Containment Control for First-Order Multi-Agent Systems with Time-Varying Delays and Uncertain Topologies

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1 Commun. heor. Phys. 66 (06) Vol. 66, No., August, 06 Contanment Control for Frst-Order Mult-Agent Systems wth me-varyng Delays and Uncertan opologes Fu-Yong Wang ( ), Hong-Yong Yang ( ), Shu-Nng Zhang ( Û), and Fu-Jun Han (ô ) School of Informaton and Electrcal Engneerng, udong Unversty, Yanta 6405, Chna (Receved Aprl 7, 05; revsed manuscrpt receved May 6, 06) Abstract Contanment control of frst-order mult-agent systems wth uncertan topologes and communcaton tmedelays s studed. Suppose system topologes are dynamcally changed, a contanment control algorthm wth tme-varyng delays s presented. he stablty of the control algorthm s studed under the assumpton that communcaton topologes are jontly-connected, and constrant condton of dstrbuted contanment control for delayed mult-agent systems s derved wth the ad of yapunov Krasovsk functon. Smulaton results are provded to prove the correctness and effectveness of the concluson. PACS numbers: Fb, Hj Key words: uncertan topologes, tme-varyng delays, mult-agent systems, contanment control, jontlyconnected Introducton Recently, dstrbuted cooperatve control of multagent systems has attracted great attenton n the felds of control theory, mathematcs, computer scence, etc. he analyss and synthess of dstrbuted cooperatve control problem has become more complex and dffcult, due to communcaton delays and uncertan topologes emergng n practcal applcatons. Meanwhle, t s practcal sgnfcance to nvestgate contanment control for mult-agent systems wth delays and uncertantes. Consensus s an mportant research problem of dstrbuted cooperatve control of mult-agent systems. Consensus of mult-agent systems means to reach agreement state of each agent through nteracton and coordnaton. Average-consensus problem n drected networks of agents wth both swtchng topology and tme-delay s studed n Ref., and the stablty analyss s performed based on a proposed yapunov Krasovsk functon. Consensus of contnuous-tme agents wth dverse tme-delays and jontly-connected topologes are nvestgated n Ref., and a seres of suffcent condtons are derved by a contradcton approach. In Ref. 3, average consensus n networks of dynamc agents wth uncertan topologes as well as tme-varyng communcaton delays s studed, and several suffcent condtons for average consensus n the exstence of both uncertantes and delays are establshed by usng the lnear matrx nequalty method. In Ref. 4, average consensus n contnuous-tme mult-agent systems wth uncertan topologes as well as multple tme-varyng communcaton delays s nvestgated n the network of both fxed and swtchng topologes. Contanment control has been pad much more attenton as a knd of extended consensus problem wth multple leaders, whch ams to allow followers eventually convergng to a target area (convex hull formed by leaders). In Ref. 5, a contanment control algorthm for any fnte dmensonal state vector s studed wth asalle s nvarance prncple n swtchng system. In Ref. 6, two asymptotc contanment controls of contnuous-tme systems and dscrete-tme systems are proposed for the second-order mult-agent systems wth dynamc leaders, and constrant condton for control gan and samplng perod s gven. In Ref. 7, dstrbuted contanment control for networked agrangan systems wth multple dynamc leaders s studed, and a dstrbuted adaptve control algorthm combned wth sldng-mode estmators s proposed. In Ref. 8, contanment control problem for second-order mult-agent systems wth tme-varyng delays s consdered, and contanment controls wth multple statonary leaders and multple dynamc leaders are nvestgated. In