Dynamic Consensus and Formation: Fixed and Switching Topologies

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Dynamic Consensus and Formation: Fixed and Switching Topologies Jun Xu

∗

Tao Li ∗∗ Lihua Xie

∗∗∗

Kai Yew Lum ∗

∗

Temasek Laboratories, National University of Singapore, Singapore 117411.Email: {tslxuj,tsllumky}@nus.edu.sg ∗∗ Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Email: [email protected]. ∗∗∗ School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Email: [email protected] Abstract: This paper addresses the dynamic consensus and formation problems of multi-agent systems for both leader-less and leader-follower cases under fixed and switched topologies. Necessary and/or sufficient conditions are presented in terms of graph topology, detectability, stabilizability, rank condition and Lyapunov inequality constraints. These conditions explicitly reveal how the intrinsic dynamics of the agents and the communication topology affect consensusability and formationability. In addition, several constructive procedures for protocol design are proposed to achieve consensus and formation. In particular, the so-called separation principle is established, which simplifies the design procedure greatly. Keywords: Co-operative control, Convergence analysis, Agents 1. INTRODUCTION

based essentially. All these DOFC-based protocols are only for fixed topologies.

In recent years, the interplay between control and comminations has attracted increasing attention. It is well-known now that the communication constraints and information flow can have significant influence on the performance of networked control systems, especially for networked multi-agent systems. The consensus problem is one of the fundamental problems for networked multi-agent systems [Ren and Beard, 2008, OlfatiSaber et al., 2006, Li and Zhang, 2009]. In [Olfati-Saber et al., 2006, Olfati-Saber and Murray, 2004, Ren and Beard, 2008, Ma, 2009], static consensus protocols for continuous-time systems are presented using a static state feedback control. For discrete-time counterpart, You and Xie [2010a] give a necessary and sufficient condition for consensusability.

In this paper, we will further consider the necessary and sufficient conditions of consensus and formation using generalized dynamic output feedback control with different information levels. Our work contains all these existing situations as special cases. Meanwhile, protocols with different properties are designed using separation principle. In addition, time-varying topologies are also considered.

However, when not all the states are available, the output feedback control has to be considered. A sufficient condition with mild assumptions for multi-agent systems with static output feedback control (SOFC) to reach consensus is presented in [Ma, 2009]. In [Wang et al., 2009], a sufficient condition for SISO structure is presented. Note that the applicability of the SOFC-based protocols is inherently limited due to their existence conditions and design procedures, e.g., the stabilibility and detectability generally cannot guarantee the existence of a SOFC (thus more difficult for simultaneous stabilization) and the design procedure is usually BMI (blinear matrix inequality)-based. Hence, dynamic output feedback control (DOFC)-based protocols have been studied in the literature [Fax and Murray, 2004, Li et al., 2010, Seo et al., 2009]. In [Fax and Murray, 2004], a necessary and sufficient condition using simultaneous stabilization for a local controller to stabilize a certain formation dynamics is given. An observer-based controller is proposed in [Li et al., 2010], where a separation principle-like condition is presented. In [Seo et al., 2009, Wang et al., 2009, You and Xie, 2010b], the DOFC is also observerCopyright by the International Federation of Automatic Control (IFAC)

The notation in this paper is standard. R and C denote the real and complex space, respectively. AT (A∗ ) is the (conjugate) transpose of A. A > 0 (A ≥ 0) means that A is positive definite (semidefinite). A⊥ is an orthogonal complement of A, i.e., the basis of the null space of AT , (A⊥ )T A = 0. ∅ is the empty set.  is the pure imaginary number such that 2 = −1. 1 is a vector with ones as its entries. ⊗ is the symbol for the Kronecker product. Lg ∈ Rn×n is the Laplacian matrix of a digraph G (0-1 digraph) with n vertexes. Denote the adjacency matrix of G as Ag = [aij ] ∈ Rn×n and the in-degree matrix as D. λi is the eigenvalue of Lg . More properties of graphs are listed in Appendix and [Ma, 2009, Ren and Beard, 2008]. 2. DYNAMIC CONSENSUS AND FORMATION 2.1 Problem Statement Consider the following multi-agent system: x˙ i = Axi + Bui

(1)

yi = Cxi , i = 1, 2, ..., n nx

r

m

(2)

where xi ∈ R ui ∈ R , yi ∈ R are the state, control input and output of the agent i, respectively.The communication topology among agents is represented by a directed graph G, which consists of a node set V = {1, 2, ..., n} and an edge set

