Adaptive multi-agent containment control with multiple parametric uncertain leaders

Automatica (

)

–

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Adaptive multi-agent containment control with multiple parametric uncertain leaders✩ Xinghu Wang a,1 , Yiguang Hong a , Haibo Ji b a

Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

b

Department of Automation, University of Science and Technology of China, Hefei, 230027, China

article

info

Article history: Received 3 May 2013 Received in revised form 8 December 2013 Accepted 29 April 2014 Available online xxxx Keywords: Multi-agent systems Multiple leaders Containment Uncertain parameter Adaptive internal model

abstract In this paper, an adaptive containment control is considered for a class of multi-agent systems with multiple leaders containing parametric uncertainties. The agents are heterogeneous though their dynamics have the same relative degree and are minimum phase, while the interconnection topology is described by a general directed graph. A distributed containment control is proposed for the agents to enter the moving convex set spanned by the leaders, based on an adaptive internal model and a recursive stabilization control law. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Recent decades have witnessed a lot of research interest in the coordination of multi-agent systems, especially the leaderfollowing coordination of linear systems (Hong, Wang, & Jiang, 2013; Li, Liu, Ren, & Xie, 2013). Distributed output regulation has been studied as a framework for such problems when the leader’s dynamics are totally different from those of the followers, and distributed control based on internal model (IM) is shown to be an effective way to track a moving leader (Hong et al., 2013; Su & Huang, 2013; Wang, Hong, Huang, & Jiang, 2010). In fact, IM has been widely used in conventional output regulation (Huang, 2004; Isidori, Marconi, & Praly, 2012). Moreover, adaptive IMs were proposed for the output regulation with uncertain exosystems (Liu, Chen, & Huang, 2009; Obregón-Pulido, Castillo-Toledo, & Loukianov, 2011; Serrani, Isidori, & Marconi, 2001), and then distributed output regulation with an uncertain leader was studied for multi-agent systems (Su & Huang, 2013). Also, multi-agent containment control attracts more and more attention by forcing follower agents to enter a given convex set

✩ This work is supported by National Natural Science Foundation of China under Grants 61174071 and 61273090. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tamas Keviczky under the direction of Editor Frank Allgöwer. E-mail addresses: [email protected] (X. Wang), [email protected] (Y. Hong), [email protected] (H. Ji). 1 Tel.: +86 10 82541824; fax: +86 10 62587343.

http://dx.doi.org/10.1016/j.automatica.2014.07.019 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

(maybe spanned by a group of leaders) in a distributed way. Various containment controllers were designed for fixed or switched interconnection topologies and related connectivity conditions for containment were discussed (e.g., Cao, Stuart, Ren, & Meng, 2011; Lou & Hong, 2012; Meng, Ren, & You, 2010; Shi & Hong, 2009; Shi, Hong, & Johansson, 2012). The objective of our paper is to study containment control with multiple leaders containing parametric uncertainties. Our contribution can be summarized as follows: (i) The leader model we considered is different from those in existing containment results for the leaders with known dynamics (Cao et al., 2011; Mei, Ren, & Ma, 2012; Meng et al., 2010), though the parametric uncertainties may exist in the followers’ dynamics (see Mei et al., 2012). For the leaders with uncertain parameters, the existing containment methods cannot be applied directly. To deal with this hard problem (noting that adaptive control design is essentially nonlinear), we propose an adaptive IM-based design for containment control. (ii) Even when there is one leader, our problem is different from the leader-following problem with the leader containing uncertain inputs (e.g., Li et al. (2013), where the leader shares the same dynamics with the followers and sends the full state information to the agents connected to it). In this paper, the leaders have uncertain parameters and their dynamics are different from those of the followers. Moreover, the containment problem is solved using agents’ relative outputs. (iii) The results are different from those given for distributed output regulation when there is one leader with parametric

2

X. Wang et al. / Automatica (

uncertainties (Su & Huang, 2013). The follower agents with relative degree one and an undirected interconnection topology among followers were considered in Su and Huang (2013). Here we develop an adaptive IM-based approach to study the leader-following/ containment when the interconnection topology is a general digraph and the followers are heterogeneous with relative degree greater than one.

