Scientific Journal of Control Engineering April 2014, Volume 4, Issue 2, PP.34-42
Output Consensus of Multi-agent Systems with Fixed or Switching Unbalanced Topology Shuyu Bao#, Chen Wei, Yan Ding School of Automatic Science and Electrical Engineering, Beihang University, Beijing 100191, China #
Email: bsyvchj3000@sina.com
Abstracts In this paper, the output consensus problems for multi-agent systems with unbalanced topology are discussed. We proposes weighted output consensus protocols and proves the asymptotic consensus of multi-agent systems with unbalanced topology is reachable using Lyapunov functions with integral type. Finally, simulations are provided to demonstrate our consensus protocols are effective for the output value consensus of multi-agent systems. Keywords: Multi-agent System; Unbalanced Topology; Output Consensus; Lyapunov Method
1 Introduction Consensus problems have a long history in the distributed cooperative control of multi-agent systems. In recent years, more and more researchers both at home and aboard pay attention to the consensus problems. This is mainly due to the broad applications of multi-agent system theory in many areas, such as formation control [5] and path following control [10]. Vicsek presented a non-balance discrete multi-agent system model for the first time from the point of view of statistics [1]. Then, based on matrix analysis method, Jadbabaie analysed Vicsek’s model and drew a conclusion that as long as the topological graph is connected, the status of all the nodes in the graph achieve consensus [2]. Olfati-Saber and Murray introduced the consensus protocols for balanced topology with and without time-delays and provided a convergence analysis with disagreement functions [6]. Furthermore, the consensus protocols for switching topology and nonlinear topology were given in [3]. Based on Lyapunov direct method, Liu et al presented and rigorously proved some sufficient conditions of nonlinear protocols guaranteeing asymptotical or exponential consensus for systems with unbalanced topology [8]. Wang introduced the notion of a “knowledge� leader and proposed a nonlinear consensus protocol for leader-follower multi-agent systems [7]. The researches above are all confined to the consensus of the states. However, in fact, output consensus is with greater universal significance. Consider a group of agents with complex dynamics or high order models, sometimes we just care about some specific parts of the system or some specific output functions. Besides, in some cases, the states of the agents may be unobservable, instead we can observe some output variables of the system. Up to now, there have been some research achievements on the output consensus. Wang et al transformed the output consensus problem with uncertainties to the classic adaptive tracking control problem and used some established control methods such as backstepping to solve the adaptive output consensus tracking with mismatched uncertainties [11]. In order to solve the output consensus problem for a class of heterogeneous uncertain linear multi-agent systems, Kim et al embedded an internal model, which plays a role of command generator, and the command is determined as time goes since it is the outcome of on-line consensus by communicating the outputs of the agents[12]. Yang and Liu discussed the output consensus problems of linear multi-agent systems with fixed topologies and agents described by homogeneous or heterogeneous linear systems and provided the conditions of output consensus with respect to a set of admissible consensus protocols [13]. Li studied nonlinear output consensus protocols for multi-agent systems with fixed and balanced topology and proves the asymptotic consensus is reachable [9]. In this paper, we will consider the output consensus problems of networks with unbalanced topology, where network means a weighted directed graph. Weighted output consensus protocols are proposed and convergence analysis - 34 http://www.sj-ce.org
based on mirror graph theory and Lyapunov function with integral form is provided. An outline of this paper is as follows. In Section 2, we review some algebraic graph definitions and come up with the main problems of this paper. In Section 3 and 4, we mainly discuss the output consensus problems of multi-agent systems with fixed and switching topology. Simulation results are given in Section 5. In Section 6, this paper is finally concluded.
2 Preliminaries and problem formulation Let G(V , , A) be a directed weighted graph of order n with the set of nodes is V {v1 , v2 ...vn } and the set of edges V V . Use e (vi , v j ) to denote an edge of graph G, where e 0 . When e 0 it means that there exist some information transmits from agent j to agent i, otherwise it means that there is no information transmutation between agent j and agent i. Sometimes we need to give weights to the edges of a graph. Assume that w is the weight of edge e , then there exist a matrix A [a ] R nn whose elements can be expressed as ij
ij
ij
ij
ij
ij
wij , eij E (G );
aij
0, other .
