Edge Agreement of Multi-agent System with Quantized Measurements via the Directed Edge Laplacian

0.32 MB·English

by  Zhiwen Zeng

Edge Agreement of Multi-agent System with
Quantized Measurements via the Directed
Edge Laplacian
6
1 Zhiwen Zenga, Xiangke Wanga, Zhiqiang Zhenga
0
2
n aCollege of Mechatronics and Automation, National University of Defense
a Technology, 410073, China
J

9
2

]
Y
Abstract
S
.
s
c This work explores the edge agreement problem of second-order nonlinear multi-
[
agentsystemunderquantizedmeasurements.Undertheedgeagreementframework,

2 we introduce an important concept about the essential edge Laplacian and also
v
obtain a reduced model of the edge agreement dynamics based on the spanning
8
7 tree subgraph. The quantized edge agreement problem of second-order nonlinear
6 multi-agent systemisstudied,inwhichbothuniformandlogarithmicquantizers are
6
considered.Wedonotonlyguaranteethestabilityoftheproposedquantizedcontrol
0
. law, but also reveal the explicit mathematical connection of the quantized interval
1
and the convergence properties for both uniform and logarithmic quantizers, which
0
5 has not been addressed before. Particularly, for uniform quantizers, we provide the
1
upper bound of the radius of the agreement neighborhood and indicate that the
:
v radiusincreaseswiththequantization interval.Whileforlogarithmicquantizers,the
i
X agents converge exponentially to the desired agreement equilibrium.In addition, we
r figure out the relationship of the quantization interval and the convergence speed
a
and also provide the estimates of the convergence rate. Finally, simulation results
are given to verify the theoretical analysis.
Key words: Edge agreement, edge Laplacian, multi-agent system, quantized
measurements.
⋆
This paper was not presented at any meeting or journal. Corresponding author
Xiangke Wang. Tel. [phone].
Email addresses: [email protected](Zhiwen Zeng),
[email protected] (Xiangke Wang), [email protected] (Zhiqiang Zheng).
Preprint submitted to Elsevier 1 February 2016
1 Introduction
Graph theory contributes significantly in the analysis and synthesis of multi-
agent systems, since it provides natural abstractions for how information are
shared among agents in a network [1,2,3,4]. Pioneering researches on edge
agreement [5,6] not only provide totally new insights that how the spanning
trees and cycles effect the performance of the agreement protocol, but also
set up a novel systematic framework for analysing multi-agent systems from
the edge perspective. In our previous work [7], the concept of edge Laplacian
was extended to more general directed graphs and the classical input-to-state
nonlinear control methods together with the recently developed cyclic-small-
gain theorem were successfully implemented to drive multi-agent system to
reach robust consensus.
Early efforts on multi-agent systems mainly focuses on the study with high
accurate data exchanging among agents. However, it is hard to be guaranteed
in the real digital networks when considering that communication channel
has a limited bandwidth, and energy used for transmission is generally re-
strained. Frankly speaking, constraints on communication have a considerable
impact on the performance of multi-agent system. To cope with the limita-
tions, the measurement data are always processed by quantizers. In practice,
to realize the quantized communication scheme, a encoder-decoder pair is em-
ployed. Generally, the quantized data is always encoded by the sender side
before transmitting and dynamically decoded at the receiver side. Recently,
the gossiping algorithms [8], the coding/decoding schemes [9] and nonsmooth
analysis [10] have been proposed to solve the coordination control problem
of first-order multi-agent systems with quantized information. However, as
known that second-order multi-agent systems have significantly different co-
ordination behaviour even if agents are coupled through similar topology. To
the best of authors’ knowledge, there are still little works explore the quan-
tization effects on second-order dynamics. Considering different quantizers,
[11] studies the synchronization behaviour of mobile agents with second-order
dynamics under an undirected graph topology. The authors also point out
that the quantization effects may cause undesirable oscillating behaviour un-