K 

E¯ ij (σ )vj

j=1

i = 1, . . . , N

(1)

where x˘ i ∈ R , ui , yi ∈ R, and σ ∈ R denotes the uncertain parameter in the agents’ dynamics. The leaders are described as ni

v˙ j = Sj (ω)vj ,

nσ

j = 1, . . . , K

(2)

where vj ∈ R and ω ∈ W ⊂ R represents the uncertain parameter in a fixed compact set W. yN +j = Fj vj ∈ R is the output of leader j. It is assumed that the initial conditions of the leaders belong to a fixed compact set V ⊂ RKnv , i.e., (v1 (0), . . . , vK (0)) ∈ V. nv

nω

Remark 2.1. There are two different types of parametric uncertainties in our problem: uncertainty σ in the followers, to be handled with robust technique (see the distributed stabilization control in Theorem 4.1), and uncertainty ω in the leaders, to be tackled using adaptive internal model. In what follows, we use A¯ i , B¯ i , C¯ i , E¯ ij by dropping σ for simplicity. With regarding the followers and leaders as nodes, a digraph (directed graph) G = (V , E ) with a node set V = {1, . . . , N + K } and an arc set E can be defined to describe the interconnection for the multi-agent system (1) with the leaders (2), where the first N nodes are associated with the N followers of system (1) and the last K nodes without any neighbors represent the leaders of (2). The arc set E contains an arc, denoted by (i, j), if node j can get the output yi of node i. The set Ni = {j ∈ V : (j, i) ∈ E } denotes the set of neighbors of node i. A path from i to j in G is a node/arc sequence i1 e1 i2 e2 · · · em−1 im of distinct nodes iκ and arcs eκ = (iκ , iκ+1 ) ∈ E for κ = 1, 2, . . . , m − 1 with i1 = i, im = j. The adjacency matrix of G is denoted as A = (aij ) ∈ R(N +K )×(N +K ) , where aii = 0, aij = 1 if (j, i) ∈ E and aij = 0 otherwise. The Laplacian of G is denoted by L = (lij ) ∈ R(N +K )×(N +K ) with lii = j=1 aij and lij = −aij for i ̸= j, i, j = 1, . . . , N + K (Godsil & Royle, 2001). Convex sets are used to describe containment problems. A set S ⊂ Rm is convex if (1 − γ )x + γ y ∈ S whenever x, y ∈ S and 0 < γ < 1 (Rockafellar, 1972). The intersection of all convex sets containing S is the convex hull of S , denoted by co(S ). The convex hull of a set of points y1 , . . . , yK ∈ Rm is a polytope, denoted by co{y1 , . . . , yK }. Denote Bρx , {x ∈ Rn : ∥x∥ ≤ ρ} for a constant ρ > 0, and Ωc (W ) , {x ∈ Rn : W (x) ≤ c } for a constant c > 0 and a smooth positive definite function W (x). For a closed convex set S , denote ∥x∥S , inf{∥x − y∥ : y ∈ S }, where ∥ · ∥ denotes the Euclidean norm. Define a relative neighbor-based output eiv :

N +K

1  eiv = (yi − yj ), |Ni | j∈N

if, for any set Bρx˘ with x˘ = (˘x1 , . . . , x˘ N )T ,2 we can find a distributed control

ς˙ i = gi (eiv , ςi , ςj , j ∈ Ni ),

ui = fi (eiv , ςi , ςj , j ∈ Ni ) (4) ς n ςi with Bρ ′ , and ς = (ς1 , . . . , ςN ), ςi ∈ R such that, for any (˘x(0), ς ς (0)) ∈ Bρx˘ × Bρ ′ and (v1 (0), . . . , vK (0), ω) ∈ V × W, the solution of the closed-loop system is bounded over [0, ∞), and i = 1, . . . , N

t →∞

In this paper, we consider a multi-agent system composed of N follower agents and K leaders. The followers’ dynamics are described as follows:

yi = C¯ i (σ )˘xi ,

–

lim ∥yi (t )∥C (t ) = 0,

2. Formulation

x˙˘ i = A¯ i (σ )˘xi + B¯ i (σ )ui +

)

i = 1, . . . , N

(3)

i

where |Ni | is the cardinality of the set Ni . Definition 2.1. The (semi-global) adaptive containment problem is solved for the multi-agent system (1) with uncertain leaders (2)

