On the Synchronization of Linear Heterogeneous ...

On the Synchronization of Linear Heterogeneous Multi-Agent Systems in Cycle-Free Communication Networks Saman Khodaverdian Abstract— This paper deals with the output synchronization problem for linear heterogeneous multi-agent systems. It is shown that in cycle-free communication networks, synchronization can be ensured by static state feedback laws under mild assumptions. Moreover, we present an observer-based strategy for the case that the agents have only access to relative output differences. While in general additional controller states must be communicated to synchronize the agents by observerbased methods, we prove that in cycle-free networks, only output differences are needed to achieve an observer-based synchronization.

I. Introduction The cooperative control of networked dynamic systems has received great attention over the past two decades. A multi-agent system can be described as a group of dynamic systems or agents that have to achieve a common goal cooperatively. Often, multi-agent systems are used if single agents are not able to cope with a given task or if the task can be solved more efficient by multiple agents. There are many applications in different research areas such as, for instance, air traffic control, formation problems, cooperative search and surveillance, and multi-robot exploration. For a detailed overview, we refer to the articles [9] and [11]. A key problem of such networked systems is the synchronization of the agents. For this purpose, the agents interact with each other by exchanging information through a communication network. The synchronization problem for multi-agent systems has been investigated intensively in recent years, where the literature has mainly been focusing on homogeneous, meaning identical, agents [2], [8], [12]. While [4], [10] and [11] have investigated agents with simple integrator dynamics, the authors in [6], [14] and [16] have considered agents with general linear dynamics. Currently, the problem of synchronizing heterogeneous, meaning non-identical, agents has been attracted more and more research attention. In the literature, this problem is known as partial-state or output synchronization problem. For identical systems it is possible to synchronize the agents’ complete state vector, but for non-identical systems, in general, it is only possible to synchronize a subset of physically comparable states. It is well known that the agents have to satisfy an internal model principle to achieve This work was gratefully supported by the German Research Foundation (DFG) within the GRK 1362 “Cooperative, Adaptive and Responsive Monitoring of Mixed Mode Environments” (www.gkmm.de). Saman Khodaverdian is with the Institute of Automatic Control and Mechatronics, Control Methods and Robotics Lab, Technische Universität Darmstadt, Landgraf-Georg-Str. 4, 64283 Darmstadt, Germany. E-mail: [email protected].

(output) synchronization [5], [17]. This means that there must be some homogeneous dynamics among all agents, which describe the synchronization trajectory. Usually, the output synchronization of heterogeneous agents requires dynamic state feedback laws. One of the first solutions has been presented in [17], where a dynamic exosystem, prescribing the synchronization trajectory, is embedded into every agent. Then, based on the classical output regulation theory [3], a tracking controller is designed to regulate the agent’s output to the given output of the exosystem. However, the embedded exosystem belongs to the controller of the agent which, regarding the overall multiagent system, leads to a highly dynamic control strategy. In this paper, we will show that in case of cycle-free communication topologies, it is possible to synchronize heterogeneous agents by static state feedback laws, provided that the internal model principle is satisfied. In that case, no controller dynamics are necessary. Recently, in [7] a static output feedback strategy has been presented for synchronizing a class of heterogeneous agents in cycle-free networks. However, it is not always possible to synchronize the agents by such an output feedback controller. Stronger conditions on the agents’ dynamics are required to achieve synchronization by static output feedback [8]. Thus, we present a state feedback strategy that is applicable under standard assumptions about the agents’ dynamics. If the agents have only output information, observer-based control laws can be applied to ensure synchronization, as shown for instance in [6] and [15] for the homogeneous case. However, additionally to the exchange of the output variable, the solution needs to exchange some controller states through the communication network. In this paper, we will show that in cycle-free communication networks it is not necessary to exchange additional variables. Using only output information, it is possible to design distributed observer-based control laws to synchronize the heterogeneous agents. An observer-based technique using only output information has also been presented in [13], but the solution is restricted to a special class of systems and uses a low gain method, which leads to a slow convergence rate for the synchronization. Both restrictions are not given in the presented approach. In this paper, we consider SISO agents, but under certain conditions the results can be extended to MIMO agents. The rest of the paper is structured as follows: In the next section, we provide some notations, mathematical tools and a detailed problem description. In Section III, a static state feedback strategy is presented to synchronize the heterogeneous agents. In Section IV, an observer-based technique

