Formation Control for Nonlinear Multi-agent Systems with Linear Extended State Observer

Transcription

1 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL Formaton Control for Nonlnear Mult-agent Systems wth Lnear Extended State Observer Wen Qn Zhongxn Lu Zengqang Chen Abstract Ths paper nvestgates the formaton control problem for nonlnear mult-agent systems wth a vrtual leader. A dstrbuted formaton control strategy based on lnear extended state observer (LESO) s proposed under the hypothess that velocty of the agent s neghbors could not be measured. Some suffcent condtons are establshed to ensure that the nonlnear mult-agent systems form a predefned formaton wth swtchng topology when the nonlnear functon s nown. Moreover, the tracng errors are bounded wth external dsturbance. Lastly, a numercal example wth dfferent scenaros s presented to demonstrate the valdty of the obtaned results. Index Terms Mult-agent system, lnear extended state observer (LESO), nonlnear functon, formaton control, uncertanty, vrtual leader. I. INTRODUCTION RECENTLY, dstrbutve cooperatve control of multagent systems, ncludng unmanned underwater vehcles and mult-robot systems, has been a hot research topc n many felds, such as formaton control [1 3, congeston control n communcaton systems, flocng [4, 5 and dstrbuted sensor networs [6. As a nd of consensus-assocated problems [7 10, formaton control of mult-agent systems has been prmarly nvestgated n many domans, such as unmanned ar vehcles, cooperatve transportaton and survellance. Compared wth consensus ssue, whch requres the states of all agents to reach an agreement, the man objectve of formaton control s to desgn approprate protocols that can mae the agents acheve consensus of ther states as well as preserve a desred geometrcal shape wth or wthout a group reference. There are many control methods rased to acheve the control targets, for nstance, the leader-followng approach [11 13, the behavour based method [14 16, and the vrtual archtecture approach [17, 18. Most exstng wor of formaton control for mult-agent systems focuses on local rules usng dstrbuted sensng and Manuscrpt receved June 23, 2013; accepted October 16, Ths wor was supported by Natonal Natural Scence Foundaton of Chna ( ), Program for New Century Excellent Talents n Unversty of Chna (NCET ), Natural Scence Foundaton of Tanjn (13JCY- BJC17400, 14JCYBJ18700). Recommended by Assocate Edtor Yguang Hong Ctaton: Wen Qn, Zhongxn Lu, Zengqang Chen. Formaton control for nonlnear mult-agent systems wth lnear extended state observer. IEEE/CAA Journal of Automatca Snca, 2014, 1(2): Wen Qn and Zhongxn Lu are wth the College of Computer and Control Engneerng, Nana Unversty, Tanjn , Chna, and also wth the Tanjn Key Laboratory of Intellgent Robotcs, Nana Unversty, Tanjn , Chna (e-mal: qnwen.wts@163.com; lzhx@nana.edu.cn). Zengqang Chen s wth the College of Computer and Control Engneerng, Nana Unversty, Tanjn , Chna, wth the Tanjn Key Laboratory of Intellgent Robotcs, Nana Unversty, Tanjn , Chna, and also wth the College of Scence, Cvl Avaton Unversty of Chna, Tanjn , Chna (e-mal: chenzq@nana.edu.cn). controllng. In practce, some nformaton of the agents n a mult-agent system, such as velocty, may not be obtaned. For example, the agent may not be equpped wth velocty sensors or the velocty measurement s not precse. Thus, the desgn of controller based on observers becomes a man trend n the research of mult-agent systems. Hong et al. [19,20 studed the leader-follower consensus problem for multagent systems wth an unnown velocty of varable leader, n whch certan neghbour-based rules ncludng dstrbuted controller algorthm and observers were developed. In addton, many physcal models n realty are nonlnear and have many uncertan terms. Several types of dsturbance evaluatng technques are unnown nput observer [21,22, perturbaton observer [23, dsturbance observer [24,25, and extended state observer (ESO) [ Dfferent from other observers, ESO s desgned dstnctvely to evaluate both extrnsc dsturbance and nner uncertanty of the system tself. The total uncertan part s regarded as an augmented state whch can be estmated va observer. ESO, worng as an estmator, plays a ey role n the actve dsturbance rejecton control. Actve dsturbance rejecton control (ADRC) was put forward based on ESO by Han n 1995, whch was desgned to estmate the total uncertantes on-lne usng ESO [26. Besdes, the nonlnear extended state observer could be smplfed nto a lnear ADRC (LADRC) whch s manly dscussed n ths paper. Motvated by the aforementoned statements, the formaton control for a class of nonlnear mult-agent systems wth a vrtual leader to be traced by followers s nvestgated n ths paper. The followers respond to the vrtual leader as they respond to ther neghbours. We emphasze that there s no real leader among the followers. The ntroducton of a vrtual leader ncreases robustness of the group n case of any sngle vehcle s falure. A dstrbuted lnear extended state observer (LESO) for second order mult-agent systems s proposed n the crcumstance that veloctes of an agent s neghbors cannot be measured. Furthermore, some local laws based on LESO are devsed to trac the actve vrtual leader, and the followers can shape the desred group formaton by swtchng topology at the same tme. The asymptotc stablty for closed systems s guaranteed by a gven model of the nonlnear couplngs and the boundary of tracng error s obtaned wthout a specfc model of the plant. The organzaton of ths artcle s as follows. In Secton II, some prelmnary nowledge s outlned. In Secton III, by desgnng a dstrbuted control based on LESO, some suffcent condtons are proposed to ensure that the agents acheve the desred confguraton wth gven or unnown nonlnear functons under the swtchng topology stuaton. Illustratve examples are presented to show valdty of the theoretcal results n Secton IV, and the concludng statements are gven n Secton V.