where C (t ) = co{yN +1 (t ), . . . , yN +K (t )}. To solve the problem, standard assumptions are listed. Assumption 2.1. For each j = 1, . . . , K , all the eigenvalues of Sj (ω) are distinct with zero real parts for all ω. Assumption 2.1 is widely used in the study of output regulation with exosystem/leader containing uncertain parameters. Under this assumption, for any (v1 (0), . . . , vK (0), ω) ∈ V × W, there is a known compact set Σ ⊂ RKnv +nω such that d(t ) , (v1 (t ), . . . , vK (t ), ω) ∈ Σ for t ≥ 0. With uncertainty ω, the leaders may produce sinusoidal signals with arbitrary unknown frequencies, amplitudes and phases. Assumption 2.2. For i = 1, . . . , N, the system (1) has the same relative degree r (that is, C¯ i A¯ ki B¯ i = 0, k = 0, 1, . . . , r − 2, and C¯ i A¯ ir −1 B¯ i ̸= 0) with a known high-gain frequency C¯ i A¯ ir −1 B¯ i , and is minimum phase. Without loss of generality, we assume C¯ i A¯ ri −1 B¯ i = 1 and r ≥ 2. Assumption 2.3. For each follower, there exists at least one leader that has a directed path to it. Remark 2.2. Under Assumption 2.3, the Laplacian L of the digraph G can be expressed as

 L=

H1 0K ×N

H2 0K ×K



,

H1 ∈ RN ×N , H2 ∈ RN ×K .

From Meng et al. (2010), all the eigenvalues of H1 have positive real parts; each entry of −H1−1 H2 is nonnegative; and each row of −H1−1 H2 has a sum equal to 1. By Theorem 2.3 of Berman and Plemmons (1979) (p. 134), there is a matrix Q = diag (q1 , . . . , qN ), ¯ 0 be the minqi > 0 such that QH1 + H1T Q is positive definite. Let λ imum eigenvalue of QH1 + H1T Q . 3. Preliminary results To solve the containment problem, we provide preliminary results in this section. At first we convert the containment problem into a leader-following problem with a virtual leader. Define the tracking error for each agent ei = yi + (H1−1 H2 )(i) F v,

i = 1, . . . , N

(5)

(i)

where (H1 H2 ) denotes the ith row of the matrix H1 H2 , F = block diag (F1 , . . . , FK ), and v = (v1 , . . . , vK )T . Thus, we can define a multi-agent system with a virtual leader as follows. −1

x˙˘ i = A¯ i x˘ i + B¯ i ui + E¯ i v,

−1

yi = C¯ i x˘ i ,

i = 1, . . . , N

(6)

where E¯ i = block diag (E¯ i1 , . . . , E¯ iK ), and

v˙ = S¯ (ω)v,

S¯ (ω) = block diag (S1 (ω), . . . , SK (ω)).

(7)

Lemma 3.1. Under Assumption 2.3, suppose that the leader-following problem of (6) with the virtual leader (7) can be solved in the sense ς that, for any set Bρx˘ , there is a distributed control (4) with Bρ ′ , such

2 For vectors x , . . . , x , (x , . . . , x )T , [xT , . . . , xT ]T . 1 n 1 n n 1

X. Wang et al. / Automatica (

ς

that, for each (˘x(0), ς (0)) ∈ Bρx˘ × Bρ ′ and (v(0), ω) ∈ V × W, the solution of the closed-loop system composed of (6) and (4) is bounded for t ≥ 0 and limt →∞ ei (t ) = 0, i = 1, . . . , N. Then the adaptive containment problem can be solved. Proof. If the leader-following problem of (6) with (7) is solved, then limt →∞ ei (t ) = 0, i = 1, . . . , N. As shown in Remark 2.2, each row of −H1−1 H2 has a sum equal to 1, so limt →∞ ∥yi (t )∥C (t ) = 0. Thus, the conclusion follows immediately.  Remark 3.1. In the traditional leader-following problem with a single leader, each follower agent tracks the same trajectory of the leader, but in the problem for (6) with the virtual leader (7), follower agents need to track different reference trajectories. Letting ev = [e1v , . . . , eN v ]T , y = [y1 , . . . , yN ]T , we have with ∆ = diag 1/|N1 |, . . . , 1/|NN | . Hence, taking eN +j = 0, j = 1, . . . , K , the relative information eiv can be rewritten as 1 

eiv =

|Ni |



(ei − ej ),

i = 1, . . . , N .

