Multi-Agent Coordination by Iterative LearningControl: Centralized and Decentralized Strategies

Hyo-Sung Ahn†, YangQuan Chen‡, and Kevin L. Moore§

Abstract— Iterative learning control (ILC), an approachto achieve perfect trajectory tracking for uncertain dynamicsystems that are periodic or repetitive, can be viewed as a kindof coordination or planning algorithm. This paper exploits thisview to provide two coordination algorithms for distributedmulti-agent systems. First we show how to achieve formationcontrol for a class of nonholonomic mobile agents thoughan iterative update of each agent’s angular velocity alongthe trajectory. The algorithm required to achieve this resultuses local measurements, but a centralized computation of thecontrol input. Second, we show a decentralized coordinationstrategy for a set of simple first-order integrator dynamicsystems. In this case the control updates are computed locallyby each agent using only local information, yet throughthe iterative update process the group achieves the desiredformation. Numerical simulations illustrate the results.

Index Terms— Iterative learning control (ILC), Centralizedcoordination, Decentralized coordination, Multi-agents, For-mation control

I. INTRODUCTION

Iterative learning control (ILC) has been widely em-ployed for the control of periodic dynamic systems [1].ILC is an efficient control algorithm to handle trackingor disturbance rejection in iteratively-operated dynamicsystems. If a system is subject to temporally- or spatially-repetitive disturbances, then ILC can compensate for thedisturbance. Likewise, since a reference trajectory can beconsidered a repetitive disturbance along the iteration axis,ILC can also be used to track a desired trajectory perfectly[2]. ILC has also been used a as scheduling controllerfor optimal trajectory tracking or control gain tuning [3],[4], ballistic control [5], and PID gain tuning [6]. In theseapplications one may think of the ILC algorithms as actingas a scheduling or coordination planner. From that point ofview, recently ILC was proposed for the formation controlof a set of nonlinear dynamic systems. Specifically in [7],it was shown that a leader-follower type formation of aset of model-uncertain nonlinear systems can be perfectlyachieved by an iterative learning control scheme. Similarly,in [8], formation flying of a set of leader-follower type

†School of Mechatronics, Gwangju Institute of Science and Tech-nology (GIST), 1 Oryong-dong, Buk-gu, Gwangju 500-712, Korea.hyosung@gist.ac.kr

‡Center for Self-Organizing and Intelligent Systems (CSOIS), Dept. ofElectrical and Computer Engineering, 4160 Old Main Hill, Utah StateUniversity, Logan, UT 84322-4160, USA. yqchen@ece.usu.edu

§Division of Engineering, Colorado School of Mines, 1610 IllinoisStreet, Golden, CO 80401, USA. kmoore@mines.edu

satellites was ensured by repetitive learning control method.Related, in [9] a motion coordination of multiple agentswas developed using iterative learning control and in [10],a group of n agents that repeatedly perform the same taskwas studied and it was found that by using ILC the jointwork performance among the agents was improved.

Though the results of [7], [8], [9], [10] are attractive,these papers have all used global information in theirproposed coordination algorithms. However, recently it hasbeen shown that formation control only using local infor-mation could be much more challenging and promising insome applications [11]. Thus here we propose new ILCschemes for a coordination of a set of distributed multi-agent systems that use only local information. We formulatetwo problems. The first problem is to coordinate the motionsof nonholonomic distributed agents in a centralized manner,but using only local information. The dynamics of thesystem is described by local relative distances and relativeorientation angles. However, the computation of the nextcontrol signal for each iteration update is computed in acentralized manner. The second problem is to update theiterative controller at distributed local computers using onlylocal information. Individual agents find next their controlsignals on the basis of their own previous information.Thus, we call this a decentralized coordination scheme.This method is developed for simple integrator agents andemploys the λ-norm in its proof, though simulation resultsshow that it is likely other norms can be employed. Thepaper is organized as follows. In Section II, a formationcontrol ILC algorithm for a set of nonholonomic mobileagents is described that uses only local relative information.In Section III, a decentralized coordination scheme usingILC for a set of distributed agents with integrator dynamicsis developed. Simulation results are presented in Section IVand conclusions are given in Section V.

