Coordination of Multi-Agent Systems under Switching Topologies via Disturbance Observer Based Approach

0.86 MB·

by  Yutao Tang

Coordination of Multi-Agent Systems under Switching Topologies
via Disturbance Observer Based Approach
Yutao Tang ∗∗
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Abstract

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a In this paper, a leader-following coordination problem of heterogeneous multi-agent systems is
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considered under switching topologies where each agent is subject to some local (unbounded) dis-

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turbances. Whiletheseunknowndisturbancesmaydisrupttheperformance of agents,adisturbance
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observerbasedapproachisemployedtoestimateandrejectthem. Varyingcommunicationtopologies

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are also taken into consideration, and their byproduct difficulties are overcome by using common
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O Lyapunovfunctiontechniques. Accordingtotheavailableinformationindifferencecases,twodistur-
. bance observer based protocols are proposed to solve this problem. Their effectiveness is verified by
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simulations.
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Keywords: multi-agent system; disturbance observer; switching topology; common Lyapunov
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function.
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v 1 Introduction
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2 In the past decades there has been a large literature in the study of multi-agent systems due to its
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0 wide applications such as cooperative controlof unmanned aerialvehicles, communication among sensor
1. networks, and formation of mobile robots (see [1, 2] and the references therein). As an important topic
0
of multi-agent systems, the leader-following problem is actively studied by many authors, e.g. [3–5]. In
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1 this formulation, one or multiple agents are selected as leaders to generate desired trajectories for those
:
v followers and lead the whole group to achieve collective tasks [6–12].
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X Ithasbeenwell-recognizedthatdisturbancerejectionisoffundamentalimportanceintheapplicability
r of designed controllers. While there are always disturbances in real applications, it is necessary to take
a
them into consideration and attenuate or eliminate them for multi-agent control design. In [13] for
first-order multi-agent systems, it was shown that under bounded unknown external disturbances the
steady-state errors of any two agents can reach a small region and is called lazy consensus. Later, the
authors of [14] proposedanH analysisapproachto investigate robustconsensus problem ofhigh-order
∞
multi-agentsystemswithexternaldisturbances. In[15],consensusofmulti-agentsystemswithexogenous
disturbances wasconsidered,and anobserverwas constructedto compensate the negativeeffect of those
disturbances. However, most of these results were obtained for special dynamic systems, and there are
fewgeneralconsensusresultsemphasizingdisturbancerejectionwithanexception[16],wheretheauthors
consideredleader-followingconsensuswith disturbancerejectionfromthe viewpointofoutput regulation
[17] and solved it for linear multi-agent systems with a fixed topology.
∗∗YutaoTangiswithSchoolofAutomation,BeijingUniversityofPostsandTelecommunications,Beijing100876,China.
E-mail:[email protected]
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Also note that, in centralized or decentralized setup, there is usually no need to partition and treat
those disturbancesandreferencesina separateway. However,inmulti-agentsystems,itwouldbe better
to distinguish those two kinds of signals and not to model them in the same manner. Intuitively, the
reference is globally set up to drive all agents to complete a common task, while the disturbances are
usually local and harmful to such cooperation. Thus, another motivation of this paper is to formulate a
problem that treats those two kinds of signals by different approaches.
Hence, we aim to investigative a coordination problem of heterogeneous multi-agent systems under
switching topologies, where the reference is given by a traditional leader. At the same time, those
followers may be subject to local disturbances modeled by some other autonomous systems. Since the
disturbances areoften unmeasurable,a disturbance observerbased(DOB) approach[18,19] is employed
to tackle this problem. DOB approach stems from feedforward control, and can be perceived as a
composite controller comprising a feedforward compensation part to reject those disturbances, based on
disturbanceobservationandafeedback ruleto regulatethe plantto achieveothergoals. Althoughithas
beeninvestigatedby many publications(see [20] andreferencestherein), there is no correspondingresult
to our knowledge for heterogeneous multi-agent systems.