–

3

where αi = (αi1 (ω), . . . , αi,ι(si ) (ω))T can be computed following Proposition 6.14 of Huang (2004). Without loss of generality, we can assume that (10) is already the polynomial of the least degree satisfying (11), namely, (10) is the minimal zeroing polynomial of Ui (ω)v (refer to Liu et al. (2009)). Taking τi = (τi1 , . . . , τisi )T with

τij (v, ω) = Ui (ω)S¯ j−1 (ω)v, Ψi = [1 0 · · · 0]    Isi −1 0   , si is odd;   0 αi1 , . . . , αi,ι(si ) 0  Φi (αi ) =      0 Isi −1   , si is even, αi1 0 · · · αi,ι(si ) 0 we have

ev = ∆(H1 y + H2 F v) = ∆H1 (y + H1−1 H2 F v)



)

(8)

j∈Ni

Under Assumption 2.2, let Ti = [Ti1 , . . . , Ti,ni −r , A¯ ri −1 B¯ i , . . . , A¯ i B¯ i , B¯ i ] with Ti1 , . . . , Ti,ni −r as a basis of the null space of matrix [C¯ , (C¯ A¯ ) , . . . , (C¯ A¯ r −1 )T ]T . It can be verified that, under the transformation (zi , xi )T = Ti−1 x˘ i with zi ∈ Rni −r and xi = (xi1 , . . . , xir )T ∈ Rr , the T

T

τ˙i (v, ω) = Φi (αi )τi (v, ω),

Ui (ω)v = Ψi τi (v, ω).

(12)

Another assumption is needed to guarantee the parameter convergence in the adaptive containment control. Assumption 3.1. The initial condition vj (0) excites all oscillatory modes of the jth leader (2) for j = 1, . . . , K , and the matrix pair (Ui (ω), S¯ (ω)) is observable for i = 1, . . . , N. This assumption was used in Marino and Tomei (2003) for adaptive output regulation when N = 1 and K = 1. Under Assumption 3.1, all oscillatory modes of leaders appear in Ui (ω)v . Following the discussion in Liu et al. (2009), all τi (t )’s are Persistence of Excitation (PE).

system (6) can be converted into z˙i = A0i zi + D0i xi1 + E0i v

4. Main results

x˙ i = Ac xi + D1i xi1 + BRi zi + E1i v + Bui yi = Cxi ,

i = 1, . . . , N

(9)

where



0 0

Ac =

Ir − 1 0



,

 B=

0(r −1)×1 1



,

 C =

1 0(r −1)×1

T

.

To solve the leader-following problem by output regulation theory, as shown in Huang (2004), we first need to solve the regulator equations described by a group of matrix equations. By Assumption 2.2, A0i is Hurwitz. Then, with Assumption 2.1, the matrix equation Zi (ω)S¯ (ω) = A0i Zi (ω) − D0i (H1−1 H2 )(i) F + E0i admits a unique solution Zi (ω) (see Proposition A.2 in Huang (2004)). As a result, the regulator equations associated with (7) and (9) can be solved with a solution (Zi (ω), Xi (ω), Ui (ω)) where Xi1 = −(H1−1 H2 )(i) F

Xi (ω) = (Xi1 , . . . , Xir (ω))T ,

Xi,j+1 (ω) = Xij (ω)S¯ (ω) − D1ij Xi1 − E1ij ,

Ui (ω) = Xir (ω)S¯ (ω) − D1ir Xi1 − E1ir − Ri Zi (ω). It can be verified that there exists a polynomial Pi (λ) = λsi − αi,ι(si ) (ω)λsi −2 − αi,ι(si )−1 (ω)λsi −4 (10)

with

ι(s) ,

s/2, (s − 1)/2,



s is even; s is odd,

dt si

− αi,ι(si ) (ω)

− · · · − αi1 (ω)

dsi −2 Ui v dt si −2

− αi,ι(si )−1 (ω)

dsi −2ι(si ) Ui v dt si −2ι(si )

=0

θ˙i = Φi (αi )θi ,

Ui (ω)v = Mi (αi )θi .

