Some Stability and Boundedness Conditions for Second-order Leaderless and Leader-following Consensus with Communication and Input Delays

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1 010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 0, 010 WeA16.4 Some Stability and Boundedne Condition for Second-order Leaderle and Leader-following Conenu with Communication and Input Delay Ziyang Meng, Wei Ren, Yongcan Cao, Zheng You Abtract In thi paper, time-domain (Lyapunov theorem) and frequency-domain (the Nyquit tability criterion) approache are ued to tudy econd-order leaderle and leaderfollowing conenu algorithm with communication and input delay under a directed network topology. We conider three different cae, namely, leaderle conenu, conenu regulation, and conenu tracking with full acce to the virtual leader, and preent tability or boundedne condition. Several intereting phenomena are analyzed and explained. I. INTRODUCTION Cooperative control of multi-agent ytem ha received ignificant reearch attention in recent year. Compared with olo ytem, additional benefit, uch a high robutne and great efficiency, can be obtained by having a group of agent work cooperatively. A fundamental approach to achieve cooperative control i conenu [1], [], [3]. Conenu mean the agreement of a group of agent on their common tate via local interaction. In a leaderle conenu problem, there doe not exit a virtual leader while in a leaderfollowing conenu problem, there exit a virtual leader which pecifie the objective for the whole group. More pecifically, conenu with a tatic virtual leader i called a conenu regulation problem and conenu with a dynamic virtual leader i called a conenu tracking problem. Since delay are inevitable in real ytem, it i neceary and beneficial to tudy leaderle and leader-following conenu algorithm in the preence of the delay. Mot exiting reference on conenu algorithm conidered input delay. The author in [] firt gave a leaderle conenu algorithm with input delay and then preented a frequency-domain approach to find the tability condition. A imilar leaderle conenu algorithm with uniform input delay wa tudied in [4], where a time-domain approach, Lyapunov-Kraovakii Theorem, wa ued to obtain the tability condition under trongly connected and balanced network topologie. Beide leaderle conenu algorithm, leader-following conenu algorithm with input delay were alo tudied. By combining the reult in [4] and [5], the author in [6] propoed Z. Meng and Z. You are with the Department of Preciion Intrument and Mechanology, Tinghua Univerity, Being, , P. R. China. mengzy06@mail.tinghua.edu.cn, yz-dpi@mail.tinghua.edu.cn. W. Ren and Y. Cao are with the Department of Electrical and Computer Engineering, Utah State Univerity, Logan, UT 843, USA. wei.ren@uu.edu, yongcan.cao@aggi .uu.edu. Thi work wa upported by a National Science Foundation CAREER Award (ECCS ). Z. Meng would like to acknowledge the Minitry of Education of P. R. China and China Scholarhip Council (CSC) for their financially upporting him a a viiting Ph.D. tudent at Utah State Univerity. a firt-order conenu tracking algorithm with input delay, where an etimator wa