graph $\tilde{\mathcal{G}}\left({t}\right)$ as,

[TABLE]

where $W\left({t}\right)\in\mathbb{R}^{m\times m}$ is a diagonal matrix which captures the time-varying nature of each interaction and $w_{ii}\geq 0\,\,\forall t$. We assume that the underlying graph $\mathcal{G}$ is connected and therefore contains a spanning-tree, i.e. the edge set $E$ of $\mathcal{G}$ can be partitioned into two subsets $E=E_{\tau}\cup E_{c}$, where $E_{\tau}$ consists of the spanning-tree edges and $E_{c}$ contains the cycle edges. We also assume that the time-varying edge weight matrix $W\left({t}\right)$ is piece-wise continuous and satisfies the persistent excitation condition with constants $\left({\mu_{1},\mu_{2},T}\right)$.

2.2 Persistence of excitation

Definition 1

(Sastry and Bodson, 2011, p. 72)* The signal $g(\cdot):\mathbb{R}^{\geq 0}\rightarrow\mathbb{R}^{n\times m}$ is Persistently Exciting (PE) if there exist finite positive constants $\mu_{1},\mu_{2},T$ such that,*

[TABLE]

A function $g(\cdot)$ that satisfies the condition (3) is said to be persistently exciting with constants $(\mu_{1},\mu_{2},T)$.

We say that a graph $\tilde{\mathcal{G}}$ is persistently exciting if its associated edge-weight matrix $W\left({t}\right)$ is persistently exciting.

2.3 Other Conventions

$\lVert\cdot\rVert$ denotes the frobenius 2-norm on vectors and the induced 2-norm on matrices. $\lambda_{min}\left({\cdot}\right)$ and $\lambda_{min}\left({\cdot}\right)$ operate on square matrices and return the minimum and maximum eigenvalues respectively of the said matrix. Boldfaced $1$ and [math] ($\mathbf{1},\mathbf{0}$) represent vectors with all ones and all zeroes respectively and $I$ is used to denote the identity matrix. Their dimensions can be inferred contextually if not mentioned explicitly.

3 Network Models

In this section we state the single integrator dynamical equations and perform a series of linear transformations, to bring them to a form that we could work with later on. This section has been taken from Chowdhury et al. (2016) and presented here for reference of the readers.

3.1 Single Integrator

Consider a multi-agent system with states $x_{i}\in\mathbb{R}$ for $i=1,2,3\cdots n$ with a connected underlying graph $\mathcal{G}$ and the Laplacian of the time-varying graphs $L\left({\tilde{\mathcal{G}}\left({t}\right)}\right)=[l_{ij}\left({t}\right)]$. The dynamics of each state with control $u_{i}\in\mathbb{R}$ can be written as,

[TABLE]

for $t>t_{0}$ with initial condition $x_{i}\left({t_{0}}\right)\in\mathbb{R}$ for $i=1,2,3.....n$ and positive control gain $k\in\mathbb{R}$. The augmented dynamics of the system can be written in terms of the state vectors $x=\left[x_{1},x_{2},\cdots x_{n}\right]^{\top}\in\mathbb{R}^{n}$ as,

[TABLE]

Taking cue from Zelazo and Mesbahi (2011), Mesbahi and Egerstedt, 2010, p. 77-81 we transform the consensus problem of equation (5) into a stabilization problem by considering the edge states instead of node states. We effect the conversion through the following transformation,

[TABLE]

The dynamics of the edge states, on differentiation of (6) yields,

[TABLE]

where $L_{e}\left(\mathcal{G}\right)\in\mathbb{R}^{m\times m}$ represents the edge-Laplacian of the graph $\mathcal{G}$ and can be expressed as, $L_{e}\left({\mathcal{G}}\right)=D\left({\mathcal{G}}\right)^{\top}D\left({\mathcal{G}}\right)$ as shown in Zelazo and Bürger (2014). Also, using the fact that the underlying graph contains a spanning-tree $\mathcal{G}_{\tau}$ we partition the edge states after a suitable permutation as follows,

[TABLE]

for $x_{\tau}\in\mathbb{R}^{p}$ and $x_{c}\in\mathbb{R}^{m-p}$ where $p$ is the number of edges in $\mathcal{G}_{\tau}$. Similarly we can partition $D\left({\mathcal{G}}\right)$, $W\left({t}\right)$ as $D\left(\mathcal{G}\right)=\left[D\left(\mathcal{G}_{\tau}\right)\,\,D\left(\mathcal{G}_{c}\right)\right]$, W\left(t\right)=\left[\begin{array}[]{cc}W_{\tau}\left(t\right)&0\\ 0&W_{c}\left(t\right)\end{array}\right]. The edge-Laplacian, in terms of the partitions of $D\left({\mathcal{G}}\right)$ is then,