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E ∈ V × V. For the agent i, denote its neighbor set as N (i). The consensus and formation are defined as follows. For the system (1)-(2), if there exists a u(t) ∈ U, where U is a set of admissible control inputs, such that lim kxj (t) − xi (t)k = 0, i, j = 1, ..., n

where 

δ2 ..  δ= . δn

t→∞

A=

then we say that the system asymptotically reaches a consensus. Similarly, if there exists a u(t) ∈ U, such that lim kxj (t) − xi (t) − (hj − hi )k = 0, i, j = 1, ..., n where H = is a given formation, then we say that the system asymptotically reaches a formation H. The purpose of this paper is to study consensus and formation using the following distributed DOFC-based consensus protocol for the agent i, i ∈ V: aij (ξj − ξi ) + H

j=1

ui = Sξi + Ryi + L

n X

aij (ξj − ξi ) + K

j=1

n X

aij (yj − yi )

aij (yj − yi )

A + BRC BS NC M



 , ςi = 

j=1

where M, N, L, K, S, R, V, H are gain matrices with proper dimensions to be determined and ξi is the internal state. When all the gain matrices are non-zero matrices, it means that the protocol of the agent i uses full local information from itself (yi and ξi ) and its neighbors (including the external output information yj and the internal state information ξj , j ∈ N (i)). However, there are some cases that not all local information can be used for design. For example, suppose that yi presents the location information of the agent i via GPS signals. When the GPS is not available, yi may be unknown. In this case, we may only have the relative information yi − yj (e.g., distance) through some on-board sensors. Thus N and R are zero matrices. We will present some basic properties of the DOFC-based consensus protocols using full local information in Subsection 2.2 and consider partial local information cases in Subsection 2.3. 2.2 Dynamic Consensus with Full Local Information Theorem 1. Suppose that G contains a spanning tree. There exists a protocol (3)-(4) such that a consensus is asymptotically reached for all initial states if and only if Kd 6= ∅, where ¯ : A¯ + B¯K ¯ C¯i is Hurwitz, i = 2, ..., n.} (5) Kd = {K where       A0 B 0 R S A¯ = , B¯ = , K¯1 = , (6) 0 0 0 I N M     C 0 K L ˆ ¯ (7) , C= K2 = 0 I H V     ˆ Cˆ ¯ = [K¯1 , K¯2 ], C¯ = C , C¯i = . (8) K λi Cˆ Cˆ Proof: Define

δi ζi







ς2 ..  , ς= . ςn

BKC BL HC V

n3

···

···

−a3n

n X

anj

j=1

Since the Laplacian matrix is Lg = 

,



··· ···





−a2n

X n

(3)

(4)



,B =

n2

j=1

n X



ζ2  , ζ =  .. . ζn

a2j −a23   j=1    n X  a12  −a32 a2j . α =  ..  , L =   j=1  ··· a1n    −a −a

(hT1 , ..., hTn )T

n X



X n

t→∞

ξ˙i = M ξi + N yi + V



          

a1j −αT

j=1

−L · 1

L

(12)



, we

can easily deduce that 1) the remaining n − 1 eigenvalues of Lg (except 0) are determined by L + 1 · αT ; 2) there exist a Jordan matrix J and an invertible matrix T such that T −1 (L + 1 · αT )T = J. Thus (T ⊗ I)−1 [I ⊗ A − (L + 1 · αT ) ⊗ B](T ⊗ I) = I ⊗ A − J ⊗ B. That means the eigenvalues ¯ C¯i . Note of [I ⊗ A − (L + 1 · αT ) ⊗ B] is governed by A¯ + B¯K that the choice of the difference between vector indices i and 1 is without loss of generality. Hence, the consensus is reached if ¯ C¯i is Hurwitz for a certain K, ¯ i = 2, ..., n. and only if A¯ + B¯K ♦ Theorem 2. If G contains any spanning tree, then for a given ¯ ∈ Kd , we have K   v1 (0)   xi  . T A  . (13) v(t) → (1r ⊗ e )  .  , t → ∞, υi = ξi vn (0)

where rT = [r1 , ..., rn ] ∈ Rn is the left eigenvector of Lg associated with the eigenvalue 0, satisfying rT 1 = 1. Proof: Define 

