An Extended Consensus Algorithm for Multi-Agent ...

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

ThB16.6

An Extended Consensus Algorithm for Multi-Agent Systems Guisheng Zhai, Shohei Okuno, Joe Imae and Tomoaki Kobayashi

Abstract— In this paper, we study an extended consensus problem for multi-agent systems, where the entire system is decentralized in the sense that each agent can only obtain information (states or outputs) from its neighbor agents. The concept extended consensus means that a combination of each agent’s state elements is required to converge to the same vector. For this extended consensus problem, we propose to reduce the problem to a stabilization problem with an appropriate transformation, and thus obtain a strict matrix inequality with respect to a Lyapunov matrix and a structured controller gain matrix. We then utilize a homotopy based method for solving the matrix inequality effectively, and show validity of the result by an example. The feature of the present algorithm is that it can deal with various additional control requirements such as convergence rate specification and actuator limitations. Index Terms—Multi-agent systems, extended consensus, graph Laplacian, matrix inequality, LMI, homotopy method.

I. I NTRODUCTION For multi-agent systems, the notion consensus means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents [1]. The theoretical framework for posing and solving consensus problems for networked dynamic systems was introduced by Olfati-Saber and Murray in [2], [3] based on the earlier work of Fax and Murray [4], [5]. In recent years, there has been much interest in problems related to multi-agent systems with close relation to consensus problems, including collective behavior of flocks and swarms, sensor fusion, random networks, synchronization of coupled oscillators, formation control of multi robots, optimization-based cooperative control, etc. For more detailed information on this line, see the survey paper [1] and the references therein. Focusing on the basic consensus problem requiring that all agents’ states converge to the same vector, the well known existing method is to describe the agents’ interconnection structure as a directed graph and to use the graph Laplacian as a state feedback gain. However, to the best knowledge of the authors, such a Laplacian based method is generally limited to the case that each agent has low dimension and the control specification is simple. To deal with the case that each agent’s dimension is higher and additional requirement is desired such as convergence rate specification and/or actuator limitations, we need to establish alternative methods. For this purpose, the authors proposed a new approach to the basic consensus problem for multi-agent systems in a recent paper [6]. The idea is to reduce the consensus problem to a stabilization problem with an appropriate transformation, This research has been supported in part by the Japan Ministry of Education, Sciences and Culture under Grant-in-Aid for Scientific Research (C) 21560471. The authors are with the Department of Mechanical Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan; Corresponding e-mail: [email protected] (G. Zhai).

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

and then to solving a strict matrix inequality with respect to a Lyapunov matrix and a controller gain matrix, which is a necessary and sufficient condition for the consensus problem. The controller gain matrix has a structure constraint corresponding to the interconnection among the agents, and thus the matrix inequality is bilinear with respect to the variables. In that context, we proposed a homotopy based method for solving the matrix inequality effectively. It turns out that the algorithm in [6] includes the existing Laplacian based one as a special case, and can deal with various additional control requirements such as convergence rate specification and actuator limitations. The result in [6] was extended to the case of decentralized dynamic output feedback for the basic consensus problem in [7]. As also mentioned in [7], the idea of reducing the consensus problem to a stabilization one was also proposed in [8], but the existence condition and the controller design in [8] was not in the form of matrix inequality, and thus may not deal with the above-mentioned additional control requirements flexibly. In this paper, we extend the problem formulation and the results in [6] to an extended consensus problem for multi-agent systems. As in [6], [7], the entire system is decentralized in the sense that each agent can only obtain information (states or outputs) from its neighbor agents, and the interconnection among the agents is described by a directed graph. Different from the existing literature [1], [2], [3], [6], we consider the extended consensus problem which requires a combination of each agent’s state elements should converge to the same vector. As can be seen later, this problem setting includes the basic consensus problem as a special one, and can deal with various consensus problems such as partial state consensus. The design procedure is the same as in [6], We first reduce the consensus problem to a stabilization problem with an appropriate transformation, and then use the Lyapunov stability theory to obtain a strict matrix inequality with respect to a Lyapunov matrix and a controller gain matrix. Since the controller gain matrix has a structure constraint corresponding to the interconnection among the agents, the matrix inequality is bilinear with respect to the variables, and we utilize a homotopy based method for solving the matrix inequality effectively. The remainder of this paper is organized as follows. In Section II we give some preliminaries about graph theory together with the existing Laplacian based method for the basic consensus problem. Section III formulates the consensus problem, establishes the matrix inequality based method by reducing the problem to solving a matrix inequality with an algorithm, and then discusses how to solve the matrix inequality effectively. Section IV gives a numerical example to show validity of the proposed algorithm, and finally Section V concludes the paper.