[TABLE]

We know that for connected graphs, $x_{c}$ can always be written as $x_{c}=Z^{\top}x_{\tau}$ as shown in Sandhu et al. (2005) where $Z=\left(L_{e}\left({\mathcal{G}_{\tau}}\right)\right)^{-1}D\left(\mathcal{G}_{\tau}\right)^{\top}D\left(\mathcal{G}_{c}\right)$. This is because the spanning-tree edges essentially capture the behaviour of all the edges. So we can focus our attention only on the spanning-tree edges. Using (7), (10) and (13), we get $\dot{x}_{\tau}=-kL_{e}\left(\mathcal{G}_{\tau}\right)RW\left(t\right)R^{\top}x_{\tau}$ where $R=\left[I_{p}\,\,Z\right]\in\mathbb{R}^{p\times m}$. The preceding transformations that helped us re-write the system dynamics in terms of $x_{\tau}$ are along the lines of Zelazo and Mesbahi (2011) and Mesbahi and Egerstedt, 2010, p. 77-81. Since $L_{e}\left(\mathcal{G}_{\tau}\right)$ is symmetric and positive definite, it can be diagonalized as $L_{e}\left(\mathcal{G}_{\tau}\right)=\Gamma\Lambda\Gamma^{\top}$ for some orthogonal matrix $\Gamma\in\mathbb{R}^{p\times p}$ and diagonal matrix $\Lambda\in\mathbb{R}^{p\times p}$. Consider a change of variable by the transformation $\Upsilon=\Gamma^{\top}x_{\tau}$. The above equation becomes,

[TABLE]

where $M\left(t\right)=\Gamma^{\top}RW\left(t\right)R^{\top}\Gamma\in\mathbb{R}^{p\times p}$. As stated earlier, the consensus problem of equation (5) is equivalent to stabilization problem of equation (14). For the sake of further reference, we denote the preceding set of transformations from $x$ to $\Upsilon$ by $\psi:=\Gamma^{\top}\left[I_{p}\;0_{m-p}\right]D\left({\mathcal{G}}\right)^{\top}$. So we have, $\Upsilon=\psi x$. We state (Chowdhury et al., 2016, Theorem 5) and use the result to obtain the rate of convergence to consensus of a single integrator system defined by equations (5).

Theorem 1

((Chowdhury et al., 2016, Theorem 5)) Consider the closed-loop consensus dynamics (5). Assume that, the underlying graph $\mathcal{G}$ is connected. The states of the closed-loop dynamics $x(t)$ with time-varying communication topology characterized by $W(t)$, achieve consensus exponentially, if there exists a spanning tree with corresponding edge-weight matrix $W_{\tau}(t)$ that is persistently exciting. Further the convergence rate $\alpha_{v}$ to consensus is bounded below by,

[TABLE]

where, $T,\mu_{1}$ and $\mu_{2}$ are the constants appearing in Definition 1 and $\Lambda$ is a diagonal matrix containing the eigenvalues of the spanning tree edge Laplacian matrix.

Further,

[TABLE]

where $\alpha_{v}$ and $m_{v}$ can be calculated from the underlying graph and control gains. Note: The relationship between $\left\|x_{e}\right\|$ and $\lVert\Upsilon\rVert$ can be established in the following way.

[TABLE]

Using the fact that $\Gamma$ is orthogonal, $x_{\tau}=\Gamma\Upsilon$,

[TABLE]

where $\rho=\left\|\Gamma\right\|\sqrt{1+\left\|Z^{\top}\right\|^{2}}$. The induced 2-norm of $Z$ can be calculated as $\lambda_{max}\left(Z^{\top}Z\right)^{\frac{1}{2}}$.