υ1 υ =  ... υn



 , T¯ = [1 ♯] ∈ Rn×n , T¯−1 = T¯ −1 Lg T¯ = J¯ =



0 0 0 J¯d



rT ♯



,



where ♯ denotes the part we are not interested in and J¯d is a Jordan matrix with the non-zero eigenvalues of Lg as its diagonal entries. It is easy to observe that υ˙ = (I ⊗ A + Lg ⊗ B)υ, which gives the solution υ(t) = e(I⊗A+Lg ⊗B)t υ(0) ¯ = (T¯ ⊗ I)e(I⊗A+J ⊗B)t (T¯−1 ⊗  I)υ(0) At e 0 = (T¯ ⊗ I) (T¯−1 ⊗ I)υ(0) ¯ 0 e(I⊗A+Jd ⊗B)t (14) By Theorem 1, (I ⊗ A + J¯ ⊗ B) is Hurwitz,

δi = x1 − xi

(9)

ζi = ξ1 − ξi

(10)

υ(t) → (1rT ) ⊗ eA υ(0), t → ∞ Thus we complete the proof. ♦

ς˙ = [I ⊗ A − (L + 1 · αT ) ⊗ B]ς

(11)

Based on (14), we can see that the convergent rate is determined ¯ However, the convergence (consensus) value is not inby K.

Then

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(15)

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

¯ 2 from (15). Note that here K ¯ is a full variable fluenced by K matrix, thus we have the following result. ˆ d1 = {K ˆ 1 : A¯ + Theorem 3. Kd 6= ∅ if and only if K ˆ 1 Cˆ is Hurwitz} 6= ∅. B¯K ˆ + ¯ C¯i = (A¯ + B¯K¯1 C) Proof: For necessity, note that A¯ + B¯K ˆ ¯¯ ˆ ¯ B( ¯ K¯1 +αi K¯2 )C)+β λi B¯K¯2 Cˆ = (A+ i B K2 C, where λi = αi + ¯ C¯i is Hurwitz βi . Based on Lemma 14 in Appendix, A¯ + B¯K ¯ ˆ ¯ ¯ ¯ only if (A + B(K1 + αi K2 )C) is Hurwitz. Since Kˆ1 is a full ˆ d1 6= ∅ implies that K ˆ d2 = {K ¯ : (A+ ¯ B( ¯ K¯1 + variable matrix, K ¯ ˆ ˆ αi K2 )C) is Hurwitz} 6= ∅. For sufficiency, since Kd1 6= ∅, we simply choose K¯1 = Kˆ1 and let K¯2 such that B¯K¯2 Cˆ sufficiently small (an extremely case is that K¯2 is the null of B¯ or CˆT ), so that Kd 6= ∅. Thus we complete the proof. ♦

topologies, as long as the switched topologies are known a priori, or at least the minimum no-zero eigenvalue of all Laplacian matrices is known a priori. Note that we may obtain some other conditions than (17)-(19) using different transformations. Now we consider some cases that only partial local information is used for design. First consider the case that N and R are zero, i.e., yi is unknown for the agent i. Corollary 5. The protocols (20) and (21) satisfy the condition of Corollary 4. n n X X aij χji aij (ξj − ξi ) + H ξi = Aξi + BL ui = L

A + BS1 + λi BL1 λi BL + BS 0 A − N C + BRC + λi (−HC + BKC)

i

j=1

aij χji

(21)

ui = Sξi where χji = yj − Cξj − yi + Cξi . Proof: The proof is similar to that of Corollary 4. Thus we omit the details. ♦



Note that the generalized DOFC protocol is reduced to observerbased protocol here, i.e, either (20), where ξi in (20) estimates xj − xi , or (21), where ξi in (21) estimates xi . In fact, (21) is suggested in Li et al. [2010] and [Wang et al., 2009] and (20) is proposed for discrete-time cases in [You and Xie, 2010a].