To sum up, the main contributions of the present paper are at least twofold:
• We extend the conventional leader-following consensus [2, 6] to general linear multi-agent systems
with local disturbances (which may be unbounded). When there are no such disturbances, these
results are consistent with existing consensus results. Here we consider heterogeneous multi-agent
systems under switching topologies,while many existing results were derivedfor only second order
dynamic systems [21] or for fixed graph cases [16].
• We extend the conventional disturbance observer based (DOB) approach [19] to its distributed
version for multi-agent systems with both reference tracking and disturbance rejection. When
there is only one agent, our problem becomes the conventional DOB formulation. Even for the
centralized case, we propose different full-order and reduced-order disturbance observers to solve
this problem without using the derivative of the plant’s states as that in [20]. It is also remarkable
thatthesedisturbanceobserverscanallowbothboundeddisturbances(e.g.,constantandharmonic
signals in existing literature) and unbounded disturbances (e.g. ramping and polynomial signals).
Therestofthispaperisorganizedasfollows. InSection2,somepreliminariesaregivenandourproblemis
formulated. ThenmainresultsarepresentedinSection3,wheretwotypesofcontrollawsareconstructed.
Finally, simulations and our concluding remarks are provided in Sections 4 and 5, respectively.
Notations: Let Rn be the n-dimensional Euclidean space, Rn×m be the set of n×m real matrices.
0n×m represents an n×m zero matrix. diag{b ,...,b } denotes an n×n diagonalmatrix with diagonal
1 n
elements b , ··· ,b ; col(a ,...,a ) = [aT,...,aT]T for column vectors a (i = 1,...,n). A weighted
1 n 1 n 1 n i
directed graph(or weighteddigraph) G =(N,E,A) is defined as follows, where N ={1,...,n} is the set
of nodes, E ⊂ N ×N is the set of edges, and A ∈ Rn×n is a weighted adjacency matrix [22]. (i,j) ∈ E
denotes anedge leavingfromnode iandenteringnode j. The weightedadjacencymatrix ofthis digraph
G isdescribedbyA=[a ] ,wherea =0anda ≥0(a >0ifandonlyifthereisanedgefrom
ij i,j=1,...,n ii ij ij
agentj to agenti). A path in graphG is an alternating sequence i e i e ···e i of nodes i andedges
1 1 2 2 k−1 k l
e =(i ,i )∈E for l=1,2,...,k. If there exists a path from node i to node j then node i is said to
m m m+1
be reachable from node j. The neighbor set of agent i is defined as N = {j: (j,i) ∈ E} for i = 1,...,n.
i
A graph is said to be undirected if a = a (i,j = 1,...,n). The weighted Laplacian L = [l ] ∈ Rn×n
ij ji ij
of graph G is defined as l = a and l =−a (j 6=i).
ii j6=i ij ij ij
P
2
2 Problem formulation
In this paper, we consider N +1 agents and N of them are followers of the form:
x˙ =A x +B u +E d
i i i i i i i
(1)
y =C x +D u , i=1,...,N
i i i i i
wherex ∈Rni,y ∈Rl,andu ∈Rmi arethestate,output,andinputoftheithsubsystem,respectively.
i i i
d ∈Rqi is the local disturbance of agent i governedby
i
d˙ =S d . (2)
i i i
The reference signal is given by a leader (denoted as agent 0) described as
r˙ =S r, y =F r, r ∈Rn0 (3)
0 0 0
Without loss of generality, we assume (F ,S ) is detectable and S ,...,S have no eigenvalues lying in
0 0 0 N
the openleft halfplane. Lete =y −y (i=1,...,N), we aimtodesignpropercontrollerssuchthatfor
i i 0
any initial condition x (0), d (0),r(0), the tracking error e will converge to zero as time goes to infinity
i i i
in spite of these disturbances.
Unlike in centralized cases, we do not assume the availability of y (hence r) to all agents in our
0
problem. An agent can get access to y unless there is an edge between this agent and the leader.