4 Event-Triggered Control

In this section we outline the event-triggered control strategies and show how it modifies the single-integrator dynamics defined by equation (5). Under the event-triggered control paradigm each agent broadcasts a (piecewise)constant value, $\hat{x}_{i}$ which is updated to the current value of the state whenever an 'event is triggered'. The control applied by each agent would then be,

[TABLE]

To define an event, we introduce error variables $e_{i}\left(t\right)=\hat{x}_{i}\left(t\right)-x_{i}\left(t\right)$ which denote the difference between the broadcasted value and the current value of the state for each agent. The trigger condition that updates the broadcasted states $\hat{x}$ effectively shapes the behaviour of the system. We define the trigger condition using a trigger function for each state $f_{i}\left({t,e_{i}}\right):\mathbb{R}\rightarrow\mathbb{R}$ . An event is said to be 'triggered' when $f_{i}>0$. Once an event is 'triggered' say at time $t^{*}$, the broadcast value is updated $\mathsf{i.e.}$ $\hat{x}_{i}\left({t}\right)=x_{i}\left({t^{*}}\right)\implies e_{i}\left({t^{*}}\right)=0$ for $t^{*}\leq t<t^{{}^{\prime}}$ where $t^{{}^{\prime}}$ is the time instant when the next subsequent event is triggered. We define two trigger functions as shown in Seyboth et al. (2011)- the static trigger and the dynamic trigger.

Static Trigger Function

[TABLE] 2. 2.

Dynamic Trigger Function

[TABLE]

where $c>0$

The static trigger function can be seen as a special case of the dynamic trigger function with $\beta=0$. So we will perform our analysis with (19) as the trigger function, and substitute $\beta=0$ when we want to evaluate the static-trigger case. The dynamics of single integrator systems under event-triggered control can be written as,

[TABLE]

where $e=[e_{1},e_{2}\cdots e_{n}]^{\top}\in\mathbb{R}^{n}$. The equivalent stabilization problem under event triggered control for single integrator will then be,

[TABLE]

where $\tilde{e}=\psi e$.

4.1 Bounds on the error variables

In this section, we obtain bounds on $\tilde{e}$ in terms of the bounds on $e$. The trigger function is so designed that each $e_{i}$ is always upper-bounded. We can see that,

[TABLE]

We can relate the bounds on $e$ to bounds on $\tilde{e}$ in the following way. We know that $\tilde{e}=\psi\,e$. Let us define $\bar{e}:=D\left({\mathcal{G}}\right)^{\top}e$. From the structure of $D\left({\mathcal{G}}\right)$, we get

[TABLE]

for $i=1,2,\cdots m$ and $j,k$ chosen on the basis of $D\left({\mathcal{G}}\right)$. Using (22), (23) can be rewritten as,

[TABLE]

Also,

[TABLE]

Using the bound on each $\bar{e}_{i}$ from (24),

[TABLE]

Defining $C:=2c\sqrt{p}\left\|\Gamma\right\|$, we can write the above inequality to be,

[TABLE]

5 Consensus Analysis

Theorem 2

Consider a multi-agent system with single integrator dynamics as defined by equation (5). Assume that the underlying graph ($\mathcal{G}$), representing the interaction between the agents be connected, with $p$ edges in the spanning-tree. If the spanning tree edge weight matrix $W_{\tau}\left({t}\right)$ is persistently exciting with constants ($\mu_{1}$,$\mu_{2}$,$T$), then

on application of event-triggered control with a static trigger function defined by equation (18), the edge states $x_{e}$ of the system exponentially converge to a ball around the origin defined by $\lVert x_{e}\rVert\leq\rho\kappa_{2}^{m}$. 2. 2.

on application of event-triggered control with a dynamic trigger function defined by equation (19), the edge states $x_{e}$ of the system exponentially converge to origin. Also the rate of convergence is lower bounded by $\beta$ as defined in (19).

where, $\kappa_{2}=\frac{k\left\|\Lambda\right\|m_{v}Ce^{\alpha_{v}t_{0}+2\alpha_{v}T}\mu_{2}}{e^{\alpha_{v}T}-1}$ and $\rho$ is defined as in (16). Also the closed loop systems in the cases of static and dynamic triggers does not exhibit zeno behaviour when $\beta$ is chosen to be greater than $\alpha_{v}$.

Note: When a connected graph has multiple spanning-trees, $\kappa_{2}^{M}$ and $\kappa_{2}^{m}$ are the largest and smallest values of $\kappa_{2}$ that can be calculated considering each different spanning-tree. {pf} Let $\phi\left({t,t_{0}}\right)$ be the state transition matrix corresponding to system defined by equation (14) . The solution of the system can be written using $\phi\left({t,t_{0}}\right)$ as,

[TABLE]

We obtain a bound on $\lVert\phi(t,t_{0})\rVert$ by the following steps.

