Tracking and Formation Control of Multiple Autonomous
Agents: A Two-Level Consensus Approach
Maurizio Porfiri a , D. Gray Roberson b , Daniel J. Stilwell
a
b,1
Department of Mechanical, Aerospace and Manufacturing Engineering,
Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201
b
The Bradley Department of Electrical and Computer Engineering,
Virginia Polytechnic Institute and State University, Blacksburg, VA 24061
Abstract
Simultaneous tracking and formation control is addressed for a team of autonomous agents that evolve dynamically in a space
containing a measurable vector field. Each agent measures the local value of the field along its trajectory and occasionally
shares relevant information with other agents, in order to estimate the spatial average obtained from averaging measurements
across all agents. Using shared information, agents control their trajectories in a cooperative manner, with the dual goals
of driving the average field measurement to a specified value and maintaining a desired formation about the average. Two
approaches to virtual leader estimation are considered. The first involves the synthesis of a common virtual leader state,
whereas the second involves decentralized estimation of the virtual leader by individual agents. Under the second approach,
control is posed as a two-level consensus problem, where agents reach agreement on the virtual leader state at one level and
reach formation about the virtual leader at the other level. The decentralized approach is effective even when communication
among agents is limited, in the sense that the associated network graph can be disconnected in frozen time.
Key words: Consensus problem; Formation flying; Switched systems; Stability; Tracking; Multi-agent systems
1
Introduction
Widespread interest in multi-agent systems has been fueled by the breadth of natural and technological systems that exhibit coordinated activity of a group of autonomous agents. In nature, the flocking of birds [32],
schooling of fish [12], and foraging of insects [30] provide insight into the emergence of group behaviors from
the coordinated actions of individuals. In the field of
robotics, autonomous vehicle platoons are modeled and
studied as multi-agent systems. Applications have been
identified in practically every environment accessible to
vehicles, from mobile sensing in oceans [1], to navigation through unknown terrestrial environments [4], to
surveillance by autonomous aircraft [2], to control of
satellite formations in space [20]. Investigations into fundamental aspects of multi-agent system behavior have
been widely reported, e.g. [14,19,21,23,26]. For example,
1
Corresponding author. Tel.: +1 540 231 3204. Fax: +1
540 231 3362.
E-mail addresses: [email protected] (M. Porfiri),
[email protected] (D. G. Roberson), [email protected] (D. J.
Stilwell)
Preprint submitted to Automatica
[23] develops a comprehensive framework of flocking that
captures the major features of natural systems, such as
animal herds and schools, and that can be applied to vehicle platoons as well. It provides mathematical understanding of collective behaviors such as agent cohesion,
separation, and alignment that have been identified as
fundamental aspects of flocking behavior in [27].
In this paper, we study combined tracking and formation control of multiple autonomous agents. The agents
evolve in a space that possesses a measurable vector field.
For example, the agents may represent vehicles traveling
in an environment that possesses a temperature, chemical, or magnetic field, e.g. [16,22]. The agents measure
the value of the field locally and communicate their measurements with other agents. Using shared information,
agents control their trajectories cooperatively, with the
dual goals of driving the spatial average of the field measurement to a desired value and maintaining a desired
formation about the average.
To compute a spatial average of field measurements
among agents would seem imply that all agents can
communicate with all other agents or to an exogenous
12 January 2007
system. Although we briefly consider the case of an exogenous virtual leader to compute the spatial average,
we are most concerned with the problem of potentially
limited or sparse communication. To address limited
communication, we consider the case that each agent is
endowed with a local observer that estimates the spatial
average given local measurements and information that
is occasionally communicated from other agents.
only tracking stability. The formation stability problem can be addressed by analyzing a collection of timeinvariant problems, while the tracking problem is inherently time-varying. In the decentralized virtual leader
estimation case, the ability to decouple tracking and formation control is dependent on the communication network topology. Although it is not possible to decouple
the two controller components for a general network, it
is possible to do so for specific network topologies. We
present a technique for decoupling tracking and formation control under a specific but practically useful class
of communication network topologies, and for subsequently designing stable controllers. In this case, vector
field properties and the graph spectrum simultaneously
affect both tracking and formation stability. The decoupling between formation and tracking problems results
in a collection of time-varying subsystems.
Our approach to tracking and formation control utilizes
the concept of a virtual leader. In contrast to [8,22],
which use artificial potentials to synthesize a virtual
leader trajectory, we adopt a graph theoretic approach.
The use of virtual leaders within a graph framework has
also been considered in [7,23]. In these works, the virtual
leader acts as an external reference signal that represents
global group objectives. The role of the virtual leader is
different in the present work, in that the virtual leader
synthesis is tied to a measured vector field in the underlying ambient environment, and is not defined by an
absolute trajectory. We retain the mathematical structure for multi-agent formation problems developed in [7],
and based on consensus theory, e.g. [24,25]. However, to
address the dependence of the tracking problem on the
ambient vector field, we extend the mathematical model
and control design by connecting the virtual leader synthesis to the agent field measurements. Furthermore, we
consider the case of limited inter-agent communication
by allowing for a time-varying communication network.
Our approach results in a coupled formation and tracking problem where system behavior is simultaneously affected by the measured field, individual agent dynamics,
and dynamic communication topology.
The decentralized estimation strategy is developed first
for systems whose communication network is static, and
is then extended to systems whose communication is dynamic. Control of spatially distributed systems under
dynamic communication is an important research area,
e.g. [11,13,18,33]. For example, in [11] simultaneous optimization of communication and control objectives is
studied. That work considers discrete-time systems with
a central decision maker that communicates with subsystems independently across limited capacity channels.
We instead focus on the decentralized case. We also limit
our attention to intermittent communication between
subsystems, and not to channel bandwidth constraints.
We model the system as a switched system, consisting
of a set of time-varying subsystems and a protocol for
transitioning between them, e.g. [17,31]. We take advantage of results from fast switching theory to derive new
stability conditions for systems with limited communication, e.g. [29].
We start by presenting the common virtual leader approach for systems whose communication network is connected and static. The approach is practically suitable
when an exogenous system is present, and is theoretically useful to present the basic tools for solving the
formation tracking problem under severely limited communication. We then build on the single virtual leader
strategy to develop a novel technique that involves the
independent estimate of the virtual leader trajectory by
