IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
663
Connectivity and Set Tracking of Multi-Agent
Systems Guided by Multiple Moving Leaders
Guodong Shi, Yiguang Hong, Senior Member, IEEE, and Karl Henrik Johansson, Senior Member, IEEE
Abstract—In this paper, we investigate distributed multi-agent
tracking of a convex set specified by multiple moving leaders with
unmeasurable velocities. Various jointly connected interaction
topologies of the follower agents with uncertainties are considered in the study of set tracking. Based on the connectivity of
the time-varying multi-agent system, necessary and sufficient
conditions are obtained for set input-to-state stability and set
integral input-to-state stability for a nonlinear neighbor-based
coordination rule with switching directed topologies. Conditions
for asymptotic set tracking are also proposed with respect to the
polytope spanned by the leaders.
Index Terms—Multi-agent systems, multiple leaders, set
input-to-state stability (SISS), set integral input-to-state stability
(SiISS), set tracking.
I. INTRODUCTION
T
HE last decade has witnessed tremendous interest devoted
to the investigation of collective phenomena in multiple
autonomous agents, due to broad applications in various fields
of science ranging from biology to physics, engineering, and
ecology, just to name a few [8]–[12]. Concerning the issues of
multi-agent systems and distributed design, the revolutionary
idea is underpinning a strong interaction of individual dynamics, communication topologies, and distributed controls.
The problem is generally very challenging due to the complex
dynamics and hierarchical structures of the systems. However,
efforts have been started with relatively simple problems such
as consensus, formation, and rendezvous, and many significant
results have been obtained.
The leader-follower coordination is an important multi-agent
control problem, where the leader may be a real leader (such
as a target, an evader, or a predefined position), or a virtual
leader (such as a reference trajectory or a specified path). In
most theoretical work, a single leader with exact measurement
is considered on multi-agent systems for each agent to follow.
However, in practical situations, multiple leaders and target
Manuscript received September 13, 2010; revised February 07, 2011 and
July 14, 2011; accepted August 03, 2011. Date of publication August 15, 2011;
date of current version February 29, 2012. This work was supported in part by
the Knut and Alice Wallenberg Foundation, in part by the Swedish Research
Council, and in part by the NNSF of China under Grants 60874018, 60736022,
and 60821091. Recommended by Associate Editor Z. Qu.
G. Shi and K. H. Johansson are with the ACCESS Linnaeus Centre, School
of Electrical Engineering, Royal Institute of Technology, Stockholm 10044,
Sweden (e-mail: [email protected]; [email protected]).
Y. Hong is with the Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TAC.2011.2164733
sets with unmeasurable variables are considered to achieve
desired collective behaviors. In [8], a simple model was given to
simulate fish foraging and demonstrate the leader effectiveness
when the leaders (or informed agents) guide a school of fish to
a particular food region. In [23], a straight-line formation of a
group of agents was discussed, where all the agents converge to
the line segment specified by two edge leaders. A containment
control scheme was proposed with fixed undirected interaction
in [24], which aimed at driving a group of agents to a given
target location and making their positions contained in the
polytope spanned by multiple stationary or moving leaders
during their motion. Region following formation control was
constructed [25], where all the robots are driven and then stay
within a moving target region as a group. Moreover, different
dynamic connectivity conditions were obtained to guarantee
that the multiple leaders (or informed agents) aggregate the
whole multi-agent group within a convex target set in [26]. Additionally, control strategies were demonstrated and analyzed to
drive a collection of mobile agents to stationary/moving leaders
with connectivity-maintenance and collision-avoidance with
fixed and switching directed network topologies in [27]. As a
matter of fact, multiple leaders are usually assigned to increase
control effectiveness, enhance communication/sensing range,
improve reliability, and optimize energy cost in multi-agent
coordination.
Connectivity plays a key role in the coordination of multiagent networks, which is related to the influence of agents and
controllability of the network. Due to mobility of the agents,
inter-agent topologies usually keep changing in practice.
Therefore, the various connectivity conditions to describe frequently switching topologies in order to deal with multi-agent
consensus or flocking [15], [16], [18], [21]. In fact, the “joint
connection” or similar concepts are important in the analysis
of stability and convergence to guarantee multi-agent coordination with time-dependent topology. Uniformly jointly
connected conditions have been employed for different problems. [28] studied the distributed asynchronous iterations,
while [22] proved the consensus of a simplified Vicsek model.
Furthermore, [14] and [6] investigated the jointly connected
coordination for second-order agent dynamics via different
approaches, while [30] worked on nonlinear continuous-time
agent dynamics with jointly connected interaction graphs.