der directed topology. To more recent literature [12], the authors address the
quantized consensus problem of second-order multi-agent systems via sampled
data under directed topology. Considering the fact that quantization intro-
duces strong nonlinear characteristics such as discontinuity and saturation to
the system, the control law designed for the ideal case may lead to instability.
The research onsecond-order multi-agent systems in the presence of quantized
measurements under directed topology is still open.
While the analysis of the node agreement (consensus problem) has matured,
work related to the edge agreement has not been deeply studied yet. In this
2
paper, we are going to explore the quantization effects on the edge agreement
problem of second-order nonlinear multi-agent systems. The main contribu-
tions are twofold. First, by introducing the essential edge Laplacian, we high-
light the role of the spanning tree subgraph and then we can obtain a reduced
model of second-order edge agreement dynamics across the spanning tree un-
der the edge agreement framework. Second, we propose a general analysis
of the convergence properties for second-order nonlinear multi-agent systems
under quantized measurements. Unlike previous works [11,12], for uniform
quantizers, we provide the explicit upper bound of the radius of the agreement
neighborhoodandalso indicatethat theradiusincreases with thequantization
interval. While for logarithmic quantizers, the agents converge exponentially
to the desired agreement equilibrium. Moreover, we also provide the estimates
of the convergence rate as well as pointing out that the coarser the quantizer
is, the slower the convergence speed.
The rest of the paper is organized as follows: preliminaries are proposed in
Section 2. The quantized edge agreement with second-order nonlinear dynam-
ics under directed graph is studied in Section 3. The simulation results are
provided in Section 4 while the last section draws the conclusions.
2 Basic Notions and Preliminary Results
The null space of matrix A is denoted by (A). Denote by I the iden-
n
tity matrix and by 0 the zero matrix in Rn×Nn. Let 0 be the column vector
n
with all zero entries. Let = ( , ) be a digraph of order N specified by
G V E
a node set and an edge set with size L. The set of neigh-
V E ⊆ V × V
bors of node i is denoted by = j : e = (j,i) . The adjacency matrix
i k
of is defined as A = [a ]N RN{×N with nonn∈egEa}tive adjacency elements
G ij
G ∈
a > 0 (j,i) ε. The degree matrix ∆ = [∆ ] is a diagonal matrix
ij G ij
⇔ ∈
with [∆ ] = N a ,i = 1,2, ,N, and the graph Laplacian of is defined
ii j=1 ij ··· G
by L ( ) = ∆ A . Denote by ( ) the L L diagonal matrix of w ,
G G P G − G W G × k
for k = 1,2 ,L, where w = a represents the weight of e = (j,i) .
k ij k
··· ∈ E
The incidence matrix E( ) RN×L for a directed graph is a 0, 1 -matrix
G ∈ { ± }
with rows and columns indexed by nodes and edges of respectively. For edge
G
e = (j,i) , [E( )] = +1,[E( )] = 1 and[E( )] = 0 ifl = i,j. The
k ∈ E G jk G ik − G lk 6
in-incidence matrix E ( ) RN×L is a 0, 1 matrix and for e = (j,i) ,
⊙ k
G ∈ { − } ∈ E
[E ( )] = 1 for l = i, [E ( )] = 0 otherwise. The weighted in-incidence
⊙ G lk − ⊙ G lk
matrix Ew( ) is defined as Ew( ) = E ( ) ( ). As thus, the graph Lapla-
⊙ ⊙ ⊙
G G G W G
cianof hasthefollowingexpression [7]:L ( ) = Ew( )E( )T.Theweighted
G ⊙
G G G G
edge Laplacian of a directed graph can be defined as [7]
G
L ( ) := E( )TEw( ). (1)
e ⊙
G G G
3
A spanning tree = ( , ) of a directed graph = ( , ) is a directed
GT V E1 G V E
tree formed by graph edges that connect all the nodes of the graph; a co-
spanning tree = ( , ) of is the subgraph of having all the
GC V E −E1 GT G
vertices and exactly those edges of that are not in . Graph is called
G GT G
quasi-strongly connected if and only if it has a directed spanning tree [13]. A
quasi-strongly connected directed graph can be rewritten as a union form:
G
= . In addition, according to certain permutations, the incidence
G GT ∪ GC
matrix E( ) can always be rewritten as E( ) = E ( ) E ( ) as well. Since
G G T G C G
(cid:20) (cid:21)
the co-spanning tree edges can be constructed from the spanning tree edges
via a linear transformation [5], such that
E ( )T( ) = E ( ) (2)
T G G C G
T −1 T
with T( ) = E ( ) E ( ) E ( ) E ( ) and rank(E( )) = N 1 from
G T G T G T G C G G −