2 172 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 II. PRELIMINARIES AND PROBLEM FORMULATION A. Notatons We frst revew some nowledge of graph theory and matrx analyss that we wll use n the followng sectons. Consder a system contanng one vrtual leader and n agent-followers depcted by graph Ḡ. Graph Ḡ ncludes n followers, whch are related to graph G, and a vrtual leader, whch sends nformaton to some follower agents va drected edges. The graph G(V, E, A) wthout self-loops s composed of the node set V = {v 1,, v n }, the edge set E V V, whose element s denoted by e j = (v v j ) ( j), and the adjacency matrx A = [a j (, j = 1, 2,, n), where a j > 0 f (v v j ) E, a j = 0 f (v v j ) / E and a = 0. The Laplacan matrx assocated wth the graph s expressed as L = D A, where D = [d j s the dagonal degree matrx, wth dagonal elements d = n j=1 a j. The set of neghbours of agent s denoted by N (t) = {v j V : e j =(v v j ) E} at tme t. B = dag{b 1,, b n } s a dagonal matrx wth the elements denotng the connecton between the vrtual leader and the follower agents, where b > 0 f the vrtual leader s a neghbour of agent, and otherwse, b = 0. The maxmal connected subgraph G 1 nduced from G, s named as a component of G. We call graph Ḡ connected graph f at least one agent of each component of G s lned to the vrtual leader va a drected edge. In addton, to descrbe the varaton of the relatonshp between agents, we defne a pecewse-constant swtchng sgnal σ : [0, ) Π = {1, 2,, N}. p = σ(t) Π swtches n dfferent tme ntervals [t j, t j+1 ), where t j+1 t j τ 1 (τ 1 > 0, j = 0, 1, ). The tme ntervals are bounded, nonoverlayng and contnuous. Therefore, N (t), a j (, j = 1, 2,, n), L p and B p related wth the swtchng nterconnecton graph Ḡp are tme-varyng (p Π). B. Lnear Extended State Observer for Mult-agent Systems wth Total Uncertanty We consder a team of n agents movng wth a vrtual leader operatng n the same worspace. The vrtual leader s descrbed as x 0 = v 0, x 0 R m, v 0 = u 0, v 0 R m, (1) y = x 0, where x 0, v 0 R m are the vrtual leader s poston and velocty state, respectvely. Input u 0 s nown to all agents. The movng vrtual leader we set s a nown group reference to some followers representng a common nterest of all the agents. The followng second-order system of n agents s modeled as { ẋ = v (2) v = f(x v ω) + u where x v R m are the poston and velocty states of agent (=1, 2,, n), respectvely. f (x v ω) R m s a nonlnear functon to portray the nner dynamcs of agent, where ω s exteror dsturbance. ADRC s developed to deal wth ths problem. Presumng that f s dfferentable, and settng z (1) = x z (2) = v z (3) = f(x v ω), z (4) = h(x v ω), where h = f, we can rewrte (2) n an augmented state space form ż (1) = z (2) ż (2) = z (3) + u ż (3) = h(z ω), x = z (1) where z = (z (1)T z (2)T z (3)T ) T R 3m. Consderng that f s an extended state n model (3), LESO for (3) s desgned and the evaluaton error of LESO s revealed n the next part. Remar 1. In system (3), f(x v ω) s the total uncertanty functon whch ncludes nternal uncertan dynamcs and external dsturbance, and thus t s not easy to obtan the exact model of f(x v ω). In the followng secton, the total uncertan part f wll be seen as an augmented state, and can be estmated usng the desgned LESO. C. Convergence of LESO 1) Convergence of LESO wth the gven model of the plant: We frst consder the case that the model of f s nown. The followng decentralzed LESO for (3) s proposed wth the gven functon h, ẑ (1) = ẑ (2) + l 1 (z (1) ), ẑ (2) = ẑ (3) + l 2 (z (1) ) + u (4) ẑ (3) = l 3 (z (1) ) + h(ẑ ω), where ẑ = (ẑ (1)T ẑ (2)T ẑ (3)T ) R 3m s the observer outputs, ẑ (1) s the estmaton of system output n system (3), ẑ (2) s used to evaluate the dfferental of system output, ẑ (3) s the extended state to estmate total dsturbance f, and l ( = 1, 2, 3) s the observer gan to be determned. In partcular, we choose parameters as [29, that s, (3) (l 1, l 2, l 3 ) = (ω 0 α 1, ω 2 0α 2, ω 3 0α 3 ) (5) wth ω 0 > 0. Moreover, let the concrete characterstc polynomal of (4) be s 3 + α 1 s 2 + α 2 s + α 3 = (s + 1) 3. Then we can choose parameters as [29,.e., α = (n+1)!!(n+1 )! ( = 1, 2, 3), so the characterstc polynomal s Hurwtz. Naturally, the characterstc polynomal of (4) s λ 0 (s) = (s + ω 0 ) 2, (6) whch maes ω 0 turn out to be the only adjustable parameter of LESO. Let z () = z () ẑ () ( = 1, 2, 3). From (3) and (4), we can get the observer estmaton error as follows z (1) = z (2) ω 0 α 1 z (1) z (2) = z (3) ω0α 2 2 z (1) (7) z (3) = h(z ω) h(ẑ ω) ω0α 3 3 z (1). Now, let δ () = z() ω 1 0 ( = 1, 2, 3). Then (7) can be shown as h(z ω) h(ẑ ω) δ = ω 0 Φ 1 δ + Φ 2 ω0 2, (8)