Also, flocking of multi-agent system with state-dependent
topology was studied with non-smooth analysis in [18], [20].
joint connection condition, which is a
What is more, the
generalized case of the uniformly joint connection assumption,
was discussed by Moreau, in order to achieve the consensus for
connectivity concept
discrete-time agents in [31]. This
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was then extended in the distributed control analysis for the
target set convergence in [26].
It is well known that input-to-state stability (ISS) is an important and very useful tool in the study of the stability and stabilization of control systems [29], [35]. Its variants such as integral input-to-state stability (iISS) were discussed in [34]. Then
few works on set input-to-state stability (SISS) were done with
respect to fixed sets in [33]. On the other hand, ISS or related
ideas can facilitate the control analysis and synthesis with interconnection conditions like small gains (referring to [29], for
example). ISS has recently been applied to the stability study of
a group of interconnected nonlinear systems [32]. Moreover, an
extended concept called leader-to-formation stability was introduced to investigate the stability of the formation of a group of
agents in light of ISS properties [19]. In fact, ISS application in
multi-agent systems is promising.
The contributions of the paper include the following.
• We propose the generalized set input-to-state stability
(SISS) and set integral-input-to-state stability (SiISS) to
handle moving sets with time-varying shapes for switching
multi-agent networks.
• We study the multi-leader coordination from the ISS viewpoint. With the help of SISS and SiISS, we give explicit
expressions to estimate the convergence rate and tracking
error of a group of mobile agents that try to enter the convex
hull determined by multiple leaders.
• We show relationships between the connectivity and set
tracking of the multi-agent system, and find that various
jointly connected conditions usually provide necessary
and/or sufficient conditions for distributed coordination.
• We develop a method to study SISS and SiISS for a moving
set and switching topology with graph theory and nonsmooth analysis. In fact, we cannot take the standard approaches to conventional ISS or iISS using equivalent ISSLyapunov functions [34], [35]. In addition, the classic algebraic methods based on Laplacian may fail due to disturbances in nonlinear agent dynamics, uncertain leader velocities, or moving multi-leader set.
This paper is organized as follows. Section II introduces some
necessary preliminaries. Section III presents problem formulation, and Section IV proposes results for the convergence estimation. Section V mainly reports a necessary and sufficient condition for the SISS with respect to the moving multi-leader set
with switching inter-agent topologies, and then presents a settracking case based on the SISS. Correspondingly, Section VI
obtains necessary and sufficient conditions for SiISS and then
shows set-tracking results related to SiISS. Finally, Section VII
gives concluding remarks.
of nodes and arcs
for
. A walk is called a path if the nodes of this walk are distinct,
. Node is called reachand a path from to is denoted as
able from if there is a path
. If the nodes
are
,
is called a (directed) cycle. A digraph
distinct and
without cycles is said to be acyclic.
and
The union of the two digraphs
is defined as
if they have the same node
set. Furthermore, a time-varying digraph is defined as
with
as a piecewise constant function,
where is the finite set which consists of all the possible diin
graphs with node set . Moreover, the joint digraph of
with
is denoted as
time interval
II. PRELIMINARIES
In this section, we introduce some preliminary knowledge for
the following discussion.
First we introduce some basic concepts in graph theory (referring to [13] for details). A directed graph (digraph)
consists of a finite set
of nodes and an arc
set , in which an arc is an ordered pair of distinct nodes of .
describes an arc which leaves and enters . A walk in
digraph is an alternating sequence
(1)
Next, we recall some notations in convex analysis (see [2]). A
set
is said to be convex if
whenever
,
and
. For any set
, the
intersection of all convex sets containing is called the convex
. Particularly, the convex hull of a
hull of , denoted by
is a polytope, denoted by
finite set of points
. In fact, we have
.
Let
be a closed convex subset in
and denote
, where
denotes the Euclidean norm
for a vector or the absolute value of a scalar ([34], [35]). Then
a unique element
we can associate to any
satisfying
, where the map
is called the
and
projector onto
(2)
Clearly,
[1])
is continuously differentiable at point , and (see
(3)
The following lemma was obtained in [26], which is useful
in what follows.
is a convex set and
Lemma 2.1: Suppose
. Then
(4)
Particularly, if
, then
(5)
Then we consider the Dini derivative for the following nonsmooth analysis. Let and
be two real numbers and
consider a function
and a point
. The
upper Dini derivative of at is defined as
It is well known that when is continuous on
, is nonif and only if
for any
increasing on
(more details can be found in [3]). The next result is given for
the calculation of Dini derivative [4], [30].