each agent, based on local measurements and shared information. Tracking and formation control takes place
in a two-level consensus framework, where agents reach
agreement on their local virtual leader estimates at one
level and achieve formation at a second level.
The theoretical results of this paper may be applied
to the problem of tracking and formation control with
a platoon of autonomous underwater vehicles (AUVs)
for environmental monitoring applications, e.g. [5,15].
Equipped with appropriate sensors, AUVs can measure
underwater fields such as scalar temperature, salinity, or
pollutant concentration fields, or vector magnetic fields
such as those produced by submarines. Furthermore,
communication within AUV platoons may be severely
bandwidth-limited, e.g. [8].
For the common virtual leader case, we present a design
technique for decoupling the centralized virtual leader
computation from the decentralized formation control,
and we give stability conditions for the system. We show
that formation stability may be studied by analyzing
reduced-order system models whose dynamics are related to the spectral properties of the communication
graph and to the vector field properties. In particular,
stability of the formation is only affected by the graph
spectral properties, while field characteristics influence
2
Model of Multi-Agent System
We consider a platoon of N agents with identical dynamics. Each agent is modeled as the linear system
ẋi = PA xi + PB ui ,
yi = PC xi
(1)
where i = 1, . . . , N identifies the agent, xi ∈ Rn represents the agent’s state, ui ∈ Rm is an input generated
2
by a control law, and yi ∈ Rk is an internal state measurement. Denoting x = (x1 , . . . , xN ), u = (u1 , . . . , uN ),
and y = (y1 , . . . , yN ) as the concatenation of the vectors
xi , ui , and yi , the state equation for the entire system is
ẋ = (I ⊗ PA )x + (I ⊗ PB )u ,
Because α and β are unknown parameters, the virtual
leader cannot be viewed as an external reference signal.
The formation is specified by a possibly time dependent
vector h(t) = (h1 , . . . , hN ) ∈ RN k , whose entries are
the desired internal state measurement offsets from the
virtual leader state for the corresponding agents, so that
the vector of desired internal state measurements is
y = (I ⊗ PC )x
where ⊗ represents the matrix Kronecker product and
all identity matrices are N × N .
ỹ(t) = 1N ⊗ q̃(t) + h(t)
In addition, we consider a set of measurements φi ∈ Rk
for each agent corresponding to an external vector field.
These measurements are dependent on the internal state
measurement through φi = ϕ(yi , t). When the distance
between agents is small relative to spatial variation in
the field, it is reasonable to assume an affine description
of the field. We therefore assume that locally in the immediate area in which the agents are operating, the field
ϕ can be represented adequately as an affine function of
the internal state measurements. The vector field measurement of agent i is then φi = αyi +β, where α ∈ Rk×k
and β ∈ Rk are time-varying parameters. The vector of
field measurements Φ = (φ1 , . . . , φN ) for all agents is
Φ(y, t) = IN ×N ⊗ α(t)y + 1N ⊗ β(t)
For consistency, the relationships hT (1N ⊗ ei ) = 0 must
hold for i = 1, . . . , k, where ei ∈ Rk is a vector whose elements are 0, except for a 1 in the ith location. These relationships indicate that the virtual leader should occupy
the geometric center of the formation. In the sequel, we
simplify notation by omitting explicit time dependence,
except when doing otherwise clarifies the presentation.
The multi-agent system and vector field model is used
in Sections 3, 4, and 5 to study tracking and formation
control design. The following cases are considered:
• A common virtual leader is computed by an exogenous
system under a static communication topology.
• A separate virtual leader estimate is computed by each
agent, and the network topology is static.
• A separate virtual leader estimate is computed by each
agent, and the network topology switches periodically
among a set of possibly unconnected topologies.
(2)
where 1N ∈ RN is a vector all of whose elements equal 1.
Following [7], graph theory is used to model the platoon
communication network, with nodes representing individual agents and edges representing active communication links. All communication is bidirectional, so that the
corresponding graph is undirected. A connected network
is one in which a direct or indirect path exists between
each pair of nodes in the associated graph. The Laplacian
matrix is used extensively in characterizing the effects
of the communication network. The Laplacian is defined
as L = D − A, where A is the graph adjacency matrix,
and D is the diagonal degree matrix whose diagonal entries equal the number of edges incident with the corresponding nodes. Defined in this manner, L is symmetric
and positive semidefinite. In addition, the vector 1N is
in the kernel of L. The total number of zero eigenvalues
of L equals the number of connected components of the
graph. The second smallest eigenvalue λ2 is termed the
algebraic connectivity of the graph [9].
3
Tracking and Formation Control: Common
Virtual Leader and Static Communication
To introduce the basic concepts of our approach, we begin with the case of a single exogenous virtual leader.
The virtual leader is modeled as a set of integrators,
q̇ = w, where w ∈ Rk is an input generated by a control
law. The information sensed by the ith agent is
zij = (yi − yj ) − (hi − hj ) ,
zii = yi − hi − q
j ∈ Ji
(5a)
(5b)
where the set Ji ⊂ {1, . . . , N } represents the set of
agents that the ith agent can sense, i.e. the graph nodes
connected to the ith node. The quantity zij represents
the error committed by the j th agent in maintaining
the desired offset with respect to the ith agent, while
zii represents the error committed by the ith agent in
maintaining the desired offset with respect to the virtual
leader. These errors
P are synthesized in a unique error
signal zi = zii + j∈Ji zij . The vector z = (z1 , . . . , zN )
is given by z = ((L + IN ×N ) ⊗ Ik×k )(y − h − 1N ⊗ q). We
also define a decentralized control law K which maps zi
to ui and has internal states vi ∈ Rs , represented by
Agents are tasked with a twofold mission: tracking a desired vector field value φ̃, and maintaining a desired formation about the desired vector. The control strategies
involve the synthesis of a virtual leader q ∈ Rk , whose
purpose is to track the desired field vector and to provide a reference about which the agents form. Under the
affine field assumption, the desired virtual leader state
q̃(t) = α−1 (t)(φ̃(t) − β(t))
(4)
(3)
corresponds to the desired location in the vector field.
v̇i = KA vi + KB zi ,
3
ui = KC vi + KD zi
(6)
The information sensed by the virtual leader
τ=
where i = 2, ..., N , and if K and V stabilize the system
1 T
(1 ⊗ Ik×k )Φ − φ̃
N N
q̇ = w
τ = α(t)y1
(7)
represents the error committed by the set of agents in
keeping their average measurement to the desired value
φ̃. We define a control law V which maps τ to w and has
internal state r ∈ Rl , represented by
ṙ = VA r + VB τ ,
w = VC r + VD τ
Proof. Since L is a symmetric matrix, it is diagonalizable and has eigenvectors ϕ1 , ϕ2 , . . . , ϕN that are orthogonal and may be chosen to be orthonormal, with
ϕ1 = 1N N −1/2 . Define the block diagonal transformation matrix Q = diag(Ik+l×k+l , T ⊗ In×n , T ⊗ Is×s ),
where T = [ϕ1 ϕ2 · · · ϕN ]. Because the column vectors
ϕi are orthonormal, T is an orthogonal matrix. Consider
the similarity transformation η̃ = Q−1 η = (q̃, r̃, x̃, ṽ),
which is the state of a system whose state equation is
Specification of (6) and (8) as dynamic control laws is not
meant to imply the superiority of dynamic control for the
tracking and formation problem. The intent is merely to
present the control design in a general framework.
We now consider the complete set of agents together with
the virtual leader and use Kronecker product to assemble the matrices governing the formation and tracking
behavior. Let v indicate the concatenation of the vectors
v1 , . . . , vN . The closed loop system dynamics becomes
η̃˙ = Q−1 AQη̃ + Q−1 Bp
(9)