for synchronizing agents that have only output information is described. The efficiency of the approach is illustrated by a numerical example in Section V, before the results are concluded in Section VI. II. Preliminaries and Problem Setup A. Notation and System Theory Vectors and matrices are represented by boldface letters and scalar values by italic letters. The identity matrix and the zero matrix of appropriate dimension are written as I and 0. The transpose of a matrix A is written as AT . If A is a square matrix, the positive (negative) definiteness of A is denoted by A ≻ (≺) 0. Furthermore, A is called Hurwitz if all of its eigenvalues have strictly negative real part, or equivalently if A ≺ 0. It is called anti-stable if all eigenvalues have non-negative real part. The state space description of a linear SISO system is given by x˙ = Ax + bu, y = cT x, and it is abbreviated as ( A, b, cT ). The system is called stable if its system matrix A is Hurwitz. We say, the pair ( A, b) is stabilizable if there exists a vector kT such that ( A − bkT ) is Hurwitz. Similarly, the pair ( A, cT ) is called detectable if there exist a vector h such that ( A − hcT ) is Hurwitz. The following result from optimal control theory [1] will be used throughout the paper: Given a stabilizable pair ( A, b), the algebraic Riccati equation PA + AT P − PbbT P + I = 0

(1)

has a unique solution P = PT ≻ 0 and the control law u = −kT x, with kT = bT P, leads to a stable closed-loop system. For the design of our distributed control laws, the following lemma, which is taken from [14], proves beneficial. Lemma 1: Let the pair ( A, b) satisfy (1) for a matrix P = PT ≻ 0. Then, with kT = bT P, the matrix ( A − σ · bkT ) is Hurwitz for all σ ≥ 1. B. Graph Theory The communication network of the multi-agent system is represented by a time-invariant directed graph or digraph G = (VG , EG ), with vertex set VG and edge set EG . The i-th agent in the network is represented by the vertex i, with VG = N < ∞. The information flow from agent i to agent j is possible if the edge (i, j) ∈ EG exists. The in-degree dGi indicates the number of vertices that send information to vertex i. A convenient way for the representation of the communication network is given by the adjacency matrix AG = [aGi j ] ∈ R N×N , with    1, if (i, j) ∈ EG , a Gi j =   0, if (i, j) < EG . P Note that dGi = Nj=1 aG ji . We say that a digraph is cycle-free if there exists no sequence of edges such that starting from a vertex i it is possible to reach again vertex i. For a cycle-free digraph, it is always possible to enumerate the vertices in a way such that AG is a lower or upper triangular matrix.

The following definition describes a fundamental property of digraphs, which is important to achieve synchronization in multi-agent systems. Definition 1: A digraph G is said to contain a directed spanning tree, if there exists at least one vertex – also called root vertex – that can reach every other vertex in the graph, using the edges given by the set EG . For a cycle-free digraph that contains a directed spanning tree, we can conclude that there exists exactly one root vertex r, with dGr = 0, and for the in-degree of the other vertices it follows that dGi ≥ 1, for all i , r. C. Problem Setup A network of N heterogeneous agents with the linear timeinvariant dynamics x˙ i = Ai xi + bi ui , T

yi = ci x i

(2a) (2b)

is considered, where xi ∈ Rni is the state vector and ui , yi ∈ R are the input and output of the i-th agent (i ∈ {1, . . . , N}). The matrix Ai and the vectors bi and cTi are assumed to be constant and known and the communication topology of the network is described by a cycle-free digraph. The aim is to find distributed control laws such that the agents’ outputs synchronize, i.e.
lim yi (t) − y j (t) = 0, ∀i, j ∈ {1, . . . , N}. (3) t→∞

Furthermore, to exclude trivial solutions, it is required that limt→∞ yi (t) , 0, meaning that we are not interested in vanishing synchronization trajectories. The dynamics of this synchronization trajectory, denoted as y s , contains no asymptotically stable modes and is described by an ordinary differential equation of order p: (p) ys

(p−1)

+ q p−1 · y s + . . . + q1 · y˙ s + q0 · y s = 0.