¯+ Proof: With the constraints (16) and  one of (17)-(19), A  A + λ BL λ BL + BS 1 i i ¯ C¯ is similar to the matrices , B¯i K 0 A − N1 C     A + BS1 λi BL + BS A + λi BL1 λi BL + BS , respective, 0 A−λ H C 0 A−λ H C 1

n X

(16)

¯ satisfies (16) Corollary 4. Assume that the controller gain K and one of the following additional constraints N = N1 + BR, H = BK, S1 = 0 (17) H = H1 + BK, N = BR, S1 = 0 (18) H = H1 + BK, N = BR, L1 = 0 (19) Then there exists a protocol (3)-(4) such that a consensus is asymptotically reached for all initial states if and only if 1) (A, B) is stabilizable; 2) (A, C) is detectable; and 3) if A is not Hurwitz, then G contains any spanning tree.

i

(20)

aij (ξj − ξi )

ξi = Aξi + BSξi + H

¯ C¯i is similar to the matrices Φi via Note that Ψi = A¯ + B¯K I 0 Π = −I I , i.e., Θi = ΠΨi Π−1 . Let

j=1

j=1

j1

2.3 Partial Local Information and Separation Principle

S = S1 − RC, L = L1 − KC, −A + N C + M − BS1 = 0, HC − BL1 + V = 0 Then Θi can be simplified as: 

n X

1

ly. There are two situations. When A is Hurwitz, the problem is trivial. Now consider the case A is not Hurwitz. In fact, according to Lemmas 12 and 15 in Appendix, we can conclude that Kd 6= ∅ if and only if 1)-3) hold. Now we only need to prove that G constrains a spanning tree if there exists at least an eigenvalue Reλ(A) ≥ 0. If G has no spanning tree when there exist Reλ(A) ≥ 0, then there are at least two eigenvalues with Re(λi ) = 0. For such a λi , i ≥ 2, A − λi H1 C (or A + λi BL1 ) is Hurwitz, A must be Hurwitz too, which contradicts the assumption. Thus we can draw the conclusion. ♦ Remark 1. The results stated in Corollary 4 together with Lemma 15 in Appendix are interesting. First, it makes the design satisfy the so-called separation principle. For example, under the constraint (16) and (17), we can design the gains L1 and N1 separately. L1 can be designed using the procedure introduced in the proof of Lemma 15. For N1 , simply choose one such that A − N1 C is stable. R and K can be arbitrarily assigned. And then L, N , S, H, M and V can be obtained accordingly. Second, using the consensability result in Corollary 4 we can easily derive a robust protocol for slowly switched connected

Second, we consider the case that besides N = 0, R = 0, the gains V and L are also zero matrices, i.e., both yi and ξj −ξi are unknown. Note that this can reduce the communication load. If (20) or (21) is used, then the consensusability is achieved only if A is stable. When A is not stable, a stronger condition may be enforced. For example, if further S = 0 (or H = 0), then ¯ = {L : A + λi BLC is Hurwitz, i = 2, ..., n} 6= ∅ is the K necessary and sufficient condition for Kd 6= ∅ (in this case, M must be stable). There are some other interesting cases. For example, in the case that yj − yi is unknown, if K = 0, R = 0, H = 0, S = 0, M = A − N C, V = BL, then the separation principle holds. If one of the following conditions holds: 1) R = 0, N = 0, H = 0, S = 0, M = A, V − BKC − BL = 0; 2) R = 0, N = 0, H = 0, L = 0, M − A − BS = 0, V = BKC,then Kd 6= ∅ if and only if L¯ 6= ∅. For the case of V = 0, L = 0, Fax and ¯ 1 is zero Murray [2004] provides some similar results. When K matrix, the problem is similar to static output protocols and can be partially solved using Lemma 16. Remark 2. Now a critical question is that what the benefits of using more information for the design are. First, we can see that the existence condition for consensusability and formationability with less information may be stronger than that with more information, e.g., the existence condition of the aforementioned protocols with N = 0, R = 0, L = 0 and V = 0 is stronger than that of (20) and (21). Second, the extra variables in the formulation may bring more freedom in pole/eigenstructure assignment, thus the system gains a fast convergent rate, which can be easily observed by (14). Third, the consensus value may be adjusted. For example, using the protocols (21) such that Kd 6= 0, xi → (rT ⊗ eA )X(0) when t → ∞, where X(0) = [x1 (0)T , ..., xn (0)T ]T . If we add a R 6= 0 to the

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

protocols (21) in (3)-(4), then xi → (rT ⊗eA+BRC )X(0) when t → ∞. 2.4 Dynamic Formation When the information exchange of internal state is allowed, we design the formation protocol as follows: ξ˙i = M (ξi − hi ) + V

n X

aij ζ¯ji + N (yi − Chi ) + H

n X

aij ζ¯ji + R(yi − Chi ) + K

j=1

ui = S(ξi − hi ) + L

n X

aij χji(22)

n X

aij χji(23)

then dii = 1, otherwise dii = 0. Denote the new graph as G¯ which contains the leader agent 0 and following agents 1, ..., n. Denote the Laplacian matrix of G¯ as L¯g . Then   Lg + D −D1 (28) L¯g = 0 0 Consider the DOFC-based consensus protocol using the relative information as follows: ξ˙i = M ξi − V di ξi + V

j=1

j=1

+H

j=1

Proof: Define (25) (26)