0
VC


0
VA

A=
−1N ⊗ PB KD 0

h
iT
p = β T φ̃T hT
1
T
N VD 1N
1
T
N VB 1N

⊗ αPC
Υ
(10)
0
0
Λi =
−KB
0
KB P C
KA
PA + PB KD PC (λi + 1) PB KC
 
q̃˙
0
  
 r̃˙  
0
  
 =
x̃˙ 1  −PB KD
  
ṽ˙ 1
−KB
(11)
Analysis of (9) begins by presenting a necessary and
sufficient condition for uniform asymptotic stability.
KB PC (λi + 1)
#
KA

VC
VD α(t)PC
VA
VB α(t)PC
0 PA + PB KD PC
0
KB P C
0

q̃
 
  r̃ 
 
 
 
PB KC 
 x̃1 
KA
ṽ1
0
which correspond to the state coefficient matrix Λ1 (t).
The closed loop dynamics obtained by applying K to
(12) are
Proposition 1 The controllers K and V stabilize the
dynamics of the system (9) if and only if K simultaneously stabilizes the set of N − 1 systems
zi = (λi + 1)PC xi
0

where all identity matrices are N × N and
ẋi = PA xi + PB ui ,
VD α(t)PC
for i = 2, . . . , N . Stability of the original system (9) may
be determined from stability of (14). Stability may therefore be determined from the time-varying homogeneous
system whose state coefficient matrix is Λ1 (t), and the
N − 1 time-invariant homogeneous systems whose state
coefficient matrices are Λi . The closed loop dynamics obtained by applying the controllers K and V to (13) are
−(L + I) ⊗ KB
Υ = I ⊗ PA + (L + I) ⊗ PB KD PC
VC
"




