:
if i 6¼ j and vj 2 Ni ,
vj 2Ni wij ,
0
if i ¼ j, otherwise.
It can be readily seen that in association with the matrix H, its off-diagonal elements are all negative or zero, and its row sums are all equal to zero. If the directed or undirected graph is unweighed, i.e. wij ¼ 1 for any j 2 Ni, then H degenerates into the Laplacian matrix L of the directed or undirected interconnection graph, and accordingly, (2) becomes x_ ¼ Lx:
ð3Þ
If x is n-dimensional, (2) would be changed into the form x_ ¼ ðH In Þx, where In is the n n identity matrix, and the symbol denotes the Kronecker product of matrices. If for any eij 2 E, the arc eji 2 E as well, the communication is said to be bidirectional. That is, if agent i can receive information from agent j, agent j can receive information from agent i as well. In this case, we use an edge (represented by —) to depict it; otherwise, the communication is said to be unidirectional and we use an arc (represented by !) to depict it. Assumption 1: The communication between leaders and followers is unidirectional, i.e. the leaders are unaffected by the followers whereas a follower may be influenced by leaders as well as other followers, and the leaders are governed by some exogenous control inputs which can drive the states of leaders to be arbitrary values. Rewrite the agents as (
4
i ¼ 1, . . . , N,
4
j ¼ 1, . . . , l:
yi ¼ xi , zj ¼ xNþj ,
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With y and z being the stack vectors of all followers yi and leaders zi, respectively, we can rewrite system (3) in the form y_ F R y 0 ¼ þ , z_ 0 L z u where F and L are N N and l l matrices corresponding to the indexes of followers and leaders, respectively; F, L and R are submatrices inherited from Laplacian matrix L, and u 2 Rl is the collection of exogenous control inputs for leaders. Then the dynamics of the followers corresponding to the y component can be extracted as
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y_ ¼ Fy Rz,
ð4Þ
with the control inputs being the leaders’ states. In what follows, to facilitate presentation, we first state the single time-delayed model. The multiple timedelayed one can be formulated in the same manner. It is assumed that the delay only affects the information transmitted from one agent to another, i.e. ii ¼ 0, i ¼ 1, . . . , N; and ij ¼ for any i 6¼ j (i, j ¼ 1, . . . , N ), where is a positive integer. It follows from (4) that y_ i ðtÞ ¼ fi1 y1 ðtÞ fi2 y2 ðtÞ fiN yN ðtÞ ri1 z1 ðtÞ ril zl ðtÞ, i ¼ 1, . . . , N, where the coefficients are the elements inherited from L. Accordingly, the dynamics with time-delays reads y_1 ðtÞ ¼ f11 y1 ðtÞ f12 y2 ðt Þ f1N yN ðt Þ r11 z1 ðtÞ r1l zl ðtÞ, .. . y_ N ðtÞ ¼ fN1 y1 ðt Þ fN2 y2 ðt Þ fNN yN ðtÞ rN1 z1 ðtÞ rNl zl ðtÞ,
ð5Þ
which can be written into the matrix form 2 32 2 3 3 y1 ðtÞ y_ 1 ðtÞ f11 6 76 6 7 7 6 76 y2 ðtÞ 7 6 y_ 2 ðtÞ 7 f22 6 76 6 7 7 76 . 7 6 . 7 ¼ 6 .. 6 7 6 .. 7 6 7 . . 4 54 . 5 4 5 fNN y_N ðtÞ yN ðtÞ 2 32 3 y1 ðt Þ 0 f12 f1N 6 76 7 6 f21 0 f2N 76 y2 ðt Þ 7 6 76 7 6 . 6 7 .. 7 .. .. .. 6 .. 7 6 7 . 54 . . . 4 5 fN1 fN2 0 yN ðt Þ RzðtÞ: Since L ¼ D A, it can be readily seen that F ¼ D A, where D and A are the N N matrix obtained from in-degree matrix D and unweighed
adjacency matrix A after deleting the last l rows and l columns, respectively. With control inputs being the leaders’ states, we have the model of multi-agent system with time-delay in state as below _ ¼ DyðtÞ þ Ayðt Þ RzðtÞ, yðtÞ
t 4 t0 :
ð6Þ
To state the problem clearly, we give the following definition for System (6). Definition 2 (Controllability with delay): System (6) is said to be controllable, if for any initial state y(t), t 2 [, t0] and any final state yf, there exist a finite time tf4t0 and an admissible input z(t) defined on [t0, tf ] such that y(tf) ¼ yf. With respect to multiple time-delays, it is assumed that ii ¼ 0, i ¼ 1, . . . , N; and there exists a positive integer max such that ij max for any i 6¼ j, (i, j ¼ 1, . . . , N ). In this case, communication delays may be different. It can be seen that all time-delays ij belong to the set {0, 1, 2, . . . , max}. Under this setup, the model with multiple time-delays in state reads _ ¼ ðD þ A0 Þ yðtÞ þ A1 yðt 1 Þ þ yðtÞ þ Amax yðt max Þ RzðtÞ, t 4 t0 ,
ð7Þ
where max4 max14 4 14 0 ¼ 0; the initial function y(t) ¼ ’(t) is given for t 2 [ max, t0]; Ak corresponds to the part of dynamics with time-delay k, and P max i ¼0 Ai ¼ A: Definition 3 (Controllability with multiple delays): System (1) is said to be controllable on [t0, tf ] if for any specified initial state y(t), t 2 [ max, t0] and any final state yf, there exist a finite time tf4t0 and an admissible input z(t) defined on [t0, tf ] such that y(tf) ¼ yf.