Then

¯˙ς = [In−1 ⊗ A − (L + 1  · αT ) ⊗ BK]¯ ς    h2 − h1 (27) A   .. + In−1 ⊗   . 0 hn − h1

T T T , ς¯ = ς¯2T · · · ς¯nT . From (27), we where ς¯i = δ¯iT ζ¯1i know that besides the conditions for the consensus in Theorem 1, A(hi − hj ), i, j = 1, 2, ..., n must be zero by considering the particularity of (25)-(26). The remaining is similar to that of Theorem 1. Thus we omit the details. ♦ Remark 3. The additional equation constraint (24) seems to be strange. However, it is automatically satisfied for the doubleintegrator dynamics discussed in [Ren and Beard, 2008]. We may also consider the choice of the controller gains for different formability. Note that besides the choice of the controller gains, whether the multiplication of the gain and state include the item hi also affects the formability. In fact, if we slightly change (23) as n n X X aij χji , aij ζ¯ji + K ui = Sξi + R(yi − Chi ) + L j=1

aij (ξj − ξi )

j=1

(29)

aij (yj − yi ) + Hdi (y0 − yi )

j=1

where ζ¯ji = ξj − ξi − hj + hi and χji is defined in (20). Theorem 6. Suppose that G contains a spanning tree. Consider the system (1)-(2). There exists a protocol (22)-(23) such that the prescribed formation is asymptotically reached for all initial states if and only if Kd 6= ∅ and A(hi − hj ) = 0, i, j = 1, ..., n (24)

δ¯i = x1 − xi − h1 + hi ζ¯1i = ξ1 − ξi − h1 + hi

n X

n X

j=1

then the formability is equivalent to Kd 6= ∅ and (A+BS)(hi − hj ) = 0, i, j = 1, ..., n. Thus it is no doubt under certain circumstance, we may relax the constraint,by importing the gains into the equation constraints via changing the protocols. Remark 4. When there exists a leader with the following dynamics: x˙ 0 = Ax0 y0 = Cx0 and the other agents are followers with dynamics (1)-(2), the consensus and formation problem can be defined similarly. Note that in this case, the topology is slightly different from previous one. Define a diagonal matrix D ∈ Rn×n with its diagonal entries dii = 1 or, 0, i.e., if the arc (0, i) exists,

u = Sξi − Ldi ξi + L +K

n X

n X

aij (ξj − ξi )

j=1

(30)

aij (yj − yi ) + Kdi (y0 − yi ).

j=1

Without giving proofs, we present some results here. Theorem 7. Suppose that G¯ contains a spanning tree rooted at the node 0. There exists a protocol (29)-(30) such that a leader-follower consensus is asymptotically reached for any ¯ d 6= ∅, where K ¯ ˜ ¯ initial condition if and only if K  : A+  d = {K

¯ KC ˜ i is Hurwitz, i = 1, ..., n}, K ˜ = [K˜1 , K˜1 ], K˜1 = B



0 −S 0 M

, K˜2 =

 −K L . λi , i = 1, ..., n are the eigenvalues of H = Lg +D. H −V

The separation principle also holds under some circumstances. Corollary 8. Suppose that G¯ contains a spanning tree rooted ˜ satisfies one of the at the node 0 and the controller gain K following conditions: K = 0, S = 0, M = A, V = BL − HC (31) K = 0, L = 0, M = A − BS, V = HC (32) Then there exists a protocol (29)-(30) such that a leaderfollower consensus is asymptotically reached for all initial states if and only if (A, B) is stabilizable, and (A, C) is detectable. 3. TIME-VARYING TOPOLOGIES When the topologies are time-varying, consider an infinite sequence of nonempty, bounded and contiguous time intervals [tk , tk+1 ), k = 0, 1, ..., with t0 = 0, tk+1 − tk ≤ T for some constant T > 0. Assume that for each [tk , tk+1 ), there is a sequence of non-overlapping subintervals mk −1 mk , tk+1 ), tk = t0k , tk+1 = [t0k , t1k+1 ), ..., [tjk , tj+1 k+1 ), ..., [tk j+1 j mk tk satisfying tk+1 − tk+1 ≥ τ , 0 ≤ j ≤ mk − 1 for some integer mk > 0 and a given constant τ > 0 such that G(t) is not j+1 changed for t ∈ [tjk , tj+1 k+1 ) and changed at time tk+1 . Index all G(t) in time sequence. Let S denote an index set for all graphs defined on the vertices {1, 2, ..., n} and S ⊂ S be the index set of actual graphs of all switched topologies of system dynamics. There is no doubt that S is a finite set. It is well-known that a switched system is asymptotically stable if all individual subsystems are asymptotically stable and the switching is sufficiently slow, i.e., τ is large enough, so as to allow the transient effects to dissipate after each switch [Liberzon, 2003]. If we assume that the dwell time τ is larger