Ref. 9, mean square contanment problem of frst-order and second-order ntegral mult-agent systems wth communcaton noses s nvestgated. In Ref. 0, contanment control for multple agrangan systems wth multple dynamc leaders n the presence of parametrc uncertantes s studed, and a fully dstrbuted adaptve sldng-mode control algorthm combned wth sldng-mode estmators s proposed. In Ref., contanment control problem of uncertan nonlnear mult-agent systems wth multple dynamc leaders s consdered, and Supported by the Natonal Natural Scence Foundaton of Chna under Grant Nos. 6735, , , the Scence Foundaton of Educaton Offce of Shandong Provnce of Chna under Grant Nos. ZR0FM07, BS05DX08 Correspondng author, E-mal: hyyang@yeah.net c 06 Chnese Physcal Socety and IOP Publshng td

2 50 Communcatons n heoretcal Physcs Vol. 66 dstrbuted adaptve contanment controllers are proposed under swtchng drected topologes. In ths paper, a dstrbuted contanment control problem for frst-order mult-agent systems wth uncertan topologes and tme-varyng delays s studed. he nnovaton of ths paper s that a dstrbuted contanment control protocol of mult-agent systems wth uncertan topologes and jontly-connected topologes s presented. By applyng yapunov Krasovsk method, the convergence of the algorthm for the uncertan mult-agent systems wth dsconnected topologes and tme-varyng delays s studed. Prelmnares et G = (V, E) be an undrected graph of order n, where V = {v, v,...,v n s the set of nodes, E = {(v, v j ) : v, v j V s the set of edges, and (v, v j ) E (v j, v ) E,, j I = {,,..., n. If (v j, v ) E, then v j s the neghbor of v. he set of the neghbors of node v s denoted by N = {v j V (v, v j ) E, j. A path from v to v j s denoted by π,j = {(v,, v, ), (v,, v,3 ),..., (v,m, v,m ), where v, = v, v,m = v j and (v,k, v,k+ ) E, k {,,..., m. If there s a path between any dstnct par of nodes then the undrected graph s sad to be connected. he unon of a collecton of graphs G, G,..., G m wth the same note set V, s defned as the graph G m wth the note set V and edge set equalng to the unon of the edge sets of all of the graphs n the collecton. Moreover, G, G,..., G m s jontly-connected f ts unon graph G m s connected. he weghted adjacency matrx A = a j R n n of undrected graph G s satsfyng a j 0, and a j > 0 f (v, v j ) E, otherwse a j = 0. he aplacan correspondng to undrected graph G s defned as = l j R n n, where l j s defned as follows { aj, j l j = j N a j, = j. emma 3 If undrected graph G s connected, then ts aplacan satsfes () Zero s a smple egenvalue of, and n s the correspondng egenvector,.e., n = 0. () he rest n egenvalues are all postve. Suppose the mult-agent systems consstng of n followng agents and m leader agents n ths paper. he nformaton topology of mult-agent systems s denoted by G, and the undrected graph of n followng agents s denoted by G F. he topology G F can be fxed or vared, and t may be unconnected. Consder an nfnte sequence of nonempty, bounded and contguous tme-ntervals t r, t r+ ), r =,,..., wth t = 0 and t r+ t r for some constant > 0. In each nterval t r, t r+ ), there s a sequence of subntervals t r,j, t r,j+ ), j =,,..., m r wth t r, = t r and t r,mr+ = t r+ satsfyng t r,j+ t r,j, j =,,..., m r for some nteger m r and gven constant > 0. Such that the communcaton topology descrbed by G r,j swtches at t r,j and t does not change durng each subnterval t r,j, t r,j+ ). et σ(t) : 0, ) Γ, Γ = {,,..., N (denotes the total number of all possble topologes), be a pecewse constant swtchng functon. G σ (t) denotes the nformaton topology of mult-agent systems at tme t, and ts aplacan s denoted by σ (t). G (t) denotes the nformaton topology of n followng agents, and ts aplacan s denoted by (t). 