(4)

For the solution of the described problem, the following assumptions have to be fulfilled for every agent. Assumption 1: The pair ( Ai , bi ) is stabilizable. Assumption 2: The pair ( Ai , cTi ) is detectable. Assumption 3: ni ≥ p. Note that Assumptions 1 and 2 are usual requirements for controlling linear dynamic systems. Assumption 3 ensures that the agents are able to generate the synchronization trajectory. If this assumption is not satisfied for an agent, this agent cannot reproduce the synchronization dynamics which are determined by (4). III. Synchronization by Static Control Laws In the synchronization literature, it is well-known that synchronization can only be achieved if the agents satisfy an internal model principle [17]. That is, the agents have to contain a common dynamic subsystem that determines the synchronization behavior. This requirement is also known as the system intersection property [7]. In the following, we assume that the internal model principle is satisfied, meaning that the agents have a dynamical subsystem which

is homogeneous among all agents in the network. If an agent does not contain this homogeneous subsystem, then, it is possible to include the missing dynamics by a dynamic control law, such that the controlled agent satisfies the internal model principle. Note that since we are considering SISO systems and since Assumption 3 is fulfilled, it is even possible to include the desired modes by a static control law. Provided that enough eigenvalues are controllable, a static state feedback law can be used to place the eigenvalues such that the desired modes are included. In the following, if we say that the dynamics of the synchronization trajectory is given by the differential equation (4), we say that every agent contains the eigenmodes of this differential equation. This means that p eigenvalues of the agents have to coincide with the eigenvalues of (4), while the remaining ni − p eigenvalues can be arbitrary. Hence, without loss of generality, the dynamics of the agents (2) can be written as #" # " # " # " S Mi x si x˙ si b = + si ui , (5a) 0 Ni xni x˙ ni bni | {z } |{z} Ai

i "x # yi = cTs 0 si , | {z } xni h

bi

(5b)

cTi

where S ∈ R p×p and cTs ∈ R1×p are identical for every agent and Ni ∈ R(ni −p)×(ni −p) , Mi ∈ R p×(ni −p) , b si ∈ R p×1 and bni ∈ R(ni −p)×1 depend on the i-th agent. Moreover, S and cTs are given by   1 ··· 0   0  . ..  .. .. h i  .. . .  , cT = 1 0 · · · 0 . (6) . S =  s   0 0 ··· 1   −q0 −q1 · · · −q p−1

From (6), it can be seen directly that the characteristic polynomial of S is the same as that of the differential equation (4). Thus, it is apparent that the internal model principle is satisfied for the agents (5). Remark 1: It is always possible to transform (2) into (5), since (2) and (5) describe the same system. This transformation is nothing but a reformulation of the internal dynamics. In the following, we say that agent 1 is the root agent to whom all other agents will synchronize. This root agent prescribes the synchronization trajectory and will not control itself to the other agents. Thus, its control input is u1 ≡ 0 and its dynamics can be represented by the uncontrolled system " # " #" # x˙ s1 S M1 x s1 = , x˙ n1 0 N1 xn1 # " h i x y1 = cTs 0 s1 . xn1 Furthermore, we assume that N1 is Hurwitz, meaning that the influence of xn1 will vanish asymptotically. In this case, the synchronization trajectory is driven by y s (t) = y1 (t) = cTs eSt x s1 (t0 ).

Note that we can also omit the assumption that N1 is Hurwitz. Then, a feedback law u1 = −kTn1 xn1 can be used to stabilize the dynamics described by N1 . Theorem 1: Given the agents in the network are described by (5) and the communication topology is given by a cyclefree digraph, which contains a directed spanning tree with root agent 1. Then, output synchronization will be achieved with the control law # " N X x − xs j , (7) aG ji si ui = −kTi xni j=1