ued to etimate the virtual leader velocity. Due to the preence of the dynamic virtual leader and the input delay, the tracking error were hown to be uniformly ultimately bounded intead of approaching zero. In the previou reference, the network topology i aumed to be either undirected, or trongly connected and balanced, which poe an obviou limitation. Except for input delay, the influence of communication delay on conenu algorithm wa alo tudied. The author in [7] howed that communication delay will not jeopardize the tability of the firt-order leaderle conenu algorithm under a directed network topology. A imilar algorithm wa dicued in [8], where the effect of initial condition wa highlighted. A econd-order conenu regulation algorithm with nonuniform communication delay wa tudied in [9], but a damping term wa ued to regulate the velocitie of all agent to zero and the network topology wa aumed to be undirected. The previou reference conidered either only the input delay or only the communication delay and hence lack completene. The cae with both the communication and input delay wa tudied in [10]. In particular, a firtorder leaderle conenu algorithm with both the communication and input delay wa tudied in a dicrete-time etting. However, a pure frequency-domain approach wa ued, thu leading the obtained tability condition to be conervative. The contribution of the current paper are threefold. Firt, we aume a general network topology, i.e., a directed network topology with a directed panning tree, intead of an undirected connected network topology or a directed trongly connected and balanced network topology a in [], [4], [6], [11]. Second, both communication and input delay are conidered in the cae of leaderle conenu, conenu regulation, and conenu tracking with full acce to the virtual leader, which guarantee the completene of the algorithm. Third, a a byproduct, we find that for the econd-order leaderle conenu problem, the final group velocity i alway dampened to zero rather than a poibly nonzero contant a in the tandard econd-order leaderle conenu algorithm in [1] when there exit the communication delay. A. Notation II. PRELIMINARIES R and C are, repectively, the et of real number and the et of complex number. 1 n and 0 n are, repectively, the n 1 all-one vector and the n 1 all-zero vector. I n and 0 n n are, repectively, the /10/$ AACC 574

2 n n identity matrix and the n n matrix with all zero entrie. λ min (A) and λ max (A) are, repectively, the minimal eigenvalue and the maximum eigenvalue of the matrix A. ρ(a) i the pectral radiu of the matrix A. tand for the Euclidean vector norm and φ c = up τ t 0 φ(t) tand for the norm of a function φ C n,τ, where C n,τ i the Banach pace of continuou vector function mapping the interval [ τ, 0] into R n with the topology of uniform convergence [13]. R{ } and I{ } are, repectively, the real part and the imaginary part of a complex number. Q < 0 mean that the matrix Q i negative-definite. B. Graph Theory Notion Uing graph theory, we can model the network topology in a multi-agent ytem coniting of n agent. A directed graph G n conit of a pair (V, E), where V = {v 1,...,v n } i a