0
This makes it much difficult to achieve collective behaviors. Associated with this multi-agent system,
a dynamic digraph G can be defined with the nodes N = {0,1,...,N} to describe the communication
topology, which may be switching. If the control u can get access to the information of agent j at time
i
instant t, there is an edge (j,i) in the graph G, i.e., a >0. Also note that a =0 for i=1,...,N, since
ij 0i
the leader won’t receive any information from the followers. Denote the induced subgraph associated
with all followers as G¯.
Wesayacommunicationgraphisconnected[6]iftheleader(node0)isreachablefromanyothernode
ofGandtheinducedsubgraphG¯isundirected. GivenacommunicationgraphG,denoteH ∈RN×N asthe
submatrix ofits LaplacianL by deleting the firstrowandfirstcolumn. By Lemma 3 in[6], H is positive
definite if the communication graph is connected. Denote its eigenvalues as λ ≥λ ≥···≥λ >0.
1 2 N
In multi-agent systems, the connectivity graph G may be time-varying. To describe the variable
interconnectiontopology,wedenote allpossiblecommunicationgraphsasG ,...,G , P ={1,...,κ},and
1 κ
define a switching signal σ : [0,∞) → P, which is piece-wise constant defined on an infinite sequence
of nonempty, bounded, and contiguous time-intervals. Assume t − t ≥ τ > 0, ∀i, where t is
i+1 i 0 i
the ith switching instant and t = 0. Here τ is often called the dwell-time. Therefore, N and the
0 0 i
connectionweighta (i,j =0,1,...,N)aretime-varying. Moreover,theLaplacianL associatedwith
ij σ(t)
the switchinginterconnectiongraphG isalsotime-varying(switchedatt , i=0,1,...), thoughitis a
σ(t) i
time-invariant matrix in each interval [t ,t ). The following assumption on the communication graph
i i+1
is often made [6].
Assumption 1 The graph G is switching among a group of connected graphs.
σ(t)
The coordination problem of these heterogeneous multi-agent systems composed of (1), (2), and (3)
under switching topologies is described as follows. Given these multi-agent systems and the communica-
tion graph G , find a proper distributed control law such that for all initial conditions of the closed-loop
σ(t)
system, we have
lim e (t)=0, i=1,...,N. (4)
i
t→+∞
3
Remark 1 While a large literature in multi-agent systems only considered a globally reference tracking
problem [23, 24] or treated those disturbances and the referenced signal in the same manner [16], both
reference tracking and disturbance rejection are considered in this formulation while local disturbances
are modeled by separate autonomous systems. When d = 0, this formulation is consistent with existing
i
consensus results for general linear dynamics, e.g., [23], which include the well-known consensus for
integrators [1] as its special case.
3 Main results
In this section, we employ a two-phase design procedure to achieve the coordination goal. First, we
constructa distributed observerfor eachagentand transformsucha coordinationproblemof this multi-
agentsystemintoN decentralizedsub-tasks. Then,thosesub-taskswillbecompletedviaDOBapproach
together with both full-order and reduced-order controllers.
Thefollowinglemmaisusefulandcanbeprovedfromtheconvergent-inputconvergent-stateproperty
of the first subsystem [25].
Lemma 1 Consider the linear time-varying system
x¯˙ =A¯x¯+B¯v¯
v¯˙ =S¯(t)v¯
e¯=C¯x¯+F¯v¯.
If A¯ is Hurwitz and the subsystem v¯˙ = S¯(t)v¯ is uniformly exponentially stable, then for any x¯(0) = x¯
0
and v¯(0)=v¯ , it holds lim e¯(t)=0.
0 t→+∞
By letting v , col(d ,r) and some mathematical manipulations, this leader-following coordination
i i
problem is equivalent to the following N sub-problems:
x˙ =A x +B u +E¯ v
i i i i i i i
v˙ =S¯v (5)
i i i
e =C x +D u +F¯v
i i i i i i i
where E¯ = [E , 0ni×l], S¯ = blockdiag{S , S }, and F¯ = [0l×qi, −F ]. v is the exogenous signal for
i i i i 0 i 0 i
agent i including its local disturbance d and the global reference r.
i
The following equations known as regulatorequations [26] play a key role in solving the coordination
problem of multi-agent systems.