T

if and only if ki is determined such that   Ai − dGi · bi kTi , i ∈ {2, . . . , N}

is Hurwitz. Note that the control law (7) requires only the exchange of the state variable x si , which corresponds to the homogeneous subsystem of the agents. Proof: Consider the state transformation # " " # X N x si − x s j z si , ∀i ∈ {2, . . . , N}, (8) = aG ji zi = xni zni j=1

and z1 = x1 . From (8), it is apparent that the states x si and therefore the outputs yi will synchronize if limt→∞ z2,...,N (t) = 0. This follows from the fact that then limt→∞ xni (t) = 0 and limt→∞ (x si (t) − x s j (t)) = 0, for all i, j ∈ {1, . . . , N}. Thus, if the states x si are synchronous, the outputs yi = cTs x si are also synchronous. First, with the transformation (8), the control law (7) reads ui = −kTi zi ,

(9)

T

where k1 = 0. The dynamics of x si are determined by x˙ si = Sx si + Mi xni + b si ui  i h 1 = Sx si + 0 dGi M j − b si kTi zi , | {z } Ri

i h with dGi = j=1 aG ji and R1 = 0 M1 . Taking this under consideration, the dynamics of the upper part of zi are given by N X z˙ si = aG ji ( x˙ si − x˙ s j ) PN

j=1

=

N X

aG ji (Sx si + Mi xni + b si ui − Sx s j − M j xn j − b s j u j )

j=1

=S

N X j=1

|

aG ji (x si − xn j ) +Mi {z

}

zsi

...+

N X

N X

aG ji xni + . . .

j=1

| {z }

aG ji b si ui −

j=1

| {z }

zni

N X j=1

aG ji (M j xn j + b s j u j ) | {z }

N X

aG ji R j z j .

dGi

= Sz si + Mi zni + dGi · b si ui −

j=1

Rj zj

For the dynamics of the lower part of zi it follows z˙ ni =

N X

aG ji x˙ ni = Ni zni + dGi · bni ui .

j=1

Stacking both vectors together, we can write " # " # #" # " # " N X R b S Mi z si z˙ si + dGi · si ui − aG ji j z j . (10) = 0 bni 0 Ni zni z˙ ni j=1

Now, including the control law (9) yields " # " # " # " # !" # X N R z si z˙ si S Mi b − aG ji j z j = − dGi · si kTi 0 zni z˙ ni 0 Ni bni j=1

  z˙ i = Ai − dGi · bi kTi zi −

N X

aG ji R j z j ,

(11)

j=1

h iT with R j = RTj 0 . Since we assume cycle-free communication structures, the agents can be sorted or enumerated in a way such that agent i uses only information from predecessor agents. This means that with aG ji = 0,

∀ j ∈ {i, i + 1, . . . , N},

the sum in (11) can be written as N X

aG ji R j z j =

i−1 X

aG ji R j z j .

(12)

j=1

j=1

Taking (12) into account, and stacking all vectors zi together, we get z˙ = Az, h iT with z = zT1 zT2 · · · zTN and   0 ··· 0  A1    T 0  aG12 R1 A2 − dG2 · b2 k2 · · ·   . (13) A =  . . . . . . . .  .  . . .   T aG1N R1 aG2N R2 · · · AN − dG N · b N k N h iT Note that R1 z1 = (M1 xn1 )T 0 . In the asymptotic case, the dynamics of z2 , . . . , zN are decoupled from z1 , due to the fact that N1 is Hurwitz or limt→∞ xn1 (t) = 0, and thus limt→∞ R1 z1 (t) = 0. Since (13) is a lower block triangular matrix, it follows that limt→∞ z2,...,N (t) = 0 if the matrices on the block diagonal   (14) Ai − dGi · bi kTi , i ∈ {2, . . . , N}, are Hurwitz. Then, the states x si and, therefore, the outputs yi will synchronize. Since ( Ai , bi ) is stabilizable, it is clear that (14) can be made Hurwitz by an appropriate choice of kTi . One possibility T T is to choose kTi = d1G ki , where ki is determined such that T

i

( Ai −bi ki ) is Hurwitz. However, the communication topology may change with time, e.g. if communication links drop out or if they are temporarily deleted due to obstacles. Then, also the in-degree dGi changes and the controller gain vector kTi needs to be updated on-line. However, with the help of