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665
is said to be jointly L-connected (JLC) if the union
is L-connected for any .
is said to be uniformly jointly L-connected (UJLC)
3)
if there exists
such that the union graph
is L-connected for any
.
Remark 3.1: Note that the L-connectedness describes the capacity for the follower agents to get the information from the
moving multi-leader set in the information flow, and an L-connected graph may not be connected since the graph with leaders
as its nodes may not be connected. In fact, if we consider the
group of the leaders as one virtual node in , then the L-connectedness becomes the quasi-strong connectedness for a digraph
[5], [30].
, is denoted as
The state of agent
, and the state of leader
, is denoted as
. Denote
and
and let the continuous function
be the weight of arc
, if any, for
,
be the weight of arc
and continuous function
, if any, for
;
.
Then we present the multi-agent model for the active leaders
and the (follower) agents
2)
graph
Fig. 1. Multiple agents (v , i
; ; ).
123
be
= 1; 2; 3; 4) with multiple leaders (v^ , i =
Lemma 2.2: Let
and
. If
is the
set of indices where the maximum is reached at , then
.
III. PROBLEM FORMULATION
In this paper, we consider the set coordination problems
follower-agents
for a multi-agent system consisting of
and leader-agents (see Fig. 1). The follower set is denoted
as
, and the leader set is denoted as
. In what follows, we will identify follower
or leader with its index (namely, agent or leader ) if
there is no confusion.
Then we describe the communication in the multi-agent netcan “see”
, there is an arc
work. At time , if
(marking the information flow) from to , and then agent
is said to be a neighbor of agent . Moreover, if
“sees”
at time , there is an arc
leaving from and enand
tering , and then is said to be a leader of agent . Let
represent the set of agent ’s neighbors and the set of agent
’s leaders (that is, the leaders which are connected to agent ),
respectively. Note that, since the leaders are not influenced by
entering .
the followers, there is no arc leaving from
as the whole agent set (including
Define
leaders and followers). Denote as the set of all possible interconnection topologies, and
as a piecewise
constant switching signal function to describe the switchings
between the topologies. Thus, the interaction topology of the
considered multi-agent network is described by a time-varying
. Correspondingly,
directed graph
is denoted as the communication graph among the follower
and
represent the
agents. Additionally, let
set of agent ’s neighbors and the set of its connected leaders
in
, respectively.
As usual in the literature [22], [26], [30], an assumption is
.
given for the switching signal
Assumption 1 (Dwell Time): There is a lower bound
between two switching instants.
We give definitions for the connectivity of a multi-agent
system with multiple leaders.
Definition 3.1:
is said to be L-connected if, for any
, there
1)
such that there is a path from
exists a leader
at time . Moreover,
is
leader to agent in
if
said to be jointly L-connected in time interval
the union graph
is L-connected.
(6)
describes the control inputs of the leader
where
, which is continuous in for fixed and piecewise continis a continuous function to deuous in for fixed , and
scribe the disturbances in communication links and individual
dynamics to follower agent . Then another assumption is given
and
.
on the weight functions
Assumption 2 (Bounded Weights): There are
and
such that
,
for any
.
and , may not be conRemark 3.2: In (6), the weights,
stant. Instead, because of the complex communication and environment uncertainties, they are dependent on time or space or
relative measurement (see nonlinear models given in [18], [26],
[30], [31]). Some models such as those studied in [26], [30] can
be written in the form of (6), while other nonlinear multi-agent
models may be transformed to this class of multi-agent systems
and
are written
in some situations. Here
in a general form simply for convenience, and global informaand
can
tion is not required in our study. For example,
, which
depend only on the state of , time and
is certainly a special form of
or
. In other
words, the control laws in specific decentralized forms are still
decentralized.
,
Without loss of generality, we assume the initial time
and the initial condition
and
.
Denote the time-varying polytope formed by the active
leaders
(7)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
and let
([33]),
The conventional SISS was given for a fixed set
while the generalized SISS or SiISS is proposed with respect to
. In the following, we still use SISS
a time-varying set
or SiISS instead of generalized SISS or SiISS for simplicity.