Assumption 2 For each i=1, ..., N, there exist constant matrices X , X , U , and U satisfying
i1 i2 i1 i2
X S =A X +B U +E
i1 i i i1 i i1 i
0=C X +D U
i i1 i i1
and
X S =A X +B U
i2 0 i i2 i i2
0=C X +D U −F .
i i2 i i2 0
Remark 2 Similar conditions have beenusedin [16,27]. Asufficient condition tothesolvability ofthese
linear matrix equations is that, for any eigenvalue of S (denoted as λ), i=0,...,N,
i
A −λI B
i i
rank =n +p.
i
" Ci Di#
4
Specially, when all agents are homogenous without local disturbances, X can be taken as the identity
i2
matrix which was implicitly used in [2] and [9].
Note that (A ,B ) is stabilizable, there exists K such that A +B K is Hurwitz. Denote K ,
i i i1 i i i1 i2
U −K X , K ,U −K X . By Theorem 1.7 in [26], the full-information controller u =K x +
i1 i1 i1 i3 i2 i1 i2 i i1 i
K d +K r trivially solves the output regulation problem of the ith subsystem (5), and hence achieves
i2 i i3
theleader-followingcoordinationgoalofthewholemulti-agentsysteminacentralizedsetup. Inspiredby
theseparateprincipleforsinglelinearsystems[17],wefollowasimilardesignandreplacethe unavailable
quantities in the full-information control law by their estimations.
Since not all agents can directly get access to the reference signal (i.e., the leader), we first construct
the following distributed observer for agent i to estimate r, and transform the original coordination
problem into several decentralized ones:
η˙ =S η +L F η , (6)
i 0 i 0 0 vi
where η = N a (t)(η −η ), η = r, i = 1,...,N, and L is a constant matrix to be designed.
vi j=0 ij i j 0 0
Letting η¯ ,η −r and denoting η¯=col(η¯ ,...,η¯ ) gives
i Pi 1 N
η¯˙ =[I ⊗S +H ⊗(L F )]η¯. (7)
N 0 σ(t) 0 0
The following lemma shows the effectiveness of this distributed observer.
Lemma 2 Under Assumption 1, there exists a constant matrix L such that the system (7) is uniformly
0
exponentially stable in the sense of ||η¯||≤c e−λ0t for some positive constants c and λ .
0 0 0
Proof. Forthispurpose,weonlyhavetodetermineanL suchthat,foreachi,thereexisttwoconstants
0
c¯ and λ¯ such that ||η −v||≤c e−λ0it.
0i 0i i 0i
Note that H is positive definite and constant during each interval [t ,t ) under Assumption 1.
σ(t) i i+1
We first consider this problem in each interval. Assume σ(t) = p for t ∈ [t ,t ), there exists a unitary
i i+1
matrix U such that Λ ,UTH U =diag{λp,...,λp }. Let ηˆ=(UT ⊗I )η¯, then,
p p p p p 1 N p N
ηˆ˙ =(I ⊗S +Λ ⊗L F )ηˆ
N 0 p 0 0
that is, ηˆ˙ = (S +λpL F )ηˆ for i = 1,...,N, where λp > 0 for i = 1,...,N are the eigenvalues of H
i 0 i 0 0 i i p
(p∈P). Since (S , F ) aredetectable, there exists [28] a positive definite symmetric matrix P satisfying
0 0
PS +STP −2FTF <0. (8)
0 0 0 0
Note that the minimum eigenvalue of H for all p is well-defined. Denoting it as λ¯ > 0 and taking
p
L =−µ∗P−1FT with µ∗ ,max{1,1} gives
0 0 λ¯
(S +λpL F )TP +P(S +λpL F )
0 i 0 0 0 i 0 0
=STP +PS −2µ∗λpFTF
0 0 i 0 0
=STP +PS −2FTF −2(µ∗λp−1)FTF
0 0 0 0 i 0 0
≤STP +PS −2FTF
0 0 0 0
Since STP +PS −2FTF is negative definite, under Assumption 1, there exists a positive constant c,
0 0 0 0
such that
(S +λpL F )TP +P(S +λpL F )≤−cP (9)
0 i 0 0 0 i 0 0
5
By letting V = N ηˆTPηˆ, we can derive V˙ ≤ −cV . Recalling the dwell-time assumption this
η i=1 i i η η
inequality holds for all t. Note that η¯Tη¯=ηˆTηˆ, it follows
P
||η −v||2 ≤ηˆTηˆ≤λ (P)−1V (t)<λ (P)−1V (0)e−ct
i min η min η
The conclusion is readily obtained.