Lemma 1, it is also possible to find a kTi for what (14) is Hurwitz independently of dGi . From Lemma 1, we know that if kTi is determined by kTi = bTi Pi , where P follows from (1), the matrices (14) are Hurwitz for all dGi ≥ 1. Since the communication graph is assumed to contain a directed spanning tree, it follows that dGi ≥ 1 for all i ∈ {2, . . . , N} and synchronization will be achieved independently of the in-degree dGi . Further Discussion and Extension to MIMO Agents As discussed in the proof of Theorem 1, in cycle-free networks the synchronization problem reduces to a simple stabilization problem for the matrices (14). In cyclic networks, considering these matrices is not sufficient for synchronization. In that case, the matrix A given in (13) is not block triangular and has to be checked for stability. However, the synchronization problem is substantially simplified if the communication network does not contain any cycles. Theorem 1 provides a control strategy (7) that ensures synchronization if the controller gain vectors kTi are determined such that the matrices (14) are Hurwitz. Moreover, with the help of Lemma 1 this control strategy is even distributed, since the controller gain vector kTi for every agent depends only on its own system ( Ai , bi ). Hence, synchronization will be achieved independently of the communication graph, as long as it contains a directed spanning tree. If there is no directed spanning tree, there exists at least one agent k ∈ {2, . . . ,N} with in-degree dGk = 0. In that case, (14)  reduces to Ak − 0 · bk kTk = Ak , which is not Hurwitz, due to the fact that S is assumed to be anti-stable. The presented solution is directly applicable to MIMO agents ( Ai , Bi , Ci ) which can be represented in the form # " # " B S Mi (15a) x + si ui , x˙ i = Bni 0 Ni i h i yi = C s 0 x i . (15b) In that case, the outputs yi will synchronize by using the same control law (7) if the controller gain matrices Ki are determined such that the matrices
Ai − dGi · Bi Ki , i ∈ {2, . . . , N},

are Hurwitz. The difficulty here is to transform the MIMO agents into the form (15). In [7], a method is presented to check if heterogeneous MIMO agents can be transformed into this or a similar form. The problem is that, in general, it is not possible to get this form for arbitrary MIMO agents, since MIMO systems have a more complex internal structure. For SISO (and MISO) agents it is always possible to get the desired form. Besides, in heterogeneous networks we are often interested in synchronizing a single variable (e.g. the velocity of different vehicles), thus considering SISO agents is probably more reasonable. It should be noted that for the given assumptions, synchronization can be guaranteed if the states x si are exchanged through the communication network. However, in the next section we will show that it is also possible to reduce the communication load to an exchange of the outputs yi .

IV. Observer-Based Synchronization Using Only Relative Output Differences If the communication between the agents is restricted to the exchange of the output variable yi or if the agents have only information about output differences, then, in general, it is not possible to achieve synchronization by control laws that use only static output couplings [8]. In that case, dynamic control laws can be applied to achieve synchronization, as shown in [6] and [15] for homogeneous agents. In summary, it can be stated that the idea of these approaches is to use observers which estimate the state differences between the agents, and to use the estimated state differences for the feedback law. However, additionally to the exchange of the outputs, communication of the observer states is required. This leads to an increased communication load. In this section, we will show that in cycle-free communication structures, only the exchange of the outputs is needed to design an observer-based control law which achieves output synchronization. The exchange of observer states or any other variables is not necessary. From Theorem 1, we know that the control law (7) achieves synchronization, where kTi is determined accordingly. Moreover, with the state transformation (8), the control law can be written as given in (9). The vectors zi can be interpreted as disagreement vectors which state the differences between the agents. From the previous section, it is clear that the agents are synchronous if zi (t) = 0 for all i ∈ {2, . . . , N}. If only the output differences ζi =

N X

aG ji (yi − y j )

(16)

j=1

are available for agent i, then it is possible to estimate the disagreement vector b zi from (16). For this purpose, we rewrite (16) to get # " N N X h iX x si − x s j T T ζi = c s aG ji (x si − x s j ) = c s 0 aG ji xni j=1

j=1

= cTi zi .

(17)

Considering (10), ζi can be interpreted as the output of the system N X (18a) z˙i = Ai zi + dGi · bi ui − aG ji R j z j , j=1

T

ζi = ci zi .