3.2 Double-integrator dynamics agents In this subsection, we consider the following continuous-time system of N þ l agents with double-integrator dynamics x_ i ¼ vi ,
v_i ¼ ui ,
i ¼ 1, . . . , N þ l,
ð8Þ
where xi 2 Rn and vi 2 Rn are, respectively, the position and velocity of agent i, and ui 2 Rn is the control input. We study the controllability problem under the following two protocols (Jiang, Wang, Xie, Ji, and Jia, 2009): one is with the feedbacks of relative velocities X X ðxi xj Þ k ðvi vj Þ, i 2 f1, . . . , N þ l g, ui ¼ j2Ni
j2Ni
ð9Þ
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International Journal of Control and the other is with the feedbacks of absolute velocity X ui ¼ ðxi xj Þ þ kvi , i 2 f1, . . . , N þ l g, ð10Þ
with the control inputs being the leaders’ states (positions). Then the double-integrator dynamics (13) with delay in state can be written as
j2Ni
where k 6¼ 0 is a feedback gain. The terminologies and notations appearing in this section have the same meaning as those in Section 3.1. So the System (8) with double-integrator dynamics under protocol (9) can be written in the form of 32 3 2 3 2 3 2 0 0 IN 0 0 y y_ 6 7 6 z_ 7 6 0 6 7 0 0 Il 7 6 z 7 6 0 7 7 6 7 6 76 7 þ 6 7, 6 7¼6 4 v_y 5 4 F R kF kR 54 vy 5 4 0 5
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v_z
0
L
0
u
vz
kL
where y ¼ ½xT1 , . . . , xTN T is the stacked vector of followers’ positions, z ¼ ½xTNþ1 , . . . , xTNþl T is the stacked vector of leaders’ positions, vy ¼ ½vT1 , . . . , vTN T and vz ¼ ½vTNþ1 , . . . , vTNþl T are the corresponding velocity vector and u 2 Rl is the collection of exogenous control inputs for leaders. Consequently, we can induce the dynamics of followers into the following LTI system: y y_ 0 IN 0 0 z ¼ þ , v_y R kR vz F kF vy ð11Þ with the control inputs being the leaders’ states (positions and velocities). By repeating the same arguments as those in Section 3.1, the doubleintegrator dynamics (11) with single time-delay in state reads _ yðtÞ yðtÞ 0 IN ¼ v_y ðtÞ D kD vy ðtÞ yðt Þ 0 0 þ A kA vy ðt Þ 0 0 zðtÞ þ , t 4 t0 : ð12Þ R kR vz ðtÞ If protocol (10) is employed, the closed-loop system of multi-agent system (8) is 2 3 2 32 3 2 3 0 0 IN y 0 0 y_ 6 6 z_ 7 6 0 7 6 7 0 0 Il 76 z 7 6 0 7 7 6 7 6 6 7¼6 76 7 þ 6 7, 4 v_y 5 4 F R kIN 0 54 vy 5 4 0 5 v_z
0
L
0
kIl
vz
u
where y, z, vy, vz, u are defined as above. This yields the following dynamics of followers: y y_ 0 IN 0 0 z ¼ þ , ð13Þ v_y F kIN vy R 0 vz
_ yðt Þ yðtÞ yðtÞ 0 IN 0 0 ¼ þ v_y ðtÞ A 0 vy ðt Þ D kIN vy ðtÞ 0 0 zðtÞ þ ð14Þ , t 4 t0 : R 0 vz ðtÞ
Definition 4: System (12) or (14) is said to be controllable if for any initial state ½ yT ðtÞ, vTy ðtÞ T , t 2 [, t0] and any final state ½ yT , vTy Tf , there exist a finite time tf4t0 and an admissible input ½zT ðtÞ, vTz ðtÞ T defined on [t0, tf ] such that ½ yT ðtf Þ, vTy ðtf Þ T ¼ ½ yT , vTy Tf . Under protocol (9), the model with multiple delays reads "
_ yðtÞ
"
# ¼
v_y ðtÞ
0
IN
#"
yðtÞ
#
vy ðtÞ D þ A0 kD þ kA0 " #" # yðt 1 Þ 0 0 þ þ vy ðt 1 Þ A1 kA1 " #" # yðt max Þ 0 0 þ vy ðt max Þ Amax kAmax " #" # 0 0 zðtÞ þ , t 4 t0 , R kR vz ðtÞ
ð15Þ
and under protocol (10), the model reads
_ yðtÞ yðtÞ 0 IN ¼ v_y ðtÞ D þ A0 kIN vy ðtÞ yðt 1 Þ 0 0 þ þ A1 0 vy ðt 1 Þ yðt max Þ 0 0 þ Amax 0 vy ðt max Þ 0 0 zðtÞ þ , t 4 t0 , R 0 vz ðtÞ
ð16Þ
where max4 max14 4 14 0 ¼ 0, the initial function y(t) ¼ (t) is given for t 2 [ max, t0], Ak corresponds to the part of dynamics with time-delay k and P max i ¼0 Ai ¼ A. Definition 5 (Controllability with multiple delays): System (15) or (16) is said to be controllable on [t0, tf ] if for any specified initial state ½ yT ðtÞ, vTy ðtÞ T , t 2 [ max, t0] and any final state ½ yT , vTy Tf , there exist a finite time tf4t0 and an admissible input ½zT ðtÞ, vTz ðtÞ T defined on [t0, tf ] such that ½ yT ðtf Þ, vTy ðtf Þ T ¼ ½ yT , vTy Tf .
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Z. Ji et al.
4. Main results 4.1 Controllability of the delayed system with single integrator dynamics It has been shown in Bellman and Cooke (1963) that the solution of (6) can be represented as Zt Kðt, sÞðRÞzðsÞds, yðt, t0 , ’, zÞ ¼ yðt, t0 , ’, 0Þ þ t0
where y(t, t0, ’, z) denotes the solution to (6) at time t corresponding to the initial time t0, y(t, t0, ’, 0) denotes the free solution and K(t, s) is the N N fundamental matrix of (6) which satisfies the following equations:
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(1) @K(t, s)/@s ¼ K(t, s)D K(t, s þ )A, s 2 [t0, t ] (2) K(t, t) ¼ I (3) K(t, s) ¼ 0, for all s4t. As presented in Chyung and Lee (1966), system (6) is controllable to the origin from time t0 if there existsR a finite value of time t14t0 such that t rank t01 Kðt1 , sÞRR0 K0 ðt1 , sÞds ¼ N, which is the controllable Gramian for the delayed system (6). Due to the difficulty of computing K(t, s), some algebraic criteria on controllability of linear systems with timedelay have been established in Chyung and Lee (1966), Kirillova and Curakova (1967), Weiss (1970), where a matrix only consisted with the coefficients of differential equations is constructed for the delayed system, and the delayed system is controllable if the matrix has full rank. So the controllability problem of the delayed system is transformed to the one without time-delay through judging the rank of the matrix. To formulate the problem clearly, we denominate the matrix as the controllable matrix. The following controllability matrix Q is constructed for the multi-agent system with state delay described by (6).