Remark 3 Although η = r appears in (6), y = F η will suffice this design. When y = r (i.e.,
0 0 0 0 0
F = I ), it means that the state of the leader can be directly obtained when some agent is connected
0 p
to it. This circumstance has been partly considered in [16]. We extend these results using only output
measurements of the leader to deal with the cases when only partial states are available. Similar control
laws were proposed in [6] when the followers are all integrators, while here we consider general linear
agents and also local disturbances.
After building distributed observers for those followers, it is natural to replace r by its estimation
η . The following lemma guarantees the validity of this substitution and shows how it transforms the
i
coordination problem of these multi-agent systems into N decentralized estimation and regulation sub-
tasks.
Lemma 3 Under Assumptions 1 and 2, u = K x +K d +K η , η˙ = S η +L F η will solve
i i1 i i2 i i3 i i 0 i 0 0 vi
the output regulation problem of system (5), and hence the leader-following coordination problem of this
multi-agent system with the selected L , where K , K , K are matrices defined in the centralized case.
0 i1 i2 i3
Proof. Under Assumption 2, letting x¯ =x −X d −X r gives
i i i1 i i2
x¯˙ =(A +B K )x¯ +B K η¯
i i i i1 i i i3 i
η¯˙ =S η¯ +L F η (10)
i 0 i 0 0 vi
e =(C +D K )x¯ +D K η¯
i i i i1 i i i3 i
or in compact form
x¯˙ =A¯x¯+B¯η¯
η¯˙ =(I ⊗S +H ⊗L F )η¯ (11)
N 0 σ(t) 0 0
e=C¯x¯+D¯η¯
where x¯=col(x ,...,x ) and
1 N
A¯=blockdiag{A +B K ,...,A +B K }, B¯ =blockdiag{B K ,...,B K },
1 1 11 N N N1 1 13 N N3
C¯ =blockdiag{C +D K ,...,C +D K }, D¯ =blockdiag{D K ,...,D K }.
1 1 11 N N N1 1 13 N N3
Since A¯ is Hurwitz, by Lemmas 1 and 2, e =y −y will converge to zero as t→∞.
i i 0
Next, we aim to solve those decentralized reference tracking and disturbance rejection problems. As
havingbeenpointedoutbeforethatthedisturbancesareoftenunmeasurable,thecontrollawinLemma3
isnotimplementable. Totacklethisproblem,weemploythedisturbanceobserverbased(DOB)approach
which has been well-studied in its centralized version by many authors [18, 20].
Basically, we seek to reconstruct an estimation of the disturbances that affect the tracking perfor-
mance,andthenuse these estimations toachievedisturbance rejection. Hence, the followingassumption
comes naturally.
A E
Assumption 3 For each i=1,...,N, the pair [C , 0l×qi], i i is detectable.
i
"0ni Si#!
6
We first consider the output feedback cases, where only y is available through measurement, and
i
propose a composite control law in the following form.