[13]. We defin(cid:16)e (cid:17)
R( ) = I T( ) (3)
G G
(cid:20) (cid:21)
and then we have
E( ) = E ( )R( ). (4)
G T G G
The column space of E( )T is known as the cut space of and the null space
G G
of E( ) is called as the flow space of E( ). Additionally, the rows of R( )
G G G
form a basis of the cut space of and the rows of T( )T I form a basis of
− G
(cid:20) (cid:21)
the flow space, respectively [13].
Lemma 1 ([7]) For a quasi-strongly connected graph , the graph Laplacian
G
L ( ) and the edge Laplacian L ( ) have the same N 1 nonzero eigenvalues,
G e
G G −
which are all in the open right-half plane.
Lemma 2 ([7]) For a general quasi-strongly connected graph = ,
G GT ∪ GC
L ( ) contains L N +1 zero eigenvalues. Moreover, if the edge set of is
e G − GC
not empty, then zero is a simple root of the minimal polynomial of L ( ).
e
G
3 Quantized Edge Agreement with Second-order Nonlinear Dy-
namics under Directed Graph
In this section, the edge agreement of second-order nonlinear multi-agent sys-
tems under quantized measurements is studied. To ease the notation, we sim-
ply use E, Ew and L instead of E( ), Ew( ) and L ( ).
⊙ e ⊙ e
G G G
4
3.1 Problem Formulation
We consider a group of N networked agents and the dynamics of the i-th agent
is represented by
x˙ (t) = v (t) (5)
i i
v˙ (t) = f (x (t),v (t),t)+u (t) (6)
i i i i
where x (t) Rn is the position, v (t) Rn is the velocity and u (t) Rn
i i i
is the contro∈l input. The nonlinear term∈f (x (t),v (t),t) : Rn Rn ∈Rn is
i i
× →
unknown and satisfies the following assumption:
Assumption 3 For a nonlinear function f, there exists nonnegative con-
stants ξ and ξ such that
1 2
f (x,v,t) f (y,z,t) ξ x y +ξ v z ,
1 2
| − | ≤ | − | | − |
x,v,y,z Rn; t 0.
∀ ∈ ∀ ≥
The goal for designing distributed control law u (t) is to synchronize velocities
i
and positions of the N networked agents.
The generally studied second-order consensus protocol proposed in [14] is de-
N N
scribed as follows: u (t) = α a (x (t) x (t))+β a (v (t) v (t)),
i ij j i ij j i
j∈Ni − j∈Ni −
for i = 1,2 ,N, where αP> 0 and β > 0 are thePcoupling strengths.
···
As in [15], we assume that each agent i has only quantized measurements
of relative position Q(x x ) and velocity information Q(v v ), where
i j i j
− −
Q(.) : Rn Rn denotes the quantization function. Therefore, the protocol
→
can be modified as
N N
u (t) =α a Q(x (t) x (t))+β a Q(v (t) v (t)) (7)
i ij j i ij j i
− −
jX∈Ni jX∈Ni
for i = 1,2 ,N. In this paper, two typical quantization operators are con-
···
sidered: uniform and logarithmic quantizer. For a given δ > 0, a uniform
u
quantizer q : R R satisfies q (a) a δ , a R; for a given δ > 0,
u u u l
→ | − | ≤ ∀ ∈
a logarithmic quantizer q : R R satisfies q (a) a δ a , a R. The
l l l
→ | − | ≤ | | ∀ ∈
positive constants δ and δ are known as quantization interval. For a vector
u l
ν = [ν ,ν , ,ν ]T Rn, Let Q (ν) =∆ [q (ν ),q (ν ), ,q (ν )]T and
1 2 n u u 1 u 2 u n
··· ⊂ ···
∆ T
Q (ν) = [q (ν ),q (ν ), ,q (ν )] . Then we obtain the following bounds:
l l 1 l 2 l n
···
Q (ν) ν √nδ , Q (ν) ν δ ν .
u u l l
| − | ≤ | − | ≤ | |
5
Considering the dynamics of the networked agents described in (5) and (6),
by directly applying the quantized protocol (7), we obtain
x˙ (t) = v (t)
i i
 N
v˙ (t) = f (x (t),v (t),t)+α a Q(x (t) x (t))
 i Ni i j∈PNi ij j − i
+β a Q(v (t) v (t))
ij j i
 j∈PNi −
To ease the difficulty of the analysis, we technically chose α = σ2 and β = σ3
(σ > 0) as in [16]. The biggest advantage to using this trick is that we can
easily construct a positive definite matrix which will be used in the proof of
the main results. As thus, the system can be collected as
x˙ (t) = v(t)
v˙ (t) = F (x(t),v(t),t) σ2(Ew I )Qˆ (ET I )x(t) (8)
⊙ n n
 σ3(Ew I )Qˆ (−ET I )v⊗(t) (cid:16) ⊗ (cid:17)
⊙ n n
− ⊗ ⊗
 (cid:16) (cid:17)
with x(t), v(t) and F(x(t),v(t),t) denoting the column stack vector of x (t),
i
ˆ
v (t)andf (x (t),v (t),t);andQrepresents thevector formofthequantization
i i i
function Q.
Define x = (ET I )x and v = (ET I )v, which denote the difference
e n e n
⊗ ⊗
of position and velocity of two neighbouring nodes respectively. We suppose
e = Qˆ(x ) x and e = Qˆ(v ) v as in [15]. Then by left-multiplying
xe e − e ve e − e
ET I of both sides of (8), we have
n
⊗
x˙ (t) = v (t)
e e
v˙ (t) = (ET I )F σ2(L I )x σ3(L I )v (9)
e n e n e e n e
 σ2(L⊗ I )−e σ3(⊗L I −)e . ⊗
− e ⊗ n xe − e ⊗ n ve