N
2 N Q ¼ Q11 , Q21 , . . . , QN 1 , Q2 , . . . , Q2 , . . . , QN ,
ð17Þ
where Q11 ¼ R, Qkj ¼ 0 for j4k or j ¼ 0, and Qkþ1 ¼ j DQkj þ AQkj1 , j ¼ 1, 2, . . . , k, k ¼ 1, 2, . . . , N. The controllability of delayed system (6) relies on the rank of controllability matrix Q. If Q has full row rank, i.e. rank(Q) ¼ N, system (6) is controllable. To simplify the computation for the rank of Q with l N(N þ 1)/2 columns, we will give sufficient conditions for the delayed system to be controllable. The following lemma will be used in the derivation of the result. Lemma 1 (Hewer 1972):
The identity
þ QNþ1 þ þ QNþ1 þ QNþ1 ðD þ AÞN ðRÞ ¼ QNþ1 1 2 N Nþ1 is valid.
If the communications between followers are bidirectional, they will be depicted by edges except those between leaders and followers. In this case, the interconnection graph is undirected and the following theorem can be derived to guarantee the delayed system to be controllable. Here the associated unweighed adjacent matrix A, in-degree matrix D and Laplacian matrix L are denoted by adding the inverseorientation links of unidirectional arcs between the leaders and followers of the original graph G. So, A, D and L are all symmetric. Theorem 1: The delayed multi-agent system (6), with undirected interconnection graph G, l leaders and N followers, is controllable for 40 if any of the following five conditions is fulfilled: (I)
(i) The eigenvalues of A are all distinct. (ii) The eigenvectors of A are not orthogonal to at least one column of R. A and A share no common eigenvalues. (i) The eigenvalues, i.e. the diagonal elements of D are all distinct. (ii) At least all the elements in one column of R are nonzero. (i) The eigenvalues of F are all distinct. (ii) The eigenvectors of F are not orthogonal to at least one column of R. L and F share no common eigenvalues.
(II) (III)
(IV)
(V)
Proof: We shall show that rank(Q) ¼ N under each of Conditions (I)–(V). Accordingly, system (6) is controllable since the controllability matrix Q is full row rank. Conditions (I) and (III): In view of the expression (17) for the controllability matrix Q, calculations show that N1 Q11 ¼ R, Q22 ¼ AR, . . . , QN R, N ¼ A
and N N1 R: Q11 ¼ R, Q21 ¼ DR, . . . , QN 1 ¼ ð1Þ D D
D
1 2 N Set Q1 ¼ ½Q11 , Q22 , . . . , QN N , Q2 ¼ ½Q1 , Q1 , . . . , Q1 , it follows that Q1 and Q2 are the controllability matrices of the following non-delayed systems, respectively,
_ ¼ AyðtÞ RzðtÞ, yðtÞ
ð18Þ
_ ¼ DyðtÞ RzðtÞ: yðtÞ
ð19Þ
The idea is that since Q consists of Q1 (or Q2) and other matrices, the controllability matrix Q is full row rank if so is for matrix Q1 (or Q2). The remaining argument in this part of proof is then to show that the non-delayed system (18) and (19) are controllable under Conditions (I) and (III), respectively, as follows.
377
International Journal of Control Let 1 2 N be the eigenvalues of A. Since A is symmetric, there exists an orthogonal matrix U such that A ¼ UUT, where U consists of the orthogonal eigenvectors of A and ¼ diag{1, 4 2 , . . . , N}. Denote UTR ¼ [r1 , . . . , rl] with 4 T ri ¼ [r1i , . . . , rNi] , i ¼ 1, . . . , l. Then Q1 ¼ U UT R, UT R, . . . , N1 UT R ¼ U ½r1 , . . . , rl , ½r1 , . . . , rl , . . . , N1 ½r1 , . . . , rl :
By repeating the same lines of arguments as those for the system (18) in the proof concerning Condition (I), it can be seen that system (19), and then system (6), is controllable if Condition (III) is fulfilled.
Consider the following matrix: e1 ¼ ½r1 , r1 , . . . , N1 r1 , . . . , ½rl , rl , . . . , N1 rl Q
With respect to the nondelayed system (18), the control input matrix R is obtained by deleting the last l rows and the first N columns of the Laplacian matrix L. This, together with the observation that L ¼ D A and D is a diagonal matrix, yields that A12 ¼ R. Noticing that A is symmetric since it is the adjacency matrix of the undirected interconnection graph G, we have A21 ¼ RT. Next, we shall prove that the non-delayed system (18) is controllable under Condition (II). We are to show this by contradiction. Suppose that system (18) is uncontrollable. It follows from the controllability PBH criteria that there exists a vector p 2 RN such that A p ¼ p for some 2 R, with RT p ¼ 0. Accordingly, q q q A R ¼ , A ¼ T 0 0 R A22 0
¼ ½diagfr11 , . . . , rN1 g, . . . , diagfr1l , . . . , rNl g ,
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ð20Þ where 2
1 6 .. ¼4.
1 .. .