ξ˙ =A ξ +E ζ +B u −L (y −yˆ)
i i i i i i i i1 i i
ζ˙ =S ζ −L (y −yˆ)
i i i i2 i i
η˙ =S η +L F η
i 0 i 0 0 vi
u =K ξ +K ζ +K η , i=1,...,N (12)
i i1 i i2 i i3 i
where yˆ =C ξ +D u , i=1,...,N, and L , L are constant matrices such that
i i i i i i1 i2
A +L C E
A , i i1 i i
ci
" Li2Ci Si#
is Hurwitz.
It is time to give our first main theorem.
Theorem 1 UnderAssumptions1-3,theleader-following coordination problemofthemulti-agentsystem
composed of (1), (2), and (3) can be solved by the control law (12).
Proof. Let xˆ = ξ −x , dˆ = ζ −d , and x¯ = x −X d −X r. Under the control law (12), the
i i i i i i i i i1 i i2
closed-loop system of agent i is of the form:
x¯˙ =(A +B K )x¯ +B K xˆ +B K dˆ +B¯ K η¯
i i i i1 i i i1 i i i2 i i i3 i
xˆ˙ =(A +L C )xˆ +E dˆ
i i i1 i i i i
dˆ˙ =L C x¯ +S dˆ (13)
i i2 i i i i
η¯˙ =S η¯ +L F η
i 0 i 0 0 vi
e =(C +D K )x¯ +D K xˆ +D K dˆ +D K η¯
i i i i1 i i i1 i i i2 i i i3 i
Letx¯=col(x ,...,x ),v¯=col{xˆ , ..., xˆ , dˆ, ..., dˆ , η¯ , ..., η¯ },andthewholemulti-agentsystem
1 N 1 N 1 N 1 N
can be put into a compact form as
x¯˙ =A¯x¯+B¯v¯
v¯˙ =S¯v¯ (14)
e=C¯x¯+F¯v¯
where A¯ = blockdiag{A +B K ,...,A +B K },S¯ = blockdiag{A ,...,A ,I ⊗S +H ⊗
1 1 11 N N N1 c1 cN N σ(t)
L F },B¯ = [B¯ ,B¯ ,B¯ ],C¯ = blockdiag{C +D K , ..., C +D K },F¯ = [F¯ , F¯ , F¯ ] and B¯ =
0 0 1 2 3 1 1 11 N N N1 1 2 3 k
blockdiag{B K , ..., B K }, F¯ = blockdiag{D K , ..., D K } (k = 1,2,3). Since A¯ and A
1 1k N Nk k 1 1k N Nk ci
are Hurwitz, by Lemmas 1 and 2, e = y − y will converge to zero as t → ∞. The proof is thus
i i 0
completed.
Remark 4 When E =0 for all agents, it reduces to the well-studied leader-following consensus problem
i
considering only reference tracking problem. Then, the relevant results in [1] and [2] are actually special
cases of this theorem for integrators. Even when the leader has a general linear dynamics as in [23], we
consider both reference tracking and local disturbance rejection problems under switching topologies.
Remark 5 When N =1, this problem becomes a centralized reference tracking and disturbance rejection
problem. While most of existing DOB results focus on rejecting those disturbances [19, 20], our design
also incorporates the reference tracking aspects. Those DOB controllers can admit not only bounded
disturbances(e.g.,constantsandsinusoidalsignals)butalsounboundeddisturbances(e.g.,rampingsignals
and polynomials) under switching topologies.
7
In many circumstances, the state x may be available for us by direct measurement or other treat-
i
ments. Thus, there exists a certain degree of redundancy in the controller (12), which still produces the
estimations of x . To remove such redundancies and save our controller’s order, we propose a reduced-
i
order DOB control law to facilitate our design.
For this multi-agent system, the reduce-order disturbance observer is given as follows.
ζ˙ =(S +L E )ζ +(L A −S L −L E L )x +L B u
i i i i i i i i i i i i i i i i
η˙ =S η +L F η
i 0 i 0 0 vi
u =K x +K (ζ −L x )+K η , i=1,...,N (15)
i i1 i i2 i i i i3 i
where K , i=1,...,N, j =1,2,3, and L are gain matrices to be determined later.
ij i
With this reduced-order controller, the following theorem can be derived.
