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Simple synchronization protocols for

heterogeneous networks: beyond passivity

(extended version)

Anton V. Proskurnikov ∗,∗∗ Manuel Mazo Jr ∗

∗Delft Center for Systems and Control (DCSC), Delft University of

Technology, The Netherlands

∗∗ ITMO University & Institute for Problems of Mechanical

Engineering (IPME RAS), St. Petersburg, Russia

Abstract: Synchronization among autonomous agents via local interactions is one of the

benchmark problems in multi-agent control. Whereas synchronization algorithms for identical

agents have been thoroughly studied, synchronization of heterogeneous networks still remains

a challenging problem. The existing algorithms primarily use the internal model principle,

assigning to each agent a local copy of some dynamical system (internal model). Synchronization

of heterogeneous agents thus reduces to global synchronization of identical generators and local

synchronization between the agents and their internal models. The internal model approach

imposes a number of restrictions and leads to sophisticated dynamical (and, in general,

nonlinear) controllers. At the same time, passive heterogeneous agents can be synchronized

by a very simple linear protocol, which is used for consensus of ﬁrst-order integrators. A natural

question arises whether analogous algorithms are applicable to synchronization of agents that

do not satisfy the passivity condition. In this paper, we study the synchronization problem

for heterogeneous agents that are not passive but satisfy a weaker input feedforward passivity

(IFP) condition. We show that such agents can also be synchronized by a simple linear protocol,

provided that the interaction graph is strongly connected and the couplings are suﬃciently

weak. We demonstrate how stability of cooperative adaptive cruise control algorithms and some

microscopic traﬃc ﬂow models reduce to synchronization of heterogeneous IFP agents.

1. INTRODUCTION

As the inﬂuential monograph (Strogatz, 2003) states, “the

tendency to synchronize is one of the most pervasive

drives in the universe, extending from atoms to animals”.

Synchrony among subsystems (agents, cells) of a complex

system is a basic principle, which explains many natu-

ral phenomena (Strogatz, 2003) and has found numer-

ous applications in engineering (Mesbahi and Egerstedt,

2010; Olfati-Saber et al., 2007; Wu, 2007; Ren and Beard,

2008). Establishing synchronization (consensus) is consid-

ered now as a benchmark problem in multi-agent control

and has been thoroughly examined in the recent decades.

Most of the attention has been paid to synchronization

among identical agents. The protocols establishing syn-

chronization among single integrators are usually based

on the idea of contraction: the convex hull, spanned by the

agents’ states, is shrinking until it collapses into a single-

ton (M¨unz et al., 2011). An alternative approach is based

on convergence criteria for inﬁnite matrix products (Ren

and Beard, 2008). The protocols for synchronization of

agents, obeying higher order equations, are similar in

spirit to ﬁrst-order algorithms. Synchronization of linear

and linearly coupled agents is often analyzed via the

spectral decomposition of the Laplacian matrix (Olfati-

Saber et al., 2007; Li et al., 2010; Ren and Beard, 2008;

?Partial funding was provided by STW (project 13712). E-mails:

anton.p.1982@ieee.org, m.mazo@tudelft.nl

Ren and Cao, 2011). Nonlinear protocols are usually

examined by Lyapunov methods (Ren and Cao, 2011),

employing, among others, the Kalman-Yakubovich-Popov

lemma (Zhang et al., 2014; Proskurnikov and Matveev,

2015), contraction theory (DeLellis et al., 2011) and the

idea of incremental passivity (Stan and Sepulchre, 2007;

Proskurnikov et al., 2015; Liu et al., 2015b).

However, in practice autonomous agents are usually het-

erogeneous. Algorithms for output synchronization of

non-identical agents have been proposed quite recently

and most of them employ the internal model princi-

ple (Wieland et al., 2011; De Persis and Jayawardhana,

2014; Isidori et al., 2014; Bidram et al., 2014; Liu et al.,

2015a), assigning to each agent a virtual copy of some

dynamical system, referred to as the internal model or the

local reference generator. The control algorithm then con-

sists of two layers: a protocol, synchronizing the (identical)

reference generators and local model-matching controllers,

synchronizing the agents to their generators.

The general internal model approach has, however, several

disadvantages. Being formally decentralized, its implemen-

tation assumes that the agents share the same internal

model and are able to match it (e.g. in the case of

linear agents the Francis regulator equations should be

solvable (Wieland et al., 2011; Liu et al., 2015a)). Hence

design of an algorithm requires to know the global infor-

mation about the network. Unlike many synchronization

arXiv:1703.02937v1 [cs.SY] 8 Mar 2017

algorithms for identical agents (Olfati-Saber et al., 2007;

Li et al., 2010; Ren and Beard, 2008; Ren and Cao, 2011)

that use only relative measurements, that is, the deviations

between an agent’s output and the outputs of its neigh-

bors, the model-matching controllers need access to the

absolute outputs of the agents. Dealing with mobile robots,

this implies that agents have to measure their positions

and/or velocities in the global frame of reference.

At the same time, synchronization among heterogeneous

passive agents (e.g. mechanical systems in the Euler-

Lagrange form) can be established by the same simplest

protocols (Pogromsky and Nijmeijer, 2001; Arcak, 2007;

Hatanaka et al., 2015) as used to synchronize single inte-

grator agents (Olfati-Saber et al., 2007). Such a protocol

does not require any knowledge of the agents’ dynamics

(except for their passivity) and uses only deviations be-

tween the agents’ outputs, but not the outputs themselves.