I ⊗ P B KC 

I ⊗ KA

0
−1N ⊗ KB
0 (L + I) ⊗ KB PC

V −VD
0
 D

V −V

0
B
 B

B=

 0
0 −(L + I) ⊗ PB KD 


0

0



0
VA
VB α(t)PC
0 


Λ1 (t) = 
 (15)
−PB KD 0 PA + PB KD PC PB KC 



⊗ αPC
(14)
The state matrix Q−1 AQ = diag Λi of (14) is block
diagonal, with
with
h
iT
η = q T rT xT v T
,
(13)
where λi are the eigenvalues of L, ordered so that 0 =
λ1 ≤ · · · ≤ λN .
(8)
η̇ = Aη + Bp
ẋ1 = PA x1 + PB u1
z1 = PC x1 − q
" # "
#" #
x̃˙ i
PA + PB KD PC (λi + 1) PB KC x̃i
=
ṽ˙ i
KB PC (λi + 1)
KA
ṽi
(12)
4
which represents the error committed by the ith agent
in maintaining the desired offset with respect to its local
virtual leader estimate qi . By synthesizing the error signals in a unique error signal zi defined as the sum of the
errors sensed by the ith agent, the error vector z becomes
which correspond to the state coefficient matrices Λi .
Therefore the stability of these closed loop systems is
equivalent to stability of the original system (9). 2
Tracking control, achieved through the virtual leader,
is associated with Λ1 in (15). Indeed, since the ambient
vector field parameter α appears in Λ1 only, tracking is
decoupled from the platoon formation control problem,
which is associated with the Λi for i ≥ 2. Because the Λi
are constant for i ≥ 2, stability of the formation problem
reduces to the requirement of Hurwitz Λi . Time-varying
methods, such as those presented in [28], are required
to determine the stability of the tracking problem, due
to the time-varying nature of Λ1 . The case where α(t)
varies slowly may be particularly relevant. Additional
investigations on the formation stability may be undertaken by using Nyquist methods, as done in [7] for formation control about static targets. We emphasize that
the offset function h, the desired measurement φ̃ and the
parameter β act only as inputs to the dynamical system, while α(t) enters directly in the state matrix and
influences the system stability.
z = ((L + IN ×N ) ⊗ Ik×k )(y − h − q)
(16)
The sensed information is used by the decentralized controllers (6). On the other hand, we replace the measurement error in (7) by the vector τ = (τ1 , . . . , τN ) comprised of the local measurement errors computed by each
agent
τ 6= (F ⊗ Ik×k )(Φ − 1N ⊗ φ̃)
(17)
where F ∈ RN ×N computes a local weighted average
and satisfies Fij 6= 0, if Aij 6= 0, i, j = 1, . . . N . Thus,
each agent computes its own average estimate by using the measurement from the communicating agents.
Consistent with its averaging function, F has 1N as an
eigenvector corresponding to the eigenvalue 1. Due to
the incomplete averaging inherent in F , the control law
does not aim at exactly regulating τ to zero. Indeed, the
control objective is not to drive each entry of τ to zero,
but to ensure that the average of the entries in τ , which
corresponds to (17), is zero.
Computation of the virtual leader trajectory from agent
measurements is a unique feature of the controller. Other
works, such as [7,14], also consider formation about a
leader or target. In such works, though, the leader or
target is modeled as an additional agent that communicates with or is sensed by the agents. In our system,
the leader trajectory cannot be sensed independently by
the agents, due to the nature of the environmental field.
Construction of the virtual leader trajectory from measurements is therefore a distinguishing characteristic of
our controller. We emphasize that our virtual leader trajectory is not known a priori, as in [23], but is rather
constructed in real time as field measurements are taken.
A possible choice for F is
F = (IN ×N + D)−1 (IN ×N + A)
(18)
which provides unweighted local average measurements.
When using (18) in (17), the elements of τ become
τi =
³
X ´
1
Φi +
Φj − φ̃
1 + |Ji |
j∈Ji
4
Tracking and Formation Control: Decentralized Virtual Leader Estimation and Static
Communication
where |Ji | is the number of agents communicating with
the ith agent. When each agent communicates with ρ
other agents, F may be written as
A second tracking and formation control strategy involves independent estimation of the virtual leader trajectory by the agents. Each agent estimates a spatial average of the field measurements based on its local measurements and information from the agents with which
it communicates. We model the virtual leader estimators
as single integrators by q̇ = w, where q = (q1 , . . . , qN )
contains the virtual leader position estimates and w ∈
RN k are inputs generated by decentralized control laws.
F = IN ×N − (1 + ρ)−1 L
In addition to the error signal zij representing the errors committed by the ith agent in keeping the required
offsets with respect to j th one or to its virtual leader estimate, we define the error signal eij = qi − qj , where
j ∈ Ji , to represent the disagreement error between the
virtual leader estimate of the ith and j th agents. For
simplicity, these errors are synthesized in a unique error
signal ei defined as the sum of the errors sensed by the
ith agent. Thus, the vector e = (e1 , . . . , eN ) is given by
e = (L ⊗ Ik×k )q. In order to drive the virtual leader estimates to the desired common value, we define a control
law V which maps τi and ei to wi and has internal state
Since we assume that each agent does not sense the actual virtual leader position but only its proper estimate,
we modify (5a) and (5b) as
zij = (yi − yj ) − (hi − hj ) − (qi − qj ) ,
zii = yi − hi − qi
(19)
j ∈ Ji
5
ri ∈ Rl , represented in state-form by
column vectors ϕi are orthonormal. Applying the similarity transformation Q to A in (21) yields
ṙi = VA ri + VB1 τi + VB2 ei
wi = VC ri + VD1 τi + VD2 ei
Q−1 AQ
(22)


I ⊗ VC F̃ ⊗ VD1 αPC
L̃ ⊗ VD2
0



 L̃ ⊗ V
0
I ⊗ VA F̃ ⊗ VB1 αPC
B2


=


−M̃ ⊗ PB KD
0
Υ̃
I
⊗
P
K
B C

−M̃ ⊗ KB
0
M̃ ⊗ KB PC I ⊗ KA
Let r indicate the concatenation of the vectors
r1 , . . . , rN . The closed-loop system dynamics becomes
η̇ = Aη + Bp
(20)
with η and p defined as in (10) and
where L̃ and F̃ are diagonal matrices whose diagonal elements are the eigenvalues λi and µi of L and F , respectively, ordered by the eigenvectors ϕi , where all identity matrices are N × N , where M̃ = L̃ + I, and where
Υ̃ = I ⊗ PA + M̃ ⊗ PB KD PC . By permuting rows and
columns, the matrix (22) may be written in block diagonal form, with the generic 2 × 2 blocks