II. CENTRALIZED COORDINATION

Let us consider a leader-follower type formation controlproblem for a group of mobile autonomous agents asdepicted in Fig. 1. The i-th agent is following the i + 1-th agent in a local coordinate frame. The relative distancebetween the two agents is given as ri+1

i , and the relativeorientation angles between agents are given as αi+1

i andαii+1 where αi+1

i is the orientation angle of the i+1-th agentwhen sensing from the i-th agent coordinate frame and αii+1

2011 IEEE International Symposium on Intelligent Control (ISIC)Part of 2011 IEEE Multi-Conference on Systems and ControlDenver, CO, USA. September 28-30, 2011

978-1-4577-1103-9/11/$26.00 ©2011 IEEE 394

is the orientation angle of the i-th agent when sensing fromthe i + 1-th agent coordinate frame. The individual agentsare rotating with angular rates of wi and wi+1, respectively.The angle βi+1

i is the relative angle between the x axes ofthe two agents. The relative dynamics can be derived usingrelative local information ri+1

i , αi+1i , and βi+1

i accordingto:

ri+1i = − cos(αi+1

i )vi + cos(βi+1i − αi+1

i )vi+1 (1)

αi+1i = − sin(αi+1

i )vi −1

ri+1i

sin(αi+1i − βi+1

i )vi+1

−wi (2)βi+1i = wi − wi+1 (3)

Note that the above dynamics has a nonholonomic con-straint. For more about dynamics of this form see [12].

vi

v i+1

wi

wi+1

rii+1

αii+1

βii+1

i+1

i

αi+1i

Fig. 1. Geometric relationships in the leader-and-follower type mobileagents.

Now, if the desired distances and the derivatives of thedesired distances between agents are given as ri+1,d

i andri+1,di , then by controlling the speed of the i agent as

vi =cos(βi+1

i − αi+1i )vi+1 − ri+1,d

i

cos(αi+1i )

, (4)

then the desired speed and desired distance could beachieved if the initial distance is equal to the desireddistance, i.e, ri+1

i (t0) = ri+1,di (t0). Inserting (4) into (2)

yields

αi+1i = − tan(αi+1

i )[cos(βi+1i − αi+1

i )vi+1 − ri+1,di ]

− 1

ri+1i

sin(αi+1i − βi+1

i )vi+1 − wi (5)

= −[tan(αi+1i ) cos(βi+1

i − αi+1i )

+1

ri+1i

sin(αi+1i − βi+1

i )]vi+1

+tan(αi+1i )ri+1,d

i − wi (6)

In (6), the velocity vi+1 is determined by the desireddistance ri+2

i+1 . If we denote −[tan(αi+1i ) cos(βi+1

i −

αi+1i ) + 1

ri+1i

sin(αi+1i − βi+1

i )]vi+1 + tan(αi+1i )ri+1,d

i =

f i+1i (αi+1

i , βi+1i , t), then we are able to simplify the for-

mation equation to be[αi+1i

βi+1i

]=

[f i+1i

0

]+

[−1 01 −1

] [wiwi+1

](7)

Then, the ILC task is to design the control sig-nals w1, w2, . . . , wn such that the desired αi+1,d

i (t) andβi+1,di (t) are achieved along the iteration.

Remark 2.1: The control scheme (4) will be singularwhen αi+1

i = π2 . This means that the dynamics for ri+1

i

(i.e., (1)) will be no longer controllable. Thus, in this paper,we consider only the limited domain for αi+1

i such asDα = [−π/2 + ε, π/2 − ε]. In our future works, we willgeneralize the problem such that it can be applicable towhole global domain. To this aim, we may have to changethe control scheme.

If we consider the domain Dα = [−π/2 + ε, π/2 − ε]for α and Dr = [ε,∞) for r where ε is a small positiveconstant, then f i+1

i (αi+1i , βi+1

i , t) is Lipschitz continuous;thus, it is differentiable twice. Since we consider n agentsthat are cyclic connected with a leader-and-follower type,we can write the connected system as:

α21

β21

α32

β32...

αnn−1

βnn−1

=

f21 (α21, β

21 , t)

0f32 (α

32, β

32 , t)

0...

fnn−1(αnn−1, β

nn−1, t)

0

+Bw, (8)

where B is the following matrix with size of 2(n− 1)× n

B =

−1 0 0 0 0 · · · 0 01 −1 0 0 0 · · · 0 00 −1 0 0 0 · · · 0 00 1 −1 0 0 · · · 0 00 0 −1 0 0 · · · 0 0...

......

......

. . ....