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ThB16.6 II. P RELIMINARIES A. Graph Laplacian Let us review some basic definitions for consensus in a network. The interconnection structure of a family of agents can be represented by using a directed graph (or digraph) G = (V, E) with the set of nodes V = {1, · · · , N } and edges E ⊂ V × V. The edge (i, j) ∈ E or i → j means that the information of the i-th agent is available for the j-th agent. The neighbor agents set of the i-th agent is defined as n o △ Ni = j ∈ V (j, i) ∈ E (2.1)

which is the index set of the agents from which the i-th agent can obtain necessary information. Then, the graph Laplacian of the agents’ structure is defined as L = [lij ]N ×N , where  j ∈ Ni   −1 |Ni | j=i (2.2) lij =   0 otherwise

and |Ni | denotes the number of neighbors of the i-th agent (or in-degree of agent i). For example, using the above definition, the graph Laplacians of the structures in Fig. 1 are respectively     2 0 −1 −1 1 −1 0  −1  1 0 0  ,  0 1 −1  . (2.3)  0 −1 1 0  −1 0 1 0 0 −1 1

It is easy to see from the definition (2.2) that all row-sums of L are zero, and thus L always has a zero eigenvalue and a corresponding eigenvector 1 = [1 1 · · · 1]T . For other spectral properties of graph Laplacian, see for example [9], [10].

To solve the consensus problem, the well known existing method is to construct the control input as X (xj − xi ) (2.6) ui = j∈Ni

which is based on the idea of proportionally reducing the errors between two agents’ states. In fact, with the definitions x = [x1 x2 · · · xN ]T and u = [u1 u2 · · · uN ]T , the control input (2.6) can be compactly written as u = −Lx .

(2.7)

In this sense, we call (2.6) or (2.7) a graph Laplacian based method. The closed-loop system composed of (2.4) and (2.6) is x˙ = −Lx

(2.8)

and in the literature LaSalle’s Invariant Principle [11] is usually used to show that all states converge to the same value as in (2.5) . C. Kronecker Product A tool that is very useful in modeling and manipulating equations governing group motion is Kronecker product ⊗, which is defined between two matrices P = [pij ] and Q as P ⊗ Q = [pij Q] .

(2.9)

For example, if x˙ i = Axi represents the dynamics of a single agent, the dynamics of N same agents can be represented as x˙ = (IN ⊗ A)x. Another important case is if A is an N × N matrix representing the manipulation of scalar data from N agents, and that the manipulation needs to be applied to each value of a vector of length n. In that case, the manipulation can be represented by concatenating the N vectors of length n into a single vector of length N n, and multiplying it by A ⊗ In . The following property of Kronecker product (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)

(2.10)

can be proved easily when all matrix manipulation is wellposed. III. E XTENDED C ONSENSUS A LGORITHM A. Problem Formulation Consider the case where all agents have the same dynamics described as x˙ i = Axi + Bui (3.1)

Fig. 1 Directed Graph Examples

B. Consensus via Graph Laplacian Here, we review the existing graph Laplacian based method for the basic consensus problem. For simplicity, consider the case where all agents are integrators described by x˙ i = ui , xi ∈ R . (2.4) The basic consensus problem is to design the control input ui , depending on states of its neighbor agents, so that all agents’ states converge to the same value, i.e., lim |xi (t) − xj (t)| = 0 ,

t→∞

∀i 6= j .