Then A is called the weighted adjacent matrix. In most situations, the weighted adjacent matrix can be used to describe the characters of a graph. Besides, another matrix which can be used to describe the characters of a graph is the Laplacian matrix L [l ] R , and the elements of L can be expressed as n n
ij
lij =-aij , i j; n l = aij . ii j =1
Besides, the eigenvalues of a Laplacian matrix also can be used to reflect the system’s convergence rate. For a connected undirected graph G ' , the following well-known property holds xT Lx
min
x 0, 1 x =0 T
x
2
=2 (L)
where x R n1 , 2 (L) is the second-smallest eigenvalue of matrix L, which is also called the algebraic connectivity of matrix L. Olfati-Saber extend the definition of algebraic connectivity to directed graphs using the concept of mirror graph[6]. Consider a network of continuous-time integrator agents with dynamics xi t =ui t ,i ,j Ni , yi h( xi )
(2.1)
with the initial condition x (s) x (0), s (-,0] ， where N {v j V ; (vi , v j ) } and ui represents the control protocol. Assume that, for h(x) , the following conditions hold. i
i
i
Assumption 1. 1. is continuous and monotone increasing; 2.; 3.There exist a positive constant l，for any , such that h x1 -h x2 x1 -x2
l.
The definition of output consensus is given as follow. Definition 1. If there exist ui t such that for any initial value xi (0) limt + h x j (t) -h xi (t) =0, i, j 1, 2,3,..., n
then we say system (2.1) achieves output consensus. - 35 http://www.sj-ce.org
(2.2)
The specific problem to be addressed in this paper is to solve the output consensus problems of multi-agent systems with fixed and switching unbalanced topology.
3 Output consensus of systems with fixed topology In this section, we will focus on the output consensus problems of multi-agent systems with fixed topology. In order to make the output value of all the agents in the system (2.1) achieve consensus, design the following consensus protocol
ui (t)= aij h x j (t) -h xi (t) j Ni
(3.1)
Given protocol (3.1), the closed-loop system can be written as xi (t)= aij h x j (t) -h xi (t) .
(3.2)
j N i
From the preliminary analysis of (3.2), it is observed that as the decrease of h x j (t) h xi (t) , the change rate of xi (t) progressively grow smaller, when h x j (t) =h xi (t) , xi (t) will no longer change. When the output values of all the nodes converge to a same value, all the nodes of the system achieve stability. We have the following theorem for output consensus under fixed topology. Theorem 1. Consider system (2.1) whose output function h(x) satisfies assumption 1. Suppose its topology graph G(V , , A) is fixed and has a directed spanning tree, then given feedback protocol (3.1), system (2.1) can achieve output consensus. Proof: Assume that H ( X (t )) (h( x1 (t )),..., h( xn (t )))T , then (3.2) can be rewritten as X (t) LH ( X (t )) .
(3.3)
Assume the left eigenvector of matrix L corresponding to eigenvalue 0 is ˆ=(ˆ ,..., ˆ ) , where ˆ >0 and T
1
n
n i =1
ˆi =1 .
Define a reference point as follow x (t)= in=1 ˆi xi (t) , from the properties of Laplacian matrix, we can get ˆ
xˆ (t)= ˆi xi (t)=- ˆi lij h x j (t) =0 . n
n
n
i =1
i =1
i =1
So, xˆ t =xˆ is time-invariant. Construct a Lyapounov function n
xi (t)
i =1
xˆ
V t = ˆi
h(s)-h(x ) ds . ˆ
(3.4)
Assume H X (t ) satisfy H X (t ) =a(t)1+H X (t )
where 1T H ( X (t ))=0 and a(t )
n
1
h( x ) . Denote n
P diag (ˆi ) and Q P ˆ ˆT , it is obvious that
i
i 1
n
V (t ) ˆi xi (t)-xˆ i =1
h(x (t))-h(x ) X (t ) i
T
ˆ
1 1 QH ( X (t )) H ( X (t ))T QH ( X (t ))= H ( X (t ))T QH ( X (t )) l l
max (Qk ) l
T
H ( X (t )) H ( X (t )).
where max Q is the largest eigenvalue of matrix Q. - 36 http://www.sj-ce.org
(3.5)
Let B=PL, it is obvious that B is a balanced Laplacian matrix. We suppose that matrix B is the Laplacian matrix of graph G . Then, assume that the mirror graph of G is Gˆ , whose Laplacian matrix is Bˆ PL L P 2 . From T
B
B
B
Theorem 7 in [6], we can get that Bˆ [bˆij ]nn is symmetrical and satisfies
n j 1
bˆij 0 .