finite, nonempty et of node and E V V i a et of ordered pair of node. An edge (v i, v j ) denote that node v j can obtain information from node v i, but not necearily vice vera. All neighbor of node v i are denoted a N i := {v j (v j, v i ) E}. A directed path i a equence of edge of the form (v i1, v i ), (v i, v i3 ),.... A directed graph ha a directed panning tree if there exit at leat one node having a directed path to all other node. For the leaderle conenu cae, the adjacency matrix A n = [a ] R n n aociated with G n i defined uch that a i poitive if (v j, v i ) E while a = 0 otherwie. Here we aume that a ii = 0, i. The (nonymmetric) Laplacian matrix L n = [ ] R n n aociated with A n i defined a ii = j i a and = a, where i j. For the leader-following cae, we aume that beide agent 1 to n, there exit a virtual leader, labeled a agent n 1, in the ytem. We ue G n1 to model the network topology in thi cae. The adjacency matrix A n1 = [a ] R (n1) (n1) aociated with G n1 i defined uch that a i poitive if (v j, v i ) E while a = 0 otherwie, and a (n1)j = 0 for all j = 1,..., n 1. Here again we aume that a ii = 0, i. III. DEFINITIONS AND LEMMAS Suppoe f : R C R n i continuou and conider retarded functional differential equation (RFDE) ẋ(t) = f(t, x t ). (1) Let φ = x t be defined a x t (θ) = x(t θ), θ [ τ, 0]. Suppoe appropriate initial condition are defined on the delay interval [t 0 τ, t 0 ]: x t0 (θ) = φ(θ), θ [ τ, 0]. Specifically, we aume that the initial condition atifie x(θ) = 0, θ [t 0 τ, t 0 ], in thi paper. Suppoe that the olution x(σ, φ)(t) through (σ, φ) i continuou in (σ, φ, t) in the domain of definition of the function, where σ R. Definition 3.1: [14] The olution x(σ, φ) of the RFDE (1) are uniformly ultimately bounded if there i a β > 0 uch that for any α > 0, there i a contant t 0 (α) > 0 uch that x(σ, φ)(t) β for t σt 0 (α) for all σ R, φ C, φ α. Suppoe D : R C R n i a linear operator on the econd variable uch that D(t, φ) = A(t)φ(0) G(t, φ), A(t) i a continuou noningular matrix, and G(t, φ) = h dµ(t, θ)φ(θ) atifie [dµ(t, θ)]φ(θ) γ(, t) φ for 0 h, where µ i an n n matrix function of bounded variation on θ, γ i continuou, and γ(0, t) = 0 for t 0. If g : R C R n i a continuou function, then the relation dd(t, x t )/dt = g(t, x t ) () i a neutral functional differential equation (NFDE) [15]. Definition 3.: [15] Conider the NFDE (). Suppoe operator D i table. It define a uniform ultimately bounded proce if there i a β > 0 uch that for any α > 0, there i a contant t 0 (α) > 0 uch that x(σ, φ)(t) β for t σ t 0 (α) for all σ R, φ C, φ α. Lemma 3.1: (Degenerate-Lyapunov-Kraovkii Stability Theorem) [16], [13] Conider the NFDE (). Suppoe operator D i table, g : R C R n take R (bounded et of C) into bounded et of R n, u(), v() and w() are continuou, non-negative and non-decreaing function with u(), v() > 0 for 0 and u(0) = v(0) = 0. If there exit a continuou functional V : R C n C n R n, uch that (i) u( D(t, φ) ) V (t, D(t, φ), φ) v( φ c ), (ii) V (t, D(t, φ), φ) w( D(t, φ) ), then the trivial olution of () i aymptotically table. Lemma 3.1 will be ued in the econd-order leaderle conenu and conenu regulation problem. Lemma 