3 Contanment Control of Uncertan Mult- Agent Systems wth me-varyng Delays Consderng frst-order mult-agent systems of n followers and m leaders, and usng F = {,,..., n and Υ = {n +, n +,...,n + m to denote, respectvely, the follower set and the leader set. Suppose that mult-agent systems have the followng dynamcs ẋ (t) = u (t), =,...,n, n +,...,n + m, () where x (t) R, u (t) R are the state of the -th agent and the control nput vector, respectvely. emma 4 For undrected graph G of mult-agent systems, and ts aplacan s denoted by = F F 0 m n 0 m m. If for each follower, there exsts at least a path to leader, then F s postve defnte, F F s a non-negatve matrx and the sum of the entres n every row equals. Assumpton he communcaton topologes generated by n followers and m leaders, n each nterval t r, t r+ ), r =,,..., are jontly connected. Assumpton here exsts a connectvty subset for mult-agent systems n each non-overlappng tme ntervals t r,j, t r,j+ ) t r, t r+ ), j =,,..., m r. For each follower, there exsts at least one leader that has a path to the follower n the connectvty subset. Consderng the case of statonary leaders, and supposng the control protocol of frst-order mult-agent systems s u (t) = Σ j N (a j + a j (t))x (t τ(t)) x j (t τ(t)), F, u (t) = 0, Υ, () where a j (t) s the uncertan parameter wth a (t) = 0, and τ(t) s the communcaton delay. Assumpton 3 he tme-varyng communcaton delay τ(t) n protocol () s bounded,.e., there exsts h > 0 satsfyng: 0 τ(t) < h, t 0. Defnton s the aplacan matrx of graph G, and (t) = l j (t) s the uncertan matrx of graph G defned by { aj (t), j, l j (t) = j N a j (t), = j,

3 No. Communcatons n heoretcal Physcs 5 where, = F F F(t) F(t) 0 m n 0 m m, (t) = 0 m n 0 m m. Accordng to Defnton, the system can be wrtten as ẋ(t) = + (t)x(t τ(t)), (3) where x(t) = x (t),..., x n (t), x n+ (t),..., x n+m (t). Defnton Norm bounded parameter uncertanty (t) and F (t) satsfyng (t) = B H(t)B, (4) F (t) = D H(t)D, (5) where B, B, D, D are the constant matrx wth the approprate dmensons, and dagonal matrx H(t) satsfyng H (t)h(t) I. (6) Suppose that uncertantes of system n ths paper satsfyng (t) (t) α I m+n, (7) F(t) F (t) α I n. (8) et x F (t) s the state of followers, and x (t) s the state of leaders, then (3) can be wrtten as ẋ(t) = Ex(t τ(t)), (9) where x(t) = x F (t), x (t), x F (t) = x (t),..., x n (t), x (t) = x n+ (t),..., x n+m (t) E, E = F E F 0 m n 0 m m, E F = F + F, E F = F + F (t). From (9), we get ẋ F (t) = E F x F (t τ(t)) E F x (t τ(t)), (0) ẋ (t) = 0. () et x F (t) = x F (t) + E F E Fx (t), Eq. (0) can be wrtten as x F (t) = E F x F (t τ(t)). () emma 3 5 For any real dfferentable vector functon x(t) R n and any n n dmensonal constant matrx W = W > 0, we have the followng nequalty h x(t) x(t τ(t)) Wx(t) x(t τ(t)) tτ(t) where t 0, 0 τ(t) h. ẋ (s)wẋ(s)ds, emma 4 6 et Ξ = Ξ, F, F, H(t) be a matrces wth approprate dmensons, and H(t) matrx satsfed H (t)h(t) I, then Ξ + F H(t)F + F H (t)f < 0, satsfed f and only f there exsts a postve constant ε > 0 satsfed Ξ + ε F F + εf F < 0. emma 5 (Schur complement 7 ). For a gven symmetrc matrx S = W C, f W > 0, then S > 0 C M M C W C > 0, where M, C, W are approprate dmensons matrces. emma 6 8 et φ(t) : R 0 R be a pecewse contnuous functon wth the followng propertes. () φ(t) s contnuous and dfferentable on each subnterval t r,j, t r,j+ ) and swtched at t r,j for all r Z 0 and j = 0,,..., m r. Moreover, φ(t) s rght contnuous and rght dfferentable at t r,j for all r Z 0. () he dervatve (ncludng the rght dervatve) of φ(t) for any t 0, + ) s bounded,.e., φ(t) ω for any t 0, + ) and some constant ω > 0. () lm t + φ(τ)dτ exsts and s fnte. 0 hen lm t + φ(t) = 0. Defnton 3 9 et X = {x, x,...,x m be a set