The eigenvalues of Bˆ satisfy
that 0 Bˆ Bˆ ... Bˆ , where Bˆ is the algebraic connectivity of graph Gˆ B . Take the derivative of 1
2
n
2
V (t ) , one can get
V t H X (t ) H X ˆ
T T ˆ X (t ) PLH X (t ) H X (t ) BH X (t ) H X (t ) BH
T
(3.6)
2 H X (t ) H X (t ) T
From (3.5) and (3.6), one can get V t wV t , where w l Bˆ Q . Then, V t O e exponential convergence. 2
max
wt
,
i.e. V t is
4 Output consensus of systems with switching topology In this section, we will focus on the output consensus problems of multi-agent systems with switching topology. Consider a multi-agent system whose topology is not fixed but time-varying. Assume this system has a time-continuous-state x and a discrete-state G that belongs to a finite collection , can be expressed as {G V , , A : rank L(G) n 1, 1T L(G) 0}
(4.1)
Under switching topology, closed-loop system can be rewritten as X t L Gk H X (t ) , k s(t ), GK
where s(t ) : R0 is a switching signal, N is the index collection associated with the elements of . Assume that Ak [akij ]nn is the weighted adjacent matrix of G , then, consensus protocol (3.1) can be expressed as
k
ui (t)= akij h x j (t) -h xi (t) jNi
(4.2)
Then, we have the following theorem for output consensus under switching topology. Theorem 2. Consider system (2.1) whose output function h(x) satisfies assumption 1. Suppose its topological graph G is discrete and belongs to a finite collection .Given a feedback control protocol (4.2), then, system (2.1) can reach output consensus if and only if there exist a directed spanning three in any Gk Γ . Proof: Assume the left eigenvector of L Gk corresponding to eigenvalue 0 is ˆ ˆ ,..., ˆ
k
n i =1
k1
kn
ˆki =1 . Define a reference point xˆk t i 1 ˆki xi (t ) , whose derivative can be expressed as
, where T
ˆk 0 and
n
xˆ t
n
n
n
ˆ x t ˆ l ki i
i 1
i 1
ki
j 1
ij
(Gk )h( x j (t )) 0 .
It is obvious that xk t xk is time-invariant when Gk is active. Then, construct the auxiliary function ˆ
ˆ
x (t ) n Vk t i 1 ˆki h( s ) h( xˆ ) ds, k s(t ). i
xˆ
k
k
Denote Pk diag ˆki and Qk Pk ˆk ˆk T , then it is obvious that n
Vk t ˆki xi (t ) xˆ i 1
1
k
h( x (t )) h( x ) X (t ) Q H ( X (t )) H ( X ) X (t ) Q H ( X (t )) T
ˆk
i
1
T
H ( X (t )) Qk H ( X (t )) H ( X (t )) Qk H ( X (t )) l l T
ˆk
k
T
max (Qk ) l
k
(4.3) T
H ( X (t )) H ( X (t )).
Let B P L G , whose mirror graph’s Laplacian matrix is Bˆ k ( P L G L G
T
k
k
k
k
- 37 http://www.sj-ce.org
k
k
Pk ) 2
, it is obvious that
Bˆ k [bˆkij ]n n
is
symmetrical
and
satisfy
that
n j 1
bˆkij 0 .
The
eigenvalues
of
Bˆ k
satisfy
0 1 ( Bˆk ) 2 ( Bˆk ) ... n ( Bˆk ) . Then
Vk (t ) H ( X (t )) H ( X ˆ ) Pk L Gk H ( X (t )) H ( X (t ))T Bk H ( X (t )) H ( X (t ))T Bˆk H ( X (t ))
(4.4)
2 Bˆk H ( X (t ))T H ( X (t )).
From (4.3) and (4.4), one can get that V (t ) w V (t ) , where w l ( B ) (Q ) . Suppose that w min kN ,1 k n wk , then V (t ) wVk (t ) , in other words, Vk (t ) is exponential convergence. k
k
k
k
2
k
max
k
k
It is a well-known fact that, for a switched system, when all individual subsystems are stable and the switching is sufficiently slow, the whole system is stable. But specifying a dwell time may be too restrictive, and it may be impossible to react to possible system failures during that time interval. So, in general, the average dwell time is enough to guarantee the astringency of the system[4]. From theorem 3.1 in [4], if for the Lyapunov functions of any two subsystems in system (2.1) V (t ) and V (t ) , there exist a constant , such that p
Vp (t ) Vq (t ) .
q
(4.5)
Then, the whole switching system is convergent. Now we suppose that
ˆ h( s) h( x ) ds n
pq (t )
Vp (t ) Vq (t )
xi ( t )
pi
xˆ
i 1
ˆ p
p
h( s ) h( x ) ds n
ˆ
.