3.: (Lyapunov-Razumikhin Uniformly Ultimately Bounded Theorem) [14] Conider the RFDE (1). Suppoe f : R C R n take R (bounded et of C) into bounded et of R n and u, v, w : R R are continuou non-increaing function, u() a. If there i a continuou function V : R R n R, a continuou non-decreaing function p : R R, p() > for > 0, and a contant H 0 uch that u( x ) V (x) v( x ), t R, x R n, and V (t, φ) w( φ(0) ) if φ(0) H, V (t θ, φ(θ)) < p(v (t, φ(0))), θ [ τ, 0], then the olution of (1) are uniformly ultimately bounded. Lemma 3. will be ued in the econd-order conenu tracking problem with full acce to the virtual leader. IV. SECOND-ORDER LEADERLESS AND LEADER-FOLLOWING CONSENSUS WITH COMMUNICATION AND INPUT DELAYS UNDER A DIRECTED NETWORK TOPOLOGY In thi ection, we model a group of agent with doubleintegrator dynamic a ṙ i (t) = v i (t), v i (t) = u i (t), i = 1,...,n, (3) where r i, v i, and u i denote, repectively, the poition, the velocity, and the control input of the ith agent. 575

3 A. Second-order Leaderle Conenu Conider the following leaderle conenu algorithm with both communication and input delay a 1 n u i (t) = n a a [r i (t τ 1 ) r j (t τ 1 τ )] γ c n a n a [v i (t τ 1 ) v j (t τ 1 τ )], i = 1,...,n, (4) where τ 1 and τ are, repectively, the input and communication delay, a, i = 1,...,n, j = 1,..., n, i the (i, j) entry of the adjacency matrix A n, and γ c i a poitive gain. Here we aume that every agent ha a neighbor, which implie that n a 0, i. The control objective in thi ubection i to guarantee that r i (t) r j (t) and v i (t) v j (t) a t when there exit both communication and input delay. Uing [ (4), ] (3) can[ be written in][ the matrix ] ṙ(t) 0n n I form a = n r(t) [ v(t) ][ 0] n n 0 n n v(t) 0n n 0 n n r(t τ1 ) [ I n γ c I n ] [ v(t τ 1 ) ] 0n n 0 n n r(t τ1 τ ), where r = A γ c A v(t τ 1 τ ) [r 1,...,r n ] T, and v = [v 1,...,v n ] T. A = [â ] R n n i defined a â = a / n a, i = 1,...,n, j = 1,..., n. Alo define L = I n A. When G n ha a directed panning tree, we know that L ha a imple zero eigenvalue and all other eigenvalue are on the open right half plane [ [17]. The Singular ] Vector Decompoition L W 1 0 LW = n 1 0 T i valid. Here, among the n 1 0 infinite option of W, we chooe the one that the lat column of W i the vector 1 n. Note that here all the eigenvalue of L are on the open right half plane. Before moving on, we need the following lemma. Lemma 4.1: [6] For any a, b R n and any ymmetric poitive definite matrix Φ R n n, a T b a T Φ 1 ab T Φb. Define r W 1 r and ṽ W 1 v. Denote r n 1 and ṽ n 1 a, repectively, the firt n 1 row of r and ṽ. Denote r and ṽ a, repectively, the lat row of r and ṽ. Sytem (3) uing (4) can be decoupled into the following equation: [ ] 0(n 1) (n 1) 0 (n 1) (n 1), and A I n 1 γ c I = [ n 1 ] 0(n 1) (n 1) 0 (n 1) (n 1). à γ c à Theorem 4.1: If the fixed directed graph G n ha a directed panning tree, every agent ha a neighbor, and γ c > γ c = max I(µi) µi 0{ R(µ i) µ i }, where µ i i the ith eigenvalue of L = I n A, there exit τ 1 and τ uch that the following three condition 1 are atified: i) γ c (τ 1 τ ) (τ1τ)τ < 1. ii) 1 λ i (A 1 ) 1 e τ 1 λ i (A ) 1 e (τ 1τ ) 0, for