Thus a visible gap exists between the problems of syn-

chronization in networks of passive heterogeneous agents,

provided by a very simple algorithm, and synchroniza-

tion among general heterogeneous agents, which requires

sophisticated model-based controllers. In this paper, we

make a step towards ﬁlling this gap and show that

the conventional synchronization algorithm for passive

agents (Hatanaka et al., 2015) is applicable also to input-

feedforward passive (IFP) (Khalil, 1996; Torres et al.,

2015) agents, provided that the couplings among them

are suﬃciently weak. The class of IFP systems is much

broader than the class of passive systems (and contains,

in particular, all asymptotically stable linear systems). We

demonstrate applications of our results to the design of

cooperative adaptive cruise control (CACC) for platoons

of automated vehicles and stability of a microscopic traﬃc

ﬂow model with delayed drivers’ responses, both of which

can be reduced to synchronization of IFP agents.

2. PRELIMINARIES

In this section, we introduce basic concepts from graph

theory and deﬁne input-feedforward passivity (IFP).

2.1 Graphs and their connectivity properties

A (weighted directed) graph is a triple G= (V,E, A),

where V={v1, . . . , vN}stands for the set of nodes,E ⊂

V × V is a set of arcs and A= (ajk )N

j,k=1 is a non-negative

adjacency matrix, such that ajk >0 if (vk, vj)∈Eand

otherwise ajk = 0. We always assume that the number of

nodes Nand their indices are ﬁxed, so V={1, . . . , N },

there is a one-to-one correspondence between such graphs

and their adjacency matrices A7→ G[A]∆

= (V, E [A], A),

where E[A]∆

={(j, k) : akj 6= 0}. Henceforth all graphs

have no self-loops ajj = 0 ∀j. A graph is called undirected

if A=A>. For any node jwe introduce the weighted in-

and out-degrees d+

j[A]∆

=PN

k=1 ajk and d−

j[A]∆

=PN

k=1 akj .

Awalk connecting nodes vand v0is a sequence of

nodes vi0

∆

=v, vi1, . . . , vis−1, vis

∆

=v0(n≥1) such that

(vik−1, vik)∈Efor k= 1, . . . , s. A graph is strongly

connected if a walk between any two distinct nodes exists.

A graph is quasi-strongly connected, or has a directed

spanning tree, if one of its nodes is connected by walks to

all other nodes. For an undirected graph these conditions

are equivalent (such a graph is simply called connected).

2.2 Passivity and input-feedforward passivity

Consider the dynamical system

˙x(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)), t ≥0,(1)

where x(t)∈Rn,u(t)∈Rmand y(t)∈Rmstand,

respectively, for the state, control and output.

The system (1) is passive (Khalil, 1996; Willems, 1972) if

there exists a storage function V(x)≥0 such that

V(x(T)) −V(x(0)) ≤ZT

0

y(t)>u(t)ds ∀T≥0 (2)

(here Tvaries in the interval where the solution exists).

Assuming Vto be C1-smooth, (2) can be rewritten as

˙

V(x, u) = ∂V

∂x f(x, u)≤h(x, u)>u∀x∈Rn, u ∈Rm.(3)

In this paper, we primarily deal with systems, satisfying a

“relaxed” passivity condition, deﬁned as follows.

Deﬁnition 1. The system (1) is IFP(α) (input-feedforward

passive with the passivity index α) if it is passive with

respect to the output ˜y=y+αu, i.e.

V(x(T)) −V(x(0)) ≤ZT

0y(t)>u(t) + α|u(t)|2dt. (4)

In the case α= 0 an IFP(α) system is passive; if α < 0

the condition (4) is referred to as the strict input passivity.

In this paper, we are primarily interested in systems that

are not passive but IFP(α) with α > 0. Examples of such

systems are discussed in Section 3.3.

Although this is not required by the formal deﬁnition, the

conditions of passivity and IFP usually hold for systems

with zero equilibrium: f(0,0) = 0 and h(0,0) = 0. For

systems without equilibria points a modiﬁcation of passiv-

ity condition exists, referred to as the incremental passiv-

ity (De Persis and Jayawardhana, 2014; Liu et al., 2015b).

Similarly, we introduce the incremental IFP condition.

Deﬁnition 2. A dynamical system is said to be iIFP(α)

(incrementally IFP(α)) if for any two solutions (x1, u1, y1)

and (x2, u2, y2) the respective deviations δx =x2−x1,

δu =u2−u1,δy =y2−y1satisfy the inequality

V(δx(T)) −V(δx(0)) ≤ZT

0δy>δu +α|δu|2dt, (5)

where Tbelongs to the interval where both solutions exist.

The function Vis called the incremental storage function.

Obviously, for linear systems IFP(α)⇐⇒iIFP(α).

3. MAIN RESULTS: SYNCHRONIZATION

PROTOCOLS FOR IFP AGENTS

Consider a group of Nagents obeying the equations:

˙xj(t) = fj(xj(t), uj(t)), yj(t) = hj(xj(t)), t ≥0,(6)

for j∈ {1, . . . , N }. Here xj(t)∈Rnj,uj(t)∈Rm,

yj(t)∈Rmstand respectively for the jth agent’s state,

control and output.