L ⊗ VD2
I ⊗ VC F ⊗ (VD1 αPC )
0



 L⊗V
I ⊗ VA F ⊗ (VB1 αPC )
0
B2


A =

−M ⊗ PB KD
0
Υ
I ⊗ P B KC 


−M ⊗ KB
0
M ⊗ KB PC
I ⊗ KA
(21)



1 ⊗ VD1 −1N ⊗ VD1
0
 N

1 ⊗ V −1 ⊗ V

0
B1
N
B1
 N

B=


0
0
−(L + I) ⊗ PB KD 


0
0
−(L + I) ⊗ KB

λi VD2
VC
µi VD1 αPC
−(λi + 1)KB
0
(λi + 1)KB PC
0



λi VB 2
VA
µi VB1 αPC
0 




−(λi + 1)PB KD 0 PA + (λi + 1)PB KD PC PB KC 


KA
(23)
where all identity matrices are N ×N , where M = L+I,
and where Υ is defined in (11).
which yields the claim.
In general, a decomposition of the system such as the
one performed in Section 3 does not simplify stability
analysis. However, it is possible to work with (20) directly and to specify properties of F and L that simplify
analysis when they are present. The following proposition provides an example.
Each subsystem (23) is time-varying, and stability must
be evaluated using time-varying system analysis methods. The commutativity of L and F in Proposition 2
is motivated by a platoon network in which each agent
communicates with a fixed set of relative neighboring
agents, and computes an unweighted average as in (19),
or in which the communication graph is circulant [9]. In
the latter case, F can be defined as a circulant matrix.
Since L is also circulant under the considered topology,
L and F commute, according to a standard property of
circulant matrices [6]. When the graph is complete (fully
connected) and F is chosen as in (19), the tracking error
is zero for any constant value of h.
Proposition 2 Suppose that L and F commute. The
controllers K and V stabilize the dynamics of the system
(20) if and only if K and V simultaneously stabilize the
set of N systems
ẋi = PA xi + PB ui
q̇i = wi
zi = (λi + 1)(PC xi − qi ) τi = µi αPC xi
e i = λi q i
5
for i = 1, . . . , N , where λi are the eigenvalues of L, ordered such that 0 = λ1 ≤ · · · ≤ λN , and µi are the eigenvalues of F , 1 = µ1 ≥ · · · ≥ µN ordered according to the
common eigenvectors.
2
Tracking and Formation Control: Periodic
Switched Communication
We now investigate tracking and formation control for
a platoon with switched communication. Our main result is that the average network determines stability of
the system if the network switches at a sufficiently fast
rate. Moreover, the average system provides a useful approximation of the switched system, as the simulations
in Section 6 illustrate. Communication may be severely
limited, so that at any point in time there may be isolated agents that do not share information. Hence the
approach of estimating the virtual leader trajectory on
each agent, as described in Section 4, is adopted.
Proof. Since L and F commute, they share the set of
orthonormal eigenvectors ϕ1 , . . . , ϕN . These eigenvectors are real due to the symmetry of L. It follows that
the eigenvalues of F are real, and thus its eigenvalues
1 = µ1 ≥ · · · ≥ µN are real. Define the block diagonal
matrix Q = diag(T ⊗ Ik×k , T ⊗ Il×l , T ⊗ In×n , T ⊗ Is×s ),
where T = [ϕ1 ϕ2 · · · ϕN ] is orthogonal because the
6
The platoon formation error z continues to be defined
by (16) with the qualification that L is time-varying.
The estimation error (17) is modified so that τ = (F ⊗
In×n )(Φ − 1N ⊗ φ̃) to account for the fact that isolated agents rely on the virtual leader estimate computed
during their most recent communication event to drive
their navigation. The time-varying matrix F is the same
as F in (17) with the exception that rows corresponding to isolated agents vanish. The matrices L and F
switch among sets of constant matrices {Lj } and {Fj }.
At every time instant, L = Lj and F = Fj for some
j ∈ {1, . . . , J}, where J is the total number of constant
matrices. To simplify the analysis, we assume that the
platoon communication network switches periodically,
i.e. that there is a T ∈ R+ such that F(t) = F(t + kT )
and L(t) = L(t+kT ) for all t ∈ R+ and k ∈ N. Each pair
Lj , Fj is active for a set percentage δj of the switching
PJ
period T , with i=1 δi = 1, so that
L(t) = Lj ,
F(t) = Fj
Lemma 3 Suppose the matrix functions S1 (·), S2 (·) :
R+ → Rn×n are right-continuous, bounded and discontinuous in a finite number of instants over every finite
interval, and the matrix function G(·) : R+ → Rn×n is
differentiable and bounded, and has a bounded derivative
for any t ∈ R+ . Also suppose there exists τ > 0 such that
S1 and S2 haveRuniform average over any interval of dut+τ
ration τ , i.e. τ1 t Si (t) = S̄i is constant for all t ∈ R+
and i = 1, 2. If the linear time-varying system
ψ̇(t) = (S̄1 + G(t)S̄2 )ψ(t) ,
ẏ(t) = (S1 (t/ε) + G(t)S2 (t/ε))y(t) ,
Using Lemma 3, we seek to characterize stability of (25)
in terms of the uniform average values
L̄ =
J
X
δj Lj ,
j=1
(25)