all C. iii) Q c = (A 0 A 1 A ) T P c P c (A 0 A 1 A )τ 1 S c (τ 1 τ )H c τ 1 [(A 0 A 1 A ) T P c A 1 Sc 1 A T 1 P c(a 0 A 1 A )](τ 1 τ )[(A 0 A 1 A ) T P c A Hc 1 A T P c (A 0 A 1 A )] < 0, where P c i a ymmetric poitive-definite matrix choen properly uch that (A 0 A 1 A ) T P c P c (A 0 A 1 A ) < 0, and S c and H c are arbitrary ymmetric poitivedefinite matrice. If the above condition are atified, τ 1 [0, τ 1 ], and τ [0, τ ], ytem (3) uing (4) reache conenu aymptotically. Specifically, r i (t) pt v(0) τ and v i (t) 0, where p R n i a non-negative left eigenvector of L aociated with the zero eigenvalue atifying p T 1 n = 1. Proof: We firt prove that the tability of ytem (3) uing (4) i guaranteed if the three condition in Theorem 4.1 are atified. Then, we how that thee three condition are indeed atified when G n ha a directed panning tree, every agent ha a neighbor, and γ c > γ c. At lat, the conenu equilibrium i explicitly preented by uing the final value theorem. We know that the tability of the following ytem d dt ( x n 1(t) A 1 x n 1 (t θ)dθ A τ x n 1 (t θ)dθ) = (A 0 A 1 A ) x n 1 (t) (6) implie the tability of ytem (5a) if the condition ii) in Theorem 4.1 i atified [13]. Conider a Lyapunov function candidate for ytem (5a) V ( x (n 1)t ) = [ x n 1 (t) A 1 x n 1 (t θ)dθ A x n 1 (t) = A 0 x n 1 (t) A 1 x n 1 (t τ 1 ) x n 1 (t θ)dθ] T P c [ x n 1 (t) A 1 x n 1 (t θ) A x n 1 (t τ 1 τ ), (5a) τ t [ ] [ ] [ ] [ ] dθ A x n 1 (t θ)dθ] x n 1 (ξ) T S c r (t) 0 1 r (t) 0 0 τ tθ = ṽ (t) 0 0 ṽ (t) 1 γ 0 t c [ ] [ ] [ ] x n 1 (ξ)dξdθ x n 1 (ξ) T H c x n 1 (ξ)dξdθ. r (t τ 1 ) 0 0 r (t τ 1 τ ) τ tθ, ṽ (t τ 1 ) 1 γ c ṽ (t τ 1 τ ) Taking the derivative of V along (6) give (5b) where x n 1 [ = [ r n 1 T, ṽt n 1 ]T, à = ] I n 1 L, V ( x (n 1)t ) x n 1 (t) T Q c x n 1 (t), 0(n 1) (n 1) I A 0 = n 1, A 0 (n 1) (n 1) 0 1 = 1 Note here that the three condition are ued to obtain the upper bound (n 1) (n 1) τ 1 and τ for allowable delay. 576

4 where Q c i defined in Theorem 4.1 and we have ued Lemma 4.1 to derive the inequality. Note that Q c < 0 atifie the condition ii) in Lemma 3.1. Alo note that α 1 D( x (n 1)t ) V ( x (n 1)t ) α x (n 1)t c [18], where D( x (n 1)t ) = x n 1 (t) A 1 x n 1 (t θ)dθ A τ x n 1 (t θ)dθ, x (n 1)t c = up θ [ τ1 τ,0] x (n 1) (t θ), α 1 = λ min (P c ), and α = λ max (P c ) τ 1 λ max (S c ) (τ 1 τ )λ max (H c ). Thi atifie the condition i) in Lemma 3.1. Thu the tability of ytem (5a) i guaranteed if the condition ii) and iii) are atified by uing Lemma 3.1. For ytem (5b), define g() (γ c 1)(e τ1 e (τ1τ) )/. By uing the Nyquit tability criterion, we know that the tability of (5b) can be guaranteed if R{g(jω)} > 1, ω (, ). Becaue R{g(jω)} = co τ 1ωco(τ 1τ )ω ω = in (τ 1 τ ) ω in τ ω ω γ c in τ 1ωγ c in(τ 1τ )ω ω γ c in τ 1ωγ c in(τ 1τ )ω ω γ c τ 1 γ c (τ 1 τ ) (τ1τ)τ, the condition i) in Theorem 4.1 guarantee the tability of ytem (5b). Next, we how that the three condition in Theorem 4.1 are indeed atified if G n ha a directed panning tree, every agent ha a neighbor, and γ c > γ c. It i traightforward to ee that there exit τ 1 and τ uch that the condition i) and ii) are atified. For the condition iii), noting that A 0 [ ] 0(n 1) (n 1) I n 1 A 1 A =, we know that the L γ c L aumption that G n ha a directed panning tree, every agent ha a neighbor, and γ c > γ c imply that all eigenvalue of A 0 A 1 A are on the open left half plane according to the proof of Lemma 4.4 in [19]. Thu there alway exit a P c to guarantee that (A 0 A 1 A ) T P c P c (A 0 A 1 A ) < 0. Note that P c can be eaily obtained by uing the LYAP olver in Matlab Toolbox. Therefore, baed on the continuity, there mut exit τ 1 and τ uch that Q c < 0 when τ 1 [0, τ 1 ] and τ [0, τ ]. Thi implie that the condition iii) i atified. For the conenu equilibrium, by uing the final value theorem, we know that the aymptotical tability of (5b) implie that lim t r (t) = lim 0 r (0)ṽ (0)γ ce r (0) γ ce (τ 1 τ ) r (0) (γ c1)(e e (τ 1 τ ) ) = ṽ(0) τ and ṽ (t) 0 a t. The aymptotical tability of (5a) implie that x n 1 0 a t. Thu, it follow that [ ] [ ] r(t) p T v(0) τ 1 n a t. v(t) 0 n Remark 4.1: Note that in Theorem 4.1 it i aumed that the fixed directed graph ha a directed panning tree and every agent ha a neighbor. Thu the concluion can be viewed a a generalization of [4], [6], [11], where the directed graph are aumed to be trongly connected and balanced. Remark 4.: The cae of a general network topology with a directed panning tree wa alo conidered in [10] in a firtorder dicrete-time etting. A pure frequency-domain approach wa ued to derive the tability condition. In contrat, we here introduce both time-domain and frequency-domain approache in a econd-order continuou-time etting. B. Second-order Conenu Regulation with a Contant Final Velocity In thi ubection, we aume that there exit a virtual leader, labeled a agent n 1 with poition r d and velocity v d. Here we aume that v d i contant. The control objective here i to guarantee that all agent can track the virtual leader under limited communication in the preence of delay. The propoed conenu regulation algorithm i given a u i = n1 1 n1 a a [r i (t τ 1 ) r j (t τ 1 τ )] n1 γ r n1 a a [v i (t τ 1 ) v j (t τ 1 τ )], i = 1,...,n, (7) where τ 1 and τ are, repectively, the input and communication delay, a, i = 1,...,n, j = 1,...,n 1, i the (i, j) entry of the adjacency matrix A n1, r n1 r d, v n1 v d, and γ r i a poitive gain. Note that the condition that G n1 ha a directed panning tree and the fact that all entrie of the lat row of A n1 are zero imply that no other row of A n1 have all zero entrie. It thu follow that n1 a 0, i = 1,..., n [0]. Uing (7), (3) can be written in the matrix form a ẋ(t) = A 0 x(t) A 1 x(t τ 1 ) A x(t τ 1 τ ) R r, (8) where A = [ā ] R n n i defined a ā = a / n1 a, i = 1,..., n, j = 1,...,n, r [r 1 r d,..., r n r d ] T, v [v 1 v d,..., v n v d ] T, x = [r T, v T ] T, [ ] [ ] 0n n I A 0 = n 0n n 0, A 0 n n 0 1 = n n, A n n I n γ r I = [ ] [ ] n 0n n 0 n n 0, and R A γ r A r = n. Alo note that τ v d 1 n here we have ued the fact that v d i contant. By letting M = (A 0 A 1 A ) 1 R r and x = x M, we can tranform (8) a: x = A 0 x(t) A 1 x(t τ 1 ) A x(t τ 1 τ ). (9) Before moving on, we need the following lemma regarding (I n A). Lemma 4.: [1] The real part of all eigenvalue of (I n A) are poitive if the fixed directed