Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 2 (September 2014), 125-129

Adaptive Consensus Control of Multi-Agent Systems withLarge Uncertainty and Time Delays *Li Liu

School of Mechanical Engineering
University of Science and Technology Beijing, Beijing 100083, China
Yu Meng**

School of Mechanical Engineering
University of Science and Technology Beijing, Beijing 100083, China
Dongxu Zou

School of Automation and Electrical Engineering
University of Science and Technology Beijing, Beijing 100083, China
Weicun Zhang

School of Automation and Electrical Engineering
University of Science and Technology Beijing, Beijing 100083, China**Corresponding author. E-mail: myu@ustb.edu.cn

Abstract
A weighted multi-model adaptive control (WMMAC) method is proposed to achieve consensus of multi-agent
system with large parameter uncertainty and communication delays, in which H ∞ control is adopted as local
control strategy to deal with smaller parameter uncertainty. Moreover a simple and effective weighting algorithm isadopted to assign weights for each local controller, based on which the global control law is obtained as a weighted
sum of all the local control at each time instant. The simulation results demonstrate the effectiveness of theproposed method.
Keywords: Adaptive consensus; Multi-agent system; WMMAC; H ∞ control.
*This work was supported by Major State Basic Research Development Program (973 Program) (No. 2012CB821200), National High-Tech Researchand Development Program of China (863 Program) (No. 2011AA060408), Beijing Higher Education Young Elite Teacher Project (No. YETP0362),
National Key Technology Research and Development Program of China (No. 2013BAB02B07)

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1251. Introduction
During the past decades, the study of decentralizedcontrol has been focused on multi-agent systems, suchas flocking or swarming behaviors [1-3], multi-robot
formation control [4,5], and path planning [6] and so on.
The consensus problem, the most important andfundamental issue in the cooperation control of multiagent system, is of theoretical value and practicalsignificance. For a multi-agent system, consensus meansthat the states of all agents tend to be identical
asymptotically under given protocols (control law)based on the communication networks.
Recently, some researchers have solved theconsensus problem of multi-agent systems with timevarying
external
disturbancesandrandom
communication delays [7]. In [8], model uncertainty, i.e.,the small parameter uncertainty, was taken into
consideration, and a robust H ∞ controller was designed
accordingly.
In this paper, the consensus problem is considered
for multi-agent systems with large parameter
uncertainty, which will be divided into smaller intervals
that can be dealt with by each local H ∞ controller.
External disturbances and communication delays arealso taken into account. WMMAC strategy showed its
priority of dealing with large parameter uncertainties [9],which consists mainly of two components, i.e. the local
controller set and the weighting algorithm. The local
controller set was designed according to a local model
set which was determined off-line to cover the large
uncertainty of system parameters. The weighting
algorithm was designed to get weights for each localcontroller, based on which the global control law isobtained as a weighted sum of all the local control ateach time instant.
Finally, simulation results indicate that under theproposed protocol, i.e. the WMMAC strategy, multiagent systems with large parameter uncertainty canreach the desired consensus performance in the presenceof communication delays.
2. Uncertain Multi-Agent Systems
Consider a second-order multi-agent system consistingof n identical agents with the i th one modeled by as
the form of (1).

xi (t ) =Axi (t ) + B1ωi (t ) + B2 ui (t ),xi (t ) [ xi1 (t ), xi 2 (t )]T , ωi (t ) ∈ L2 (0, ∞]=(1)

xi (t ) is the state of agent i , while ωi (t ) represents
appropriate external disturbance that belongs
to L2 (0, ∞] . ui (t ) denotes the control protocol.
If system matrices A, B1 , B2 are uncertain, then they
are supposed to follow the forms of (2).
In (2), A0 , B10 , B20 are constant matrices andare time-varying matrices.
∆A(t ), ∆B1 (t ), ∆B2 (t )E and Fi , i = 1, 2,3 are constant matrices withappropriate dimensions and Σ(t ) is time-varying
uncertain matrix satisfying Σ(t )T Σ(t ) ≤ I . It is also
assumed that ( A0 , B20 ) is stabilizable.