In this paper, we study distributed protocols, synchroniz-

ing the outputs yjasymptotically or in L2-norm.

Deﬁnition 3. Solutions {(xj(t), uj(t), yj(t))}N

j=1 of the sys-

tems (6), deﬁned on t∈[0; ∞), are output synchronized if

|yi(t)−yj(t)| −−−→

t→∞ 0∀i, j = 1, . . . , N. (7)

More speciﬁcally, the solutions are output synchronized

with a predeﬁned reference signal ¯y: [0; ∞)→Rmif

|yi(t)−¯y(t)| −−−→

t→∞ 0∀i= 1, . . . , N. (8)

Deﬁnition 4. Solutions {(xj(t), uj(t), yj(t))}N

j=1 of the sys-

tems (6), deﬁned for t≥0, are output L2-synchronized if

Z∞

0

|yi(t)−yj(t)|2dt < ∞ ∀i, j = 1, . . . , N. (9)

The solutions are output L2-synchronized with a prede-

ﬁned reference signal ¯y: [0; ∞)→Rmif

Z∞

0

|yi(t)−¯y(t)|2dt < ∞ ∀i= 1, . . . , N. (10)

In practice, the diﬀerence between the asymptotical

and L2-synchronization is minor. Mathematically, none

of these conditions implies the other one. However, in

some special situations it is possible to prove that L2-

synchronization implies asymptotical synchronization.

Proposition 5. Let yj(t) be absolutely continuous and

( ˙yi−˙yj)∈Lp[0; ∞] for some p > 1 and for any i, j.

Then (9) implies (7). If, additionally, ¯y(t) is absolutely

continuous and (˙yi−˙

¯y)∈Lp[0; ∞]∀ithen (10) entails (8).

Proposition (5), as well as all other statements of this

paper, is proved in Appendix. In the following subsections

we examine synchronization algorithms.

3.1 Synchronization without reference signal

We start examinating the linear controller:

uj(t) =

N

X

k=1

ajk (yk(t)−yj(t)),(11)

where ajk ≥0 are the coupling gains. The matrix A=

(ajk ) determines the interaction graph (or the network’s

topology)G[A], where node kis connected to node jby

an arc if and only if ajk 6= 0, that is, the control input of

agent jis directly inﬂuenced by the output of agent k.

It is widely known (Olfati-Saber et al., 2007; Ren and

Beard, 2008; M¨unz et al., 2011) that single integrators ˙yj=

uj, coupled via the protocol (11) reach consensus (that

is, a common limit y∗= limt→∞ yj(t) exists) whenever

G[A] has a directed spanning tree. Output synchroniza-

tion (7) is retained replacing single integrators by general

passive systems (6) and assuming strong connectivity of

G[A] (Hatanaka et al., 2015, Theorem 8.3). Our ﬁrst result

extends this to IFP agents.

Theorem 6. Assume that agent j(for j= 1, . . . , N ) is

IFP(αj) with a storage function Vj(xj)≥0. Let G[A] be

strongly connected and the couplings be “weak”, i.e.

αjd+

j[A] = αj

N

X

k=1

ajk <1/2∀j= 1, . . . , N . (12)

Then the following statements hold.

(1) Any solution of the system (6),(11), which is pro-

longable to ∞, is output L2-synchronized (9);

(2) Suppose that for any jthe function Vjis radially

unbounded lim|xj|→∞ Vj(xj) = ∞, the map fjis

continuous and hjis C1-smooth. Then, any solution

of the closed-loop system (6),(11) is prolongable to

∞, bounded, and output synchronized (7).

The proofs of Theorem 6 and other results of this section

are given in Section A. Note that in the case of αj= 0 the

inequalities (12) hold for any matrix A, and Theorem 6

coincides with Theorem 8.3 in (Hatanaka et al., 2015). We

proceed with two remarks, regarding the assumptions.

Remark 7. Unlike passive agents, for general IFP agents

the requirement of weak coupling (12) cannot be disre-

garded, as demonstrated by the following example. For

any p, q > 0 the system:

...

yj(t) + p¨yj(t) + q˙yj(t) = uj(t)∈R, t ≥0,(13)

is IFP(α) with some α=α(p, q)>0 (c.f. Subsect. 3.3).

Applying the protocol (11) with all-to-all coupling aij =

κ>0∀i, j to a group of identical agents (13), output

synchronization is guaranteed (Olfati-Saber et al., 2007;

Li et al., 2010) only when the polynomial s3+ps2+qs +

κ(N−1) = 0 is Hurwitz. Accordingly to the Routh-

Hurwitz criterion, this is possible only if κ(N−1) < pq,

i.e. the gain κis small.

Remark 8. Dealing with general heterogeneous agents, the

condition of strong connectivity cannot be replaced by the

existence of a directed spanning tree in G[A]. Consider, for

instance, a pair (N= 2) of harmonic oscillators

¨

ξ1+ω2

1ξ1=u1,¨

ξ2+ω2

2ξ2=u2, ω16=ω2.

that are passive with respect to the outputs y1=˙

ξ1and

y2=˙

ξ2. Consider the protocol u1=k(˙

ξ2−˙

ξ1), u2=

0, which corresponds to the graph with N= 2 nodes

and the only arc 2 7→ 1. It can be shown that the

system has a family of solutions ξ1(t) = Re[W(ıω2)ceıω2t],

ξ2=Re[ceıω2t], where c∈Cis constant and W(s) =

ks/(s2+ks +ω2

1). The corresponding outputs are y1(t) =

Re[ıω2W(ıω2)ceıω2t] and y2(t) = Re[ıω2ceıω2t]. Since

|W(ıω2)|=

kıω2

(ω2

1−ω2

2) + kıω2

<1,

the outputs are harmonic signals with the same frequency

ω2but diﬀerent amplitudes and cannot be synchronous.