The edge agreement dynamics (9) describes the evolution of the edge states
T
z = xT vT , which depends on its current state and its adjacent edges’
e e
(cid:20) (cid:21)
states. In comparison to the node agreement (consensus), the edge agreement,
rather than requiring the convergence to the agreement subspace, expects
the edge dynamics (9) to converge to the origin, i.e., lim x (t) = 0 and
t→∞ e
| |
lim v (t) = 0.
t→∞ e
| |
6
3.2 Main Results
For the quasi-strongly connected graph , the incidence matrix can be written
T G
as E = E E . Let z = xT vT denotes the states across the spanning
T C T T T
(cid:20) (cid:21) (cid:20) (cid:21)
T
tree with x = (ET I )x,v = (ET I )v andz = xT vT denotes the
GT T T ⊗ n T T ⊗ n C C C
(cid:20) (cid:21)
states across the cos-spanning tree with x = (ET I )x, v = (ET I )v,
GC C C ⊗ n C C ⊗ n
respectively. Notice that E T( ) = E as mentioned in (2); therefore the
T G C
co-spanning tree states can be reconstructed through matrix T, i.e., x (t) =
C
(T( )T I )x (t) and v (t) = (T( )T I )v (t). Moreover, based on the ob-
G ⊗ n T C G ⊗ n T
servation that E = E R( ) form (4), then we can obtain x = RT( ) I x ,
T G e G ⊗ n T
v = RT( ) I v and
e G ⊗ n T
RT( ) I 0¯
n
z = G ⊗ z (10)
  T
0¯ RT( ) I
n
 G ⊗ 
 