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

than the lower-bound/critial dwell time as stated in Liberzon [2003]. Then there exists a protocol (20)-(21) such that a leader-less consensus is asymptotically reached for all initial states if (A, B) is stabilizable, and (A, C) is detectable, and the finite switching topology G(t) always contains a spanning tree. We can also obtain a similar result for leader-follower cases. However, this strong connection condition and slow switching condition are conservative. In the following, we relax the requirement by using the concept of the so-called joint spanning tree. We call that the graph G(t) for t ∈ [tk , tk+1 ) has a joint spanning tree if the union of the graphs G(t) for t ∈ [tjk , tj+1 k+1 ) has a spanning tree. Suppose for t ∈ [tk , tk+1 ), σ(t) ∈ S. The asson ¯ σ(t) 1 . ciated graph is Gσ(t) , whose components are Sσ(t) , ...., Sσ(t) Without loss of generality, we assume that there exist some nσ(t) 1 components Sσ(t) , ...., Sσ(t) , nσ(t) ≤ n ¯ σ(t) , whose Lj + T

1αj , j = 1, ..., nσ(t) , have positive real-parts eigenvalues, where Lj and αj are defined o similar to (12). Define ℓ(σ(t)) = n Snσ(t) j j ) is the index set of v : v ∈ j=1 V(Sσ(t) ) , where V(Sσ(t)

j all vertexes of the corresponding components Sσ(t) . The following lemma is straightforward, as if not, it will contradict the fact that G(t) contains a joint spanning tree. Lemma 9. If G(t) contains a joint spanning tree for each interS val [tk , tk+1 ), then t∈[tk ,tk+1 ) ℓ(σ(t)) = {1, ..., n}.

Before presenting the main result, a hypothesis is proposed: for any undirected (0-1) graph, the matrix L + 1 · αT is similar to a diagonal matrix. As the choice of elements of the Laplacian matrix Lg for a given dimension is finite (1 for connection and 0 for disconnection), it is possible to use greedy method to check n(n−1) all the 2 2 combinations. Simulation shows that for n=2 to 8, there is no exception. Theorem 10. Assume that G(t) is undirected and its corresponding L(t) + 1 · αT (t) is diagonable. There exists a protocol (20) (or (21)) such that a leader-less consensus is asymptotically reached for all initial states if (A, B) is stabilizable, and (A, C) is detectable, and the switching topology G(t) contains a joint spanning tree for each interval [tk , tk+1 ). Proof: We only consider the case (21). A constructive method similar to Lemma 15 is applied. Let λσı = ασı + βıσ be the eigenvalues of all Laplacian matrices Lg(σ) of system dynamics, σ(t) ∈ S. Arrange all the non-zero ασı for all σ(t) ∈ S into a set A. ¯ C¯l , Analogous to Corollary 5, we can see that Ψl = A¯ + B¯K A + BS BS . Construct the l ∈ S, is similar to Θl = 0 A − λ HC l

H using the method in Lemma 15, H = τ2¯ P −1 C T , where P is the solution of AT P + P A − τ C T C < 0, P > 0, τ > 0, τ¯ = ατ and α = min A. Thus, for certain Wl > 0, (A − λl HC)T P + P (A − λl HC) = −Wl < 0. Construct ¯ −1 , where Q ¯ is the solution of AQ ¯ + QA ¯ T − S = −B T Q T ¯ BB < 0, Q> 0. Now  the task is to find a common Lyapunov Q 0 0 φP

for all Θl such that the Lyapunov ¯ −1 . That inequality holds, i.e., PΘl + ΘT P < 0, where Q = Q matrix P =