Under some additional assumptions, synchronization over

a quasi-strongly connected graph may be established by

using the internal model control (Isidori et al., 2014).

3.2 Reference-tracking synchronization

We now consider the more complex problem of output

synchronization with a reference signal (8). In this paper

we conﬁne ourselves to a special situation: when the

desired trajectory is generated as the output of an agent

for some appropriate control input and initial condition.

Assumption 9. For any jsystem (6) has a solution

(¯xj(t),¯uj(t),¯yj(t)) such that ¯yj(t)≡¯y(t)∀t≥0 (in partic-

ular, the solution is prolongable to ∞). At any time agent

jis aware of the value 1¯uj(t), however the reference ¯y(t)

may be available only to a few “dedicated” agents.

1If ¯uj(·) is not unique, agent jknows one of such solutions.

Assumption 9 is often adopted implicitly or explicitly

in reference-tracking synchronization problems. For linear

agents (Li et al., 2010; Liu et al., 2015a) the reference

signal ¯y(t) is usually supposed to be an output of a

reference system, whose model is known and included

by the models of other agents. Dealing with ﬁrst-order

integrator agents ˙yj=uj, Assumption 9 implies that the

agents know the derivative ˙

¯y(t); this holds e.g. if ¯y(t) =

t¯v+ ¯y(0), where ¯vis known, but the initial condition ¯y(0)

is uncertain. A practical example of this type is discussed

in Section 4. Note that the solution (¯xj(t),¯uj(t),¯yj(t)) is

not assumed to be asymptotically stable, so the control

uj(t) = ¯uj(t)does not guarantee the reference signal

tracking (8). In general, only some of the agents are able

to measure the tracking error ¯y(t)−yj(t), whereas the

remaining agents measure only deviation between theirs

and their neighbors’ outputs.

Consider the following modiﬁcation of the algorithm (11)

ui(t) = ¯ui(t)+bi(¯y(t)−¯yi(t))+

N

X

j=1

aij (yj(t)−yi(t)).(14)

Here bi>0 if agent ihas access to the reference signal,

and otherwise bi= 0. The following result is a counterpart

of Theorem 6 for reference-tracking synchronization.

Theorem 10. Let Assumption 9 hold and further assume

that: for all j∈ {1, . . . , N }agent jis iIFP(αj), G[A] is

strongly connected, at least one agent has access to the

reference signal, i.e. Pibi>0, and the couplings are

suﬃciently weak, i.e.

αj(d+

j[A]+2bj)<1/2∀j= 1, . . . , N . (15)

Then, the following two statements hold:

(1) Any solution of the system (6),(14), prolongable to

∞, is output L2-synchronized with the reference

signal (10); in particular, R∞

0|¯u(t)−uj(t)|2dt < ∞.

(2) If for all jthe functions Vjare radially unbounded,

the maps fjare C1-smooth, the Jacobians ∂fj

∂xj,∂fj

∂uj

are uniformly bounded, and the maps hjare linear:

hj(ξ1−ξ2) = hj(ξ1)−hj(ξ2); then, any solution of the

closed-loop system (6),(14) is prolongable to ∞and

output synchronized (8) with the reference signal.

3.3 Examples of IFP agents

In this subsection, examples of IFP agents are provided.

SISO agents with a pole at zero

Consider a SISO system

sρ(s)ζ(t) = u(t)∈R, s ∆

=d

dt, ρ(λ) =

r

X

k=0

ρkλk;

y(t) = η(s)ζ(t), η(λ) =

r

X

k=0

ηkλk

(16)

Lemma 11. Assume that ρ(s) is a Hurwitz polynomial and

η0ρ0≥0. Then the system (16) is IFP(α) for suﬃciently

large α≥0. Denoting the transfer function from uto yby

W(λ) = η(λ)/(λρ(λ)), the passivity index can be found as

α=−inf

ω∈RRe W (ıω).(17)

For instance, Lemma 11 implies that the system (13) is

IFP (in this case, ρ(λ) = λ2+pλ +qis Hurwitz since

p, q > 0 and y(t) = ξ(t)).

First-order delayed integrators

Consider now a delayed system:

˙y(t) = u(t−α)∈Rm.(18)

Here α≥0 is a constant delay and we assume, by

deﬁnition, that u(t)≡u0(t) for t∈[−α; 0], where

u0∈L2([−α; 0] →Rm) is a given function. The vector

y(0) and the function u0are the initial conditions for

the system (18). Formally, our deﬁnition of IFP deals

with ordinary diﬀerential equations (1) only and is not

applicable to delay systems. However, the following weaker

condition holds for (18), see (Proskurnikov, 2016, p.141,

proof of Lemma 7.3).

Lemma 12. For any solution of (18) one has

ZT

0

(y(t)>u(t) + α|u(t)|2)dt ≥ −V ∀T≥0,(19)

where V=V(y(0), u0(·)) ≥0 is independent of T.