Theorem 2 UnderAssumptions1–3,thereexistconstantmatrices K andL ,i=1,...,N, j =1, 2, 3,
ij i
such that the leader-following coordination problem of this multi-agent system is solved by the control law
(15).
Proof. The proofis similarwith thatofTheorem1. Letdˆ =ζ −L x −d , andx¯ =x −X d −X r.
i i i i i i i i1 i i2
Under the control law (15), the closed-loop system of agent i is with the form of
x¯˙ =(A +B K )x¯ +B K dˆ +B¯ K η¯
i i i i1 i i i2 i i i3 i
dˆ˙ =(S +L E )dˆ
i i i i i
(16)
η¯˙ =S η¯ +L F η
i 0 i 0 0 vi
e =(C +D K )x¯ +D K dˆ +D K η¯
i i i i1 i i i2 i i i3 i
Let x¯ = col(x ,...,x ), v¯ = col{dˆ, ..., dˆ , η¯ , ..., η¯ }. By some mathematical manipulations, the
1 N 1 N 1 N
whole multi-agent system can be put into a compact form as
x¯˙ =A¯x¯+B¯v¯
v¯˙ =S¯v¯ (17)
e=C¯x¯+F¯v¯
where
A¯=blockdiag{A +B K ,...,A +B K }
1 1 11 N N N1
B¯ =blockdiag{B K , ..., B K },
1 13 N N3
C¯ =blockdiag{C +D K , ..., C +D K },
1 1 11 N N N1
F¯ =blockdiag{D K , ..., D K },
1 13 N N3
S¯=blockdiag{S +L E ,...,S +L E ,I ⊗S+H ⊗L F }.
1 1 1 N N N N σ(t) 0 0
According to Lemma 1 and 2,we only have to find proper K and L such that A¯ and S +L E are
ij i i i i
all Hurwitz. Then, by similar arguments as that in Theorem 1, e = y −y will converge to zero as
i i 0
t→∞. Infact,suchgainmatricesindeedexist. TakeK asdefinedinTheorem1,andthe detectability
ij
of (E , S ) will suffice the selection of L , which is obvious by PBH-test under Assumption 3. Thus the
i i i
conclusion follows readily.
Remark 6 As having been pointed before, unlike in existing cooperative output regulation result [16], the
disturbances are locally modeled by different autonomous systems from that of the global reference. A
8
0 1 2 3 0 1 2 3
(a) ThegraphG1 (b) ThegraphG2
Figure 1: The communication graphs.
similar setup has been used in [29] and [30]. This separate modeling method results in two dissimilar
treatments, distributed observers for the global reference and DOB approach for the local disturbances,
which may enhance the effectiveness of our design and bring a better performance. Also, even for the
case when N =1, we proposed different reduced-order disturbance observers from that in [20] to solve this
problem without using the derivative of the plant’s states.
4 Simulations
Asanexample,weconsiderthecoordinationproblemforamulti-agentsystemconsistingofthreefollowers
and one leader. The follower agents are the mass-damper-spring systems with unit mass described by:
y¨ +g y˙ +f y =u +E d , i=1,2,3
i i i i i i i i
where d is the local disturbance. Those disturbances are modeled by S = [0,1;0,0],E = [1,0];
i 1 1
S = 0,E = 1; S = [0,1;−1,0], E = [1,0]. The leader is specified by a harmonic oscillator: r˙ =
2 2 3 3 1
r , r˙ =−r , y =r . WeassumeheretheinterconnectiontopologyisswitchingbetweengraphG andG
2 2 1 0 1 1 2
describedbyFig.1. Theswitchingsareperiodicallycarriedoutinthefollowingorder{G ,G ,G ,G ,···}
1 2 1 2
with switching period t=5.
Letting x =y , x =y˙ , then,
i1 i i2 i
x˙ =x , x˙ =−f x −g x +u +E d , y =x .
i1 i2 i2 i i1 i i2 i i i i i1
Apparently, the coordination laws in [6] and [23] will not work for these agents. Even the discontinuous
rule ([13]) fails to solve this problem because of unbounded disturbances in Agent 1. Nevertheless, as
Assumptions 1-3 are satisfied, it can be solved by the methods given in last sections.