with zero matrix 0¯ of compatible dimension.
To simplify the subsequent analysis, the essential edge Laplacian will be em-
ployed,whichhelpsustoobtainareducedmodeloftheclosed-loopmulti-agent
system based on the spanning tree subgraph .
GT
Before moving on, we introduce the following transformation matrix:
T −1
R( )R( ) R( )
S ( ) = R( )T θ ( ) S ( )−1 = G G G
e G (cid:20) G e G (cid:21) e G (cid:16) θe( )T(cid:17) 
 G 
 
where R( ) is defined via (3) and θ ( ) denotes the orthonormal basis of the
e
G G
flow space, i.e., Eθ ( ) = 0. Since rank(E) = N 1, one can obtain that
e
G T −
dim(θ ( )) = (E) and θ ( ) θ ( ) = I . Applying the above similar
e e e L−N+1
G N G G
transformation lead to
Lˆ ETEwθ ( )
S ( )−1L S ( ) = e T ⊙ e G (11)
e e e  
G G 0¯ 0¯
 
 
where Lˆ = ETEwR( )T is referred to as the essential edge Laplacian. For the
e T ⊙ G
essential edge Laplacian, we have the following lemma.
Lemma 4 The essential edge Laplacian Lˆ contains exactly N 1 nonzero
e
−
eigenvalues of L .
e
PROOF. Based on the similar transformation S ( ) and S ( )−1, the eigen-
e e
G G
7
values of the block matrix (11) are the solution of
λ(L−N+1)det λI Lˆ = 0
e
−
(cid:16) (cid:17)
which shows that Lˆ contains exactly all the nonzero eigenvalues of L from
e e
Lemma 1 and 2.
Meanwhile, we can construct the following Lyapunov equation as
HLˆ +LˆTH = I (12)
e e − N−1
where H is positive definite.
Next, we will provide a reduced multi-agent system model in terms of the
corresponding dynamics across . For edge dynamics (9), we make use of the
GT
following transformation
x v
(S−1 I )x = T (S−1 I )v = T .
e ⊗ n e   e ⊗ n e  
0 0
   
   
ET
Since E = E R( ) and L = ETEw, we have S−1ET = T and S−1L =
T G e ⊙ e   e e
0
 
 
ETEw T
T ⊙ . Then we define ω = e T e T and let Lˆ = ETEw. By using
  xe ve O T ⊙
0 (cid:20) (cid:21)
 
thesimilartransformation(11),system (9)finallycanberecast intoacompact
matrix form as follows:
z˙ = +( I )z +( I )ω (13)
T FT LT ⊗ n T LT1 ⊗ n
0 I 0 0 0
N−1 N−1 N−1×L N−1×L
with = , = and = .
LT  σ2Lˆ σ3Lˆ  LT1  σ2Lˆ σ3Lˆ  FT (ET I )F
− e − e − O − O  T ⊗ n 
     
Remark 5 The decomposition of the spanning tree and co-spanning tree sub-
graph has been wildly applied to solve many magnetostatic problems, such as
tree-cotree gauging [17], finite element analysis [18]. As is well known, the
spanning tree plays a vital role in the stability analysis of networked multi-
agent system. Under the edge agreement framework, we reveal the connection
of the algebraic properties and the graph structure and highlight the role of the
spanning tree subgraph by introducing the essential edge Laplacian.
8
To further look at the relation between the quantization interval and the edge
agreement, we propose the following theorem.
Theorem 6 Considering the quasi-strongly connected directed graph as-
G
sociated with the edge Laplacian L , suppose = + T with
e Q − PLT LTP
(cid:16) (cid:17)
σH H
= , where H is obtained by (12). If σ > λmax(H) +1 and
P   2
H σH
q
 