3 QIN et al.: FORMATION CONTROL FOR NONLINEAR MULTI-AGENT SYSTEMS WITH LINEAR EXTENDED STATE OBSERVER 173 where δ = (δ (1)T Φ 1 = δ (2)T δ (3)T α α α ) R 3m,, Φ 2 = (0 0 1) T, where Φ 1 s Hurwtz accordng to the selecton of α ( = 1, 2, 3). Consderng the above analyss, the followng lemma quoted from [29 wll be useful. Lemma 1 [29. Assumng h(z ω) s globally Lpschtz wth respect to z there exsts a constant ω 0 > 0, such that lm z() (t) = 0 ( = 1, 2,, n, = 1, 2, 3). Remar 2. In ths subsecton, LESO for system (3) s presented, where the model of f s completely nown. Consequently, the dervatve of functon f s added n LESO (4). Remar 3. Compared wth the nonlnear extended state observer n [27 and other nds of observers, ω 0 turns out to be the only adjustable parameter of LESO accordng to (5). In addton, parameter ω 0 can be determned by ω 0 > 1+ P Φ 2 c 2, where P s a postve defnte matrx satsfyng Φ T 1 P + P Φ 1 = I, c s the Lpschtz constant related to h [29. 2) Convergence of LESO wth the unnown functon model: In ths part, we wll study the case that the nonlnear functon ncludng uncertanty,.e., the dynamcs of plant f s largely unnown. Dfferent from (4), LESO n (4) now taes the form of ẑ (1) ẑ (2) = ẑ (2) + l 1 (z (1) ), = ẑ (3) + l 2 (z (1) ) + u ẑ (3) = l 3 (z (1) ). The observer estmaton error n (7) becomes z (1) = z (2) ω 0 α 1 z (1) z (2) = z (3) ω0α 2 2 z (1) z (3) = h(z ω) ω0α 3 3 z (1) and (8) can be expressed as (9) (10) h(z ω) δ = ω 0 Φ 1 δ + Φ 2 ω0 2. (11) Smlar to the above analyss, we can get Lemma 2. Lemma 2 [29. Assumng h(z ω) s bounded, there exst a constant σ > 0 and a fnte T 1 > 0 such that z () (t) σ ( = 1, 2,, n, = 1, 2, 3), t T 1 > 0, ω 0 > 0. Furthermore, σ = O ( 1 / ) ω0, for some postve nteger. Remar 4. Dfferent from the case that the model of plant s gven n Secton II-C-1, LESO (9) for system (3) s presented n the face of unnown nonlnear functon n Secton II-C- 2. Therefore, the dervatve of functon f does not appear n LESO (9). Remar 5. What maes LESOs (4) and (9) dfferent from other observers s that t s desgned to estmate not only the state but also the comprehensve uncertanty by extended state ẑ (3) n (4) and (9). LESO needs the least nformaton of the system plant. The above fact hghlghts the advantages of LESO. III. MAIN RESULTS A. ADRC for Mult-agent Systems wth Gven Model of Plant Our control goal here s to coordnate all the agents to acheve the prescrbed spatal pattern, and to mantan the consstent pace of the vrtual leader wth the velocty of all agents convergng to v 0,.e., z (1) x 0 c z (2) v 0 as t, where c ( = 1, 2,, n) s the expected constant relatve poston vector between agent and the vrtual leader. Wth the assurance that our LESO s convergent, the LESObased local law wll have the form of u = 1 [ 2 [ j N (t) j N (t) a j (ẑ (1) ẑ (1) j c +c j )+b (ẑ (1) x 0 c ) a j (ẑ (2) ẑ (2) j )+b (ẑ (2) v 0 )+u 0 ẑ (3) (12) for = 1, 2,, n, wth constants 1, 2 > 0 to be determned, along wth LESO (4). The lemmas used n the proofs of the followng theorems are gven hare. Consder ϑ(t) = W ϑ(t)+y(t), where ϑ(t) = (ϑ 1 ϑ n ) T R n, y(t) = (y 1 y n ) T R n, and W s an n n matrx. Lemma 3 [29. If W s Hurwtz and lm y(t) = 0, then lm ϑ(t) = 0. Lemma 4 [19. If graph Ḡp, p Π = {1, 2,, N} s connected, then the symmetrc matrx L p +B p related wth Ḡ p s postve defnte. Consderng system (3) and LESO (4), we have Theorem 1. Theorem 1. Assumng that h(z ω) s globally Lpschtz regardng to z f the entre graph Ḡp (p Π = {1, 2,, N}) s connected n each tme nterval [t, t +1 ), there exst constants ω 0 > 0 and 1, 2 > 0, such that controller (12) wth LESO (4) together yelds lm z (1) z (2) x 0 c = 0, lm v 0 = 0, namely, the agents can attan and preserve pre-descrbed formaton, and reach velocty consensus under arbtrary swtchng rule among the possble topologes. Proof. e 1 and e 2 are set as e 2 = (v T 1 e 1 = (x T 1 x T n ) T x 0 1 (c T 1 c T n ) T = (z (1)T 1 z (1)T n ) T x 0 1 (c T 1 c T n ) T, v T n ) T v 0 1 = (z (2)T 1 z (2)T n ) T v 0 1, where 1 = (1 1) T R n. By usng (3) and (4) we can get { ė1 = e 2, ė 2 = 1 H p e 1 2 H p e H p z (1) + 2 H p z (2) + z (3), (13) where H p = (L p +B p ) I m. Let e = (e T 1 e T 2 ) T R 2nm, z = ( z (1)T z (2)T z (3)T ) T R 3nm,