Remark 3.3: Similar to the study of conventional ISS, local
SISS and SiISS can be defined. In this paper, we focus on the
global SISS and SiISS. In fact, it is rather easy to extend research
ideas of global set tracking to study local cases.
be the maximal distance for the followers away from the moving
.
multi-leader set
The following definition is to describe the convergence to the
.
moving convex set
Definition 3.2: The (global) set tracking (ST) with respect to
for system (6) is achieved if
(8)
and
.
for any initial condition
For a stationary convex set , set tracking can be reduced
to set stability and attractivity, and methods to analyze
were proposed in some existing works [26]. In fact, [24] and [27]
discussed the convergence to the static convex set determined by
stationary leaders with well designed control protocols. Moreover, if we assume that the target set is exactly the polytope with
the positions of the stationary leaders (or informed agents) as its
vertices, then the convergence to the polytope, treated as a target
set, can be obtained straightforwardly based on the results and
limit-set-based methods given in [26].
Input-to-state stability has been widely used in the stability
analysis and set input-to-state stability (SISS) for a fixed set
has been studied in [33]. To study the multi-leader set tracking
in a broad sense, we introduce a generalized SISS with re, a moving set with a time-varying shape, for
spect to
multi-agent systems with switching interaction topologies. De,
,
,
note
and
with
([35]).
A function
is said to be a -class function
. Moreover,
if it is continuous, strictly increasing, and
is a
-class function if
a function
is of class for each fixed
and
decreases to 0 as
for each fixed
.
Definition 3.3: System (6) is said to be globally generalwith
ized set input-to-state stable (SISS) with respect to
-function and a -function such
input if there exist a
that
IV. CONVERGENCE ESTIMATION
For the set tracking with respect to a moving multi-leader
set of system (6), we have to deal with the estimation of
when
is a time-varying convex set, where
is a trajectory of the moving leaders in system (6) with
. Define
initial condition
and
Obviously,
(11)
The following result is given to estimate the changes of the
distance between an agent and the convex hull spanned by the
leaders.
and
,
Lemma 4.1: For any
Proof: Suppose
where
for
with
. Define
, and then
(9)
for
and any initial conditions
and
.
Integral-input-to-state stability (iISS) was introduced as an
integral variant of ISS, which has been proved to be strictly
weaker than ISS [34]. We also introduce a definition of (generalized) set integral-input-to-state stability (SiISS) with respect
to a time-varying and moving set.
Definition 3.4: System (6) is (globally) generalized set inteif there
gral-input-to-state stable (SiISS) with respect to
-function and a -function such that
exist a
Moreover,
(10)
(12)
for any initial conditions
and
.
SHI et al.: CONNECTIVITY AND SET TRACKING OF MULTI-AGENT SYSTEMS GUIDED BY MULTIPLE MOVING LEADERS
Also, similar analysis leads to
667
Therefore,
(13)
Therefore, (12) and (13) lead to the conclusion.
For simplicity, define
,
and
which is locally Lipschitz but may not be continuously differ,
and
entiable. Clearly,
.
Then, we get the following lemma to estimate the set convergence.
Lemma 4.2:
.
Proof: It is not hard to see that
Then, according to (3), we obtain
(14)
denote the set containing all the agents that
Moreover, let
at time . Then,
reach the maximal distance away from
, according to (2), one has
for any
(15)
for any
. Furthermore, in light of Lemma 2.1, since
for any
. Therefore, the conclusion follows since
according to Lemma 2.2.
V. CONNECTIVITY AND SISS
Furthermore, according to Lemma 4.1
In this section, we study the SISS with respect to the convex
set spanned by the moving leaders in an important connectivity
case, uniformly jointly L-connected (UJLC) topology. Without
in the sequel.
loss of generality, we will assume
A. Main Results
and then it is easy to find that
in this section. Then we have
Suppose
the main result on SISS.
if
Theorem 5.1: System (6) is SISS with respect to
and only if
is UJLC.
The main difficulties to obtain the SISS inequalities in the
UJLC case are how to estimate the convergence rate in a time
interval by “pasting” time subintervals together and how to estimate the impact of the input to the agent motion.
To prove Theorem 5.1, we first present two lemmas to estimate the distance error in the two standard cases during
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
(s) and to agent in
in
with
each arc exists for at least . This completes the proof.
Remark 5.2: If there is a convex set such that
,
, that is, is a positively invariant set for the leaders, then
. By Theorem 5.1, system (6) is SISS
is UJLC.
with respect to with as the input if
Sometimes, the velocities of the moving leaders and uncertainties in agent dynamics (maybe because of the online estimation) may vanish. To be strict, consider the following condition:
(s).
uniformly for
for
and a constant
with
as the
dwell time of switching.
leaving from follower
Lemma 5.1: If there is an arc
entering
in
for all
,
then there exist a continuous function
and a constant
such that
, we have
Proof: See Appendix A.1.