where xi ∈ Rn is the state, ui ∈ Rm is the control input, and A, B are constant matrices of appropriate dimension. The discussion will be extended to the case where the agents have different dynamics later in Remark 6. Since it is known that consensus among agents is possible if and only if the interconnection graph includes a directed spanning tree [1], we assume that the interconnection graph of the present system also has this property. The existing basic consensus problem requires that the states xi converge to the same vector. Here, we extend the problem by considering consensus of the following vector

(2.5)

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zi = Exi , i = 1, · · · , N ,

(3.2)

ThB16.6 where zi ∈ Rr (r ≥ 1) is the consensus vector, E ∈ Rr×n is a constant matrix with full row rank. In other words, our problem is to design the decentralized state feedback in the form of X fb Kj xj + Kif b xi (3.3) ui = j∈Ni

where Kjf b ’s are gain matrices to be determined, so that all zi ’s converge to the same vector, or equivalently, lim kzi (t) − zj (t)k = 0 ,

t→∞

∀i 6= j .

(3.4)

Here, the symbol k·k denotes the Euclidean norm of a vector. Remark 1: When E = Ir with r = n, the above consensus   problem shrinks to the basic one. When E = Ir 0 with r < n, the above consensus problem can be regarded as the basic consensus problem for the first r elements of the state vector x. In other cases, it actually requires the consensus of a combination of each agent’s state elements. Thus, the above extended consensus problem is the extension of the basic consensus problem. To describe and deal with all the agents in a compact form, we write the entire system as x˙ = AD x + BD u ,

x ∈ RnN , u ∈ RmN

(3.5)

where x = [xT1 xT2 · · · xTN ]T ∈ RnN is the group state and u = [uT1 uT2 · · · uTN ]T ∈ RmN is the group control input, and AD = IN ⊗ A , BD = IN ⊗ B . The consensus vectors are also collected as z = ED x (3.6) T T ] ∈ RrN is the group consensus where z = [z1T z2T · · · zN vector and ED = IN ⊗ E .

B. Solvability Condition and Algorithm In this section, we propose a matrix inequality based method for the above-mentioned consensus problem. The basic idea is to reduce the consensus problem to a stabilization one. First, in order to consider the extended consensus problem, we need the dynamics equation of the consensus vector z. Differentiating (3.6) results in z˙ = ED x˙ = ED (AD x + BD u) .

and thus A¯ = M1−1 Aˆ11 M1, Aˆ12 = 0 with the definition Aˆ11 Aˆ12 M2−1 AM2 = . Although the requirement of Aˆ21 Aˆ22 A12 = 0 or Aˆ12 = 0 is a little restrictive at present, we made the assumption so that the extended consensus problem can be dealt with in the same framework as in [6], and we will make some discussion later on the possibility of relaxing Assumption 1. Under Assumption 1, the dynamics of z is rewritten as z˙ = A¯D z + ED BD u ,

¯ Notice that the control input has a where A¯D = IN ⊗ A. decentralized structure as in (3.3). In other words, the control input of the i-th agent only depends on states of its neighbor agents and itself. To meet this requirement, we propose the following control input ¯ C ED x , ¯ C z = KD L u = KD L

¯ C )z . z˙ = (A¯D + ED BD KD L

(3.12)

Note that the objective of designing KD is not to drive all zi ’s to zero, but to drive all zi ’s to the same vector. For this purpose, we define ¯C z , zC = L (3.13) which is based on the observation that if zC converges to zero, then the consensus is achieved among zi ’s. Therefore, the control problem is now reduced to considering the stability/stabilization of zC whose dynamics is described as ¯ C )z ¯ C (A¯D + ED BD KD L z˙C = L ¯ C ED BD KD zC . ¯ C A¯D z + L = L At this point, we need the following lemma. Lemma 1: ¯C . ¯ C A¯D = A¯D L L