n a real vector space V R. he convex hull CO(X) of the set X s denoted as CO(X) = { m = α x x X, α R, α 0, m = α =. emma 7 et x F = x,...,x n, x = x n+,..., x n+m, f x F E F E Fx, then the contanment control of the mult-agent system can be acheved. Proof From emma and Defnton 3, we can see E F E F s a non-negatve matrx and the sum of the entres n every row equals, and E F E Fx s located n the convex hull formed by those statonary leaders, where x = x n+,..., x n+m. hus, f x F E F E Fx, then the contanment control of mult-agent systems can be acheved. Suppose the communcaton graph G σ on subnterval t r,j, t r,j+ ) has n σ connected subgraphs G σ, =,,..., n σ, and each connected subgraph G σ has d σ = d +d σ nodes, where d represents the number of followers, d σ represents the number of leaders. he aplacan matrx of subgraph G σ s denoted by σ, and the uncertan matrx of subgraph G σ s denoted by σ. hen there exsts an orthogonal matrx U σ such that U σ σu σ = dag{ σ, σ,..., nσ σ, (3) U σ σ (t)u σ = dag{ σ(t), σ(t),..., nσ σ (t), (4) Uσ E σ U σ = dag{eσ, Eσ,...,Eσ nσ, (5) x F (t)u σ = x (t), x (t),..., x n σ. (6) Accordng to the above descrbed, under the dynamc swtchng topology () can be wrtten as x F (t) = E x F (t τ(t)), (7) where E = + (t). he system dynamcs (7) can be transformed nto the followng subsystems dynamc n each subnterval t r,j, t r,j+ ) x (t) = E x (t τ(t)), =,,..., n σ, (8)

4 5 Communcatons n heoretcal Physcs Vol. 66 where x (t) = x (t) + E E x σ (t), x (t) = x σ(t),..., x σd (t), x σ (t) = x σd +(t),..., x σd σ (t), E = + (t), =,,..., n σ. heorem Consder a frst-order dynamc system of n followers and m leaders wth dynamcs () under swtchng topologes. Suppose Assumpton, Assumpton and Assumpton 3 holt, the control protocol () can solve the contanment control of frst-order problems wth tme-varyng delays and uncertan topologes, f there exsts a constant ε > 0 for any tme t (0, + ) λ n ( ) h < α + λ n ( ), max{hα, β β 4γ < ε < β + β 4γ, (9) = where λ n ( ) s the maxmum egenvalue of, α > 0, β = hλ n( ) λ n ( ), γ = α h α 4 α hλ n ( ) + α h λ n ( ). Proof Defne a common yapunov Krasovsk functon for system (7) as follows V (t) = x F(t) x F (t) + (s t + h) x F(s) x F (s)ds. (0) From (6), the functon V (t) can be rewrtten as n σ V (t) = x (t) x (t) = + (s t + h) x (s) x (s)ds. () akng the dervatve of V (t) along the trajectores of (8), t yelds V (t) = E x (t τ(t)) x (t) + x (t)e x (t τ(t)) + h x (t) x (t) From Assumpton 3, snce τ(t) < h, h > 0, we get x (s) x (s)ds. () Accordng to emma 3, we can get tτ(t) x (s) x (s)ds x (s) x (s)ds. (3) tτ(t) x (s) x (s)ds h x (t) x (t τ(t)) x (t) x (t τ(t)). (4) From (3) and (4), we have V (t) x (t)e x (t τ(t)) x (t τ(t))e x (t) + h x (t τ(t))e E x (t τ(t)) = h x (t) x (t τ(t)) x (t) x (t τ(t)) = h x (t) x (t) h x (t τ(t)) x (t τ(t)) + x (t τ(t))he E x (t τ(t)) = + x (t) h I d E x (t τ(t)) + x x (t τ(t)) h I d E (t), where E = + (t). Accordng to Defnton V (t) h x (t) x (t) + x (t τ(t)) h I d + h x (t τ(t)) = + h x (t τ(t)) (t) (t) x (t τ(t)) + h x (t τ(t)) (t) x (t τ(t)) + h x (t τ(t)) (t) x (t τ(t)) + x (t) h I d ( + (t)) x (t τ(t)) + x (t τ(t)) h I d ( + (t)) x (t) ( h x (t) x (t) + x (t τ(t)) hα ) I h d + h x (t τ(t)) = + h x (t τ(t)) (t) x (t τ(t)) + h x (t τ(t)) (t) x (t τ(t))