(4.6)
xi ( t )
qi
ˆ
xˆ
i 1
q
q
It is obvious that pq (t ) is continuous on R. From L'Hospital's rule, one can get that lim pq (t ) lim
t
where
Vp (t )
t
Vq (t )
Vp (t ) Vq (t )
t
H ( X (t ))T B p H ( X (t )) H ( X (t ))T Bq H ( X (t ))
t
2 Bˆ p H ( X (t )) H ( X (t )) H ( X (t )) Bk H ( X (t )) max Bp H ( X (t )) H ( X (t )) T
T
T
2 Bˆ q H ( X (t )) H ( X (t )) H ( X (t )) Bq H ( X (t )) max Bq H ( X (t )) H ( X (t )) T
T
T
So 2 ( Bˆ p ) max ( Bq )
lim pq (t ) t
max ( Bp ) 2 ( Bq )
.
which shows that pq (t ) has a bounded limit, in other words, pq (t ) is bounded on R. Suppose the upper bound of pq (t ) is pq max , and the lower bound is pq min , then pq min pq (t ) pq max , which means that Vp (t ) pq maxVq (t ) . Assume max p , q pq max , it is obvious that Vp (t ) Vq (t ) .
(4.7)
That is to say the whole switching system is convergent.
5 Simulation Consider a multi-agent system with 4 agents whose topology graph is strongly connected. Fig.1 shows the topological structure of it.
FIG.1. THE TOPOLOGICAL STRUCTURE OF (4.1). - 38 http://www.sj-ce.org
Assume this system has a leader and 3 followers, and the motion equation of the leader (agent 0) is p0 t q0 t 3; q0 t 3 p0 t . T Rewrite the states of the system into vector form as x0 p0 q0 and xi pi equation of the leader and followers are x0 f x0 , t xi f xi , t u i
qi , where i=1,2,3. Then the T
where ui is the cooperation term. Based on theorem 1 in [7], ui is usually constituted by consensus protocol. Then, the whole function of the system is x0 f x0 , t (5.1) x f x ,t Kij h( x j ) h xi Ki 0 h( x0 ) h( xi ) . i i jNi First consider the linear situation. When the output function is 1 0 h1 ( x) x, 0 0 which can theoretically make the leader and follows satisfy that for any i and j, pi p j . The simulation results of system (5.1) with output function h1 can be seen in fig.2.
(a)
(b)
FIG.2.SIMULATION RESULTS UNDER h1 : A SHOWS THE OUTPUT VALUES; B SHOWS THE MOTION TRAILS.
Next consider another linear output function 1 1 h2 ( x) x, 1 1 which can theoretically make the agents system (5.1) satisfy that for any i and j, pi qi p j q j . Fig.3 shows the simulation results of system (5.1) with output function h2 .
(a)
(b)
FIG.3.SIMULATION RESULTS UNDER h2 : A SHOWS THE OUTPUT VALUES; B SHOWS THE MOTION TRAILS. - 39 http://www.sj-ce.org
It can be seen from the two simulations above that the output values kept consistent meanwhile the motion trails kept consistent. It is because of that the dimension of the system is relatively low and the function on p-axis and the function on q-axis are correlative, so when the output function is simple, the result of output consensus protocol equate to the state consensus. Then consider the nonlinear situation, suppose the output function is 1 h3 ( x) xT x . 1
In theory, h3 can let the agents of system (5.1) satisfy that for any i and j, pi2 qi2 p 2j q 2j . With output function h3 , the simulation results of system (5.1) are shown in fig.4.
(a)
(b)
FIG.4.SIMULATION RESULTS UNDER h3 : A SHOWS THE OUTPUT VALUES; B SHOWS THE MOTION TRAILS.