Lemma 5.2: If there is an arc
entering
in
for all
for constants
uous function
such that
and
leaving from
, and
(16)
, then there exist a continand a positive constant
, we have
(17)
Clearly, (17) yields that for any
, there is
such
that
, where
is the truncated part of defined
. Suppose (17) holds and
is UJLC. Based on
on
, there is
such that
Theorem 5.1, for any
Hence, the set tracking for system (6) with respect to set
is achieved easily. On the other hand, similar to the proof of Theorem 5.1, the necessity of the global set tracking for system (6)
with condition (17) can also be simply proved by counterexammay be large and the distance error may accuples since
mulate to a very large value over a sufficiently long period of
time. Therefore, we have the following result.
Corollary 5.1: The global set tracking with respect to
is achieved for all
satisfying (17) if and only if
is
UJLC.
B. Proof of Theorem 5.1
Proof: See Appendix A.2.
Remark 5.1: The following properties of
and
are
quite critical in the study of the set tracking with jointly L-connected topology (see Fig. 2):
.
1)
2)
and
are strictly decreasing during
.
and
are strictly increasing during
3)
, and
,
.
Next, we introduce the following lemma to state an important
property for UJLC graphs.
is UJLC, then, for any
and
,
Lemma 5.3: If
there is a path
from some leader
to follower in
with
, and each arc of
exists
at least during
.
in a time interval with length
Proof: Denote as the first moment when the interaction
(suppose there are switchtopology switches within
ings without loss of generality). If
, then, for any
, there is a path
from some leader with index
to agent in
, where each arc stays there for at least
during
due to the definition of
the dwell time
. On the other hand, if
,
.
, there is also a path
from some leader
Then, for any
We are now in a position to prove Theorem 5.1: “If” part:
Denote
with
. Then we estimate
at subintervals
for
.
Based on Lemma 5.3, in
, there must be an arc
leaving from a leader
to a
and this arc remains for at least . Suppose
follower
for
. According
to Lemma 5.1, for any
, we have
where
and
were defined in Lemma 5.1. Take
. Since
, we obtain
(18)
for any
. Furthermore, in
,
there must be a follower
,
, such that there
for some
, or an arc
in
exists an arc
.
There are two cases:
for
,
1) If
one also has that for any
SHI et al.: CONNECTIVITY AND SET TRACKING OF MULTI-AGENT SYSTEMS GUIDED BY MULTIPLE MOVING LEADERS
2) If
for
According to (11) and Lemma 4.2, one has
.
Thus, (18) will lead to that for any
Then, by Lemma 5.2, if we take
obtain that for any
Because
, we
,
, we see that
for
and
.
Repeating the above procedure yields
and
such that, there exists
669
and
where
denotes the largest integer no greater than
,
which implies the conclusion.
is not UJLC, there is a time sequence
“Only if” part: If
such that
is not L-conwith
.
nected for
,
and
Taking
,
with
,
, and
,
,
. Since
we obtain
is not L-connected, there is
such that agent
is reachable from no leader. Define
is
from
.
Since
contains no leader and there is no arc entering
, no agent in
leaves
when
. Moreover, none of the followers can enter
in finite time. Therefore, for any
, we have
,
satisfying
Thus, the SISS with respect to
cannot be achieved.
VI. CONNECTIVITY AND SIISS
(19)
where
. Moreover, the nodes
are distinct.
, and then
Denote
for any
Therefore,
, which leads to
, we have
Again by Lemma 4.2, one has
with
,
. Thus, (19) leads to
In this section, we aim at the connectivity requirement to ensure the set integral-input-to-state stability (SiISS) when
is jointly L-connected (JLC).
A. Main Results
Theorem 5.1 showed an equivalent relationship between SISS
and UJLC. However, this is not true for SiISS. Here, we propose
a couple of theorems about SiISS. The proofs of these conclusions can be found in the following subsection.
First of all, we propose a sufficient condition.
if
Theorem 6.1: System (6) is SiISS with respect to
is UJLC.
(i.e.,
is L-connected for
Remark 6.1: JLC of
any ) is necessary for the SiISS, though it is not sufficient. If
is not L-connected for some
, there is a subset
such that no arcs enter
in
. Hence, the
may not be SiISS for some initial conditions since
agents in
they will not be influenced by the convex leader-set after .
UJLC, which is a special case of JLC, provides a sufficient
condition for SiISS, but UJLC is not necessary to ensure SiISS.
In fact, there are other cases of JLC to make SiISS hold. Here
we consider two important special JLC cases, i.e., bidirectional
graphs and acyclic graphs.
is called a bidirectional graph when is a
A digraph
neighbor of if and only if is a neighbor of , but the weight
of arc
may not be equal to that of arc
. The next result
shows a necessary and sufficient condition for the bidirectional
case.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
Theorem 6.2: Suppose that
is bidirectional for all
.
is JLC.