1
N
3 N1 1 .. 7 . 5 N1 N
e1 is obtained by is a Vandermonde matrix. Since Q rearranging the columns of Q1 and U is an orthogonal matrix, we have e1 Þ: rankðQ1 Þ ¼ rankðQ
ð21Þ
Condition (II): Since A is obtained by deleting the last l rows and l columns of A, the adjacency matrix A can be partitioned as A A12 : A¼ A21 A22
By (i) of Condition (I), 1, . . . , N are all distinct. As a consequence, is nonsingular. By (ii) of Condition (I), it can be assumed, without loss of generality, that r1 is not orthogonal to the eigenvectors of A. Accordingly, r11, . . . , rN1 are nonzero and then diag{r11, . . . , rN1} is nonsingular. Therefore, it follows from (20) and (21) e1 and then Q1 is full row rank if Condition (I) is that Q fulfilled. So, under Condition (I), the controllability matrix Q is full row rank, that is, the state delayed system (6) is controllable. With respect to the Condition (III), we denote R ¼ [ 1, . . . , l]. Then Q2 can be written as Q2 ¼ ½1 , . .. , l ,D½1 ,. . . ,l , . .. , ð1ÞN DN1 ½1 , . .. , l ,
implying that is a common eigenvalue of A and A. Hence, the fulfilment of Condition (II) leads to the controllability of system (18). By following the arguments in the first part of proof for Condition (I), the original system (6) is then controllable.
e2 , where which has the same rank as Q e2 ¼ ½1 , D1 , . . . , ð1ÞN DN1 1 , . . . , Q ½l , Dl , . . . , ð1ÞN DN1 l e ..., ¼ diagf11 , . . . , ð1ÞN N1 g, e, diagf1l , . . . , ð1ÞN Nl g
Since elementary column operations do not affect the rank of a matrix, Q^ 3 has the same row rank with Q. By Lemma 1, we have Q^ 3 ¼ ðRÞ, ðD þ AÞðRÞ, . . . , ðD þ AÞN1 N ðRÞ, Q22 , . . . , QN 2 , . . . , QN ¼ R, FR, F2 R, . . . , ð1ÞN N FN1 R, Q22 , . . . , QN 2 , . . . , QN :
4
i ¼ [ 1i, . . . , Ni]T, i ¼ 1, . . . , l, and 2 3 1 d1 dN1 1 .. .. 7: . e¼6 4 .. . . 5 1 dN
dN1 N
Condition (IV): Consider the following matrix Q^ 3 derived from the controllability matrix Q of the delayed system (6) by elementary column operations: N N Q^ 3 ¼ Q11 , Q21 þ Q22 , . . . , QN 1 þ Q2 þ þ QN1 2 N N þ QN , Q , . . . , Q , . . . , Q N 2 2 N :
4
Note that Q3 ¼ [R, FR, . . . , (1)NFN1R] is the controllability matrix with respect to the following nondelayed system: _ ¼ FyðtÞ RzðtÞ: yðtÞ
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Z. Ji et al.
The controllability matrix Q is full row rank if so is matrix Q3. The remaining argument is a repetition of the same part in the proof of Condition (I), with the only difference being to replace A by F. Condition (V): The proof in this part can be carried on in the same way as that for Condition (II) by replacing A with F. The proof is completed. œ The advantage of Conditions (I)–(III) in Theorem 1 consists in the independence of matrices D and A.
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Corollary 1: The nondelayed system (4) with multiple leaders is controllable if the eigenvalues of F are all distinct and the eigenvectors of F are not orthogonal to at least one column of R. Proof: The assertion is a direct consequence of proof with respect to Condition (I) in Theorem 1. œ In practice, the information channels among agents are influenced by environment, obstacles along the trajectories of agents, etc. So, not all the communications between followers are bidirectional. Some agents could only send information to others or receive information from others. Accordingly, there only exist unidirectional communications between some followers. In this case, we use directed graph to describe the system. The following result is on the controllability under directed interconnection graph. Corollary 2: The delayed multi-agent system (6), with undirected interconnection graph G, l leaders and N followers, is controllable for 40 if any of the two matrices Q1, Q3 has full row rank or Condition (III) in Theorem 1 is fulfilled, where 2
N1
Q1 ¼ ½R, AR, A R, . . . , A 2
N
R ,
N1
Q3 ¼ ½R, FR, F R, . . . , ð1Þ F
R :
Proof: Set Q2 ¼ [R, DR, D2R , . . . , (1)NDN1R], it follows from the proof of Theorem 1 that the controllability matrix Q of the state delayed system (6) is full row rank if so is any of the matrices Q1, Q2 and Q3. Furthermore, if Condition (III) in Theorem 1 is fulfilled, Q2 is full row rank. Hence, the assertion holds. œ Since the solution to (7) can beR written in the form t of y(t, t0, ’, z) ¼ y(t, t0, ’, 0) þ t0 Kðt, sÞðRÞzðsÞds, the results in Buckalo (1968) can be employed to cope with the controllability problem with multiple time-delays. Theorem 2: The delayed multi-agent system (7), with undirected interconnection graph G, l leaders and N followers, is controllable for 140, . . . , max40 if the
following conditions are fulfilled: (I) (i) The eigenvalues, i.e. the diagonal elements of D þ A0 are all distinct. (ii) At least all the elements in one column of R are nonzero. (II) Amaxy(t max) Rz(t) 0 admits to a piecewise continuous solution z(t) for t 2 [t1, t1 þ max]. Proof: The matrix [R,(D þ A0)R,... , (D þ A0)N1R] has full row rank of N if (i) and (ii) of Condition (I) are fulfilled. The rest of the proof is a direct consequence of the result in Buckalo (1968). œ 4.2 Controllability of the delayed system with double integrator dynamics The terminologies and notations in this section have the same meanings as those in Section 4.1. The controllable matrices for the delayed systems (12) and (14) have the same structure as those for system (6). The only difference consists in the following specific forms of the associated matrices in the controllability matrix: 0 0 0 IN Q11 ¼ ¼ Qk , Qkþ1 j D kD j R kR 0 0 þ Qk ; A kA j1 0 0 0 IN 1 kþ1 Q1 ¼ , Qj ¼ Qkj R 0 D kIN 0 0 k þ Q : A 0 j1 Theorem 3: For the state-delayed system described by (12) or (14), with undirected interconnection graph G, l leaders and N followers, the dimension of the controllable subspace of system (12) or (14) is not less than N if the following conditions are both fulfilled: (i) The eigenvalues of A are all distinct. (ii) The eigenvectors of A are not orthogonal to at least one column of R. Proof: Consider the following nondelayed system:
_ yðtÞ yðtÞ 0 0 ¼ v_y ðtÞ A kA vy ðtÞ 0 0 zðtÞ þ , R kR vz ðtÞ
t 4 t0 :
ð22Þ
By following the same reasonings as those in the proof of Condition (I) in Theorem 1, we see that the controllable matrix of the nondelayed system (22) constitutes part of that of the state-delayed system (12). So the dimension of the controllable subspace
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International Journal of Control associated with system (12) is always greater than that of system (22). Computations show that the controllability matrix C1 of system (22) is C1 ¼
0
0
0
R kR kAR
0 k2 AR
Accordingly, the rank of C1 equals that of matrix e C1 with e C1 ¼ ½e C11 , ke C11 , where
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e C11 ¼ ½R, kAR, k2 A2 R, . . . , k2N1 A2N1 R : Since A is symmetric, it can be assumed that A ¼ UUT, with U being an orthogonal matrix. 4 4 Denote UTR ¼ [r1, . . . , rl] with ri ¼ [r1i, . . . , rNi]T, i ¼ 1, . . . , l. There exists a permutation matrix P such that e C11 ¼ U ½r1 , . . . , rl , k½r1 , . . . , rl , k2 2 ½r1 , . . . , rl , . . . , k2N1 2N1 ½r1 , . . . , rl ¼ U ½r1 , kr1 , . . . , k2N1 2N1 r1 , . . . , ½rl , krl , . . . , k2N1 2N1 rl h i e . . . , diagfr1l , . . . , rNl g e P, ¼ U diagfr11 , . . . , rN1 g, where 2
1 k1 6. D .. e ¼6 . . 4. 1 kN
3 k2N1 2N1 1 7 .. 7: . 5 2N1 2N1 k N
It can be seen from the preceding arguments that under C1 is full row rank, which Condition (I), e C11 and then e leads to rank C1 ¼ N. Consequently, the dimension of the state-delayed system (12) is not less than N under the two conditions. œ It can be seen from the proof of Theorem 3 that the velocities of the followers are controllable states. Theorem 4: The state-delayed system described by (12) or (14) with undirected interconnection graph G, l leaders and N followers, is controllable for any 40 if G1 has nonzero eigenvalues andL and F share no common 0 IN : eigenvalues, where G1 ¼ F kF 0 Proof: Denote H1 ¼ 0R kR , and consider the system y y_ z ¼ G1 þ H1 : ð23Þ vy v_y vz
By repeating the same lines of proof for Condition (IV) in Theorem 1, we see that system (12) is controllable if so is system (23). 0
0
0
k2N1 A2N1 R k2N A2N1 R
:
Next, we are to show that if G1 has nonzero eigenvalues and L and F share no common eigenvalues, system (23) is controllable. We show this by contradiction. Suppose system (23) is uncontrollable. By the controllability Popov–Belevitch–Hautus (PBH) criteria, there exist an eigenvalue of G1 and an associated left eigenvector [T, T] such that ½T , T G1 ¼ ½T , T
and
½T , T H1 ¼ 0:
The first equality of (24) gives rise to T F ¼ T , T IN kT F ¼ T :
ð24Þ
ð25Þ
Then 2 T ¼ T ð1 kÞF:
ð26Þ
We claim that 1 þ k 6¼ 0. Otherwise, if 1 þ k ¼ 0, it follows from 6¼ 0 and (26) that T ¼ 0. This, together with the second equality of (25) results in T ¼ 0, which contradicts to the fact that the eigenvector [T, T] is nonzero. Accordingly, one has TF ¼ T with 2 : So, is an eigenvalue of F. On the ¼ 1þk other hand, the second equality of (24) implies TR ¼ 0. We have F R ¼ ½T , 0 : ½T , 0T L ¼ ½T , 0T RT L22 That is, is also an eigenvalue of L. This contradicts to the assumption that L and F share no common eigenvalues. The above analysis shows that if G1 has nonzero eigenvalues and L and F share no common eigenvalues, system (23) and then system (12) is controllable. œ Remark 1: The latter part of proof in this theorem is inspired by the proof of Theorem 3 in Jiang et al. (2009). Next, we present a result on multiple time-delays. Theorem 5: The state delayed system described by (12) or (14), with undirected interconnection graph G, l leaders and N followers, is controllable for 140, . . . , max40 if the following conditions are fulfilled: (1) The matrix C~ ¼ , , N1 has full row rank.
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(2)
yðtmax Þ
zðtÞ
0 admits to a piecewise zðtÞ continuous solution vz ðtÞ for t 2 [t1, t1 þ max], vy ðtmax Þ
vz ðtÞ
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where for system (15), 0 IN ¼ , D þ A0 kD þ kA0 0 0 ¼ R kR 0 0 ; ¼ Ak kAk and for (16), 0 ¼ D þ A0 0 0 ¼ : A k 0
IN kIN
,
¼
get the indirect parent vertex of agents 3 and 4, which e4 ¼ f2, 3, 4g: So, e3 ¼ f2, 3, 4g and S are, respectively, S with respect to Figure 1, agents 3 and 4 have the same indirect neighbour set of parent vertices. At the same time, it can be readily seen that agents 1 and 2 have the same direct neighbour set of parent vertices, which is S1 ¼ S2 ¼ {5, 6}. Theorem 6: The multi-agent system with or without delay in state, described by (4) or (6), is uncontrollable if at least two followers have the same direct or indirect neighbour set of parent vertices.
0
0
R
0
,
Proof: The proof can be conducted in the same way as that in Theorem 2. œ The controllability problem has been studied hereinbefore for the delayed multi-agent system (12). The same idea can be employed in the investigation of controllability for system (14) with the establishment of similar results. In the sequel, we focus on a graphbased perspective for the controllability of system (4) and (6). To this end, we first introduce the following definition. Definition 6: For a multi-agent system with directed or undirected interconnection graph G, any two agents are said to have the same direct neighbour set of parent vertices if the two parent vertex sets associated with these two agents are identical; and are said to have the same indirect neighbour set of parent vertices if the two sets obtained by adding themselves to each set of their parent vertices are identical. For example, in Figure 1, the direct parent vertex sets of agents 3 and 4 are, respectively, S3 ¼ {2, 4} and S4 ¼ {2, 3}. By adding agent 3 to its direct parent vertex set S3 and 4 to its direct parent vertex set S4, one can
Figure 1. Agents 3 and 4 have the indirect same set of parent vertices.