(18b)

Thus, we use the following observer for agent i ∈ {2, . . . , N}: ˙zi = Aib b zi + dG · bi ui + hi (ζi − b ζi ), (19a) i

b zi . ζi = cTi b

(19b)

Note that agent 1 is the root agent which is uncontrolled and, hence, does not need an observer. Furthermore, it should be noted that we are interested in distributed solutions such that P agent i does not have the information Nj=1 aG ji R j z j . For this reason, in (19a), this sum is not available for the observer. However, we will show that it is not necessarily needed in cycle-free communication networks.

Theorem 2: Given the conditions of Theorem 1 are satisfied, the observer (19) combined with the control law zi , ui = −kTi b

i ∈ {2, . . . , N},

(20)

achieves output synchronization if kTi is determined as described in Theorem 1 and hi such that ( Ai − hi cTi ) is Hurwitz. Proof: We define the observer error ei = zi −b zi , whose dynamics are given by e˙i = ( Ai − hi cTi ) ei −

N X

aG ji R j z j .

(21)

j=1

Combining (21) with (18) and (20), we get " # " A − dGi · bi kTi z˙ i = i 0 e˙i

  #" # N  R  dGi · bi kTi zi X − aG ji  j  z j . T Ai − hi ci ei Rj j=1

Due to its block triangular structure, the above matrix is Hurwitz if the block diagonal elements are Hurwitz. As shown in (12), the sum on the right side depends only on the information send by predecessor agents, meaning that     N i−1 X X  R   R  j aG ji   z j = aG ji  j  z j . Rj Rj j=1 j=1

Considering limt→∞ R1 z1 (t) = 0 and stacking the vectors h ithat T T T νi = zi ei together, similar to the proof of Theorem 1, it follows that limt→∞ νi (t) = 0 for i ∈ {2, . . . , N} if the matrices ( Ai − dGi · bi kTi ) and ( Ai − hi cTi ) are Hurwitz. This proves that the observer (19) delivers an appropriate estimate for the disagreement vector, i.e. limt→∞ (zi − b zi ) = 0, such that control law (20) achieves synchronization. It should be noted that for the design of the observer (19), the in-degree dGi must be available for agent i. Assuming that the in-degree is known or can be derived from the incoming signals, Theorem 2 shows that in cycle-free communication structures it is possible to synchronize the agents using only the output differences (16). The exchange of any other signals is not necessary. The authors in [13] have also presented an observerbased solution for agents that can only communicate their outputs through the network. However, the solution relies on a low gain feedback approach which can lead to a slow synchronization process. Moreover, the solution requires that the eigenvalues of the agents are located in the closed lefthalf of the complex plane, which is not necessary here. V. Numerical Example To demonstrate the efficiency of the presented approach, the output synchronization problem for a heterogeneous network of N = 5 agents is considered. The synchronization trajectory is given by a sinusoidal signal, whose dynamics are represented by the system (S, cTs ), with " # h i 0 1 S= , cTs = 1 0 . −1 0

3

2 4 Fig. 1.

5

 0  0 AG = 0  0 0

1 0 0 0 0

1 1 0 0 0

0 1 1 0 0

 0  0  1  1 0

4 2 yi

1

0 −2 y1

−4

y2

y3

y4

y5

Communication graph and corresponding adjacency matrix. 0

According to (5), every agent contains this dynamics and the remaining non-identical parts of the agents are given by " # " # " # 0.00 0.00 0.00 0.00 M1 = , M2 = , M3 = , −2.00 1.00 −0.90 1.00 " # " # 0.00 0.00 0.00 0.00 0.00 0.00 M4 = , M5 = , 0.65 −1.80 1.00 1.00 0.00 0.00    0.00 1.00 0.00 N1 = −1.00,   0.00 1.00 , N4 =  0.00   N2 = 1.20, −5.00 −9.50 −5.50   " # 0.00 1.00 0.00 0.00 1.00   0.00 1.00 . N3 = , N5 = 0.00 −2.88 −3.40   0.87 −3.53 −4.00

Their input vectors are h iT h b2 = 0 0 1 , b3 = 0 0 h iT b4 = b5 = 0 0 0 0 1 .