...0 0 0 0 0 · · · −1 00 0 0 0 0 · · · 1 −1

and w is the length n vector with elements w =[w1, w2, . . . , wn]

T . The output measurements are given as

y = Cx (9)

where C is the 2(n−1)×2(n−1) diagonal identity matrixand x = [α2

1, β21 , . . . , α

nn−1, β

nn−1]

T . Thus, given desiredoutput trajectories (i.e., yd(t)) and their derivations (i.e.,yd(t)), when we update the ILC controller at the (k+1)-thiteration such as

wk+1 = wk + Lk(t)[yd(t)− yk(t)] (10)395

the convergence condition can be derived as

‖In×n − Lk(t)CB‖ < 1 (11)

Theorem 2.1: Given the matrix B, there always exists acontrol gain matrix Lk(t) such that (11) holds.

Proof: The matrix B is full column rank (i.e., thecolumns of B are linearly independent); thus, by Moore-Penrose pseudoinverse property, BTB is invertible andthere exists the pseudoinverse matrix B+ = (BTB)−1BT ,which yields B+B = In×n. Thus, if we select Lk(t) = B+,then ‖In×n − Lk(t)CB‖ = 0.

The overall procedure to coordinate the set of distributedagents to maintain desired trajectories among the agents issummarized in Algorithm 1.

Algorithm 1: Procedure for centralized ILC1: Given initial orientations αi+1

i , βi+1i ,

- set initial default angular rates as w0 = 0

- set initial distances as ri+1i (t0) = ri+1,d

i (t0)- set vn = 0,- compute vi, i = 1, · · · , n− 1 using (4)

performe the 1-st iterationmeasure the outputs and compute errors

2: Reset and- calculate wk+1 using (10),- set vn = 0,- compute vi, i = 1, · · · , n− 1 using (4)

performe the 2-nd iterationmeasure the outputs and compute errors

3: Repeat the above procedure 2 until convergence

In the update control of signals by (10), the desiredtrajectories of all agents, i.e., yd(t), and actual achievedtrajectories of all agents, i.e., yk(t) are utilized. To computethe next update at k+1-th iteration, a centralized computermay gather all these data (i.e., yd(t) and yk(t)) from allagents. Then, using (10), after computing angular rateswi,= 1, . . . , wn, the centralized computer sends the controlsignals to the distributed agents. Since the coordinationscheme defined by (10) cannot update the control signalsin the distributed local agents, it is called centralized coor-dination.

However, it is certainly beneficial, depending on ap-plications, if the control signals could be updated in thedistributed local agents individually, using only local infor-mation. The next section is devoted to this problem.

III. DECENTRALIZED COORDINATION

Let us consider a set of mobile agents with integratordynamics moving in a 2-dimensional Euclidean space underthe following simple dynamics:

pk,i = uk,i, (12)

which is represented in a global inertia frame at the k-th iteration. A desired formation between agents is definedon a follower with respect to a leader. That is, the desired

formation between a follower and a leader is given on alocal coordinate frame of the follower by

pid,i,i+1 = [pi+1 − pi]id, (13)

where the superscript i means that the coordinate is rep-resented in the i-th frame attached to the i-th agent, andpid,i,i+1 is the desired relative formation of the leader(here, without notational confusion, the (i + 1)-th agent isconsidered the leader of the i-th agent), represented in thei-th coordinate frame. Since the coordinate of the i-th agentin the i-th coordinate frame is the origin, we can rewritethe above equation as

pid,i,i+1 = pid,i+1, (14)

where pid,i+1 is the desired position of the i+1-th agent withrespect to the i-th agent. The error of the desired formation,in the i-th coordinate frame, now can be defined as

eik,i,i+1 = pid,i,i+1 − pik,i+1, (15)

where pik,i+1 is the actual position of the i + 1-th agentwith respect to the i-th agent at the k-th iteration. Assumethat the i-th agent’s coordinate frame is related to theinertia frame by pk,i = Qip

ik,i + di, where Qi denotes the

rotation and di denotes the translation of the i-th coordinateframe with respect to the inertia frame. Then, we knowthat pk,i = Qip

ik,i and uk,i = Qiu

ik,i. Likewise, we also

have pk,i+1 = Qipik,i+1 + di. Let us suppose that the

orientation of the (i + 1)-th agent with respect to i-thcoordinate frame and the orientation of i-th agent withrespect to (i+ 1)-th coordinate frame are measured. Then,using these orientation angles, the orientation between the iand the (i+1)-th agents can be transformed. Let us denotethe orientation transformation as Qii+1 (it is the orientationrepresentation of the (i+1)-th agent to the i-th agent). Then,we have pk,i+1 = Qi+1p

i+1k,i+1, and uik,i+1 = Qii+1u

i+1k,i+1

and uk,i+1 = QiQii+1u

i+1k,i+1. Thus, from (12) and since Qi

is nonsingular and is Euclidean, we can obtain:

Qipik,i = Qiu

ik,i ⇐⇒ pik,i = uik,i (16)

and

Qi+1pi+1k,i+1 = QiQ

ii+1u

i+1k,i+1 (17)

We further can write (17) as

(Qi)TQi+1p

i+1k,i+1 = Qii+1u

i+1k,i+1

⇐⇒ pik,i+1 = uik,i+1 (18)

From (15), assuming no initial reset error, we have

eik+1,i,i+1 = eik,i,i+1 − (pik+1,i+1 − pik,i+1)

= eik,i,i+1 − (pik+1,i+1(t0)− pik,i+1(t0))

−∫ t

t0

pik+1,i+1(τ)− pik,i+1(τ)dτ

= eik,i,i+1 −∫ t

t0

pik+1,i+1(τ)− pik,i+1(τ)dτ

396

= eik,i,i+1 −∫ t

t0

uik+1,i+1(τ)− uik,i+1(τ)dτ

(19)

Now, let us update the control signal according to

uik+1,i+1(t) = uik,i+1(t) + γik+1,i+1(t)eik,i,i+1(t) (20)

In the sense of physical motion, uik,i(t) is the control signalof the i-th agent and uik,i+1(t) is the control signal of thei+ 1-th agent with respect to the i-th agent. It is assumedthat the relative orientation between the i-th agent and i+1-th agent are available. That is, Qii+1 is assumed available.Then, after updating the control signals by (20), the controlinput to the i+ 1-th agent is computed as

ui+1k,i+1 = Qi+1

i uik,i+1 (21)

where Qi+1i = [Qii+1]

T . The error is now propagated asfollows:

eik+1,i,i+1 = eik,i,i+1 − γik+1,i+1(t)eik,i,i+1(t)

+

∫ t

t0

dγik+1,i+1(τ)

dτeik,i,i+1(τ)dτ (22)

Let us denote maxγ = supt0,tend‖dγ

ik+1,i+1(τ)

dτ ‖, then wehave the following inequality

‖eik+1,i,i+1‖ ≤ ‖1− γik+1,i+1(t)‖‖eik,i,i+1(t)‖

+maxγ

∫ t

t0

‖eik,i,i+1(τ)‖dτ (23)

Taking the λ norm on both sides of the above equationyields

‖eik+1,i,i+1‖λ ≤ ‖1− γik+1,i+1(t)‖‖eik,i,i+1(t)‖λ

+maxγ‖eik,i,i+1(t)‖λ

1− e−λ(t−t0)

λ(24)

Thus, taking λ large enough, we can make the secondterm of the above inequality zero and following result isimmediate:

Theorem 3.1: Consider the dynamics (12). If the controlsignal of the i + 1-th agent is updated by (20) with thelearning gains γik+1,i+1(t) satisfying the condition ‖1 −γik+1,i+1(t)‖ < 1, then the desired trajectories can beachieved as the iteration increases (i.e., eik,i,i+1 → 0 ask →∞).

The distributed coordination algorithm presented in thissection is conducted in a local sense. In (20), the controlsignal for the i + 1-th agent is updated on the basis ofrelative information between the i-th agent and i + 1-thagent. For the computation, it only uses relative orientationand relative position of the i+1-th agent with respect to thei-th agent. Thus, a centralized computer is not necessaryin this task; instead decentralized controllers are used todecide the next actions of agents. The overall procedure ofthe decentralized coordination scheme is summarized in the

Algorithm 2. Note that our result is based on the λ-norm,a common approach used in early ILC research. Use of theλ-norm is often criticized as it is a bounding result that canbe conservative. However, it is the case that most resultsderived using the λ-norm can be derived for an arbitrarynorm. In this work we have not been able to do so, but notethe numerical results presented below do in fact convergeto zero error in relatively-few iteration. Thus we believeare results to be useful and valid despite their use of theλ-norm.

Algorithm 2: Procedure for decentralized ILC1: Given initial relative positions, pi0,i+1

- set initial default control as ui0,i = 0- perform the 1-st iteration

measure the relative orientaionsmeasure the outputscompute errors ei0,i,i+1

2: Reset and- update the control signals using (20),- compute ui+1

k,i+1, i = 1, · · · , n− 1 using (21)- performe the 2-nd iteration

measure the relative orientaionsmeasure the outputscompute errors eik,i,i+1

3: Repeat the above procedure 2 until convergence

IV. SIMULATION RESULTS

This section provides simulation results to illustrate thecentralized and decentralized coordination schemes devel-oped in the previous sections.