[TABLE]

Comparing (28) and (15) we can conclude that,

[TABLE]

The last statement holds true because $\Upsilon\left({t_{0}}\right)$ can be arbitrary. Consider the single integrator multi-agent system with event-triggered control defined by the equation (21). The solution of this system can be expressed as ,

[TABLE]

We get the following inequality from the above equation,

[TABLE]

Using the bounds on $\phi$ from equation (29) and $\tilde{e}\left({\tau}\right)$ from equation (27) we get,

[TABLE]

To obtain a bound on the integral in the above inequality, we divide the interval $\left[t_{0},t\right]$ into partitions of size $T$ i.e. $\left[t_{0},t_{0}+T\right]$, $\left[t_{0}+T,t_{0}+2T\right]$ and so on . The number of such partitions of length $T$ possible will be given by $\theta=\left\lfloor\frac{t-t_{0}}{T}\right\rfloor$, where $\left\lfloor\cdot\right\rfloor$ is the floor function . The last partition can then be written as $\left[t_{0}+\theta T,t\right]$. The integral in (31) can then be written as,$\int_{t_{0}}^{t_{0}+T}e^{-(\beta-\alpha_{v})\tau}\left\|M(\tau)\right\|d\tau+\int_{t_{0}+T}^{t_{0}+2T}e^{-(\beta-\alpha_{v})\tau}\left\|M(\tau)\right\|d\tau\cdots+\int_{t_{0}+\theta T}^{t}e^{-(\beta-\alpha_{v})\tau}\left\|M(\tau)\right\|d\tau$. Applying H$\ddot{o}$lder inequality with $p=\infty$ and $q=1$ and using the fact that $M\left({\tau}\right)$ is persistently exciting and $\beta<\alpha_{v}$ we can obtain the following bound on each interval,

[TABLE]

Using this upper bound on each integral we get

[TABLE]

Using the property of the sum of terms in a geometric progression, we get

[TABLE]

Using inequality (32) in inequality (31),

[TABLE]

We define $\kappa_{1}=e^{\alpha_{v}t_{0}}m_{v}\left\|\Upsilon(t_{0})\right\|-\kappa_{3}$, $\kappa_{2}=e^{\left({\alpha_{v}+\beta}\right)T}\kappa_{3}$,

[TABLE]

to be able to express (33) as,

[TABLE]

Using the properties of floor function, we can simplify the above inequality further,

[TABLE]

From the above expression it can clearly be seen that for the dynamic trigger case, $lim_{t\rightarrow\infty}\left\|x_{e}(t)\right\|=0$. This guarantees consensus of the states $x$. Also as $\beta<\alpha_{v}$ by choice, the rate of decay of the RHS of (36) will be dominated by $\beta$. For the static trigger case, $\beta=0$

[TABLE]

It can be seen that the edge-states converge to a ball around the origin $\lVert x_{e}\rVert\leq\rho\kappa_{2}$ exponentially with a rate of convergence bounded below by $\alpha_{v}$.

**Ruling out zeno behaviour in closed loop systems

**The system states $x_{i}$ and error states $e_{i}$ together form a hybrid system, which makes it necessary for us to ensure that zeno behaviour does not occur. Say that for the $i^{th}$ agent, an event is triggered at a time instant $t_{1}$ and another consecutive event is triggered at time $t_{2}$ for some $t_{0}\leq t_{1}<t_{2}$. To rule out the occurrence of zeno behaviour, it is sufficient to show that there exists a positive, non-zero lower bound on $\gamma:=t_{2}-t_{1}$. We have $\dot{e}_{i}=-\dot{x}_{i}$. Consider the following set of inequalities for time $t_{1}\leq t<t_{2}$.

[TABLE]

Using (36) we can rewrite the above inequality as,

[TABLE]

Also,

[TABLE]

We substitute $\lVert e_{i}\left({t_{2}}\right)\rVert=ce^{-\beta t_{2}}$ as an event is triggered at $t_{2}$. Integrating the inequality (37) with limits $t_{1}$ and $t_{2}$ and using the fact that the edge weights are bounded such that $\lVert W(t)\rVert\leq\omega$ and (38) we get,

[TABLE]

Rearranging the above inequality we get,

[TABLE]

It is evident that $\gamma>0$ as the RHS of (39) is strictly positive. The minimum value of $\gamma$ is a solution of the following equation,

[TABLE]

This proves that zeno behaviour does not occur in the dynanic trigger case. When $\beta=0$, the lower bound on gamma is the RHS of equation (40), which rules out zeno behaviour in static trigger case as well. The preceding arguments on ruling out zeno behaviour are similar to those provided in Seyboth et al. (2011) for time-invariant graphs.