(3.14)

(3.15)

Proof: Using (2.10), we obtain ¯ ¯ C A¯D = (L ⊗ Ir )(IN ⊗ A) L ¯ = (IN L) ⊗ (AI ¯ r) = (LIN ) ⊗ (Ir A) ¯C . ¯ = (IN ⊗ A)(L ⊗ Ir ) = A¯D L

To proceed, we make an assumption for the system. Assumption 1: There exists a constant matrix A¯ ∈ Rr×r satisfying ¯ . EA = AE (3.8)

¯ −1 )(M1 EM2 ) , (3.9) (M1 EM2 )(M2−1 AM2 ) = (M1 AM 1

(3.11)

¯ C = L ⊗ Ir , KD = diag{K1 , · · · , KN }, and where L Ki ∈ Rm×r (i = 1, · · · , N ) are gain matrices to be determined. It is easy to see that the controller (3.11) satisfies the requirement in (3.3). The closed-loop system composed of (3.10) and (3.11) is

(3.7)

Remark 2: When E = Ir with r = n (as mentioned in Remark 1), the above assumption always holds with A¯ = A. When considering partial  consensus problem with E =   A11 A12 Ir 0 , defining A = where A11 ∈ Rr×r A21 A22 ¯ A12 = 0. and substituting it in (3.8) results in A11 = A, More generally, since E is full row rank, we can always find matrices M1 and M2 such that M1 EM2 =  nonsingular  Ir 0 . Then, we obtain from the assumption that

(3.10)

(3.16)

Applying the above lemma to (3.14), we obtain the dynamic equation of zC as ¯ C ED BD KD )zC . z˙C = (A¯D + L

(3.17)

If all elements of zC are independent, we can use any existing design method (Lyapunov equation, matrix inequality, etc.) for the above system and obtain a solvability condition with respect to the unknown gain matrix KD . However, since ¯ C is not full rank either, and the matrix L is not full rank, L thus the elements of the vector zC are not independent. For example, consider the right graph structure in Fig. 1 with

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ThB16.6 all agents’ dynamics dimension being one and E = I for illustration. Then, n = 1, N = 3, z = x and     z1 − z2 1 −1 0 ¯C z =  0 1 −1  z =  z2 − z3  . zC = L z3 − z1 −1 0 1 (3.18) Obviously, z1 −z2 → 0 with z2 −z3 → 0 leads to z3 −z1 → 0, which means we only need to take care of two elements of the vector zC . Based on the above observation, we extract the linear ˜ Then, let L ˜C = independent rows of L and denote it by L. ˜ ˜ L ⊗ Ir , and z˜C = LC z. It is easy to see that z˜C is in fact a sub-vector of zC and the elements of z˜C are independent. Therefore, we focus our attention on stabilization of z˜C from now on. The dynamics equation of z˜C is ˜ C ED BD K ˜ D )˜ z˜˙ C = (A˜D + L zC ,

(3.19)

¯ and K ˜ D is computed from KD where A˜D = IN −1 ⊗ A, ¯ C . The procedure is as follows. ˜ C = KD L ˜DL satisfying K ˜ is extracted from L, suppose that the extracted row Since L number is j1 , j2 , · · · , jN −1 . Defining ei = the i-th column of an identity matrix, we obtain   eTj1  eT  △   j2 ˜ = FC L , FC = (3.20) L .   ···  eTjN −1

Then, according to (2.10),

˜C = L ˜ ⊗ Ir = (FC L) ⊗ (Ir × Ir ) L ¯C . = (FC ⊗ Ir ) (L ⊗ Ir ) = (FC ⊗ Ir ) L

(3.21)

˜ D is obtained from Thus, the relation between KD and K ˜ ˜ ¯ KD LC = KD LC that ¯C , ˜ D (FC ⊗ Ir ) L ¯ C = KD L K

(3.22)