4 174 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 where z () =( z ()T 1 z n ()T ) T R nm, z () =z () ẑ () (= By Lemma 1, we have that lm z=0, then we can get 1, 2, 3). It follows that V (t)=e T (Ω T e +Ω e )e. ė=ω e e+ω z z, (14) Snce Ω T e +Ω e s symmetrc and we label ts 2n nonpostve egenvalues as µ(p)= { } where µ 1 ( ) ( ) p, µ 2 p,, µ 2n p, there exsts an 0nm I Ω e = nm 0nm 0, Ω z = nm 0 nm orthogonal matrx W p such that. 1 H p 2 H p 1 H p 2 H p I nm W p (Ω T e +Ω e )W T p =Λ p =dag{µ 1 p, µ 2 p,, µ 2n p }. From Lemma 4, we get L p +B p (p Π) s postve defnte, Then we can obtan by translaton ē=w p e, then 1, 2 can be selected such that Ω e s Hurwtz. Snce h(z ω) s globally Lpschtz wth respect to z we V (t)=ē T W p (Ω T e +Ω e )W ptē=ē T Λ p ē= get from Lemma 1 that lm Ω z z () (t) =0. Snce Ω e s Hurwtz, from Lemma 1 and Lemma 3, we can choose constants 2n 2n 2n µ pē T ē µ max p ē T ē µ m ē T ē 0, =1 =1 =1 ω 0 >0 and 1, 2 >0, such that lm e (t)=0 (=1, 2). Ths completes the proof. where µ max p =max { } { } µ 1 p, µ 2 p,, µ 2n p, µm =max µ max p. p In the above theorem, the connected condton n each tme Hence, lm V (t) exsts. nterval [t, t +1 ) (=0, 1, ) s somehow strong. Then we 2) Next, we want to show lm e(t)=0 by usng Cauchy s wll dscuss the case wth jontly connected condton. convergence crtera. Assume that n each nterval [t, t +1 ) (=0, 1, ) there For arbtrary ε>0, there exsts a constant K such that s a sequence of bounded, nonoverlappng subntervals [t 0, t 1 ),, [t j, tj+1 ),, [t l 1, t l ), t =t 0, t +1 =t l, where t j+1 t j τ 2>0 (j=0, 1,, l 1). The graph n each subnterval [t j, tj+1 ) (=0, 1,, j= 0, 1,, l 1) s fxed, allowed to be connected or dsconnected. However, the graphs across nterval [t, t +1 ) (= 0, 1, ) are requred to satsfy the followng condton. Assumpton 1. The graphs over each tme nterval [t, t +1 ) (=0, 1, ) are jontly connected. We can now that matrx H p =L p +B p (p Π) has n nonnegatve egenvalues related to graph Ḡp, then we can get that the real part of the egenvalues of Ω e n (14) are nonpostve by selectng approprate 1 and 2. We label { } λ 1 p,, λ 2n p as the 2n egenvalues of Ω e. In addton, we defne λ (p)= { l : λ l p 0, l=1, 2,, 2n }. Accordng to the above denotaton, and by Lemma 5 n [30, we can derve that the graphs are jontly connected across [t, t +1 ) (=0, 1, ), f and only f t [t,t +1 ) λ(σ(t))={1, 2,, 2n}. Wth the prevous dscusson, we wll gve the followng theorem. Theorem 2. Wth the assumpton that h(z ω) s globally Lpschtz regardng to z f the nteracton graph satsfes Assumpton 1, then there exst constants ω 0 >0 and 1, 2 >0, such that controller (12) wth LESO (4) together yelds lm z (1) =0, z (2) =0, x 0 c lm v 0 namely, the agents can attan and preserve desred formaton, and reach velocty consensus. Next, we wll prove Theorem 2. Proof. Consder the Lyapunov functon V (t)=e T (t)e(t) based on (14). 1) We wll frst show that V 0. Wthout losng generalty, we assume that the pth graph s actve at tme t, then V (t)=ė T e+e T ė= e T (Ω e T +Ω e )e+2e T Ω z z. K, t+1 t Then we have ε> [ 1 t 0 [ 1 µ m l t l 1 V (t)dt = V (t +1) V (t ) <ε. l t V (t)dt+ + t l 1 t 0 V (t)dt λ(σ(t 0 )) ē T ē dt+ + λ(σ(t l 1 )) [ 0 +τ2 µ m t 0 l 1 +τ 2 t l 1 ē T ē dt > λ(σ(t 0 )) ē T ē dt+ + λ(σ(t l 1 )) ē T ē dt, where t +1 t τ 2>0 (=0, 1,, l 1). Thus, for nonnegatve nteger >K, we have Thus, +τ2 µ m t λ(σ(t )) ē T ē dt<ε, =0, 1,, l 1. +τ2 [ lm ē T (s)ē (s)+ + t λ(σ(t 0 )) ē T (s)ē (s) ds=0. λ(σ(t l 1 )) By Lemma 5 n [30 and the prevous statements, we have t [t,t +1 ) µ(σ(t))={1, 2,, 2n}, and consequently, lm β [ +τ2 2n t ē T (s)ē (s) ds=0, where β (= =1 1, 2,, 2n) are some postve ntegers.