> 0

l

is possible since Wl = Wc − φ1 QBSWl−1 S T B T Q < 0 where Wc = φ((A + BS)T Q + Q(A + BS)). By adjusting φ, i.e., making φ large enough, Wl < 0 always holds for ∀l ∈ S as S

is a finite set. Thus, PΘl + ΘTl P < 0 holds for ∀l ∈ S. Define ¯ l + ΨT P¯ = ΠT (PΘl + ΘT P)Π < 0. For P¯ = ΠT PΠ. PΨ l l ¯ l + ΨT P¯ + ϕI < 0. ∀l ∈ S, there exists a ϕ > 0 such that PΨ l There exist invertiable matrices Tl such that (Tl ⊗I)−1 [I ⊗A− T T (Ll +1·αl )⊗B](Tl ⊗I) = I ⊗Θl . TlT (Ll +1·αl )Tl = Λl = (1) (l ) (j) diag{λl , ..., λl n }, where λl are nonnegative real scalars. Thus we define ς¯ = (Tl ⊗ I)ς. Since Tl is invertible, instead of proving the stability of ς, we can prove the stability of ς¯. Now we can define the Lyapunov function for the switching system ς¯˙ similar to (11). ¯ ς V (t) = ς¯T (I ⊗ P)¯

Clearly, V (t) > 0 for ς¯ 6= 0.


¯ +(I ⊗ P)(I ¯ V˙ (t) = ς¯T (I ⊗ A − Λl ⊗ B)(I ⊗ P) ⊗ A − Λl ⊗ B) ς¯ X
T T ¯ T ¯ = ς¯ I ⊗ (P Ψl + Ψ P) ς¯ < −ϕ ς¯ ς¯j j

l

j∈ℓ(l)

Now we can use a similar technique introduced in [Ni and Cheng, 2010]. The main tools are Cauchy’s convergence criteria and Barbalat’s lemma [Khalil, 2002]. The extended version is available upon request. ♦ Remark 5. For the leader-follower case, suppose for t ∈ [tk , tk+1 ), σ(t) ∈ S¯ (S¯ is similarly defined as S) and the associated graph is G¯σ(t) . Assume that there are some componσ(t) 1 of G¯σ(t) satisfying 1) the leader agent 0 , ...., Sσ(t) nents Sσ(t) is included; and 2) Ljg(σ) + Dj , j = 1, ..., nσ(t) , have positive real-parts eigenvalues, where Ljg(σ) and Dj are defined similar n o Snσ(t) ¯ to (28). Define ℓ(σ(t)) = v:v∈ V(S j ) . Thus, if j=1

σ(t)

¯ contains a joint spanning tree rooted at the node 0 for each G(t) S ¯ = {0, 1, ..., n}. interval [tk , tk+1 ), then t∈[tk ,tk+1 ) ℓ(σ(t)) ¯ Theorem 11. Assume that G is undirected. There exists a protocol (29)-(30) under the equation constraint (31) or (32) such that a leader-follower consensus is asymptotically reached for all initial states if (A, B) is stabilizable, and (A, C) is detectable, ¯ contains a joint spanning tree and the switching topology G(t) rooted at 0 for each interval [tk , tk+1 ). 4. CONCLUSION In this paper, some necessary and sufficient conditions for dynamic consensability and formability have been presented. Most of these conditions were proved by constructive methods. We show that by the constructive methods, the design protocol can be used for not only the time-invariant communication topologies but also the time-varying communication topologies as long as these the minimum non-zero eigenvalue of the Laplacian matrices of these topologies are known a priori. Our future work will continue to study the properties of these protocols, such as the specified convergence region. We will also address the corresponding problems for the constrained input situation and partial consensus and formation situations. Appendix A. SOME USEFUL LEMMAS Lemma 12. Olfati-Saber and Murray [2004], Ma [2009] The Laplacian matrix Lg of a directed graph G has no eigenvalues with negative real part and at least one zero eigenvalue. Lg has only one zero eigenvalue if and only if G contains a spanning tree. In the latter case, assume that λi , i = 2, ..., n are the nonzero eigenvalues of Lg and 0 ≤ λ2 ≤ · · · ≤ λn .