A. Centralized Coordination

For simulation purposes, let us consider three agents thatare moving in 2-dimensional space. To satisfy the conditionof Theorem 2.1, we select the control gain matrix to be

Lk =1

2B+

=1

2

−0.3333 0.1667 −0.1667 0−0.1667 −0.1667 −0.3333 0−0.1667 −0.1667 −0.3333 −0.5000

Fig. 2 shows the desired and achieved trajectories for r21 .The desired trajectory and achieved trajectory are exactlysame because the exact inverse controller is used as shownin (4). Fig. 3 shows the desired (solid line) and achievedtrajectories (dashed lines) for α2

1 as iteration increases.As shown in this figure, the ILC controller successfullyachieves the trajectory tracking for the leader-follower typeformation of a set of nonholonomic mobile agents. Fig. 4shows the perfect trajectory tracking in α2

1, β21 , α3

2, and β32 .

As seen in this figure, as the number of iterations increases,all the desired formations have been perfectly achieved.

397

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Iteration number

Nor

m o

f er

rors

|| α12 ||

|| β12 ||

|| α23 ||

|| β23 ||

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54.75

4.8

4.85

4.9

4.95

5

Time (seconds)

Mag

nitu

de o

f r 12

Fig. 2. Desired r21 and achieved trajectory.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Time (seconds)

Mag

nitu

de o

f a 12

As iteration increases

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Time (seconds)

Mag

nitu

de o

f b 12

As iteration increases

Fig. 3. Desired α21 and achieved trajectories as iteration increases.

B. Decentralized Coordination

For the task of decentralized coordination, the desiredrelative distances are given as p1d,1,2 = [sin(t), cos(t)]T andp2d,2,3 = [sin(0.5t), cos(0.7t)]T . The ILC gain matrix isselected as γ = diag[0.5]. Fig. 5 shows the desired andachieved trajectories for d23(2). As shown in this figure, asthe number of iterations increases, the desired trajectoryhas been perfectly achieved. Fig. 6 shows the errors of theachieved trajectories along the iteration domain. By the timethe number of iterations has reached 10, the trajectories havebeen perfectly achieved.

V. CONCLUSION

This paper has developed two coordination algorithmsfor distributed multi-agent formation using iterative learningcontrol. A centralized coordination strategy for a group ofnonholonomic mobile agents was developed in Section IIand a decentralized coordination scheme for a group of

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Iteration number

Nor

m o

f er

rors

|| α12 ||

|| β12 ||

|| α23 ||

|| β23 ||

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54.75

4.8

4.85

4.9

4.95

5

Time (seconds)

Mag

nitu

de o

f r 12

Fig. 4. Errors of α21, β2

1 , α32, and β3

2 along the iteration domain.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (seconds)

Mag

nitu

de o

f p 32 (2

)

As iteration increases

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Iteration number

Nor

m o

f er

rors

|| e21(1) ||

|| e21(2) ||

|| e32(1) ||

|| e32(2) ||

Fig. 5. Desired d23(2) and achieved trajectories.

simple first-order integrator mobile agents was developed inSection III. It is a tough and challenging problem to achievea perfect trajectory tracking for the nonholonomic mobileagent systems described in (1)-(3). This paper shows that theperfect trajectory tracking is still achievable if the motionsof agents are coordinated by ILC along the iteration domain.In Section III, it is shown that even though we use localinformation in distributed local agents, a perfect trajectorytracking can be ensured. The main contribution of this paperis to show that coordination control for trajectory tracking ofdistributed multi-agent systems using only local informationcan be effectively achieved using iterative learning controlscheme. In our future efforts, we will further considercoordination of a group of mobile agents when they aredefined by a general random topology, rather than theleader-follower topology presented here.

398

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (seconds)

Mag

nitu

de o

f p 32 (2

)

As iteration increases

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Iteration number

Nor

m o

f er

rors

|| e21(1) ||

|| e21(2) ||

|| e32(1) ||

|| e32(2) ||

Fig. 6. Errors of the achieved trajectories along the iteration domain.

VI. ACKNOWLEDGEMENTS

This research was supported by the National ResearchFoundation of Korea (NRF) (No. 2011-0002847), by theMKE(The Ministry of Knowledge Economy), Korea, underthe ITRC(Information Technology Research Center) sup-port program supervised by the NIPA(National IT IndustryPromotion Agency) (NIPA- 2010-C1090 -1031- 0006), andby a grant from the iMSE(Institute of Medical SystemEngineering) in the GIST.

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