Lemma 12 allows to extend the synchronization criteria to

ensembles of agents (20).

Theorem 13. For a group of linear delayed agents

˙yi(t) = ui(t−αi), i = 1, . . . , N , (20)

the protocol (11) provides output synchronization (7)

and L2-synchronization (9), whenever the graph G[A] is

strongly connected and (12) holds.

Remark 14. In the monograph Tian (2012) a more general

result is formulated without a complete proof (Theorem

7.10), stating that under assumptions of Theorem 13

synchronization is retained if the graph is not strongly

connected but has a directed spanning tree.

Remark 15. Theorem 10 also holds for agents (20). How-

ever, Assumption 9 becomes impractical since each agent

has to be aware of ¯ui(t) = ˙

¯y(t+αj) at time t, which makes

the controller (14) non-causal. The protocol (14) may still

be used in the case where the reference signal is linear

¯y(t) = v0t+ ¯y(0) and v0is known, but ¯y(0) is uncertain.

4. SYNCHRONIZATION IN VEHICLE PLATOONING

AND TRAFFIC FLOW MODELING

In this section, we consider two practical applications of

the synchronization criteria from Section 3.

4.1 Stability of a microscopic traﬃc ﬂow model

A basic problem in vehicular traﬃc is the prevention of

congestions and accidents. Microscopic traﬃc ﬂow models

are often employed to represent the traﬃc ﬂow as a result

of cooperation between individual drivers. Since the pio-

neering work of Chandler et al. (1958), the delay in drivers

reaction has been recognized as a crucial factor participat-

ing into the overall ﬂow dynamics. The simplest model of

this kind (Chandler et al., 1958; Sipahi et al., 2007) deals

with Nvehicles, indexed 1 through N, traveling along a

common straight or circular single lane road (their order

remains unchanged since overtaking is not possible). Each

Fig. 1. Platoon of vehicles with bidirectional coupling.

driver is aiming to equalize his velocity of his own vehicle

with that of its predecessor:

˙vi(t) = ui(t−α), ui(t) = K(vi−1(t)−vi(t)).(21)

Here vi(t) is the speed of the i-th vehicle, αis the

delay in its driver’s action, and Kstands for the driver’s

“sensitivity” to alterations of the relative velocity of the

predecessor vehicle. In the case of straight road, v0(t)≡v0

is the desired velocity with respect to the leading vehicle

1; for a circular road, v0(t)≡vN(t), i.e. vehicle 1 follows

vehicle N. A key issue addressed via this model Chandler

et al. (1958) is that of the stability of the “synchronous”

manifold: v1=. . . =vN.

For the straight road case a necessary and suﬃcient

condition for such a synchronization: 2αK < 1 was found

in (21). We extend this classical result to the traﬃc ﬂow

model with a general directed interaction topology and

heterogeneous delays and sensitivities of the drivers.

˙vi(t) = ui(t−αi), ui(t) =

N

X

j=1

aij (vj(t)−vi(t)),∀i. (22)

The model (22) allows, in particular some drivers to

respond to the change not only in the predecessor’s, but

also in the follower’s velocity, or use the information about

several predecessors and followers. The following theorem

gives a criterion of velocity synchronization in (22) under

the assumption of a strongly connected topology, which

holds e.g. for uni- and bidirectional ring coupling (circular

road). The gain aij ≥0 in (22) stands for the sensitivity of

driver ito changes in the speed of vehicle j. Theorem 13,

applied to yi=vi, yields in the following corollary:

Corollary 16. Suppose that the graph G[A] is strongly

connected and (12) holds. Then the vehicles’ velocities are

asymptotically synchronized vi(t)−vj(t)−−−→

t→∞ 0.

4.2 An application to cooperative adaptive cruise control

In this subsection we demonstrate an application of The-

orem 10 to the stability of a platoon of vehicles (Fig. 2),

constituted by the leading vehicle 0 and Nfollower vehi-

cles, indexed 1 through N(Fig. 1). Cooperative adaptive

cruise control (CACC) system implements a control algo-

rithm, making each vehicle keep the safe distance to the

predecessor and, provided that this safety constraint is sat-

isﬁed, follow the leader’s velocity. The interaction topology

between the vehicles may be diﬀerent (Zheng et al., 2016);

the most studied is a unidirectional topology, where each

vehicle has information only about the predecessor.

In this subsection, we examine a CACC algorithm with

bidirectional interactions. The advantages of bidirectional

platooning algorithms over unidirectional ones are dis-

cussed e.g. in (Zhang et al., 1999; Barooah et al., 2009;

Zheng et al., 2016) (see also references therein); in many

senses such algorithms are more robust against distur-

bances propagating through the platoon (“string-stable”).

