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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).