5 QIN et al.: FORMATION CONTROL FOR NONLINEAR MULTI-AGENT SYSTEMS WITH LINEAR EXTENDED STATE OBSERVER 175 In addton, from V (t) 0, we have e(t) and ė(t) are bounded, and thus 2n β ē T (s)ē (s) s unformly contnuous, =1 then we can get by the Barbalat s lemma that lm e (t)=0, =1, 2,, 2n. From 1) and 2), the proof s completed. Remar 6. It should be noted that the formaton control problem n Theorem 1 s studed wth the condton that the graphs n each nterval [t, t +1 ) are connected, whch s somehow strong, whle swtchng between all the possble topologes s random wthout restrcton. However, n Theorem 2, we relax the condton to jontly connected graphs across [t, t +1 ), whch mposes certan requrements on desgnng the rule of swtchng among the possble topologes. B. ADRC for Mult-agent Systems wth Functon Model Mostly Unnown In the followng, we nvestgate the problem wthout detaled model of nonlnear functon by LESO (9). Theorem 3. Wth the assumpton that h(z w) s bounded, f the entre graph Ḡp (p Π={1, 2,, N}) s connected n each tme nterval [t, t +1 ), there exst constants ξ >0, ω 0 >0 and 1, 2 >0 such that controller (12) wth LESO (9) together yelds e (t) ξ =1, 2,, 2nm, t T 3 >0, namely, the tracng error s bounded. Proof. Solvng (14), we get e(t)=e Ωet e(0)+ 0 e Ωe(t τ) Ω z z(τ)dτ. (15) From (14) and Lemma 2, we can get for any t T 1, (Ω z z(t)) =1,,nm =0, (Ω z z(t)) =nm+1,,2nm Ω z σ =γ 0. (16) Smlar to Theorem 1, we choose approprate 1 and 2 such that Ω e s Hurwtz, and set Ψ=(0 0 0 γ 0 ) T and ϕ(t)= 0 eωe(t τ) Ω z z(τ)dτ, where ϕ (t) (Ω 1 e Ψ) + (Ω 1 e e Ωet Ψ) (17) s the th element of ϕ(t), and (Ω 1 e Ψ) =1,2,,nm γ 1 (γ 0 ), (Ω 1 e Ψ) =nm+1,,2nm =0. (18) Snce Ω e s Hurwtz, we have ( e Ωet) γ j 2, (19) for t T 2,, j=1, 2,, 2nm. Let T 3 =max {T 1, T 2 }. It follows that (e Ω et Ψ) γ2 γ 0, (20) for t T 3, =1, 2,, 2nm, and { (Ω 1 e e Ωet Ψ) (Ωe ) γ 2 γ 0 =1,2,,nm, (21) γ 2 γ 0 =nm+1,,2nm, for t T 3. Accordng to (17), (18) and (21), we have ϕ (t) { γ1 (γ 0 )+ (Ω e ) γ 2 γ 0 =1,2,,nm, γ 2 γ 0 =nm+1,,2nm, (22) for all t T 3. Let e sum (0)= e 1 (0) + e 2 (0) + + e 2nm (0). The followng nequalty s obtaned ( e Ωet e(0) ) γ 2 e sum (0), (23) for t T 3, =1, 2,, 2nm. From (15), t follows that e (t) ( e Ωet e(0) ) + ϕ (t). (24) Accordng to (15), (22) (24), we have e (t) { γ2 e sum (0)+γ 1 (γ 0 )+ (Ω e ) γ 2 γ 0 =1,2,,nm γ 2 γ 0 =nm+1,,2nm ξ for t T 3, =1, 2,, 2nm, where ξ =max {γ 2 e sum (0)+γ 1 (γ 0 )+ (Ω e ) γ 2 γ 0, γ 2 γ 0 }. (25) Ths completes the proof. Remar 7. In Secton III-A, ADRC based on LESO (4) s put forward for the exact model of the plant whch s completely nown. Dfferently, n Secton III-B, ADRC based on LESO (9) s desgned when the nonlnear uncertan functon model s mostly unnown. Remar 8. Snce the strong estmaton and compensaton ablty, ADRC based on LESO s wdely appled due to ts marvellous merts, especally n engneerng projects. 