A= A0 + ∆A ( t ) , B=B10 + ∆B1 (t )1B=
B20 + ∆B2 (t )2(2)[ ∆A(t ) ∆B1 (t ) ∆B2 (t )] =E Σ ( t ) [ F1 F2 F3 ]The consensus of multi-agent system is that thestates of agents satisfy (3) under control protocol ui (t ) .

lim ( xi ( t ) − x j ( t ) ) =0, ∀i, j ∈ nt →∞(3)

3. Weighted Multi-model Adaptive Control
Weighted multi-model adaptive controller is made up of
the following components.
3.1. Model set
Considering the large uncertainty of plant parameters
cannot be dealt with by a single H ∞ controller, thenwe need to use multiple models, i.e. a model set, tocover the uncertainty of the plant. The model set isdesigned as=Ω {M=1, 2, , N } . Obviously, thei |iperformance of multi-agent control system depends onthe model set and the local control strategy, as well as
the right convergence of the weighting algorithm.
3.2. Controller set
The controller set C is shown as the form of=
C {=Ci | i 1, 2, , N }in which the controller Ci is designed according tomodel M i according to H ∞ control strategy or otherfeasible control strategy.
In this paper, robust H ∞ method is selected ascontrol strategy. Considering the multi-agent system (1),the output is supposed to follow (4). We can reformulate
the system (1) as the form of (5) that satisfies (6) withthe help of Kronecker product.

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1,..., n∑ x j (t ), i =n j =1(4)x (t ) = ( I n ⊗ A) x(t ) + ( I n ⊗ B1 )ω (t ) + ( I n ⊗ B2 )u (t )(5)z (=t ) ( Lc ⊗ I m ) x(t )=x(t ) [ x1 (t )T , x2 (t )T ,..., xn (t )T ]T ∈ R mn

ω (t ) [ω1 (t )T , ω2 (t )T ,..., ωn (t )T ]T ∈ R m1n=

where =
λσ i , i 1, , n − 1 is the positive eigenvalues ofLσ , for details please be referred to Reference [8].
If LMI (9) is feasible for the maximum and minimum
eigenvalues of all the connected topological graphs, thefeedback matrix can be defined by K = QP −1 .
If LMI (9) is not feasible, then we need to divide theuncertainty of parameters, i.e. ∆A(t ), ∆B1 (t ), ∆B2 (t ) into
limited number of smaller intervals, such that for eachsmaller interval, LMI(9) is feasible.
And then, we have the modified protocol of agent i=u (t ) [u1 (t )T , u2 (t )T ,..., un (t )T ]T ∈ R m2 n

(6)=z (t ) [ z1 (t ) , z2 (t ) ,..., zn (t ) ] ∈ R=ui ( t )T Lc  ∈ R
Lc = ij Tn× n

T Tmn

( n − 1) n , i =j, Lc ij =−1 n , i ≠ j=i 1j∈ N i ( t )(7)where 0 ≤ d (t ) ≤ d , d > 0 is the time-varying
communication delay, N i (t ) ≥ 0 is the neighbor set ofagent i at time instant t , aij (t ) are adjacency elementsof the corresponding interaction graph, and K is anundetermined feedback matrix.
Substituting control protocol (7) into system (5), weobtainx ( t ) = ( I n ⊗ A ) x ( t ) + ( Lσ ⊗ B2 K ) x ( t − d ( t ) )dB1T−α P
0ijij

The method based on probability-weighted is that theweighted sum of controller set is taken as the globalcontrol input. The weight of controller can be calculated
by partition theorem.
Suppose the control output of controller Ci is ui ( k ) ,the weight is pi ( k ) , the global control input is u ( k ) , thesystem output is y ( k ) , the output of i th model
is yi ( k ) .
Let the output difference between i th model andplant be ei=( k ) y ( k ) − yi ( k ) . Then the weighting
algorithm can be designed as in [9].

1, ifN

there are N models in the model set, i.e.

(8)
Finally, the consensus problem of multi-agent
system is converted to calculate the feedback matrix K .
Under control protocol (7), system (1) achievesconsensus with a given H ∞ index γ, if there are a scalar
α>0, a positive define matrix P ∈ R m×m and a matrix
Q ∈ R m2 ×m that satisfiesd Ψσ id αλσ i QT B2Tj∈ N i ( t )First, weights of all models are initialized to+ ( I n ⊗ B1 ) ω ( t )，z ( t ) = ( Lc ⊗ I m ) x ( t ) Ψσ i + Ψσ i αλσ i B2 QB1
T T−α P0αλσ i Q B2
−γ 2 IB1T0d αλσ i B2 Q dB1
 d Ψσ iP00Ψσ i= A0 P + λσ i B2 Qi

3.3. Weighting Algorithmui ( t ) K ∑ aij ( t )  xi ( t − d ( t ) ) − x j ( t − d ( t ) ) =Ti

where pi ( k ) can be calculated according to theweighting algorithm described in the next subsection.