We examine the CACC algorithm, proposed in (Barooah

et al., 2009). The leader’s speed v0(t)≡v0is broadcasted

to every follower (Fig. 1). Besides this, the vehicles 1

through N−1 measure the distances to both their pre-

decessors and followers, and the rear vehicle Nmeasures

the distance to its predecessor. Denoting the position of

vehicle i’s rear bumper by qi∈R(see Fig. 2), the goal of

the CACC algorithm is to keep the desired distance to the

predecessor and the desired velocity, i.e.

qi−1(t)−qi(t)−−−→

t→∞ si, vi(t) = ˙qi(t)−−−→

t→∞ v0.(23)

Fig. 2. Platoon of vehicles. Notation used in the text

As usual in CACC problems (Zhang et al., 1999; Zheng

et al., 2016), the follower vehicles obey linear models

τi...

qi+ ¨qi=ai,des (t),(24)

where ai,des is the desired acceleration and τiis a time con-

stant, depending on the vehicle’s powertrain. The vehicles

1 through N−1 apply the following controller:

ai,des(t) = µi(v0−vi(t)) + ηi(qi−1(t)−qi(t)−si)

+νi(qi+1(t)−qi(t) + si+1 ),1≤i≤N−1,(25)

Vehicle Nis controlled similarly, but has no follower

aN,des(t) = µN(v0−vN(t)) + ηN(qN−1(t)−qN(t)−sN).

(26)

Theorem 17. Let µiτi<1

2and ηi, νi>0 satisfy

µ2

i

2>

ηi+νi,1< i < N;

2η1+ν1, i = 1;

ηN, i =N.

∀i(27)

Then the algorithm (25), (26) provides (23).

The result of Theorem 17 can be extended to some

cases of nonlinear vehicles’ dynamics, where the inner-loop

engine and torque controllers (Zhang et al., 1999) fail to

attenuate the nonlinearities. Notice that Theorem 17 does

not address the string stability problem, i.e. the robustness

of CACC against small disturbances in measurements as

Nbecomes large; the analysis of string stability is based

on other techniques and is beyond the scope of this paper.

5. CONCLUSIONS AND FUTURE WORK

In this paper, we oﬀer simple distributed protocols

for synchronization of heterogeneous non-passive agents

that satisfy an IFP property. We apply the obtained

results to analysis of microscopic traﬃc ﬂow models

and CACC algorithms for heterogeneous platoons. The

results can be extended to nonlinearly coupled net-

works, where the couplings satisfy the conditions of anti-

symmetry and sector inequalities (Hatanaka et al., 2015;

Proskurnikov, 2016; Proskurnikov and Matveev, 2015),

time-varying graphs and antagonistic interactions among

the agents (Proskurnikov and Cao, 2016). The results may

be extended to discrete-time IFP agents. The robustness

of synchronization against measurements noises and com-

munication delays are subjects of ongoing research.

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Appendix A. TECHNICAL PROOFS

We start with several technical lemmas, used to prove

Theorems 6 and 10, and then proceed with proofs of

Proposition 5, Theorems 6, 10, 13 and 17 and Lemma 11.

A.1 Auxiliary lemmas

The proofs of Theorems 6,10 are based on the following

lemma. Consider two groups of vectors y1, . . . , yN∈Rm

and u1, . . . , uN∈Rm, such that

ui=

N

X

j=1

aij (yj−yi)∀i= 1, . . . , N. (A.1)

Lemma 18. Let the graph G[A] be strongly connected.

Then the system of equalities (A.1) entails that

N

X

i=1

piy>

iui=−1

2

N

X

i,j=1

piaij |yj−yi|2,(A.2)

where (pi)N

i=1 are constants, determined 2by A= (aij ).

The proof of Lemma 18 is a part of the proof of Theo-

rem 8.5 in (Hatanaka et al., 2015) and omitted here.

Corollary 19. Let the graph G[A] be strongly connected

and the conditions (12) hold. Then (A.1) entails that

N

X

i=1

piy>

iui+αi|ui|2≤ −ε

N

X

i,j=1

|yj−yi|2,(A.3)

where ε > 0 is a constant, determined by the matrix A.

Proof. Using the Cauchy-Schwartz inequality, one has

d+

i[A]αi

N

X

j=1

aij |yj−yi|2=αi

N

X

j=1

aij

N

X

j=1

aij |yj−yi|2≥

≥αi

N

X

j=1

a1/2

ij a1/2

ij (yj−yi)

2

(A.1)

=αi|ui|2∀i.

By multiplying these inequalities by pi, summing them up

over iand using (A.2), it can be easily shown that

N

X

i=1

piy>

iui+αi|ui|2≤ −

N

X

i,j=1

κiaij |yj−yi|2,(A.4)

where κi

∆

=pi(1/2−d+

i[A]αi)(12)

>0. The inequality (A.3)

with some ε=ε(A)>0 now follows from (A.4) since the

matrix (κiaij )N

i,j=1 corresponds to a strongly connected

graph (Olfati-Saber and Murray, 2004, Theorem 8).

The proof of Theorem 10 also will use the following lemma.

Lemma 20. Suppose that the system (1) is IFP(α) with

some α≥0 and storage function V. Consider the constants

b∈(0; 1/(2α)) and ˆα∆

=α/(1 −2αb). Then the system (1)

is IFP(ˆα) for the same output yand the new input ˆu=u+

by with the storage function ˆ

V(x) = V(x)/(1 −2αb).

Moreover, for γ∆

=b(1 −αb)/(1 −2αb)≥0 one has

2In fact, p>= (p1,...,pN) stands for the non-negative left eigen-

vector of the weighted Laplacian matrix L=L[A], corresponding

to zero eigenvalue p>L= 0. The strong connectivity of the graph

implies that piare strictly positive (Hatanaka et al., 2015).