(13)

η˙ i = (Ac − LC )ηi + Bui ζ˙i = Φi (αˆ i )ζi − L¯ i (C ηi + µi ζi ) α˙ˆ i = −Γi Ωi (ζi )(C ηi + µi ζi )

(14)

with (ηi (0), ζi (0), αˆ i (0)) = (0, 0, 0), where L¯ i = (0(si −2)×1 , 1, ¯li )T ∈ Rsi with ¯li > 0, and

Γi = diag (γi1 , . . . , γi,ι(si ) )

whose roots are distinct with zero real parts for ω ∈ Rnω , such that dsi Ui v

s −1

(Mi Φi i )T ]T is nonsingular. Performing the transformation θi = Oi−1 τi , we obtain

Based on (13), motivated by Obregón-Pulido et al. (2011), we consider a distributed adaptive internal model

j = 1, . . . , r

− · · · − αi1 (ω)λsi −2ι(si )

The containment design will be given in two steps: distributed internal model design and distributed stabilization control design. Let us first construct a distributed adaptive internal model. To this end, an observable pair is constructed first. Take µi = [µi1 , . . . , µi,si −1 , 1] to make the polynomial P¯i (λ) = λsi −1 + µi,si −1 λsi −2 + · · · + µi2 λ + µi1 stable, and L = [l1 · · · lr ]T ∈ Rr to make the matrix (Ac − LC ) Hurwitz. Define Υi (αi ) = [Υi1T , . . . , ΥirT (αi )]T and Mi (αi ) = −Υi,r +1 (αi ) with Υi1 = µi and Υi,j+1 (αi ) = Υij (αi )Φi (αi ) + lj µi , j = 1, . . . , r. By Popov–Belevitch–Hautus test, it can be verified that the pair (Mi (αi ), Φi (αi )) is observable. Hence, the matrix Oi = [MiT , . . . ,

dsi −4 Ui v

Ωi (ζi ) = [ζi,si −2ι(si )+1 , . . . , ζi,si −3 ζi,si −1 ]T and performing another coordinate transformation z¯i = zi − Zi (ω)v,

x¯ i = xi − Xi (ω)v η¯ i = x¯ i − Υi (αi )θi − ηi , ζ¯i = θi − ζi

dt si −4 (11)

α¯ i = αˆ i − αi ,

u¯ i = ui − Mi (αˆ i )ζi .

(15)

4

X. Wang et al. / Automatica (

Then, with ϵi = (η¯ i , ζ¯i , α¯ i )T , we have

–

It can be verified that the constant ℓ1 > 0 can be chosen to make Vi1′ (¯zi , ϵi ) be positive definite and satisfy

ϵ˙i = gi (¯zi , ϵi , ei ) z˙¯ i = A0i z¯i + D0i ei , ˙x¯ i = Ac x¯ i + D1i ei + B(¯ui + ϕi (zi , ϵi ))

(16)

¯ 1 ∥(δi , ζ¯isi )T ∥2 V˙ i1′ ≤ −λ

(20)

¯ 1 > 0. for a constant λ Consider the following system

where x¯ i1 = ei and

  i (Ac − LC )η¯ i + D1i ei + BRi z¯i + Le  (Φi (αi ) − L¯ i µi )ζ¯i − L¯ i (C η¯ i − ei )   gi (¯zi , ϵi , ei ) =    +(Φi (αi ) − Φi (αˆ i ))ζi Γi Ωi (ζi )(C η¯ i − ei + µi ζ¯i ) ϕi (zi , ϵi ) = Ri zi − Mi (αi )ζ¯i + (Mi (αˆ i ) − Mi (αi ))(θi − ζ¯ ).

ζ˙¯ isi = −¯li ζ¯isi − Ωi (θi )⊤ α¯ i ,

It can be seen that ϕi (0, 0) = 0. Remark 4.1. The distributed internal model (14) consists of three parts: the ηi subsystem is a stable filter to convert the signal ui to ηi with the state ηi viewed as the input of the other two subsystems; the ζi subsystem reproduces the steady-state information of the system (13); the αˆ i subsystem estimates the uncertain parameter αi . This structure of internal model benefits us with an asymptotical stability property of the (¯zi , ϵi ) subsystem in (16) (see Lemma 4.1). The role of the coordinate transformation (15) is to convert the leader-following problem of (9) with (7) into a distributed stabilization problem of (16) (see Lemma 4.2). Let us first introduce a result for the following system z˙¯ i = A0i z¯i ,