˜ D . Note that the matrix LC is which we can use to obtain K singular and can not be canceled from the above equation. To see the above procedure more precisely, consider again the right interconnection graph structure in Fig. 1 with all agents’ dynamics dimension being one and E = I for ˜ C from LC and FC as illustration. We can easily obtain L     1 −1 0 1 0 0 ˜ LC = , FC = . (3.23) 0 1 −1 0 1 0 Since the original gain matrix is KD = diag{K1 , K2 , K3 }, where Ki ’s ∈ R, we obtain from (3.22) that     1 −1 0 1 0 0 ˜D  0 1 −1  K 0 1 0 −1 0 1   1 −1 0 ˜ ¯ (3.24) = KD = KD LC 0 1 −1   K1 −K1 0 0 K2 −K2  = −K3 0 K3

and thus 

˜D =  K 



 = 

  K1 −K1 0 1 1  0 K2 −K2  0 1  0 0 −K3 0  K3 K1 0  0 K2  . −K3 −K3

(3.25)

˜ D has different dimension from KD , but it Notice that K inherits all the variables in KD in a linear form. To summarize the above discussion, we have reduced the extended consensus problem to stabilizing the system (3.19) ˜ D . The original feedback gain by the feedback gain matrix K ˜ D and thus can be extracted matrices Ki are included in K easily. Applying the matrix inequality based Lyapunov stability theory for the closed-loop system (3.19), we obtain the following theorem immediately. Theorem 1: The controller (3.11) solves the extended consensus problem for the system (3.1) or (3.5) with (3.6) if and only if there is a positive definite matrix P satisfying ˜ D )T P ˜ C ED BD K (A˜D + L ˜D) < 0 . ˜ C ED BD K +P (A˜D + L

(3.26)

Remark 3: When E = I and thus ED = I, the condition (3.26) shrinks to the one established in [6], and thus Theorem 1 is the extension for the basic consensus problem to the extended one. As mentioned in [6], the existing Laplacian based method is a special case of the main theorem in [6], and thus is further a special case of the present theorem. Remark 4: As also remarked in [6], although the convergence rate issue is mentioned in the literature using the name of algebraic connectivity, there is no practical design method for it. In contrast, Theorem 1 can design the convergence rate of the agents’ vectors zi by for example specifying an appropriate positive scalar ζ in the condition (3.26) as ˜ C ED BD K ˜ D + ζI)T P (A˜D + L ˜ D + ζI) < 0 . ˜ C ED BD K +P (A˜D + L

(3.27)

Remark 5: Theorem 1 can also deal with actuator limitations directly by for example specifying an appropriate positive scalar γ and then constructing an additional linear matrix inequality (LMI) # " ˜D γI K > 0. (3.28) ˜ T γI K D Remark 6: Although it is assumed in the problem formulation that all agents have the same dynamic differential equation, it can be seen from the above discussion that the result is the same even if Bi ’s are different. In that case, BD = diag{B1 , · · · , BN }, where Bi is the control input matrix of the i-th agent. Concerning the system matrix, it ¯C ¯ C A¯D = A¯D L can be relaxed to the assumption that L holds with A¯D = diag{A¯1 , · · · , A¯N }, where A¯i is the matrix

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ThB16.6 satisfying EAi = A¯i E and Ai is the system matrix of the i-th agent. Remark 7: In the case where the interconnection of the agents may change among several structures (graphs), we denote the graph Laplacian matrix of each possible structure by ˜ (1) , · · · , L ˜ (M ) correspondingly. L(1) , · · · , L(M ) , and obtain L C C Then, to design a fixed decentralized controller (3.11) solving the extended consensus problem on hand with varying interconnection structure, one method is to solve the matrix inequalities ˜ (i) ED BD K ˜ D )T P (A˜D + L C ˜D) < 0 . ˜ (i) ED BD K +P (A˜D + L

(3.29)