iT 0 1 ,

Note that Agent 1 is the uncontrolled leader of the network (u1 ≡ 0), hence, its input vector is not considered. Furthermore, it should be noted that the system matrices of Agent 2 and 5 are not Hurwitz, since N2 and N5 contain unstable eigenvalues. The communication topology of the network is described by the cycle-free graph shown in Fig. 1. The agents can only exchange their outputs through the network, meaning P that only the relative output differences Nj=1 aG ji (yi − y j ) can be used for control and no other information are available. Hence, the observer-based control strategy of Section IV is applied. For the calculation of the controller gain vectors kTi , we have used Lemma 1 and the observer gain vectors hTi are determined such that the observer eigenvalues are about 5 times faster than the eigenvalues of the controlled agents. The simulation results of the agents’ outputs are depicted in Fig. 2, with random initial state values. It can be seen that synchronization is achieved successfully. VI. Conclusion This paper considers the output synchronization problem for linear heterogeneous multi-agent systems, under cyclefree communication networks. A distributed static state feedback law is presented that guarantees synchronization under mild assumptions. In contrast to cyclic networks, where usually dynamic state feedback laws are needed, it is proved that in cycle-free networks static controllers are sufficient. In addition, the problem that the agents have only access

2

4

6

8

10 t (s)

12

14

16

Fig. 2. Synchronization of the agents, based on distributed observers using only relative output differences.

to relative output differences is considered and an observerbased method is proposed that ensures synchronization. It is shown that unlike the cyclic case, where additional variables must be communicated, in cycle-free networks it is possible to get an appropriate estimation of the state differences using only output differences. The efficiency of the approach is demonstrated by a numerical example. References [1] B. D. Anderson and J. B. Moore. Optimal Control: Linear Quadratic Methods. Prentice Hall, 1990. [2] J. A. Fax and R. M. Murray. Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49(9):1465–1476, 2004. [3] B. A. Francis and W. M. Wonham. The internal model principle of control theory. Automatica, 12(5):457–465, 1976. [4] F. Jiang, L. Wang, and Y. Jia. Consensus in leaderless networks of high-order-integrator agents. In Proceedings of the American Control Conference, pages 4458–4463, 2009. [5] H. Kim, H. Shim, and J. H. Seo. Output consensus of heterogeneous uncertain linear multi-agent systems. IEEE Transactions on Automatic Control, 56(1):200–206, 2011. [6] Z. Li, Z. Duan, G. Chen, and L. Huang. Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint. IEEE Transactions on Circuits and Systems I: Regular Papers, 57(1):213–224, 2010. [7] J. Lunze. Synchronization of heterogeneous agents. IEEE Transactions on Automatic Control, 57(11):2885–2890, 2012. [8] C.-Q. Ma and J.-F. Zhang. Necessary and sufficient conditions for consensusability of linear multi-agent systems. IEEE Transactions on Automatic Control, 55(5):1263–1268, 2010. [9] R. Olfati-Saber, J. A. Fax, and R. M. Murray. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1):215–233, 2007. [10] W. Ren. On consensus algorithms for double-integrator dynamics. IEEE Transactions on Automatic Control, 53(6):1503–1509, 2008. [11] W. Ren, R. W. Beard, and E. M. Atkins. Information consensus in multivehicle cooperative control. IEEE Control Systems Magazine, 27(2):71–82, 2007. [12] L. Scardovi and R. Sepulchre. Synchronization in networks of identical linear systems. Automatica, 45(11):2557–2562, 2009. [13] J. H. Seo, H. Shim, and J. Back. Consensus of high-order linear systems using dynamic output feedback compensator: Low gain approach. Automatica, 45(11):2659–2664, 2009. [14] S. E. Tuna. LQR-based coupling gain for synchronization of linear systems, 2008. [Online] http://arxiv.org/abs/0801.3390. [15] J. Wang, Z. Liu, and X. Hu. Consensus of high order linear multiagent systems using output error feedback. In Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pages 3685–3690, 2009. [16] P. Wieland, J.-S. Kim, H. Scheu, and F. Allgöwer. On consensus in multi-agent systems with linear high-order agents. In Proceedings of the 17th Triennial IFAC World Congress, pages 1541–1546, 2008. [17] P. Wieland, R. Sepulchre, and F. Allgöwer. An internal model principle is necessary and sufficient for linear output synchronization. Automatica, 47(5):1068–1074, 2011.