I ⊗ VC F ⊗ VD1 αPC
0


 L⊗V

I ⊗ VA F ⊗ VB1 αPC
0
B2


A=

−M ⊗ PB KD
0
Υ
I ⊗ PB KC 


−M ⊗ KB
0
F̄ =
J
X
δ j Fj
(29)
j=1
of L(t) and F(t) over the period T and the corresponding
average state coefficient matrix


L̄ ⊗ VD2
I ⊗ VC F̄ ⊗ VD1 αPC
0


 L̄ ⊗ V

I ⊗ VA F̄ ⊗ VB1 αPC
0
B2


Ā = 

−M̄ ⊗ PB KD
0
Ῡ
I ⊗ PB KC 


−M̄ ⊗ KB
0
M̄ ⊗ KB PC I ⊗ KA
(30)
where all identity matrices are N × N , M̄ = L̄ + I, and
Ῡ = I ⊗ PA + M̄ ⊗ PB KD PC . Although Ā incorporates
the constants L̄ and F̄, it is nevertheless time-varying
due to the parameter α. The following proposition provides a useful control design tool for T -periodic systems.
is obtained from (21) upon substituting F with F, i.e.
L ⊗ VD 2
y(0) = y0 (28)
is uniformly exponentially stable.
´
h
Pj
Pj−1
for t ∈ kT + T i=1 δi , kT + T i=1 δi . Therefore,
the state matrix of the switched linear system

(27)
is uniformly exponentially stable, then there exists ε∗ > 0
such that for all fixed ε ∈ (0, ε∗ ),
(24)
ẏ(t) = A(t)y(t) + B(t)p(t)
ψ(0) = ψ0
M ⊗ KB PC
I ⊗ KA
(26)
where L and F are defined in (24), where all identity
matrices are N × N , where M = L + I, and where Υ
is defined as in (11). Note that A is time-varying, due
to the time-varying nature of L, F, M , Υ, and α. The
time-varying input vector is
Proposition 4 Let α be a bounded differentiable matrix
function with a bounded derivative on the entire time
axis. If the time-varying average system


F1N ⊗ VD1 −F1N ⊗ VD1
0


F1 ⊗ V −F1 ⊗ V

0
B1
N
B1
 N

B=



0
0
−M
⊗
P
K
B D

0
0
−M ⊗ KB
ψ̇(t) = Ā(t)ψ(t) ,
ψ(0) = ψ0
since in general F1N 6= 1N . To obtain stability results,
we study the response of the time-averaged system and
relate it to the response of a platoon whose communication network switches at a sufficiently fast rate.
with Ā(t) defined in (30), is uniformly asymptotically
stable, then there exists a T ∗ > 0 such that the switched
system
ẏ(t) = A(t)y(t) , y(0) = y0
with A(t) given in (26), is uniformly asymptotically stable
for any period T < T ∗ .
A new result for analyzing the stability of time-varying
switched systems is required, the proof of which is found
in the Appendix.
Proof. Define auxiliary functions L∗ (t/T ) = L(t) and
F ∗ (t/T ) = F(t). We note that the period of L∗ and F ∗
7
is τ = 1. Using L∗ and F ∗ , the homogenous linear differential equation associated with (25) has state matrix
A(t) = S1 (t/T ) + G(t)S2 (t/T ), where
An example platoon with switched communication
whose average system meets the conditions of Proposition 5 is one in which switching occurs in a circulant
manner. That is, the commutativity of L̄ and F̄ are
satisfied if, over the course of the switching period,
each agent communicates with a common set of relative
neighbors for a common amount of time.
S1 (t/T )


L(t) ⊗ VD2
I ⊗ VC
0
0


 L(t) ⊗ V

I ⊗ VA
0
0
B2


=

−M (t) ⊗ PB KD

0
Υ(t)
I
⊗
P
K
B C

−M (t) ⊗ KB
0 M (t) ⊗ KB PC I ⊗ KA
6
In this section we illustrate tracking and formation control, by considering a platoon of agents moving in a twodimensional space possessing a two-dimensional affine
and time-invariant vector field. Each agent is modeled
as a point mass with viscous-like damping. The internal measurement corresponds to a position-like variable,
and the control input corresponds to a force-like variable. Therefore the agent model in (1) is given by
S2 (t/T ) = F(t) ⊗ I(s+l+n+k)×(s+l+n+k)