ˆ

V(x(T)) −ˆ

V(x(0)) ≤ZT

0y>ˆu+ ˆα|ˆu|2−γ|y|2dt (A.5)

for any solution and T≥0. The statement retains its

validity for iIFP system; in the latter case the vectors

x(t), y(t), u(t) in (A.5) should be replaced by the respective

deviations δx(t), δy (t), δu(t) between two solutions.

Proof. The proof is straightforward from Deﬁnition 1,

substituting u= ˆu−by into (4) and noticing that

y>u+α|u|2= (1 −2αb)y>ˆu+ ˆα|ˆu|2−γ|y|2.

The statement for iIFP system is proved in the same way,

substituting δu =δˆu−bδy into (5).

A.2 Proof of Proposition 5

Note ﬁrst that an absolutely continuous function ξ, such

that ˙

ξ∈Lp[0; ∞] with p > 1, is uniformly continuous

on [0; ∞]. This follows e.g. from the H¨older inequality,

entailing that for t≥0 and t0≥t

|ξ(t)−ξ(t0)|=

Zt0

t

˙

ξ(s)ds

≤(t0−t)qk˙

ξkLp[t;t0],

where q=p/(p−1) (by deﬁnition, q= 1 if p=∞). If,

additionally, ξ∈L2, the Barbalat lemma (Khalil, 1996)

implies that ξ(t)−−−→

t→∞ 0. Proposition 5 now follows,

applying this to, respectively, ξ=yj−yiand ξ= ¯y−yi.

A.3 Proof of Theorem 6.

Let the conditions of Theorem 6 hold. Introducing the

stack vector X(t) = [x1(t)>, . . . , xN(t)>]>and denoting

V(X)∆

=PN

i=1 piVi(xi), the IFP(αi) property of the

agents (6) and Corollary 19 imply that

V(X(T)) −V(X(0)) ≤

N

X

i=1 ZT

0

piy>

iui+αj|ui|2dt ≤

−ε

N

X

i,j=1 ZT

0

|yj(t)−yi(t)|2dt ≤0.

(A.6)

This implies statement (1): if the solution is deﬁned for any

t≥0, the solution is output L2-synchronized. To prove

statement (2), notice that radial unboundedness of all

storage functions Vi(xi) implies that V(X) is also radially

unbounded since pi>0∀i. Since V(x(T)) ≤V(x(0)) for

any T, the state vectors xi(t) are uniformly bounded, in

particular, the solution does not escape to inﬁnity in ﬁnite

time and thus is inﬁnitely prolongable. Recalling that the

maps hiare continuous, the outputs yi(t) are bounded;

the same holds for ui(t) due to (11). Since the maps fiare

continuous, ˙xi(t) are bounded. Recalling that hjis C1-

smooth, ˙yj(t) = h0(xj(t)) ˙xj(t) is also bounded. Output

synchronization (7) now follows from Proposition 5.

A.4 Proof of Theorem 10

Denoting ˜xi(t)∆

=xi(t)−¯xi(t), ˜ui(t)∆

=ui(t)−¯ui(t),

˜yi(t)∆

=yi(t)−¯yi(t), where ( ¯xi(t),¯ui(t),¯yi(t)) is the solution

from Assumption 9. Recalling that ¯yi(t)≡¯y(t), one has

˜ui(t) + bi˜yi(t)(14)

=X

j=1

aij (˜yj(t)−˜yi(t)).(A.7)

Denoting ˆui

∆

= ˜ui+bi˜yiand applying Lemma 20 to the

system (6), α=αjand b=bj, one arrives at

ˆ

Vi(˜xi(T)) −ˆ

Vi(˜xi(0)) ≤ZT

0˜y>

iˆui+ ˆαi|ˆui|2−γi|˜yi|2dt,

(A.8)

where ˆαi=αi/(1 −2biαi) and γi>0 if and only if bi>0.

The inequalities (15) imply that d+

j[A]ˆαj<1/2. Thus

retracing the proof of Corollary 19, (A.7) implies that

N

X

i=1

pi˜y>

iˆui+ ˆαi|ˆui|2≤ −ε

N

X

i,j=1

|yj−yi|2,(A.9)

where pi>0 and ε > 0 depend on A,αiand bi.

Introducing the stack vector ˜

X(t) = [˜x1(t)>,...,˜xN(t)>]>

and the storage function ˆ

V(˜

X) = Pipiˆ

Vi(˜xi), we obtain

ˆ

V(˜

X(T)) −ˆ

V(˜

X(0)) ≤ −ε

N

X

i,j=1 ZT

0

|˜yj(t)−˜yi(t)|2dt−

−X

i

piγiZT

0

|˜yi(t)|2dt =−ZT

0

F(˜y1(t),...,˜yN(t))dt.

(A.10)

Here F(y1, . . . , yN)∆

=εPN

i,j=1 |yj−yi|2+Pipiγi|yi|2is a

quadratic form; since γi>0 for at least one i, this form is

positive deﬁnite and thus F(y1, . . . , yN)≥ε0(|y1|2+. . . +

|yN|2) for suﬃciently small constant ε0>0. The end of

the proof retraces the proof of Theorem 6. If a solution

exists, (A.10) implies that ˜yi(t) = ¯y(t)−yi(t) is L2-

summable and hence ˜ui(t) = ¯u(t)−ui(t) is L2-summable,

which proves statement (1). To prove statement (2), notice

that since Viare radially unbounded, (A.10) implies that

the deviation of the solutions ˜

X(t) remains bounded; since

¯xi(t) is globally deﬁned, X(t) cannot grow unbounded

in ﬁnite time. Thus the solution is prolongable to ∞.