1) In practce, the uncertantes are usually derved from two sources, nternal uncertanty (ncludng parameter and structure uncertanty) and external dsturbance. There are many control methods amng at these two ssues, such as proporton ntegraton dfferentaton (PID), robust control and dsturbance observer. However, ownng to less dependence on system model, as the estmaton mechansm n ADRC, LESO s used to not only estmate all the system states but also evaluate the total uncertanty onlne. 2) Accordng to real-tme estmated values as stated n 1) above, ADRC can compensate the nternal and external dsturbance actvely. In sum, ADRC method, whch depends on less nformaton of system model, can estmate and offset the total uncertanty comng from nternal and external dsturbance. Combnng wth the dstrbuted control dea, t has the advantage of strong robustness aganst many nds of uncertanty. IV. NUMERICAL SIMULATIONS In the rest of the paper, numercal smulatons are gven to verfy effectveness of the protocols proposed above. A mult-agent system wth four followers and a vrtual leader n a two-dmensonal space s studed. Four possble varable topologes of the mult-agent system are depcted n Fg. 1. The networ starts wth topology Ḡ1 and swtches to another topology randomly chosen wthn {Ḡ1, Ḡ 2, Ḡ 3, Ḡ 4 } at tme T (=1, 2, ), wth the swtchng tme T satsfyng mod (T eps) 0.1 (eps s chosen wthn [1, 1.5).

6 176 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 Fg. 1. Four possble networ topologes assocated wth graphs Ḡ p (p=1, 2, 3, 4). Fg. 3. Trajectores of relatve dstance between the vrtual leader and follower (=1, 2, 3, 4) n Case 1(a). Case 1(a). The mult-agent system wth the gven plant model satsfes the condtons of Theorem 1 and the topologes are shown n Fg. 1. Here, u 0 =(sn(t) 0) T, f( )=(sn(x 11 ) sn(x 12 )) T +ω, ω=(0.3 sn(x 11 ) 0.1 sn(x 12 )) T, x 1 =(x 11 x 12 ) T. Fg. 4 Trajectores of poston errors e 1 n Case 1(a). We choose c 1 =( 6 3) T, c 2 =(6 3) T, c 3 =(6 3) T, c 4 = ( 6 3) T, ω 0 =8 and 1 = 2 =1. The orgnal postons are chosen wthn [ 5, 5 at random, the ntal veloctes are v 1 = v 2 = v 3 = v 4 = ( ) T. It can be easly verfed that the condtons of Theorem 1 are satsfed. The poston errors and the velocty errors of smulaton results are shown n Fgs. 2 6, respectvely. We can see that the followers have acheved the desred formaton and velocty consensus has been reached, where the fnal postons of the followers and the vrtual leader are mared by and o, respectvely, n Fg. 2. Fg. 2 shows that the followers asymptotcally form the pre-descrbed formaton and reach velocty consensus. Fg. 5. Trajectores of velocty errors e 2 n Case 1(a). In Fg. 6, the networ starts wth topology Ḡ 1 and swtches to another topology randomly chosen wthn {Ḡ1, Ḡ 2, Ḡ 3, Ḡ 4 } at tme T (=1, 2, ) wth the swtchng tme T satsfyng mod (T eps) 0.1 (eps s chosen wthn [1, 1.5). Fg. 2. Trajectores of agents n Case 1(a). Fg. 3 shows that the relatve dstances between the vrtual leader and follower (=1, 2, 3, 4) are Fg. 6. Random swtchng among 4 possble topologes n Case 1(a). Case 1(b). The mult-agent system wth the gven plant model satsfes the condtons of Theorem 2, and we use the same parameters as n Case 1(a). Four possble topologes of c 1 =( 6 3) T, c 2 =(6 3) T, c 3 = (6 3) T, c 4 =( 6 3) T. the mult-agent system are shown n Fg. 7.