graph G n1 ha a directed panning tree. Theorem 4.: If the fixed directed graph G n1 ha a I(µi) directed panning tree and γ r > γ r = max µi { R(µ i) µ i }, where µ i i the ith eigenvalue of I n A, there exit τ 1 and τ uch that the following two condition are atified: i) 1 λ i (A 1 ) 1 e τ 1 λ i (A ) 1 e (τ 1τ ) 0, for all C. ii) Q r = (A 0 A 1 A ) T P r P r (A 0 A 1 A )τ 1 S r (τ 1 τ )H r τ 1 [(A 0 A 1 A ) T P r A 1 Sr 1 A T 1 P r (A 0 A 1 A )](τ 1 τ )[(A 0 A 1 A ) T P r A Hr 1 A T P r(a 0 A 1 A )] < 0, where P r i a ymmetric poitive-definite matrix choen properly uch that (A 0 A 1 A ) T P r P r (A 0 A 1 577

5 A ) < 0, and S r and H r are arbitrary ymmetric poitivedefinite matrice. 0n where r, v, x, [ A, A] 0, A 1, A are defined a in Subection IV-B, R t =, and R In addition, if the above condition are atified, τ 1 [0, τ 1 ], R 1 = 1 n 1 τ v d (tθ)dθ and τ [0, τ ], ytem (3) uing (7) guarantee that 1 n lim t r(t) τ v d (I n A) 1 τ v d (t θ)dθ γ t 1 n τ v d (t θ)dθ by uing 1 n and lim t v(t) 0 n Leibniz-Newton formula [14]. aymptotically a t. Proof: Similar to the analyi given in Subection IV-A, conider a Lyapunov function candidate Theorem 4.3: If the fixed directed graph G n1 ha a directed panning tree and γ t > γ r, where γ r i defined a in Theorem 4., there exit τ 1 and τ uch that Q t = (A 0 A 1 A ) T P r P r (A 0 A 1 A )τ 1 (P r A 1 A 0 Pr 1 A T 0 AT 1 P r V ( x t ) = [ x(t) A 1 x(t θ)dθ A x(t θ)dθ P r A 1 A 1 Pr 1 A T 1 AT 1 P r P r A 1 A Pr 1 A T AT 1 P r 3qP r ) τ (τ 1 τ )(P r A A 0 Pr 1 A T 0 A T P r P r A A 1 Pr 1 A T 1 A T P r 0 ] T P r A A Pr 1 A T P r [ x(t) A 1 x(t θ)dθ A x(t θ)dθ] P r 3qP r ) < 0, where P r i the ame matrix given in Theorem 4. and q > 1. In addition, if τ t t Q t < 0, τ 1 [0, τ 1 ], and τ [0, τ ], ytem (3) uing (10) x(ξ) T S r x(ξ)dξdθ x(ξ) T H r x(ξ)dξdθ. guarantee that all r i r d and v i v d are uniformly ultimately tθ τ tθ bounded. In particular, the ultimate bound of x i given Taking the derivative of V give by λ max(p r)a λ, where a = [ P min(p r)κλ min( Q t) r P r A 1 τ 1 V (x t ) x(t) T Q r x(t), where Q r i defined a in Theorem 4.. By following a imilar analyi to that in Subection IV-A, we can prove the tability of (9) and the exitence of τ 1 and τ uch that the two condition in Theorem 4. are atified baed on the fact that all eigenvalue of A 0 A 1 A are on the open left half plane becaue I n A ha all eigenvalue with poitive real part if G n1 ha a directed panning tree (Lemma 4.) and γ r > γ r. Then, becaue x(t) 0 n, a t, and M = [τ v d [(I n A) 1 1 n ] T, 0 T n] T, it follow that lim t r(t) τ v d (I n A) 1 1 n and lim t v(t) 0 n aymptotically a t. C. Second-order Conenu Tracking with Full Acce to the Virtual Leader In thi ubection, the reference tate r d, v d, and v d are aumed to be time-varying and v d i aumed to be available to all agent and are bounded. The following conenu tracking algorithm with both communication and input delay i propoed a u i = v d (t τ 1 τ ) n1 1 n1 a a {[r i (t τ 1 ) r j (t τ 1 τ )] γ t [v i (t τ 1 ) v j (t τ 1 τ )]}, i = 1,, n, (10) where τ 1 and τ are, repectively, the input and communication delay, a, i = 1,...,n, j = 1,...,n 1, i the (i, j) entry