According to the neighbors’ states, the protocol ofagent i can be designed as (7) to achieve the aboveconditions.

TN∑ p K ∑ a ( t )  x ( t − d ( t ) ) − x ( t − d ( t ) )

pi=, i 1, 2,..., N( 0 ) ri=( 0 ) 1=N(10)Then, it is conductive to improve the anti-jamming
capability by selecting the cumulative mean squared
error as the performance index shown in (11).

li ( k ) = 1 +P00 <00
− I 

1 k 2∑ ei ( k )k j =1(11)

Finally, the computational process of weights isdenoted by (12). This method is much simpler thanprobability-weighted method. The performance indexes
are forced to compete at every step in order to make theweights converge.
(9)

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lmin ( k ) = min li ( k ) 0 −1
 0 1
0 1
0 1
=A1  =, A2 =, A3 =, A4 
2 1 
 4 1
6 1
8 1
(15)According to the LMI (9), the feedback matrices K i
can be determined as (16) with the help of Matlab LMI
toolbox.

i =1,..., N=ri ( k )lmin ( k )pi ( k ) =li ( k )ri ( k − 1)ri ( k )∑ r (k )Ni =1 i(12) −2.2365 −0.3528  −1.4127 0.2621 =
K1 =
 , K 2  0.4986 −2.3349 −0.1269
2.7761

4. Simulation results
 −2.0759 0.9033 
 −2.4147 1.3164 =K 3 =In this section, we will illustrate the consensus of multi , K 4  1.9986 −2.9659  1.4598 −3.0850 agent system under the method of WMMAC.
Considering a multi-agent system of four agents, theparameter matrices are designed as (13).
 0 −1
==A0 
 , B10
2 1 0
∆A(t ) =
−t 6 (1 − e )0
1 0 
= , B20 0 2  ,1 0
0, ∆B2 (t ) =0
 , ∆B1 (t ) =0 (16)The simulation results are shown in figures 2-3, in
which Fig. 2 represents the trajectories of states x1
and x2 , Fig. 3 represents the energy relationshipbetween the system output z (t ) and the externaldisturbance ω (t )(13)
The communication delay is assumed to be constant
0.05. The external disturbances are supposed to beband-limited white noises shown as (14).

ω ( t ) = ω1 ( t ) ω2 ( t ) ω3 ( t ) ω4 ( t ) T(14)T=  2ω ( t ) −1.5ω ( t ) 1.2ω ( t ) 1.8ω ( t ) For simplicity, the H ∞ performance index is chosenas γ =1 , the constant scalar α =0.1 , λ =1 and all thenonzero weighting factors of adjacency matrix areassumed to be 1. The simplified undirected interaction
graph is shown in Fig. 1.

Fig. 2(a) State curve of x1

Fig. 1 simplified undirected graphWe can get the maximum and minimum nonzero=
λσ ∗i∗ 0.5858,=
λσ ∗i∗ 4 by calculating theeigenvalues
Laplace matrices corresponding to the network graph asshown in Fig. 1.
It is assumed that there are four models in themodel set, i.e.

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Fig. 2(b) State curve of x2

References
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Fig. 3 Energy of z (t ) and ω (t )4.

If the system states are asymptotically stable, theconsensus of multi-agent system is all up to the steadystate values of each agent. Therefore, the states ofagents are designed to be divergent in order to display
the consensus of multi-agent system significantly.
It is easy to say that the multi-agent system achievesconsensus according to Fig.2
When the state matrix A changes to 0 1A=
7.5 1the weights of all models are displayed in
Fig.4.

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Fig.4 weights of models
In Fig.4, the weight of the 4th model is convergent
to 1. Then the controlled quantity of the 4th model ischosen to be the global control input of the multi-agentsystem.
5. Conclusions
By applying the weighted MMAC methodology, wehave addressed the adaptive consensus problem ofmulti-agent system with large parameter uncertainty andcommunication delays. According to the states’
differences between the plant and each model, we canobtain the weight of each local controller with theweighting algorithm mentioned. Finally, the controller
corresponding to the closest model to the plant will bechosen as the right controller.

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