0 0 VD1 α(t)PC 0


0 0 V α(t)P 0
B1
C 

G(t) = I ⊗ 

0 0
0
0


00
0
0
"
where all unlabeled identity matrices are N × N , M =
L + I, and Υ is defined as in (11). By applying Lemma
3 with ε = T , the claim follows. 2
PA =
0 1
#
,
0a
PB =
" #
0
1
,
h i
PC = 1 0
(31)
We apply the control design procedures presented in
Sections 3 and 5 and simulate the controlled motion of
a platoon of four agents. As example control laws, we
use purely static feedback. For the common and virtual leader approach we use scalar K and V of the
type ui = KD zi and w = VD τ . For the decentralized virtual leader approach, we use ui = KD zi and
wi = VD1 τi + VD2 ei .
Proposition 4 guarantees that nonzero stabilizing
switching periods exist under the given hypotheses and
provides an upper bound estimate for the stabilizing
periods. The estimate is typically conservative, and for
certain systems it may even be the case that stability
is obtained for all switching periods. Other switching
techniques involving dwell time or state feedback, e.g.
[31], may be effective under appropriate conditions.
For the simulation, the configurable parameter in the
agent model (31) is set to a = −3. The vector field is
constant with parameters
In general, L̄ in (29) is not a graph Laplacian matrix,
i.e. there is no graph whose Laplacian is L̄. Nevertheless, it is easy to show that it is symmetric and has 1N
in its kernel. By application of the Geršgorin disc theorem [10], it is straightforward to demonstrate that L̄ is
positive semidefinite. By combining Propositions 2 and
4 the following result holds.
"
α=
0.1 0.02
0.02 0.2
#
"
,
β=
#
0
0.1
The control objectives are given by
h
iT
φ̃ = −0.2 −0.4 ,
Proposition 5 Let α be a bounded and differentiable
matrix function with a bounded derivative for t ∈ R+ ,
and suppose that L̄ and F̄ commute. Suppose that the
controllers K and V stabilize the dynamics of the systems
ẋi = PA xi + PB ui
q̇i = wi
zi = (λ̄i + 1)(PC xi − qi ) τi = µ̄i αPC xi
Tracking and Formation Control Simulation
h
iT
h = 1 1 −1 1 −1 −1 1 −1
The control law is defined as KD = −0.5, VD = −1,
VD1 = −1, and VD2 = −1.
ei = λ̄i qi
The first simulation illustrates the common virtual
leader approach with a static and fully connected communication network, where the communication graph is
complete. Figure 1 shows the agent trajectories.
where i = 1, . . . , N , the eigenvalues λ̄i of L are ordered
such that 0 = λ̄1 ≤ · · · ≤ λ̄N , and the eigenvalues µ̄i of
F̄ are ordered such that µ̄1 ≥ · · · ≥ µ̄N according to the
common eigenvectors. Then there exists T ∗ > 0, such
that for any T < T ∗ the controllers K and V stabilize
the dynamics of the switched system (25).
The second simulation illustrates the switched communication case. Most of the time all agents do not
share information. At periodic time intervals, three
agents communicate, and the communication pattern
8
-4
-3
-2
4
4
3
3
2
2
1
1
1
-1
2
3
4
-4


0 1 0

,
1 0 0

0 0 0
1
-1
-1
-2
-2
-3
-3
-4
-4

1/4

3/8

F2 = 
3/8

0
2
3
4
Fig. 2. Agent trajectories, decentralized virtual leader estimation with switched communication.
repeats in a circulant manner. Over the course of a
switching period, eight communication events occur.
The corresponding adjacency matrices for the timevarying network are such that A1 , A3 , A5 , A7 are 4 × 4
zero matrices, and A2 , A4 , A6 , A8 represent all possible
three-vehicle communication patterns. The averaging
matrices are defined so that F1 , F3 , F5 , F7 are 4 × 4 zero
matrices, and F2 , F4 , F6 , F8 reflect the active communication network. So for example
1 1 0
-2
-1
Fig. 1. Agent trajectories, common virtual leader with static
and fully connected communication.