C

˜ D and P > 0 for i = simultaneously with respect to K 1, · · · , M . We now summarize the consensus algorithm as follows. Extended Consensus Algorithm: Step 1 Extract the linear independent rows of L and ˜ Let L ¯ C = L ⊗ Ir , L ˜C = L ˜ ⊗ Ir , denote it by L. ˜ ¯ AD = IN −1 ⊗ A. ¯ C or K ˜ D (FC ⊗ Ir ) L ¯C = ˜ C = KD L ˜DL Step 2 Solve K ¯ C to obtain K ˜ D including the unknown feedKD L back gains as parameters. Step 3 Solve the matrix inequality (3.26) or (3.27) (and ˜ D and (3.28) when necessary) with respect to K P > 0. ˜D. Step 4 Extract the controller gain Ki ’s from K In the end of this subsection, we discuss the possibility of relaxing Assumption 1. First, we show there always exist matrices A¯ ∈ Rr×r and G ∈ Rr×n such that ¯ + G. EA = AE

(3.30)

In fact, using the same matrices M1 , M2 and defining the same matrix M2−1 AM2 as in Remark 2, we can easily obtain that (3.30) holds with   A¯ = M1−1 Aˆ11 M1 , G = M1−1 0 Aˆ12 M2−1 . (3.31)

Next, using (3.30), we obtain the dynamic equation of z as z˙ = A¯D z + ED BD u + GD x ,

(3.32)

where GD = IN ⊗G. Now, the difficulty is how to delete the part of GD x and obtain an autonomous differential equation of z. This is possible in some cases. For example, if the matrix ED BD is a nonsingular matrix, which tacitly assumes that the control input dimension m is equal to the consensus vector dimension r, then defining a new control input u ¯ = u + (ED BD )−1 GD x

(3.33)

z˙ = A¯D z + ED BD u ¯,

(3.34)

leads to for which we can make the same discussion as before. Similar idea can be used in the case where ED BD is full row rank, but it does not work for the case where ED BD is neither nonsingular nor full row rank.

C. Discussion on Solving (3.26) Theorem 1 gives a necessary and sufficient condition under which the extended consensus problem is solved. However, the matrix inequality (3.26) is a bilinear matrix inequality ˜ D and P > 0. If there (BMI) with respect to the variables K ˜ is no constraint on KD , we can transform (3.26) into an LMI ˜D equivalently [12]. This is not the case right now, since K has fixed structure as discussed in the previous subsection. Thus, as pointed out in the literature, it is generally difficult to solve (3.26) globally and effectively. Here, we first propose a two stage method for solving (3.26). Although A˜D is not stable, we can always find a positive scalar λ such that A˜D − λI is stable. For example, we can choose λ larger than the largest real part of A˜D ’s eigenvalues, i.e., λ > max{Reλ(A˜D )}. Then, there exists a positive definite matrix Pλ satisfying (A˜D − λI)T Pλ + Pλ (A˜D − λI) < 0 . (3.35) The next stage is to solve (3.26) with P fixed as Pλ , i.e., ˜ D )T Pλ ˜ C ED BD K (A˜D + L (3.36) ˜ C ED BD K ˜D) < 0 +Pλ (A˜D + L ˜ D and thus easily solved which is an LMI with respect to K by any existing LMI software such as LMI Control Toolbox in Matlab [13]. If the above method does not provide a solution, it means that we need to consider how to fix the variable P so that the resultant LMI is feasible. For this purpose, we propose to use the homotopy based method in [14]. More precisely, we introduce a real number µ varying from 0 to 1 and define a matrix function △ ˜ D , µ) = ˜D) F (P, K (1 − µ)F1 (P ) + µF2 (P, K

(3.37)

where F1 (P ) = (A˜D − λI)T P + P (A˜D − λI) ˜ D ) = (A˜D + L ˜ C ED BD K ˜ D )T P F2 (P, K ˜D) ˜ C ED BD K +P (A˜D + L and λ is computed as in the above. Then, ( F1 (P ) ˜ D , µ) = F (P, K the left side of (3.26)