For simulations, the system parameters are taken as f =1,g =1,f =0,g =1 and f =1,g =0.
1 1 2 2 3 3
By solving the regulator equations in Assumption 2 and also the Lyapunov inequality (8), we choose
propergainmatricesforthe controllersasshowedinTable1. While theinitialsfortheplantisgenerated
between[−1, 1]2,theinitialsforcontrollersaresetattheirorigins. Thesimulationresultsusingfull-order
and reduced-order disturbance observer based control are showed in Fig. 2 and Fig. 3, respectively. We
list the outputs of agents at serval time points in Table 2. It can be found that after 21s,the agents can
track the leader and reject those disturbances with an error tolerance 2×10−3.
5 Conclusions
A leader-following coordination problem was solved for a class of heterogeneous multi-agent systems
subjecttolocaldisturbancesunderswitchingtopologies. Bydevisingadistributedobserver,thisproblem
wastransformedinto severaldecentralizedestimationandregulationsub-tasks,andeventually solvedby
twodisturbanceobserverbasedcontrollaws. Ourfuture workwillinclude nonlinearcasesandwithmore
general graphs.
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Simulation1 Simulation2
K11 =[0,0] K12 =[−1,0] K13 =[0,1] L0 =[−1.3522;−0.4142]
Agent1
L11 =[−1.9813;−1.4628] L12 =[−3.1445;−1.0000] L1 =[0,−1;0,−1]
K21 =[−1,0] K22 =−1 K23 =[0,1] L0 =[−1.3522;−0.4142]
Agent2
L21 =[−1.7321;−1.0000] L22 =−1.0000 L2 =[0,−1]
K31 =[0,−1] K32 =[−1,0] K33 =[0,1],L0 =[−1.3522;−0.4142]
Agent3
L31 =[−2.1607;−1.8343] L32 =[−0.8860;1.1023] L3 =[0,−1;0,0]
Table 1: Gain matrices for each agent in simulations.
3 3
Referece Agent 1
Agent 1 Agent 2
Agent 2 Agent 3
2 Agent 3 2
Output of all agents-011 Tracking errors of all agents-011
-2 -2
-3 -3
0 5 10 15 20 25 30 0 5 10 15 20 25 30
t(s) t(s)
(a) (b)
Figure 2: Tracking performance with output feedback control law (12).
3 3
Referece Agent 1
Agent 1 Agent 2
Agent 2 Agent 3
2 Agent 3 2
Output of all agents-011 Tracking errors of all agents-011
-2 -2
-3 -3
0 5 10 15 20 25 30 0 5 10 15 20 25 30
t(s) t(s)
(a) (b)
Figure 3: Tracking performance with reduced-order DOB feedback control law (15).
Samplingtime t=6s t=12s t=15s t=18s t=21s t=24s t=27s t=30s
Referenceoutput -0.2794 -0.5366 0.6503 -0.7510 0.8367 -0.9056 0.9564 -0.9880
Agent 1 -0.8507 -0.6008 0.6295 -0.7574 0.8347 -0.9064 0.9562 -0.9882
Simulation1 Agent 2 -0.6961 -0.5803 0.6395 -0.7532 0.8365 -0.9057 0.9565 -0.9881
Agent 3 -0.3552 -0.5100 0.6400 -0.7494 0.8364 -0.9058 0.9565 -0.9882
Agent 1 -0.3054 -0.6109 0.6292 -0.7574 0.8347 -0.9064 0.9562 -0.9882
Simulation2 Agent 2 -0.6476 -0.5807 0.6395 -0.7532 0.8365 -0.9057 0.9565 -0.9881
Agent 3 -0.1382 -0.4240 0.6205 -0.7466 0.8363 -0.9058 0.9566 -0.9882
Table 2: Outputs of each agent in simulations.
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