Recalling that hiis a linear map, the function ˜yi(t) =

yi(t)−¯y(t) = hi(xi(t)−¯xi(t)) = hi(˜xi(t)) is uniformly

bounded for any i. Thus ˜ui(t) is bounded due to (A.7)

and

|˙

˜xi(t)|=|fi(xi(t), ui(t)) −fi( ¯xi(t),¯ui(t))| ≤

≤ | ˜xi(t)|sup

xi,ui

∂fi(xi, ui)

∂ui

+|˜ui(t)|sup

xi,ui

∂fi(xi, ui)

∂ui

(here suprema are taken over the space of all possible

vectors xi∈Rni,ui∈Rm). Thus the functions ˙

˜xi(t) are

also uniformly bounded; using the linearity of hi, the same

holds for ˙

˜yi(t) = hi(˙

˜xi(t)). Applying Proposition 5, the

solutions are output synchronized (8).

A.5 Proof of Lemma 11

Lemma 11 is immediate from a more general result, which

in turn is implied by the standard positive real lemma.

Consider a linear SISO system

˙x=P x +Qu, y =Rx +Su (A.11)

and let W(λ)∆

=S+R(λI −P)−1Qstand for its scalar

transfer function.

Lemma 21. Let the system (A.11) be controllable and

observable. Then it is IFP(α) for some αif and only if

the following conditions hold

(1) the matrix Phas no strict unstable eigenvalues:

det(λI −P)6= 0 when Re λ > 0;

(2) all imaginary eigenvalues (if they exist) are simple,

at any such eigenvalue λ=ıω0the residual is non-

negative limλ→ıω0(λ−ıω0)W(λ)≥0;

(3) Re W (ıω) + α≥0 for any ω∈Rsuch that det(ıωI −

P)6= 0.

Proof. As was noticed in Section 2, the IFP(α) condition

is equivalent to passivity of system (A.11) with respect to

the new input ˜y=y+αu. The statement of Lemma 21 is

immediate, applying the result of (Willems, 1972, Theo-

rem 1 in Part 2) (the positive real lemma) to the respective

transfer function ˜

W(ıω) = αIm+W(ıω).

Notice that if the condition (1) and (2) in Lemma 21 hold

then (3) is valid for suﬃciently large α > 0. Indeed, the

function Re W (ıω) = [W(ıω) + W(−ıω)]/2 (where ω∈R)

is bounded as ω→ ∞ and in the vicinity of any imaginary

pole due to statement (2), thus this function is bounded

and, in particular, semi-bounded from below.

Proof of Lemma 11 is now obvious from Lemma 21 since

the system (16) can be rewritten as a controllable and

observable system (A.11) with a single imaginary pole

λ= 0. Such a system satisﬁes condition (1) in Lemma 21,

and (2) also holds since limλ→0λW (λ) = η0/ρ0≥0.

A.6 Proof of Theorem 13

Any solution of the linear time-invariant closed-loop sys-

tem (20), (11) is inﬁnitely prolongable. A closer look at

the proof of statement (1) in Theorem 6 reveals that the

IFP(αi) condition can be replaced by a weaker condition

ZT

0

(yi(t)>ui(t) + αi|ui(t)|2)dt ≥ −Vi∀T≥0,

where Vi≥0 is some constant (for an IFP(αi) agent (6)

with storage function Vi, the latter inequality holds for

Vi=Vi(xi(0))). Thanks to Lemma 12, statement (1) is

valid for the agents (20) and thus any solution is output L2

synchronized. Since ui∈L2[0; ∞] for any i, ˙yi∈L2[0; ∞]

due to (20) and thus outputs are also asymptotically

synchronized due to Proposition 5.

A.7 Proof of Theorem 17

Introducing the control inputs ui

∆

=ai,des +µiviand

outputs yi=qi+ (si+si−1+. . . +s1), the closed-loop

system (24), (25), (26) boils down to a group of agents

τi...

yi+ ¨yi+µi˙yi=ui, i = 1, . . . , N, (A.12)

coupled through (14). Here ¯ui(t) = µiv0, ¯y(t) = q0(t) and

bi=η1, i = 1

0, i > 1, aij =

ηi, i > 1, j =i−1

νi, i < N, j =i+ 1

0,otherwise.

Using Lemma 11, the agent (A.12) is IFP(1/µ2

i) since

Re 1

τi(ıω)3+ (ıω)2+µi(ıω)=−1

µ2

i+ (1 −2τiµi)ω2+ω4.

A straightforward computation shows that (27) im-

plies (15). Obviously, the graph G[A] is a bidirectional

chain and thus is strongly connected. Hence, the outputs

are L2-synchronized (10) and ui−¯ui∈L2. Denoting

εi

∆

=vi−v0, one has ˙yi=vi=εi+v0, ¨yi= ˙εi,...

yi= ¨εiand

hence

τi¨εi+ ˙εi+µiεi=ui−µv0=ui−¯ui∈L2[0; ∞],

so that εi= ˙yi−˙

¯y∈L2and ˙εi∈L2. Applying

Proposition (5), one proves that (7) holds and εi(t)→0

as t→ ∞, which implies (23).