Then system (6) is SiISS if and only if
The next lemma shows an important property for an acyclic
digraph, that is, a digraph without cycles.
is acyclic and
Lemma 6.1: Assume that
is L-connected. Then there is a partition of
by
,
such that in graph
, all the
are from ; and all the arcs entering
arcs entering node set
,
are from
.
node set
Proof: First we prove
exists by contradiction.
does not exist, every agent
has neighIf
bors within
in
. Denote
there is an arc leaving from
entering
. Clearly
. Take
. Then, there is
such that
. Moreover, we can associate
with
( cannot be , of course) such that there is a
in
(
if
). Hence, a
path
path
in
is found. Regarding
as
and
repeating the above procedure yields the existence of
in
with
. In this way, we obtain a path
in
with
,
. Since
are finite, there has to be
for some
the nodes in
, which lead to a directed cycle in
.
to make the conclusion hold.
Therefore, there is
with
in
, with the
Next, by replacing
to make the conclusion hold.
same analysis we can find
Repeating this procedure, since the number of all the agents is
finite, there will be a constant
such that
.
This completes the proof.
Then we have a SiISS result for the acyclic graph case.
is acyclic. Then
Theorem 6.3: Assume that
is JLC.
system (6) is SiISS if and only if
Furthermore, consider the following inequality
also consistent with Theorems 15 and 17 in [26], respectively,
. However, different from the limit-set-based
when
approach given in [26], the proposed method by virtue of (22)
and (27) also provides the estimation of the convergence rate.
Remark 6.4: Theorems 5.1 and 6.1 with Remark 6.1 proved
that for system (6), SISS is equivalent to UJLC, which implies
SiISS, while JLC is a necessary condition, namely,
(20)
It is not hard to obtain the following results based on Theorems 6.1, 6.2, and 6.3. The proofs are omitted for space limitations.
Corollary 6.1: System (6) achieves the set tracking if (20)
holds and
is UJLC.
being
Corollary 6.2: Suppose (20) holds with either
bidirectional for all
or
being acyclic. Then
system (6) achieves the global set tracking if and only if
is JLC.
Remark 6.2: In general, the condition (17) does not imply
and is not implied by the condition (20). In fact, the considered
leaders converge to some points with (20), but the leaders can go
is uniformly continuous
to infinity with (17). However, if
(which can be guaranteed once
is bounded for
in
), (17) will then be implied by (20) according to
Barbalat’s Lemma.
Remark 6.3: Corollaries 6.1 and 6.2 are consistent with
Proposition 6 in [34], where (20) and integral-ISS together
resulted in the state stability. Moreover, the two corollaries are
Thus,
, which is consistent with Corollary 4
of [34], where ISS implies iISS. Moreover, Theorems 6.2 and
6.3 show that, in either bidirectional or acyclic case,
Remark 6.5: As for set tracking (ST), Corollary 5.1 shows
that
satifying
Moreover, Corollaries 6.1 and 6.2 show that as long as (20)
holds,
in general directed cases, and
in either bidirectional or acyclic case. Usually, SISS goes with
(17) and SiISS with (20), consistent with discussions on ISS
and iISS [34], [35]. Additionally, it is worth pointing out that
the differences between Corollaries 5.1 and 6.1 result from the
fact that UJLC is necessary for SISS, but not for SiISS.
Although our results are consistent with the results on conventional ISS or iISS, the analysis methods given in [34] and
[35] are mainly based on an equivalent ISS-Lyapunov function,
which cannot be applied to our cases with a moving set and
switching topologies.
B. Proofs
To establish the SiISS in the JLC case, we will analyze the
impact of the integral of input
in a time interval and estimate the convergence rates during this time interval by “pasting”
different time subintervals together within the interval. The following lemmas are given to estimate the convergence rates in
different cases.
leaving from
Lemma 6.2: If there is an arc
entering
in
for
, then there exists a
with
strictly decreasing function
such that for any
SHI et al.: CONNECTIVITY AND SET TRACKING OF MULTI-AGENT SYSTEMS GUIDED BY MULTIPLE MOVING LEADERS
Proof: According to Lemma 4.2,
,
for any
Since there is an arc
with
,
in
and
671
, then there is a strictly increasing function
with
such that
.
for
where
.
The proofs of Lemmas 6.3 and 6.4 are similar to that of
Lemma 6.2, and therefore, omitted.
is an nonempty subset.