7 QIN et al.: FORMATION CONTROL FOR NONLINEAR MULTI-AGENT SYSTEMS WITH LINEAR EXTENDED STATE OBSERVER 177 Fg. 11. Trajectores of velocty errors e 2 n Case 1(b). Fg. 7. Four possble networ topologes assocated wth graphs Ḡ p (p=1, 2, 3, 4) n Case 1(b). The possble topology graphs are swtched as Ḡ1 Ḡ2 Ḡ 3 Ḡ4 Ḡ1 Ḡ2. We set t +1 t = 1 s and each graph s actve for 0.5 s n the nterval [t, t +1 ). Here, we choose c 1 =( 3 3) T, c 2 =(1 1) T, c 3 = (3 3) T, c 4 = ( 1 1) T, and 1 = 2 =5. From Fg. 7, we can see that graphs Ḡ1 Ḡ2 and Ḡ3 Ḡ4 are connected, whch satsfes the condtons of Theorem 2. The ntal states are chosen randomly. Then the followers have acheved the desred formaton and reached velocty consensus, as shown n Fgs Fg. 12. Desgned swtchng sgnal among 4 possble topologes n Case 1(b). Fg. 8. Trajectores of agents n Case 1(b). Case 2(a). The formaton control for mult-agent system wth plant dynamcs largely unnown s studed n ths case where the observer bandwdth ω 0 =8. We use the same parameters as Case 1 and the topologes used are shown n Fg. 1. Then we can choose c 1 =(0 3) T, c 2 =(8 0) T, c 3 = (0 3) T, c 4 = ( 8 0) T, ω 0 =8, and 1 = 2 =1. Thus t s easy to show that the condtons of Theorem 2 are satsfed. The smulatonbased results presented n Fgs show that the tracng error s bounded wth LESO (9). Case 2(b). The formaton control for mult-agent system wth plant dynamcs largely unnown s nvestgated n ths part where the observer bandwdth ω 0 =50. Fg. 9. Trajectores of relatve dstance between the vrtual leader and follower (=1, 2, 3, 4) n Case 1(b). The topologes used are shown n Fg. 1. We choose c 1 =(0 8) T, c 2 =(3 0) T, c 3 = (0 8) T, c 4 =( 3 0) T, and ω 0 =50, then we can see that the follower agents can form the desred formaton and the tracng error wll be decreased n Fgs Fg. 10. Trajectores of poston errors e 1 n Case 1(b). Fg. 13 Trajectores of agents n Case 2(a).

8 178 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 Fg. 14. Trajectores of relatve dstance between the vrtual leader and follower (=1, 2, 3, 4) n Case 2(a). LESO, an actve dsturbance rejecton control s developed that can mae the followers acheve and preserve the desred formaton under two stuatons. The suffcent condton s that the graph s connected wth the gven nonlnear functon, whch s then relaxed to the case that the graphs are jontly connected. However, n the face of large model uncertantes, the tracng errors are shown to be bounded. For some condton restrctons, t s somehow dffcult to apply the jontly connected condton to the model wth large uncertanty, thus further research wll be consdered n future. Fnally, smulaton examples are provded. Fg. 15. Trajectores of poston errors e 1 n Case 2(a). Fg. 19. Trajectores of poston errors e 1 wth ω 0=50 n Case 2(b). Fg. 16. Trajectores of velocty errors e 2 n Case 2(a). Fg. 20. Trajectores of velocty errors e 2 wth ω 0=50 n Case 2(b). ACKNOWLEDGEMENT The authors acnowledge Mngwe Sun for the valuable suggestons on the theory and applcaton of actve dsturbance rejecton control (ADRC) and the anonymous revewers. Fg. 17. Trajectores of agents wth ω 0=50 n Case 2(b). REFERENCES [1 Zheng Jun, Yan Wen-Jun. A dstrbuted formaton control algorthm and stablty analyss. Acta Automatca Snca, 2008, 34(9): (n Chnese) [2 Xao F, Wang L, Chen J, Gao Y P. Fnte-tme formaton control for mult-agent systems. Automatca, 2009, 45(11): [3 Chen F, Chen Z Q, Lu Z X, Xang L Y, Yuan Z Z. Decentralzed formaton control of moble agents: a unfed framewor. Physca A: Statstcal Mechancs and ts Applcatons, 2008, 387(19 20): Fg. 18. Trajectores of relatve dstance between the vrtual leader and follower (=1, 2, 3, 4) wth ω 0=50 n Case 2(b). Comparson between Fgs. 15, 16 and Fgs. 19, 20 demonstrates that f the parameter s selected approprately, the tracng errors wll be decreased. V. CONCLUSION The paper nvestgated formaton control for a class of nonlnear mult-agent systems wth a vrtual leader. Based on [4 Wen G H, Duan Z S, L Z K, Chen G R. Flocng of mult-agent dynamcal systems wth ntermttent nonlnear velocty measurements. Internatonal Journal of Robust and Nonlnear Control, 2012, 22(16): [5 Chen Z Y, Zhang H T. Analyss of jont connectvty condton for multagents wth boundary constrants. IEEE Transactons on Cybernetcs, 2013, 43(2): [6 Tu Zh-Lang, Wang Qang, Shen Y. A dstrbuted smultaneous optmzaton algorthm for tracng and montorng of movng target n moble sensor networs. Acta Automatca Snca, 2012, 38(3): (n Chnese) [7 Yan J, Guan X P, Luo X Y, Yang X. Consensus and trajectory plannng wth nput constrants for mult-agent systems. Acta Automatca Snca, 2012, 38(7):