0

1

A2 = 
1

0
-3
4
3
2
1
-4

3/8 3/8 0

1/4 3/8 0


3/8 1/4 0

0 0 0
-3
-2
1
-1
2
3
4
-1
-2
-3
-4
Fig. 3. Agent trajectories, average model of switched communication.
This choice of F matrices does not correspond to (18),
but rather yields an unweighted F̄ = 0.075(1N 1T
N ). The
overall switching period is T = 5, and by setting δ1 =
δ3 = δ5 = δ7 = 0.15 and δ2 = δ4 = δ6 = δ8 = 0.1, each
agent is isolated for 80% of the time. That is, each agent
communicates with no other agent 80% of the time. This
communication pattern yields the average graph Laplacian L̄ = 0.2Lf , where Lf = N IN ×N − 1N 1T
N is the
Laplacian of a fully connected communication network.
ory of averaging, see e.g. [3], the error for the switched
communication case is bounded and may be constrained
to be arbitrary small by decreasing the switching period.
With the choice of controller gains, the formation error is regulated faster than the tracking error. Roughly
speaking, agents reach formation quickly, then move as
a rigid group to satisfy the tracking objective.
Figure 2 shows the agent trajectories under switched
communication. Figure 3 displays the trajectories computed using the average model. From a comparison between the switched and average cases, the average model
provides a reasonable approximation of the switched
communication model. Indeed, the average and switched
system trajectories in Figures 2 and 3 are nearly aligned.
The simulations illustrate the effectiveness of the theoretical results and the control design procedure. In the
switched communication case, the agent trajectories approximately follow those achieved by the average system.
Acceptable performance is obtained in the switched communication case with reasonable communication network switching rates. This means that stability can be
assessed even if the network is disconnected in frozen
time. The simulations represent an idealized system. For
practical applications, other factors such as obstacle and
interagent collision avoidance must be considered.
The convergence rate of the decentralized systems is
slower than the rate of the system possessing a common
virtual leader. This slower rate is directly attributable to
infrequent communication. As one expects from the the-
9
7
Conclusion
Φ̄, we obtain
Z
There are several notable features of the control problem
considered in this paper and the strategies used to accomplish control objectives. In contrast to typical tracking problems, e.g. [7,34], the entity being tracked is not
a moving target or a path defined in spatial coordinates.
The location of the target evolves in time with the agent
motions and is tied to an unknown process external to
the agents, introducing additional complexity. Tracking
and formation control are coupled and influenced by
time-varying, state-dependent measurements. Second,
the control and estimation architecture is placed in a
consensus framework. When a common virtual leader is
estimated, formation about the virtual leader is posed as
a consensus problem. In the decentralized virtual leader
case, the control and estimation is addressed in a twolevel consensus framework, with agents reaching consensus on the virtual leader at one level, and forming about
the virtual leader at the other level. Finally, the possibility of simplifying control design by decoupling the
tracking and formation control has been explored, and
design criteria that guarantee system stability have been
identified in cases where decoupling is possible.
t+ετ
∆Φ(t, t + ετ ) = I +
+
∞ Z
X
i=2
(S1 (ξ/ε) + G(ξ)S2 (ξ/ε))dξ
t
Z
t+ετ
ξ1
(S1 (ξ1 /ε) + G(ξ1 )S2 (ξ1 /ε))
Z
···
t
t
ξi−1
(S1 (ξi /ε) + G(ξ1 )S2 (ξi /ε))dξi · · · dξ1
t
t+ετ
Z
−I −
−
(S̄1 + G(ξ)S̄2 )dξ
∞ Z
X
i=2
t
Z
t+ετ
ξ1
(S̄1 + G(ξ1 )S̄2 )
t
···
t
Z
ξi−1
(S̄1 + G(ξi )S̄2 )dξi · · · dξ1
(34)
t
where I is the identity matrix. Consider the term
Z
t+ετ
(S1 (ξ/ε) + G(ξ)S2 (ξ/ε) − S̄1 − G(ξ)S̄2 )dξ (35)
t
By definition of S̄1 , (35) may be rewritten as
8
Appendix
Z
t+ετ
t
£
¤
S1 (ξ) + α(ξ)S2 (ξ) − S̄1 − α(ξ)S̄2 dξ
Z t+ετ
£
¤
=
G(ξ) S2 (ξ/ε) − S̄2 dξ
t
Proof of Lemma 3. Since the average system (27) is
uniformly asymptotically stable, there is a matrix function Q(t) and positive scalars η, ρ, and µ such that the
Lyapunov function V (ψ(t), t) = ψ T (t)Q(t)ψ(t) satisfies ηkψ(t)k2 ≤ V (ψ(t), t) ≤ ρkψ(t)k2 and V̇ (ψ(t), t) ≤
−µkψ(t)k2 for all t in the Euclidean norm. To establish uniform asymptotic stability of (28) we show that
V (y(t), t) is also a Lyapunov function for (28) if ε is sufficiently small, by showing that for small ετ
Define r(ξ, σ) = G(ξ) − G(σ). Since the first derivative
of G(t) is bounded, there exists a positive constant m
such that kr(ξ, σ)k ≤ |ξ − σ|m in the Euclidean norm
for any ξ and σ. Therefore, (35) becomes
Z
t+ετ
(S1 (ξ/ε) + G(ξ)S2 (ξ/ε) − S̄1 − G(ξ)S̄2 )dξ
Z t+ετ
£
¤
=
(G(t) + r(ξ, t)) S2 (ξ/ε) − S̄2 dξ
t
∆V (y, t + ετ, t) = V (y(t + ετ ), t + ετ ) − V (y(t), t) (32)
t
£
¤
R t+ετ
which equals t
r(ξ, t) S2 (ξ) − S̄2 dξ by definition
of S̄2 . Therefore, the following bound holds
is negative definite for all t, see e.g. [29].
Let Φ and Φ̄ be the transition matrices of (28) and (27),
respectively, and let
°Z
°
°
°
t
∆Φ(s, t) = Φ(s, t) − Φ̄(s, t)
(33)
t+ετ
°
°
(S1 (ξ/ε) + G(ξ)S2 (ξ/ε) − S̄1 − G(ξ)S̄2 )dξ °
°
Z t+ετ
°
°
≤
kr(ξ, t)k °S2 (ξ/ε) − S̄2 ° dξ
t
≤ M (ετ )2
By using the Peano-Baker series representation of Φ and
10
(36)
°
°
where M = m supt≥0 °S2 (t) − S̄2 °. Letting
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¶
µ
Ω = max sup kS1 (t)k, kS̄1 k
t≥0
µ
¶
+ sup kG(t)k max sup kS2 (t)k, kS̄2 k
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k∆Φ(t, t + ετ )k ≤ f (ε)
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(37)
where f (ε) = 2(exp(Ωετ ) − 1 − Ωετ ) + M (ετ )2 . Note
df
that f (0) = 0 and dε
(0) = 0. By using (33), (32) may
be expressed as
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T
= [Φ(t, t + ετ )y(t)] Q(t + ετ ) [Φ(t, t + ετ )y(t)]
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£
¤
= y T (t) Φ̄T (t, t + ετ )Q(t + ετ )Φ̄(t, t + ετ )−Q(t) y(t)
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µ
¶
¶
h µ
µετ
∆V (y, t + ετ, t) ≤ ρ exp −
−1
(38)
ρ
µr
µ
¶¶
i
ρ
µετ
+2
exp −
ρf (ε) + ρ2 f 2 (ε) ky(t)k2
η
2ρ
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∂ε |ε=0 =
−µτ ky(t)k2 . Thus, since g(ε, z) → +∞ as ε → ∞ and
g(ε, z) is a continuous function of ε, there exists a ε∗ such
that g(ε∗ , z) = 0 and g(ε, z) < 0 for any ε < ε∗ . Thus
∆V (y, t + ετ, t) is negative definite for any ε < ε∗ . 2
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The authors gratefully acknowledge the support of the
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