Lemma 6.5: Suppose
entering
If there are no arcs leaving from
in
for a given constant
and
,
for
and
, then
constants
Based on Lemma 2.1, when
Therefore,
Taking
gives
for
by virtue of the analysis given for Lemma
4.2. Then Lemma 6.5 can be obtained straightforwardly.
Now we are ready to prove Theorems 6.1, 6.2, and 6.3.
with
Proof of Theorem 6.1: Denote
defined in Lemma 5.3. If
is L-connected, there
for
has to be an arc
leaving from a leader
entering
and this
arc is kept there for a period of at least . Invoking Lemmas
6.2 and 6.4, we see that for any
or equivalently
where
for
. Thus, for any
,
where
.
, there must be a
Furthermore, when
follower
,
such that there exists an arc
for some
, or an arc
when
. According to Lemmas 6.3 and 6.4, for any
, we have
with
,
, which
implies the conclusion.
Lemma 6.3: Suppose there is an edge
leaving from
entering
in
and
with constants
and
when
. Then there is a strictly decreasing
with
such that
function
where
with
.
Repeating the above procedure yields
for
,
, where
(21)
Lemma 6.4: Given a constant
, if there is
with
for constants
and
nodes of
. Moreover, the
,
are distinct.
672
Denote
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
from (21). Then we obtain
,
Furthermore, we define
there is an edge leaving from
entering
, and
there is an edge leaving from
entering
.
Noting that
is L-connected, thus, according to
Lemma 6.5, one has
It follows immediately that for any
Further, by Lemma 6.4,
Based on Lemma 4.2 and (11), we have
(25)
for
to Lemma 6.3,
, where
. Moreover, according
(22)
where
Hence, (10) holds with
since
, which completes the proof.
Proof of Theorem 6.2: The “only if” part is quite obvious, so
we focus on the “if” part.
is JLC, there exists a sequence of time instants
Since
(26)
for
. Because
, (25) and (26) lead to
(23)
such that
(24)
is L-connected for
. Moreover,
and
will be kept for at least the dwell
each arc in
time
during the time interval
,
;
.
during
. Since
Then we estimate
is L-connected, there is a time interval
such that there is an edge
between a leader
and a follower
for
. Based on Lemma 6.2,
where
.
,
Next, define
there is an edge leaving from
entering
, and
there is an edge leaving from
entering when
.
Similarly, from Lemma 6.5, by
,
, one has
Repeating the process gives
where
.
SHI et al.: CONNECTIVITY AND SET TRACKING OF MULTI-AGENT SYSTEMS GUIDED BY MULTIPLE MOVING LEADERS
for
for some
until
such that
673
with a constant
. According to Lemma 6.1, for any
, we have that for any
,
Again by Lemmas 6.3 and 6.1, for any
, we see that
Hence,
where
. Similarly, with
and
,
, we have
According to Lemma 6.5, we obtain
It is obvious to see that
and
Therefore, denote
, then for
.
for any
leads to
and
,
, which
,
Similar to the proof of Theorem 6.2, SiISS can be obtained.
VII. CONCLUSION
Thus, similar to the proof of Theorem 6.1, we also have
(27)
where
and
when
,
,
Then it is obvious to see that (27) leads to Theorem 6.2 immediately.
Proof of Theorem 6.3: We also focus on the “if” part since
the “only if” part is quite obvious.
is JLC, there is an infinite sequence in the form
Because
is L-connected for
of (23) with (24) such that
.
, there is
such that there
Then, for any
entering in
. Hence, recalling
is an arc leaving from
Lemma 6.2,
This paper addressed multi-agent set tracking problems with
multiple leaders and switching communication topologies. At
first, the equivalence between UJLC and the SISS of a group of
uncertain agents with respect to a moving multi-leader set was
shown. Then it was shown that UJLC is a sufficient condition
for SiISS of the multi-agent system with disturbances in agent
dynamics and unmeasurable velocities in the dynamics of the
leaders. Moreover, when communication topologies are either
bidirectional or acyclic, JLC is a necessary and sufficient condition for SiISS. Also, set tracking was achieved in special cases
with the help of SISS and SiISS.
Multiple leaders, in some practical cases, can provide an
effective way to overcome the difficulties and constraints in
the distributed design. On the other hand, ISS-based tools were
proved to be very powerful in the control synthesis. Therefore,
the study of multiple active leaders and related ISS tools
deserves more attention.