9 QIN et al.: FORMATION CONTROL FOR NONLINEAR MULTI-AGENT SYSTEMS WITH LINEAR EXTENDED STATE OBSERVER 179 [8 Su Y F, Huang J. Cooperatve output regulaton of lnear mult-agent systems by output feedbac. Systems and Control Letters, 2012, 61(12): [9 L Z K, Lu X D, Ren W, Xe L H. Dstrbuted tracng control for lnear multagent systems wth a leader of bounded unnown nput. IEEE Transactons on Automatc Control, 2013, 58(2): [10 Ren W. Consensus based formaton control strateges for mult-vehcle systems. In: Proceedngs of the Amercan Control Conference. Mnneapols, MN: IEEE, [11 Dong Y, Huang J. Leader-followng rendezvous wth connectvty preservaton of a class of mult-agent systems. In: Proceedngs of the st Chnese Control Conference (CCC). Hefe, Chna: IEEE, [12 Qn W, Lu Z X, Chen Z Q. Impulsve formaton control algorthms for leader-followng second-order nonlnear mult-agent systems. In: Proceedngs of the 13th IFAC Symposum on Large Scale Complex Systems: Theory and Applcatons. Shangha, Chna: IFAC, [13 Wang J L, Wu H N. Leader-followng formaton control of mult-agent systems under fxed and swtchng topologes. Internatonal Journal of Control, 2012, 85(6): [28 Gao Z Q. Scalng and bandwdth-parameterzaton based controller tunng. In: Proceedngs of the 2003 Amercan Control Conference. Denver, USA: IEEE, : [29 Zheng Q, Gao L Q, Gao Z Q. On stablty analyss of actve dsturbance rejecton control for nonlnear tme-varyng plants wth unnown dynamcs. In: Proceedngs of the th IEEE Conference on Decson and Control. New Orleans, LA: IEEE, [30 N W, Cheng D Z. Leader-followng consensus of mult-agent systems under fxed and swtchng topologes. Systems and Control Letters, 2010, 59(3 4): Wen Qn Receved her M. S. degree from the School of Tanjn Polytechnc Unversty, Chna, n She s now pursung the Ph. D. degree n the College of Computer and Control Engneerng, Nana Unversty, Chna. Her research nterest covers formaton control for mult-agent systems. [14 Arn R C. Behavor-Based Robotcs. Cambrdge, MA: MIT Press, 1998 [15 Balch T, Arn R C. Behavor-based formaton control for multrobot teams. IEEE Transactons on Robotcs and Automaton, 1998, 14(6): [16 Lumelsy V J, Harnarayan K R. Decentralzed moton plannng for multple moble robots: The coctal party model. Autonomous Robots, 1997, 4(1): [17 Lews M A, Tan K H. Hgh precson formaton control of moble robots usng vrtual structures. Autonomous Robots, 1997, 4(4): [18 Hernandez-Martnez E G, Aranda Brcare E. Non-collson condtons n mult-agent vrtual leader-based formaton control. Internatonal Journal of Advanced Robotc Systems, 2012, 9: 100 [19 Hong Y G, Hu J P, Gao L X. Tracng control for mult-agent consensus wth an actve leader and varable topology. Robotcs and Autonomous Systems, 2006, 42(7): Zhongxn Lu Receved hs B. S. degree n automaton and Ph. D. degree n control theory and control engneerng from Nana Unversty, Chna n 1997 and 2002, respectvely. He s now a professor at the College of Computer and Control Engneerng, Nana Unversty. Hs research nterest covers mult-agent system, complex dynamcal networs and control engneerng. Correspondng author of ths paper. [20 Hong Y, Chen G R, Bushnell L. Dstrbuted observers desgn for leaderfollowng control of mult-agent. Automatca, 2008, 44(3): [21 Basle G, Marro G. On the observablty of lnear, tme-nvarant systems wth unnown nputs. Journal of Optmzaton Theory and Applcatons, 1969, 2(6): [22 Chen J, Patton R J, Zhang H Y. Desgn of unnown nput observers and robust fault detecton flters. Internatonal Journal of Control, 1995, 63(1): [23 Kwon S, Chung W K. Combned synthess of state estmator and perturbaton observer. ASME Journal of Dynamc Systems, Measurement, and Control, 2003, 125(1): Zengqang Chen Receved hs B. S. degree n mathematcs, M. S. degree and Ph. D. degree n control theory from Nana Unversty, Chna, n 1987, 1990, and 1997, respectvely. He has been worng at the College of Computer and Control Engneerng, Nana Unversty, where he s currently a professor. Hs research nterest covers complex networs, adaptve control, ntellgent predctve control, and chaos system. [24 Yang H Y, Guo L, Han C. Tracng trajectory of heterogenous multagent systems wth dsturbance observer based control. In: Proceedngs of the th World Congress on Intellgent Control and Automaton (WCICA). Bejng, Chna: IEEE, [25 Yang H Y, Guo L, Zou H L. Robust consensus of mult-agent systems wth tme-delays and exogenous dsturbances. Internatonal Journal of Control, Automaton and Systems, 2012, 10(4): [26 Han Jng-Qng. A class of extended state observers for uncertan systems. Control and Decson, 1995, 10(1): (n Chnese) [27 Gao Z Q, Han Y, Huang Y Q. An alternatve paradgm for control system desgn. Decson and Control, In: Proceedngs of the 40th IEEE Conference on Decson and Control. Orlando, FL: IEEE, :