Proof: For the convenience of presentation, we assume that there are two followers, say the i-th and j-th followers, having the same direct or indirect neighbour set of parent vertices. Since F is obtained by deleting the last l rows and l columns of the Laplacian matrix L and R is obtained by deleting the last l rows and the first N columns of L, it follows that 3 2 6 .. .. 7 .. .. 6. .7 . . 6 7 6 7 6 " 7i 6 7 .. 7 .. .. 6. F ¼ 6 .. 7 . . . 6 7 , 6 " 7j 6 7 6. .. 7 .. .. 6. 7 .5 . . 4. i j 3 2 6 .. .. 7 6. .7 7 6 7 6 6 7i 7 6 .. 7 6. R ¼ 6 .. 7 , . 7 6 6 7j 7 6 6. .. 7 7 6. .5 4. where " ¼ 0 if followers i and j have the same direct neighbour set of parent vertices and " ¼ 1 if they have the same indirect neighbour set of parent vertices. Furthermore, the i-th row in F is identical with the j-th row if the i-th and j-th elements are exchanged from each other. Since the i-th and j-th followers have the same direct or indirect neighbour set of parent vertices, the i-th and j-th rows in R are also identical. Recall that the controllability matrix of system (4) is C ¼ [R, FR, . . . , (1)NFN1R]. It can be seen from the specific structure of F and R that the i-th and j-th rows of the controllability matrix are identical. As a consequence, the controllability matrix is rank deficient, i.e. the system is uncontrollable. œ
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International Journal of Control The above theorem presents a method to decide the uncontrollability directly from the interconnection graph itself by searching for some followers with the same direct or indirect neighbour set of parent vertices.
4.3 Controllability under switching topology Since the interconnection graph G is time-variant, the dynamic (4) can be viewed more reasonably as a system in switching networks, which can be written in the form y_ ¼ F ðtÞ y R ðtÞ z,
ð27Þ
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4
where (t) : Rþ ! M ¼ {1, 2, . . . , M} is the switching signal/sequence to be designed. Given a switching signal (t) : [t0, tf ] ! M, we refer to t0, t1, . . . , ts1 with t0 5 t1 5 5 ts1 as the switching time sequence, and
(t0) ¼ i0, (t1) ¼ i1, . . . , (ts1) ¼ is1 as the switching 4 index sequence. Let hi ¼ tiþ1 ti, i ¼ 0, 1, . . . , s 1, and 4 4 ts ¼ tf. We denote by ¼ {(i0, h0). . .(is1, hs1)} a switching signal. (F (t), R (t)) is said to be the system and control input matrix pair (in short matrix pair) of the multi-agent system (1) with switching topology. In particular, (F, R) is said to be the matrix pair of system (1) with fixed topology. Definition 7: The multi-agent system (1) is said to be controllable under leaders xNþj, j ¼ 1, . . . , l and switching topology if system (27) is controllable. The system (27) is controllable if for any nonzero state y 2 RN, there exist a switching sequence and input z such that y(0) ¼ y, and y(tf) ¼ 0. We denote by C the controllable state set of system (27). To derive conditions for the controllability of multi-agent systems under switched dynamic networks, results on the controllability of switched linear systems (27) are first recalled in the sequel. Given a matrix A 2 RNN, and a D linear subspace W RN, we denote hAjWi ¼ P N i1 W: It follows that hA|Wi is a minimum i¼1 A A-invariant subspace that contains W. Given B 2 RNp, let Im B denote the image space of B. For notational simplicity, we denote by hA|Bi the hA|ImBi. For system (27), we denote by C the set of all its controllable states and consider the nested subspace sequence defined by W1 ¼
M X k¼1
hFk j Rk i,
Wsþ1 ¼
M X
Proof: It is a direct consequence of Theorem 1 in Sun, Ge, and Lee (2002) and Ji, Wang, and Guo (2008d). œ To calculate the controllable subspace C, the following subspace sequence E0, E1, . . . is defined: E0 ¼
M X
ImðRk Þ, Esþ1 ¼ E0 þ
k¼1
s ¼ 0, 1, 2, . . . :
D ¼ B1 , . . . , BM , H1 B1 , . . . , H1 BM , . . . , HM B1 , . . . ,
HM BM , . . . , H1 B1 , . . . , H1 BM , H1 1 H2 B1 , . . . , H1 ð30Þ 1 H2 BM , . . . , HM B1 , . . . , HM BM , 4
4
where Hi ¼ Fi, Bi ¼ Ri, i ¼ 1,. . . , M. Below is a Kalman-type rank criteria. Theorem 7: The interconnected system with nl leaders and switching networks described by (27) is controllable if and only if matrix is full row rank, i.e. rank ¼ N: Proof: By the proof of Proposition 1 in Ji, Lin, and Lee (2008c), the subspace W coincides with the image space of , i.e. W ¼ Im . The result then follows from Lemma 2. œ Since N dim E0, the theorem still holds if is replaced by N. In the sequel, we derive directly from the nested subspace sequence (28) rather than (29). To facilitate statement, we begin with the situation nl ¼ 2, ¼ N ¼ 3 and M ¼ 2. The remainder is to calculate W. By (28), W ¼W2 þ H1 W2 þ H21 W2 þ H2 W2 þ H22 W2 ¼W1 þ W2 , where D
W1 ¼hH1 jB1 i þ H2 hH1 jB1 i þ H22 hH1 jB1 i þ H1 H2 hH1 jB1 i þ H1 H22 hH1 jB1 i
hFk jWs i,
For system (27), C ¼ W ¼ WN ¼ W.
ð29Þ
Let ¼ min {s|Es ¼ Esþ1, s ¼ 0, 1, 2, . . .}, and
þ H21 H2 hH1 jB1 i þ H21 H22 hH1 jB1 i,
ð28Þ
Lemma 2:
ðFk ÞEs ,
k¼1
k¼1
with s ¼ 1, 2, . . . . Let W ¼ lims!1 Ws , and ¼ min{s| Ws ¼ Wsþ1, s ¼ 1, 2, . . . }. It follows that N dim W1 þ 1. The following result is on the controllability of system (27).