5 No. Communcatons n heoretcal Physcs 53 = + x (t) h I d x (t τ(t)) + x (t) (t) x (t τ(t)) n σ ỹ σ Ω σ + Ψ σ ỹ σ, = where ỹσ = x (t), x (t τ(t)), Ω h σ = I d h I d hen h I d (hα h )I d + h, Ψ σ = 0 (t) (t) h (t) + (t) Ψ σ = 0 αi d H (t)i d M αi d H (t)id = 0 I d H (t) αi d h αid, =, (t) = αi d H (t)i d. αi d h αid H (t)0 I d where, M = h{ αid H (t)i d + αi d H (t)i d. Accordng to emma 4, Ω σ + Ψ σ < 0 equvalent to αid Ω σ + ε αid + ε0 I d 0 I d = Λ σ + Π σ < 0, where Λ σ = Ω σ = h I d αh h I d s postve defnte. αh h I d ( h hα )I d h, Π σ = ε α I d ε α h ε α h εid + ε α h, and ε > 0,.e., Π σ Next, we wll prove Λ σ s postve defnte and Λ σ > Π σ. Frstly, accordng to emma 3, snce /h > 0, we wll just prove ( Φ σ = h hα) I d h ( ) ( h I d h I d ) h I d = hα I d h + s postve defnte,.e., f hλ g( ) + λ g( ) hα > 0, then Φ σ s postve defnte; Secondly, we wll prove ( h σ = ε α )I d ( + ε α h) h I d ( + ε α h) h I > 0, d Θ Θ = (/h hα ε)i d (h + (/ε)α h ), just /h (/ε)α > 0, σ > 0,.e., ε > hα, ε +bε+c < 0, where b = hλ d ( )λ d ( ) < 0, c = α h α 4 α hλ d ( ) + α h λ ( d ) > 0. λ g ( ) s the egenvalue of, g =,,...,d. We get λ d h < ( ) α + λ ( d ), { max hα, b b 4c < ε < b + b 4c, where λ d ( ) s the maxmum egenvalue of. From (3), t s obvously that λ d ( ) λ n( ), then h < λ n ( )/(α + λ n( )) for any t (0, + ). herefore, f Eq. (9) holds, then Ω σ + Ψ σ < 0,.e., V (t) < 0. he system (7) s asymptotcally stable. Snce the system (7) s asymptotcally stable, then x F (t) and x F (t τ(t)) are bounded. From (), x F (t) s also bounded. Consequently the dervatve of nσ = ỹ σ Ω σ + Ψ σỹσ s bounded from (6) and (8). Snce V (t) s nonncreasng and bounded below by 0, then V (t) must approach a lmt as t. Note that ( n σ ) 0 ỹ σ (s)ω σ + Ψ σ ỹ σ (s) ds 0 0 = V (s)ds = V (t) V (0), and n σ = ỹ σ Ω σ + Ψ σ ỹ σ < 0. It follows that 0 ( n σ = ỹ σ (s)ω σ +Ψ σ ỹ σ (s))ds exsts and fnte. hen by emma 6, we have ( n σ ) ỹ σ Ω σ + Ψ σ ỹ σ = 0. (5) lm t + = We have ỹ σ = 0 as t from (5),.e., x (t) = x (t τ(t)) = 0, then x (t) = x (t τ(t)) = E E x σ (t).

6 54 Communcatons n heoretcal Physcs Vol. 66 Furthermore, n any subnterval t r,j, t r,j+ ) t r, t r+ ), j =,,..., m r, lm t + x (t) = lm t + x (t τ(t)) = E E x σ (t), =,,..., n σ. herefore, n the connected porton of mult-agent systems n the subnterval t r,j, t r,j+ ) and t r,j+, t r,j+ ),.e., n the subnterval t r,j, t r,j+ ), lm t + x (t) = lm t + x (t τ(t)) = E E x σ (t) stll holds, =,,..., n σ, t +. hen by nducton, accordng to Assumpton, lm x F (t) = lm x F (tτ(t)) = E t + t + F E Fx (t), snce all agents are jontly-connected n each t r, t r+ ). From emma 7, the contanment control of the frst-order multagent systems wth uncertan topologes can be acheved. 4 Smulatons Consder two dynamc swtchng topologes shown n Fgs. and, where the connecton weght of each edge s. Suppose the communcaton topology of the multagent system randomly s swtched among G to G 4 at t = K, k = 0,,..., t = 0.5 s. can be obtaned, and the maxmum egenvalue of s by calculatng. Accordng to the constrants of heorem n ths paper, the upper bound of the allowed delays s et τ(t) = sn t s the tme-varyng delays of mult-agent systems n the experments. he ntal state of followers and leaders are taken x (0) = 8.0, x (0) = 6.0, x 3 (0) = 4.0, x 4 (0) = 6.0, x 5 (0) = 4.0, x 6 (0) =.0; x 7 (0) = 0.0, x 8 (0) =.0, respectvely. For = α sn(t) wth α = 0., smulaton results n 0 seconds are gven n Fg. 3, whch shows that those followers state can asymptotcally converge to the regon formed by two leaders,.e., contanment control of mult-agent systems can be acheved, where the samplng nterval s 0. s. Fg. Communcaton topology of mult-agent systems wth two leaders (agent 7,8) and sx followers. We consder sx followers guded by two leaders, whch are movng n a horzontal lne, and use F = {,, 3, 4, 5, 6 and Υ = {7, 8 to denote, respectvely, the follower set and the leader set. From the communcaton topology of the unon of a collecton of smple graphs G G 4 n Fg., the system matrx = Fg. Communcaton topology of mult-agent systems wth three leaders (agent 7,8,9) and sx followers. Fg. 3 State trajectores of mult-agent systems wth the topologes shown n Fg..