It can be seen from fig.4 that the output values kept consistent but the motion trails are not completely consistent. That is because, within a certain range of error, the followers’ output value will always choose the shortest way to catch up with the leader. But in output consensus, we just need to consider the output value, without the consideration of the motion trails, so there are some deviations in motion trails of the agents is a normal phenomenon. The previous simulations are all under fixed topology, now consider the switching situation. Fig.5 shows 4 different networks, whose topology structures are all strongly connected. A finite automaton is shown with the set of states {G , G , G , G } , representing the discrete-states of a network with switching topology as a hybrid system. a
b
c
d
(a)
(b)
(c)
(d)
FIG.5. FOUR STRONGLY CONNECTED GRAPHS:(a) Ga , (b) Gb , (c) Gc , (d) Gd .
Fig.6 and Fig.7 shows the simulation results of system (5.1) with output functions h1 , h2 and h3 under the topology shown in Fig.5. - 40 http://www.sj-ce.org
(a)
(b)
FIG.6. SIMULATION RESULTS UNDER LINEAR OUTPUT FUNCTIONS: A UNDER h1 ; B UNDER h2 .
FIG.7. THE OUTPUT VALUES UNDER NONLINEAR OUTPUT FUNCTION h3
From fig.6 and Fig.7, we can see that the output values of the agents under switching topology are very similar to the situation under fixed topology. These simulation results also demonstrate the availability of output consensus protocol (4.2) under switching topology.
6 Conclusion In this paper, output consensus problems were considered for multi-agent systems with unbalanced topology. First, weighted output consensus protocols were proposed to extend the applicable range of the protocols from 0-1 topology to all kinds of networks. Then, based on Lyapunov method, mirror graph theory and switching system theory, output consensus was rigorously proved for the proposed consensus protocol. Finally, simulations are proved to demonstrate the effectiveness of this consensus protocol. The study of output consensus problems in this paper is under the ideal situation without the consideration of time-delay and noise, so future research is needed.
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Transactions Automatic Control, 2004, 49(9):1520-1533 [7] W. Wang, E. S. Jean-Jacques. A theoretical study of different leader roles networks [J] .IEEE Transactions on Control Systems Technology, 2006, 51(7): 1156-1161 [8] X. Liu, T. Chen, W. Lu. Consensus problem in directed networks of multi-agents via nonlinear protocols[J].Physics Letters A, 2009, 373:3122-3127 [9] Y. M. Li, X.P. Guan. Nonlinear consensus protocols for multi-agent systems based on centre manifold reduction[J]. Chinese Physics B, 2009, 18(8):3355-3366 [10] Ni W, Cheng D. Leader-following consensus of multi-agent systems under fixed and switching topologies. Systems and Control Letters 2010, 59(3-4):209-217 [11] Z. Wang, W. L. Zhang, Y. Guo. Adaptive Output Consensus Tracking of Uncertain Multi-agent Systems[R]. American Control Conference, San Francisco, USA: 2011, 3387-3392 [12] H. Kim, H. Sim, J. H. Seo. Output Consen- sus of Heterogeneous Uncertain Linear Multi- Agent Systems[J]. IEEE Transactions Automatic Control, 2011, 56 (1):200-206 [13] X. R. Yang, G. P. Liu. Output Consensus of Linear Multi-agent Systems[R].UKACC International Conference on Control, Cardiff, UK: 2012, 1094-1099
AUTHORS 1
Shuyu Bao was born in Shenyang,
University of Science and Technology. She is currently a
Liaoning province, China in April 12,
associate professor at the School of Automation Science and
1988. He got his B.S. in automatic control
Electrical Engineering, Beihang University. Her research
engineering from Beihang University in
interests include control theory and control engineering,
2011.
navigation and guidance, nonlinear control, fuzzy logic control
He is currently a master students in control theory and control engineering at the
and time-delay systems. 3
Yan Ding was born in Huaian, Jiangsu province, China in July
School of Automation Science and Electrical Engineering,
9, 1988. He got his B.S. in information science and technology
Beihang University. His research interests include applied
from Beijing University of Chemical Technology in 2011.
nonlinear control, cooperative control and the consensus theory of Multi-agent systems.
He is currently a master students in control theory and control engineering at the School of Automation Science and Electrical
2
Chen Wei was born in Shandong province, China on August,
Engineering, Beihang University. His research interests include
1971. She got her Ph.D. degrees in control theory and control
theory of systems with time-delay, the consensus theory of
engineering from Institute of Systems Science, AMSS in 1997.
Multi-agent systems and the formation control of multiple
From 1998 to 1999, she was a post-doctoral in Hong Kong
UAVs.
- 42 http://www.sj-ce.org