M X
D
W!2 ¼hH2 jB2 i þ H1 hH2 jB2 i þ H21 hH2 jB2 i þ H2 H1 hH2 jB2 i þ H2 H21 hH2 jB2 i þ H22 H1 hH2 jB2 i þ H22 H21 hH2 jB2 i:
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Since Fi is symmetric, it can be expressed as D
bi UT ¼ Hi , Fi ¼ Ui Di UTi ¼ Ui D i
where
i ¼ 1, . . . , M,
h b T b T b2 T ð1Þ 1 ¼ U1 , U1 ,U2 D2 U21 ,U2 D2 U21 , U2 D2 U21 ,
ð31Þ
b2 UT U1 D b1 U21 D b2 UT , U1 D b1 U21 D b2 UT , U2 D 2 21 21 21
D
bi ¼ Di , Ui is an orthogonal matrix. Denote where D 4 (Fi, Ri) ¼ (Hi, Bi), one has hHi jBi i ¼
N X
N X
Hij1 Im Bi ¼ Ui
j¼1
bj1 Im B bi , D i
j¼1
D D D bi ¼ bi ¼ bi ¼ where B UTi Bi : Set D diag d^1i , . . . , d^Ni , B ð1Þ T D ðNÞ bbi1 , bbi2 , and bbik ¼ b^ik , . . . , b^ik , i ¼ 1, . . . , M; k ¼ 1, . . . , nl. Some calculations show that Downloaded By: [National University Of Singapore] At: 10:28 30 July 2010
hHi jBi i ¼ Im i ,
ð32Þ
b1 U21 D b2 UT , U1 D b1 U21 D b2 UT U1 D b2 U21 D b2 UT , U1 D 2 21 2 21 1 21
i b2 U21 D b2 UT , U1 D b2 U21 D b2 UT , U1 D b2 U21 D b2 UT , U1 D 1 21 1 2 21 1 2 21 ð2Þ 1 ¼ diag 11 , 12 , 11 , 12 , 11 , 12 , 11 , 12 , 11 , 12 , 11 , 12 , 11 , 12 , , ð3Þ 1 ¼ diag 1 , 1 ,1 , 1 , 1 ,1 , 1 , 1 , 1 ,1 , 1 , 1 ,1 , 1 ,
and
where 22
b^ ð1Þ 66 i1 66 ^ ð2Þ 66 bi1 D 66 i ¼ Ui 66 66 .. 66 . 44 2 6 6 6 6 6 6 6 4
d^1i b^i1ð1Þ
d^2i b^i1ð2Þ
.. .
..
b^i1ðNÞ
d^Ni b^i1ðNÞ
b^i2ð1Þ
d^1i b^i2ð1Þ
b^ð2Þ i2
d^2i b^ð2Þ i2
.. .
.. .
..
b^i2ðNÞ
d^Ni b^i2ðNÞ
"
¼½Ui , Ui
.
.
i
i2 2
6 6 ik ¼ 6 4 2
i
h ð1Þ b1 UT , U1 D b1 UT , U1 D b2 UT , U1 D b2 UT , 2 ¼ U2 , U2 , U1 D 12 12 1 12 1 12 b2 U12 D b1 UT , U2 D b2 U12 D b1 UT , U2 D b2 U12 D b2 UT , U2 D 12 12 1 12 b2 U12 D b2 UT , U2 D b2 U12 D b1 UT , U2 D b2 U12 D b1 UT , U2 D 1 12 2 12 2 12 i 2 2 T 2 2 T b U12 D b U , U2 D b U12 D bU U2 D 2 1 12 2 1 12 n ð2Þ ¼ diag 21 , 22 , 21 , 22 , 21 , 22 , 21 , 22 , 2 o 21 , 22 , 21 , 22 , 21 , 22 n ð3Þ ¼ diag 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 o 2 , 2 , 2 , 2 , 2 , 2 , 2 :
,
3
b^ð1Þ ik ..
. b^ðNÞ ik
1 6 61 6 i ¼ 6 6. 6 .. 4
where
7 7 d^2iN1 b^i1ð2Þ 7 7 7, .. 7 7 . 5 ðNÞ N1 ^ ^ dNi bi1 33 d^1iN1 b^i2ð1Þ 77 7 ^ð2Þ 7 77 b d^N1 2i i2 77 77 .. 77 77 . 55 ðNÞ N1 ^ ^ dNi bi2 #
#"
i1
d^1iN1 b^i1ð1Þ
ð2Þ ð3Þ 2 ¼ ð1Þ 2 2 2 ,
3
d^1i d^2i
.. . ^ 1 dNi
i ¼ 1, . . . , M;
Using (32), it can be verified that
7 7 7, 5
d^N1 1i
W1 ¼ Im 1 , Accordingly,
3
W ¼ Im ,
7 7 d^N1 7 2i 7, .. 7 .. . 7 . 5 N1 ^ dNi k ¼ 1, . . . , nl :
D
with h i n o n o D ð1Þ ð2Þ ð2Þ ð3Þ ð3Þ ¼ ð1Þ , , , diag diag 1 2 1 2 1 2 :
ð33Þ
Let UTij ¼ UTi Uj : Then Uij is an orthogonal matrix since Ui, Uj are orthogonal. To express W further, the following two matrices are introduced: ð2Þ ð3Þ 1 ¼ ð1Þ 1 1 1 ,
W2 ¼ Im 2 :
By Lemma 2, the interconnected system (27) with nl ¼ 2, N ¼ 3 and M ¼ 2 is controllable if and only if is full row rank, i.e. rank ¼ N: The derivation of general situation can be carried out in the same way. The difference consists in the expression of , which becomes much more complex
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International Journal of Control as the state dimension N and the number of subsystems M increase. In general case, is expressed as h i n o ð1Þ ð2Þ ð2Þ , . . . , , . . . , ¼ ð1Þ diag M M 1 1 n o ð3Þ ð3Þ diag 1 , . . . , M , ð34Þ where
being leaders. The associated fixed topology is shown in Figure 2. The objective is to move the two leaders in a two-dimensional space so that the followers can be successively steered into the desired configuration, i.e. a straight line in the vertical direction with time delay ¼ 1 s. The directed interconnection graph associated with the system has the following
ð1Þ i , . . . , Ui , . . . , i ¼½U |fflfflfflfflfflffl{zfflfflfflfflfflffl} nl
bN1 Ui2 i1 D bN1 . . . D bN1 Ui2 i1 D bN1 . . . D bN1 UT , . . . , Ui1 D bN1 UT , Ui1 D i1 i2 iM iM i i1 i2 iM iM i |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
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nl
with i1 , . . . , iM 2 {1, . . . , M} iM 6¼ i; Ui, Uij are orthogobij are diagonal matrices defined in nal matrices, D (31), and ð3Þ ð2Þ i ¼ diag i1 , . . . , inl , i ¼ diagfi , . . . , i g, i ¼ 1, . . . , M, ik, i are diagonal and Vandermonde matrices defined, respectively, in (33). The following conclusion is a summary of the previous arguments. Theorem 8: The interconnected system with l leaders and switching networks described by (27) is controllable if and only if matrix is full row rank, i.e. rank ¼ N, where is defined in (34). Remark 2: Comparing with Theorem 8, the controllability matrix expression in Theorem 7 is more laconic and the controllability can be therein verified directly via computing . The advantage of Theorem 8 consists in the isolation of eigenvalues and eigenvectors, which ð2Þ is incorporated, respectively, in ð3Þ i and i in (34) and accordingly results in a specific expression for controllability matrix . But no such information is extracted in the expression of . In this sense, Theorem 8 provides further information on controllability for a multi-agent system although it seems that is not as simple as . The difference between Theorems 7 and 8 originates from the starting point of derivation. Theorem 7 is deduced from the viewpoint of subspace-based algorithm for the controllability of switching linear systems, while Theorem 8 is deduced directly from the iteratively defined subspace sequence.