7 No. Communcatons n heoretcal Physcs 55 Fg. 4 State trajectores of mult-agent systems wth the topologes shown n Fg.. We consder sx followers guded by three leaders, whch are movng n a plane, and use F = {,, 3, 4, 5, 6 and Υ = {7, 8, 9 to denote, respectvely, the follower set and the leader set. From the communcaton topology of the unon of a collecton of smple graphs G G 4 n Fg., the system matrx = can be obtaned, and the maxmum egenvalue of s by calculatng. Accordng to the constrants of heorem, the upper bound of the allowed delays s et τ(t) = snt s the tme-varyng delays of mult-agent systems n the experments. he ntal poston of followers and leaders are taken x (0) = (, ), x (0) = (, 3), x 3 (0) = (3, ), x 4 (0) = (3, 5), x 5 (0) = (5, 3), x 6 (0) = (6, 6), x 7 (0) = (8, 0), x 8 (0) = (0, 8), x 9 (0) = (0, 0), respectvely. For = α sn(t) wth α = 0., smulaton results are gven n Fg. 4, whch shows that those followers can asymptotcally convergence to the trangle formed by three leaders,.e., contanment control of mult-agent systems can be acheved. 5 Concluson In ths paper, a dstrbuted contanment control for uncertan mult-agent systems wth multple statonary leaders s studed, and control algorthm of frst-order mult-agent systems wth tme-varyng delays and jontlyconnected topologes s proposed. By applyng modern control theory and algebrac graph theory, the convergence of mult-agent systems for the contanment control algorthm s analyzed on yapunov Krasovsk method. Some smulaton examples are gven to verfy the effectveness of the concluson for one-dmensonal and twodmensonal uncertan mult-agent systems wth tmevaryng delays, respectvely. he contanment control problem for dscrete-tme systems wll be our future work. Acknowledgements he authors would lke to thank the revew experts. References P. n and Y. Ja, Physca A 387 (008) 303. P. n and Y. Ja, Automatca 47 (0) Y. Sun, J. Frankln Ins. 349 (0) Y. Shang, near Algebra Appl. 459 (04) 4. 5 G. Notarstefano, M. Egerstedt, and M. Haque, Automatca 47 (0) H. u, G. Xe, and. Wang, Automatca 48 (0) J. Me, W. Ren, and G. Ma, Automatca 48 (0) K. u, G. Xe, and. Wang, Syst. Control ett. 67 (04) 4. 9 Y. Wang,. Cheng, Z.G. Hou, et al., Automatca 50 (04) 9. 0 D. Yang, W. Ren, and X. u, Syst. Control ett. 7 (04) 44. W. Wang, D. Wang, and Z. Peng, Neurocomputng 5 (05). P. n and Y. Ja, IEEE Automatc Control 55 (00) C.D. Godsl, G. Royle, and C.D. Godsl, Algebrac Graph heory, Sprnger, New York (00). 4 Z. Meng, W. Ren, and Z. You, Automatca 46 (00) Y.G. Sun,. Wang, and G. Xe, Syst. Control ett. 57 (008) I.R. Petersen and C.V. Hollot, Automatca (986) B. Boyd,. Ghaou, E. Feron, et al., near Matrx Inequaltes n System and Control heory, SIAM, Phladelpha (994) 8. 8 P. n, K. Qn, H. Zhao, et al., J. Frankln Ins. 349 (0) Y. Cao and W. Ren, Contanment Control wth Multple Statonary or Dynamc eaders under a Drected Interacton Graph, Proceedngs of the 48th IEEE Conference on Decson and Control, Shangha (009)