µ=0 µ=1

(3.38)

(3.39)

and the problem of finding a solution to (3.26) is embedded in the parametrized family of problems ˜ D , µ) < 0 , µ ∈ [0, 1] . F (P, K (3.40) Next, we start solving (3.40) with µ = 0, which is very easy by using the LMI software. Then, we increase µ gradually (for example, let µ = k/M (k = 1, 2, · · · , M ) with a large ˜D integer M ) and solve (3.40) gradually with P and K being fixed alternatively, until µ reaches 1. For more detailed description of the homotopy based algorithm for solving BMIs, refer to [14]. Remark 8: Although for simplicity we used λI in (3.35) and (3.38) which makes A˜D − λI stable, it can be replaced by any matrix W provided that A˜D − W is stable. In that ˜D) case, F1 (P ) = (A˜D −W )T P +P (A˜D −W ), and F2 (P, K is the same.

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ThB16.6 IV. N UMERICAL E XAMPLE We provide an example showing effectiveness of the condition and the algorithm. Example 1: Consider the right interconnection graph structure in Fig. 1 with all agents’ dynamics being a double integrator, described as        x˙ i1 0 1 xi1 0 = + ui , (4.1) x˙ i2 0 0 xi2 1 and the consensus vector being zi =



0 1





xi1 xi2



.

(4.2)

Obviously, the extended consensus problem here requires that the agents’ velocities converge to the same value, which is a very practical setting in real applications. 8

(5)

6

(1) : (2) : (3) : (4) : (5) : (6) :

4

xi−xj

(6) 2 0

x11− x21 x −x 12 22 x21− x31 x22− x32 x31− x11 x −x 32

12

(4) (2)

With the above feedback gains in the decentralized state feedback controller (3.11) , the elements of xi − xj ∈ R2 (i, j = 1, 2, 3, i 6= j) in the closed-loop system are described in Fig. 2, which shows that the extended consensus among all agents has been achieved, since |zi − zj | = |xi2 − xj2 | → 0, ∀i, j = 1, 2, 3, i 6= j. The control inputs of all agents are depicted in Fig. 3. V. C ONCLUSION In this paper, we have extended the basic consensus problem by requiring a combination of each agent’s state elements to converge to the same vector. For this extended consensus problem, we have proposed to reduce the consensus problem on hand to a stabilization problem with an appropriate transformation. Then, we used the Lyapunov stability theory for the stabilization problem to obtain a strict matrix inequality with respect to a Lyapunov matrix and a structured controller gain matrix, and furthermore proposed a homotopy based method for solving the matrix inequality effectively. The discussion is an important extension of our previous paper [6] and the existing graph Laplacian method, and the results are practical since they can deal with various additional control requirements such as convergence rate specification and actuator limitations.

(3)

−2

R EFERENCES −4

(1) −6 0

2

4

6

8

10

t[sec]

Fig. 2 Consensus Achieved in Example 1 2

(1) : u1 (2) : u 2 (3) : u3

1.5

(1) 1 0.5

(2)

u

0 −0.5

(3) −1 −1.5 −2 −2.5 0

2

4

6

8

10

t[sec]

Fig. 3 Control Inputs of All Agents in Example 1

It is easy to confirm that Assumption 1 holds with A¯ = 0. Then, we compute the decentralized state feedback controller according to the algorithm proposed in the previous section, and additionally we incorporate the convergence rate by setting ζ = 1.0 in (3.27) and the actuator limitation by setting γ = 10 in (3.28). We first obtain λ = 1 such that A˜D − λI is stable, and then solve (3.27) and (3.28) by using the homotopy based method. As a result, we obtain the solution at M = 300 that   # " −0.99 0 59.82 10.61  ˜D =  0 −0.64  .(4.3) , K P =  10.61 66.92 0.54 0.54 ˜D Thus, the desired controller gain Ki ’s are extracted from K as K1 = −0.99 ,

K2 = −0.64 ,

K3 = −0.54 .

(4.4)

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