coefficient matrices: 1 0 1 0 , R¼ , F¼ 0 1 1 2 1 0 0 0 D¼ , A¼ : 0 2 1 0 The system moving in a two-dimensional space is described by the following equation:
0 0 1 0 _ ¼ I2 yðtÞ I2 yðtÞ þ 1 0 0 2
1 0 yðt Þ I2 zðtÞ: 0 1 Figure 3 shows the trajectories of the two followers with the initial state y(t) ¼ [1, 4, 0, 3]T, t 2 [1, 0]. At the time instant tf ¼ 11 s, the two followers are steered into a straight line in a vertical direction. Example 2: Consider a multi-agent system with the interconnection graph depicted by Figure 1, where nodes 5 and 6 are chosen to take the leaders role. Calculations show that system matrices are 2 3 3 2 1 1 2 0 0 0 6 1 1 7 60 2 0 0 7 6 7 6 7 F¼6 7: 7, R ¼ 6 4 0 4 0 1 2 1 5 0 5 0
1
1
2
0
0
5. Illustrative examples Example 1: We consider a directed interconnection graph consisting of four vertices, with nodes 3 and 4
Figure 2. The directed interconnection graph of four agents, where agents 3 and 4 are leaders.
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1 –1
0
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Figure 3. A straight line in the vertical direction with ¼ 1s.
0
5
10
15
20
25
30
Figure 5. An rectangle configuration.
Figure 6. Three interconnection graphs associated with three multi-agent systems. Figure 4. The directed interconnection graph of six agents, where agents 1 and 3 have the same topology.
It can be verified that the controllability matrix C has the row rank of 2, i.e. the system is uncontrollable, which coincides with Theorem 1. In the preceding sections, we discussed the controllability problem with respect to unweighed directed or undirected graph. It should be noted that communication weights distinct from 1 are ubiquitous in practice. In what follows, we will show by an example that it is possible to turn an uncontrollable system with fix topology into a controllable one by selecting appropriate weights for the communication links between leaders and followers. Example 3: For the multi-agent system with directed interconnection graph depicted by Figure 4, agents 5 and 6 are assigned to take the leader roles and agents 1 and 3 have the same input channel. If the digraph is unweighed, we have 2 3 2 3 1 0 1 0 0 0 6 0 1 7 6 1 3 1 0 7 6 7 6 7 F¼6 7: 7, R ¼ 6 4 1 0 5 4 0 0 1 05 0
0
0
1
0
1
The controllability matrix has the rank 3. The system is uncontrollable. If different weight, say weight value 2, is imposed between the leader 5 and the follower 1 (or between 5 and 3), then 2 3 2 3 2 0 2 0 0 0 6 0 1 7 6 1 3 1 0 7 6 7 6 7 F¼6 7: 7, R ¼ 6 4 1 0 5 4 0 0 1 05 0
0
0
1
0
1
The rank of the controllability matrix is 4, implying that the uncontrollable system is turned to be controllable. Figure 5 shows the motion trajectories of the followers 1–4. The four followers start from arbitrary initial positions in the plane represented by , and reach a desired rectangular configuration depicted by *’s. Example 4: The example is employed to show that under switching topologies, the desired configuration can be achieved for a multi-agent system with timedelay in state. Consider a multi-agent system with switching topology described by Figure 6, where agent 4 is the leader and the others are followers. Our objective is to move the leader in a two-dimensional space so that the followers are successively steered into a straight line with time delay ¼ 1 s under switching (a) ! (b) ! (c). Starting from a random initial position
International Journal of Control 8 7 6 5 4 3 2 1 0
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4
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Figure 7. The trajectories of the three followers in the plane with time-delay ¼ 1s under switching topology.
(1, 4), (2, 3) and (3, 5), Figure 7 shows the evolution of the three followers, which forms a configuration of straight line at time t ¼ 6 s.
6. Conclusion As an attractive issue of research, the controllability of networks of dynamic agents has recently aroused increasing attention. In this article, the controllability problem is formulated and studied for continuous-time multi-agent systems with time-delay in state and switching topology. Sufficient algebraic conditions are derived for the interconnected system to be controllable, as well as a graph-based uncontrollability topology structure is constructed. The results are analysed for both single and double integrator dynamics. For switching topology, two necessary and sufficient algebraic conditions are derived for the controllability of networked multi-agent system. The work provides insight into the effect of the interaction pattern on self-organised motion in a multi-agent system.
Acknowledgements This work was supported by the Royal Society K.C. Wong Education Foundation Postdoctoral Fellowship of the United Kingdom and the National Natural Science Foundation of China (Nos. 60604032, 10601050, 60704039).
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