5.1 Consensus in Switching Topologies

Theorem 2 is not valid for switching topologies because our spanning tree and thereby the edge states $x_{e}$ itself can be changing. The following corollary extends the aforementioned theorem to agents interacting via switching topologies.

Corollary 1

Let $t_{1},t_{2},\cdots$ be the infinite time sequence of graph switching instants with, $t_{i+1}-t_{i}\geq t_{L}$ for some positive $t_{L}$, and $i=0,1,\cdots$. Consider the agent dynamics (20) corresponding to a single integrator system with event-triggered control. If there exists an infinite sequence of contiguous, non-empty and uniformly bounded time intervals $\tau\left({j,1}\right)$ $\left({Let\;\tau\left({j,l}\right)=\left[t_{i_{j}},t_{i_{j+l}}\right)}\right)$; $j=1,2,\cdots$ starting at $t_{i_{1}}=t_{0}$, with the property that the union of the undirected graphs across each such interval has a spanning tree, then

with static-trigger, the edge-state $x_{e}$ converges to a ball of radius $\lVert x_{e}\rVert\leq\rho\kappa_{2}^{M}$. 2. 2.

with dynamic-trigger, the edge-state $x_{e}$ converges exponentially to origin at rate greater than $\beta$.

{pf}

The case of switching topologies differs from the setting of theorem 2 by the fact that the union of graphs in each time interval $\tau\left({j,1}\right)$ contains a different spanning-tree compared to the occurrence of the same spanning tree in theorem 2. We show that Corollary 1 can be treated as an extension of Theorem 2 using the results of Van der Waerden's theorem ((Graham et al., 1980, p. 29)). The collection of possible spanning-tress forms a finite, non-empty set. Taking each possible spanning-tree to be a different colour, by Theorem (Graham et al., 1980, p. 29) it is possible to find $N$ such that in the interval $\left({t_{i_{1}},t_{i_{N}}}\right)$, the same spanning-tree occurs in each $\tau\left({1+(j-1)d,1}\right)$ for $j=1,2,\cdots k$ and some $d>0$. This allows us to select a persistence window $T>(d+1)t_{max}$ where $t_{max}=max_{j}\;\tau\left({j,1}\right)$ for $j=1,2,\cdots N$. With the aforementioned selection of $T$, Theorem 2 can be invoked to prove the convergence of the edge states $x_{e}$ to a ball around origin in the case of static trigger and to origin in the case of dynamic trigger. As we cannot predict which particular spanning-tree repeats in each interval $\tau\left({1+(i-1)d,1}\right)$, the least conservative bounds are chosen.

6 Simulations

A multi-agent system with four agents under switching graphs was simulated using Matlab© for the static and dynamic trigger cases. The underlying communication topology considered is shown in figure 1 and the different spanning-trees that were switched between are shown in figure 2.

The edge-weights were chosen as $g_{i}$ = square$(4*t,20-(i-1)0.1\pi)+1).*sin(5*t)$ for $i=1,2\cdots 4$ and $g_{6}=0$, where the function square$\left({at,b}\right)$ for $a,b\in\mathbb{R}$ generates a square wave of unit amplitude, period $\frac{2\pi}{a}$ and duty-cycle $\left({\frac{T_{\text{on}}}{T_{\text{off}}+T_{\text{on}}}}\right)$ $b$. The aforementioned $g_{i}$ are defined to emulate real scenarios where there might be instances when no edges are active. The initial value of the states was taken to be $x_{0}=\left[1\>2\>0.3\>0.4\right]^{\top}$ Plots 3 and 4 show the evolution of the system states $x$ and the norm of the edge states $\lVert x_{e}\rVert$ under the static trigger with $c=0.5$ (refer equation (18)). The evolution of the states with dynamic trigger function with $\beta=0.06$ and $c=0.5$ (refer equation (19)) is plotted in figures (5) and (6). The bounds were calculated using inequalities presented in Corollary 1 and plotted along with the norm of the edge states. These plots show the convergence of the edge states $x_{e}$ to a ball around the origin in case of static trigger and consensus of states $x$ in the case of dynamic trigger.

7 Conclusions

The application of event-triggered control to classical consensus algorithms with time-varying, persistently exciting topologies guarantees consensus with dynamic trigger function. Under the more practically implementable static trigger function, the edge-states converge to a ball around the origin. For switching topologies, we utilize the work of Chowdhury et al. (2016) to show that we can extend the results of the persistent, continuously varying graphs to the case of switching topologies. The convergence bounds thus obtained depend on the 'slowest' spanning tree.