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Lemma 13. The matrix H = Lg + D is positive stable (i.e.,its all eigenvalues have positive real-parts) if and only if G¯ contains a spanning tree rooting at node 0. If G¯ is undirected, the matrix H is positive stable if and only if G¯ is connected. Proof: By applying Lemma 12, we can conclude that G¯ contains a spanning tree if and only if the Laplacian matrix L¯g of G¯ has only one zero eigenvalue. Thus the eigenvalues of Lg + D have positive real-parts. Reversely,if Hhave eigenvalues of positive   x x , we can easily = λ real-parts, then based on L¯g 0 0 obtain the eigenvalue of L¯g . Meanwhile, node 0 should be the root. Otherwise, there exist more than one zero eigenvalues. ♦ Remark 6. A similar result with different notation has been stated in Hu and Hong [2007], where the sufficient and necessary ¯ condition is that the node 0 is the global reachable in G. Lemma 14. Boyd et al. [1994] For a complex matrix Φ, Φ < 0 if and only if   Re(Φ) Im(Φ) 0 A unified viewpoint. IEEE Transactions on Circuits and (A − BF )T P + P (A − BF ) < 0 (A.2) Systems I: Regular Papers, 57(1):213–224, 2010. D. Liberzon. Switching in systems and control. Birkh¨auser (A.2) can be restated as Boston, 2003. T Q(A − BF ) + (A − BF )Q < 0 (A.3) C. Q. Ma and J. F. Zhang, Necessary and sufficient conditions −1 where Q = P > 0. Note that (A.3) can be rewritten as for consensusability of linear multi-agent systems IEEE Trans. Autom. Control, 55 (2010), 1263-1268. QAT + AQ − (Y T B T + BY ) < 0 (A.4) by letting F Q = Y . By Finsler’s lemma Boyd et al. [1994], W. Ni and D. Cheng. Leader-following consensus of multiagent systems under fixed and switching topologies. Systems there exists a matrix Y satisfying (A.4) if and only if there exists and Control Letters, 59:209–217, 2010. a scalar τ > 0 such that R. Olfati-Saber and R. M. Murray. Consensus problems in T T QA + AQ − τ (BB ) < 0 (A.5) networks of agents with switching topolgy and time-delays. τ Now we choose τ¯ = α , where α = min{αi }, then τ¯αi ≤ τ . IEEE Trans. on Automatic Control, 49(9):1520–1533, 2004. R. Olfati-Saber, J. A. Fax, and R. M. Murray. Consensus and As BB T ≥ 0, (A.5) implies that cooperation in networked multi-agent systems. Processings T T QA + AQ − τ¯αi (BB ) < 0 of The IEEE, 95(1):215–233, 2006. Thus we can choose Y = τ2¯ B T , i.e., F = τ2¯ B T Q−1 . W. Ren and R. W. Beard. Distributed consensus in Multi-vehicle cooperative control: theory and applications. Now if βi 6= 0, we have Springer, 2008. 0 > Q(A − αi BF )T + (A − αi BF )Q J. H. Seo, H. Shim, and J. Back. Consensus of high-order linear = Q(A − (αi + βi )BF )∗ + (A − (αi + βi )BF )Q systems using dynamic output feedback compensator: low gain approach. Automatica, 45:2659–2664, 2009. i.e., (A − (αi + βi )BF )∗ P + P (A − (αi + βi )BF ) < 0. J. Wang, Z. Liu, and X. Hu. Consensus of high order linear Then we can conclude that (A, −λi B) is stabilizable. ♦ multi-agent systems using output error feedback. In Proc. Lemma 16. Suppose that G contains a spanning tree and there Conference on Decision and Control, pages 3685–3690, exist P > 0 and W such that one of the following conditions Shanghai, China, Dec 2009. holds. K. You and L. Xie. On consensusability of linear discrete-time T T T T T ⊥ A P − C W + P A − W C < 0, B P (C ) = 0 multi-agent systems and quantization effects. Accepted by −1 T IEEE Trans. on Automatic Control, 2010a. P A − B T W T + AP −1 − BW < 0, B ⊥ P −1 C T = 0 K. You and L. Xie. Coordination of discrete-time multi-agent Then K = {L : A + BLC is Hurwitz} 6= 0 if and only if systems via relative output feedback. Accepted by Interna¯ = {L : A + λi BLC is Hurwitz, i = 2, ..., n} 6= 0. K tional Journal of Robust and Nonlinear Control, 2010b. Proof: The proof is based on Corollary 1 in Geromel et al. [1996]. We omit the details here. 9193