of the adjacency matrix A n1, r n1 r d (t), v n1 v d (t), and γ t i a poitive gain. We alo aume that v d < δ v, v d < δ a, and v d < δȧ, where δ v, δ a and δȧ are poitive contant. Uing (10), (3) can be written in the matrix form a ẋ(t) = A 0 x(t) A 1 x(t τ 1 ) A x(t τ 1 τ ) R t, (11) 578 P r A (τ 1 τ )][(τ 1 τ )δȧτ δ v γ t τ δ a ] and 0 < κ < 1. Proof: Uing Leibniz-Newton formula [14], we tranform (11) to the following ytem: d dt x(t) = (A 0 A 1 A )x(t) A 1 A 0 A 1 A A 0 τ 1 x(t θ)dθ A 1 A τ τ x(t θ)dθ A A 1 τ A x(t θ)dθ R t A 1 τ 1 τ A R t (t θ)dθ. τ x(t θ)dθ x(t θ)dθ x(t θ)dθ R t (t θ)dθ Conider a Lyapunov function candidate V (x) = x T P r x. Taking the derivative of V (x) along (11) give V (x) x T [(A 0 A 1 A ) T P r P r (A 0 A 1 A )]x τ 1 x T P r A 1 A 0 P 1 r A T 0 AT 1 P rx dθ τ 1 x T P r A 1 A 1 P 1 r A T 1 AT 1 P rx x T (t θ)p r x(t θ) x(t θ)dθ τ 1 x T P r A 1 A P 1 r A T A T 1 P r x τ 1 x T (t θ)p r τ x T (t θ)p r x(t θ)dθ (τ 1 τ )x T P r A A 0 P 1 P r x r A T 0 AT τ x T (t θ)p r x(t θ)dθ (τ 1 τ )x T P r A A 1 P 1 r A T 1 AT P rx x T (t θ)p r x(t θ)dθ (τ 1 τ )x T P r A A P 1 r A T A T P r x τ τ 1 τ x T (t θ)p r x(t θ)dθ x P r [(τ 1 τ )δȧ τ δ v γ t τ δ a ] x P r A 1 τ 1 [(τ 1 τ )δȧ τ δ v γ t τ δ a ] x P r A (τ 1 τ )[(τ 1 τ )δȧ τ δ v γ t τ δ a ],

6 where we have ued Lemma 4.1 and the fact that v d < δ v, v d < δ a, and v d < δȧ to derive the inequality. Take p() = q for ome contant q > 1. If V (x(t θ)) qv (x(t)) for τ 1 τ θ 0, we have that V (x) x T [(A 0 A 1 A ) T P r P r (A 0 A 1 A )]x τ 1 x T (P r A 1 A 0 P 1 r A T 0 A T 1 P r qp r )x τ 1 x T (P r A 1 A 1 P 1 r A T 1 A T 1 P r qp r )x τ 1 x T (P A 1 A P 1 r A T A T 1 P r qp r )x (τ 1 τ )x T (P r A A 0 P 1 r A T 0 AT P r qp r )x (τ 1 τ )x T (P r A A 1 P 1 r A T 1 AT P r qp r )x (τ 1 τ )x T (P r A A P 1 r A T A T P r qp r )x x [ P r P r A 1 τ 1 P r A (τ 1 τ )] [(τ 1 τ )δȧ τ δ v γ t τ δ a ] x(t) T Q t x(t) a x, where Q t and a are defined a in Theorem 4.3. Becaue the fact that I n A ha all eigenvalue with poitive real part if G n1 ha a directed panning tree (Lemma 4.) and γ t > γ r imply that all eigenvalue of A 0 A 1 A are on the open left half plane, there exit τ 1 and τ uch that Q t < 0 if P r i choen uch that (A 0 A 1 A ) T P r P r (A 0 A 1 A ) < 0. Moveover, we have that λ min ( Q t ) > 0. For 0 < κ < 1, if x a κλ min( Q t), we can obtain that V (x) (1 κ)λ min ( Q t ) x κλ min ( Q t ) x a x (1 κ)λ min ( Q t ) x. The uniformly ultimate boundedne of x then follow from Lemma 3.. Moreover, we can obtain that λ max(p r)a λ min(p r)κλ min( Q t) i the ultimate bound of x by following a imilar analyi to that in [], pp Remark 4.3: Note that if τ 1 = τ = 0, lim t x = 0. Alo note when τ 1 and τ are larger, the bound will be larger. V. CONCLUDING REMARKS Leaderle conenu, conenu regulation, and conenu tracking problem for econd-order integrator were dicued under a directed network topology with communication and input delay. By uing decoupling technique, we preented the tability condition for the leaderle conenu problem. The conenu regulation problem can be viewed a a direct extenion of the leaderle conenu problem. 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