APPENDIX A
PROOF OF LEMMA 5.1
Due to
obtain
by Lemma 4.2 and (11), we
(28)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
Since there is an arc
with
and
for
, based on (15), one has
in
,
, because
where
and
.
for
no matter
Then we evaluate
whether there is any connection between the followers and the
leaders. Similar analysis gives
Thus, with (14) and the fact that
, we obtain
(29)
.
for
Then, by Lemma 2.1, if
for
, then
,
which is equivalent to
(30)
On the other hand, if
Lemma 2.1 and (28),
,
, from
Denote
we have
. From (32), when
,
(31)
. Therefore, with (29), (30) and (31), it follows
that
where
(33)
, or equivalently,
where
,
based on (32) and (33)
for
have
. As a result, for any
, we
with
(32)
which is continuous. Thus, the conclusion follows.
and
. Therefore,
SHI et al.: CONNECTIVITY AND SET TRACKING OF MULTI-AGENT SYSTEMS GUIDED BY MULTIPLE MOVING LEADERS
675
APPENDIX B
PROOF OF LEMMA 5.2
in
for
If there is an arc
based on (28), Lemmas 2.1 and 4.1, it is easy to see
, then
(35)
where
and
because
,
,
With (34) and (35), we have
with
which is continuous. Thus, the conclusion follows.
ACKNOWLEDGMENT
for
. Then, if (16) holds, as done in the proof
of Lemma 5.1, we can obtain
The authors would like to thank the Associate Editor and the
anonymous reviewers for their valuable suggestions.
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Here are two cases.
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(34)
where
,
and
.
• when
similarly, we have
: Denote
. By (34),
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Guodong Shi received the B.Sc. degree in applied
mathematics from Shandong University, Jinan,
China, in 2005 and the Ph.D. degree in systems
theory from the Institute of Systems Science,
Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing, China, in
2010.
Currently, he is a Postdoctoral Researcher at the
ACCESS Linnaeus Centre, School of Electrical
Engineering, Royal Institute of Technology, Stockholm, Sweden. His research interests include control
of multi-agent systems and distributed optimization methods.
Yiguang Hong (M’99–SM’02) received the B.Sc.
and M.Sc. degrees from Peking University, Beijing,
China, and the Ph.D. degree from the Chinese
Academy of Sciences (CAS), Beijing.
He is currently a Professor in the Institute of
Systems Science, Academy of Mathematics and
Systems Science, CAS, and serves as the Director of
the Key Lab of Systems and Control, CAS, and the
Director of the Information Technology Division,
National Center for Mathematics and Interdisciplinary Sciences, CAS. His research interests include
nonlinear dynamics and control, multi-agent systems, robotics, and reliability
of software and communication systems.
Prof. Hong is also serving as Deputy Editor-in-Chief of Acta Automatica
Sinica, Editor-at-Large of the Journal of Control Theory and Applications, and
Associate Editors for several journals including the IEEE TRANSACTIONS ON
AUTOMATIC CONTROL, Journal of Systems Science and Complexity, and Kybernetika. He is a recipient of the Guangzhaozhi Award of Chinese Control Conference (in 1997), the Young Author Prize of IFAC World Congress (in 1999), the
Young Scientist Award of CAS (in 2001), and the Youth Award for Science and
Technology of China (in 2006), and National Natural Science Prize of China (in
2008, jointly with D. Cheng, et al.).
Karl Henrik Johansson (SM’08) received the
M.Sc. and Ph.D. degrees in electrical engineering
from Lund University, Lund, Sweden, in 1992 and
1997, respectively.
He is a Director of the ACCESS Linnaeus Centre
and a Professor in the School of Electrical Engineering, Royal Institute of Technology, Stockholm,
Sweden. He is a Wallenberg Scholar and has held
a Senior Researcher Position with the Swedish
Research Council. He has held visiting positions
at UC Berkeley (1998–2000) and the California
Institute of Technology (2006–2007). His research interests are in networked
control systems, hybrid and embedded control, and control applications in
automotive, automation and communication systems.
Prof. Johansson is the Chair of the IFAC Technical Committee on Networked
Systems. He has served on the Executive Committees of several European research projects in the area of networked embedded systems. He is on the editorial board of IET Control Theory and Applications and the International Journal
of Robust and Nonlinear Control, and previously of the IEEE TRANSACTIONS
ON AUTOMATIC CONTROL and Automatica. He was awarded an Individual Grant
for the Advancement of Research Leaders from the Swedish Foundation for
Strategic Research in 2005. He received the triennial Young Author Prize from
IFAC in 1996 and the Peccei Award from the International Institute of System
Analysis, Austria, in 1993. He received Young Researcher Awards from Scania
in 1996 and from Ericsson in 1998 and 1999.