λ ( ) 2max(ξ ,ξ ) > 0. Then, under the quantized protocol (7),
min 1 2
Q − kPk
system (13) has the following convergence properties:
(1): With uniform quantizers, the agents converge to a ball of radius
2√2nLδ
z ukPLT1k (14)
T ≤ λ ( ) 2max(ξ ,ξ )
min 1 2
(cid:12) (cid:12) Q − kPk
(cid:12) (cid:12)
(cid:12) (cid:12)
which is centred at the agreement equilibrium;
(2): With logarithmic quantizers, the agents converge exponentially to the de-
sired agreement equilibrium, provided that δ satisfies
l
λ ( ) 2max(ξ ,ξ )
min 1 2
δ < Q − kPk. (15)
l
2 RT
kPLT1kk k
The estimated trajectories of the edge Laplacian dynamics (13) is as
λmax( ) − π t
z (t) P e λmax(P) z (0) for t 0
| T | ≤ λ ( ) | T | ≥
min
P
with
π = λ ( ) 2max(ξ ,ξ ) 2δ RT .
min Q − 1 2 kPk− lkPLT1k
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
PROOF. For the edge Laplacian dynamics (13), we choose the following
Lyapunov function candidate:
V (z ) = zT( I )z (16)
T T P ⊗ n T
σH H
inwhich = ,whereH canbeobtainedfrom(12)and ispositive
 
P P
H σH
 
definite whilechoosingσ > 1.
9
By taking the derivative of (16) along the trajectories of (13), we have
V˙ (z ) =zT I + T I z +2zT( I )
T T PLT ⊗ n LTP ⊗ n T T P ⊗ n FT
+2(cid:16)zT( I )ω (cid:17)
T PLT1 ⊗ n
= zT( I )z +2zT( I ) +2zT( I )ω
− T Q⊗ n T T P ⊗ n FT T PLT1 ⊗ n
in which
σ2I σ3I σH
= + T = N−1 N−1 − .
Q − PLT LTP σ3I σH σ4I 2H
(cid:16) (cid:17) N−1 N−1
 − − 
 
1 2
Let = Q Q with = σ2I , = σ3I σH and =
  1 N−1 2 N−1 3
Q T Q Q − Q
Q2 Q3
σ4I 2H . According to Schur complements theorem [14], by selecting
N−1
−
λ (H)
max
σ > +1
s 2
then we have > 0 and T −1 = H(2(σ2 1)I H) > 0, so
Q1 Q3−Q2Q1 Q2 − N−1 −
that is positive definite.
Q
In the meanwhile, we notice that
= (ET I ) max(ξ ,ξ ) z . (17)
|FT| T ⊗ n F ≤ 1 2 | T|
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
For uniform quantizers, we can calculate the upper bound of the quantization
error as
ω √2nLδ . (18)
u
| | ≤
By combining (17) and (18), one can obtain
V˙ (z ) λ ( ) z 2 +2max(ξ ,ξ ) z 2
T ≤ − min Q T 1 2 kPk| T|
+2√2nLδuk(cid:12)(cid:12)(cid:12)PL(cid:12)(cid:12)(cid:12)T1k|zT|
= z ( λ ( )+2max(ξ ,ξ ) ) z
min 1 2
T − Q kPk T
(cid:12) (cid:12)(cid:18) (cid:12) (cid:12)
+2(cid:12)(cid:12)√2(cid:12)(cid:12)nLδukPLT1k . (cid:12)(cid:12) (cid:12)(cid:12)
(cid:19)
Clearly, the edge agreement can be reached and the radius of the agreement
neighbourhood is as (14).
For logarithmic quantizers, according to Q (a) a δ a and equation (10)
l l
| − | ≤ | |
10