)

(17)

(21)

Under Assumption 3.1, τi (t ) is uniformly bounded and PE, so is θi (t ). Then the equilibrium point (ζ¯isi , α¯ i ) = (0, 0) is global exponentially stable (see Lemma B.2.3 in Marino and Tomei (1995)). Therefore, by the converse Lyapunov function theorem of linear time-varying systems (see Theorem 3.10 in Khalil (1996)), there is a smooth Lyapunov function Vi1′′ (t , ζ¯isi , α¯ i ) with a3 ∥(ζ¯isi , α¯ i )T ∥2 ≤ Vi1′′ (t , ζ¯isi , α¯ i ) ≤ a4 ∥(ζ¯isi , α¯ i )T ∥2 V˙ i1′′ |(21) ≤ −∥(ζ¯isi , α¯ i )T ∥2

 ′′   ∂ Vi1 ∂ Vi′′    ≤ a5 ∥(ζ¯is , α¯ i )T ∥ , i  ∂ ζ¯ ∂ α¯ i  isi

(22)

for suitable constants a3 , a4 , a5 > 0. Hence, 1 a2 V˙ i1′′ |(19) ≤ − ∥(ζ¯isi , α¯ i )T ∥2 + 5 (∥βi ∥2 + ∥Γi Ωi β¯ i ∥2 )∥δi ∥2 2 2 a5 T 2 2

+

ϵ˙i = gi (¯zi , ϵi , 0).

α˙¯ i = Γi Ωi (θi )ζ¯isi .

2

(∥Ωi (ζ¯i )α¯ i ∥ + ∥Γi Ωi (ζ¯i )ζ¯isi ∥ ).

Because ∥Ωi (ζi )∥2 ≤ 2(∥θi ∥2 + ∥δi ∥2 ), ∥Ω (ζ¯i )∥ ≤ ∥δi ∥ and θi (t ) is bounded for t ≥ 0, we have

Lemma 4.1. Under Assumptions 2.2 and 3.1, there is a smooth function Vi1 (t , z¯i , ϵi ) such that

1 V˙ i1′′ |(19) ≤ − ∥(ζ¯isi , α¯ i )T ∥2 + a¯ (1 + Vi1′ )∥δi ∥2 2

V i1 (¯zi , ϵi ) ≤ Vi1 (t , z¯i , ϵi ) ≤ V i1 (¯zi , ϵi )

for a positive constant a¯ . Take Vi1 (t , zi ) = 22a¯λ¯+1 (2Vi1′ + Vi1′2 ) + 2Vi1′′ . Then we obtain (18)

V˙ i1 |(17) ≤ −∥(¯zi , ϵi ) ∥    ∂ Vi1 ∂ Vi1  T T 3   ,  ∂ z¯ ∂ϵ  ≤ ν0 (∥(¯zi , ϵi ) ∥ + ∥(¯zi , ϵi ) ∥ ) i i T

2

1

by a straightforward calculation.



(18)

The next lemma can be obtained by Lemma 3.1 and the analysis of output regulation (refer to Huang (2004)).

for smooth positive definite and radially unbounded functions V i1 , V i1 and a positive constant ν0 .

Lemma 4.2. Consider the system (16). Suppose that, for any sets Bρz¯ ,

Proof. Considering (17) with δi = (¯zi , η¯ i , ζ¯i1 , . . . , ζ¯i,si −1 ) , we have T

δ˙i = A¯ i δi ,

ζ˙¯ isi = −¯li ζ¯isi − ΩiT (ζi )α¯ i + βi δi

α˙¯ i = Γi Ωi (ζi )(ζ¯isi + β¯ i δi )

(19)

where



A0i A¯ i =  BRi 0



0

0 0 ,

Ac − LC

βˆ i

Ξi

Ξi =



0(si −2)×1

Isi −2



−µi1 , . . . , −µi,si −1

and βi , β¯ i , βˆ i can be obtained directly from (17). Since A¯ i is Hurwitz by a suitable choice of µi and L, the equilibrium point δi = 0 of the δi (t ) subsystem is exponentially stable. Hence, there is a Lyapunov function Vi0′ (δ) satisfying a1 ∥δi ∥2 ≤ Vi0′ (δ) ≤ a2 ∥δi ∥2 and V˙ i0′ ≤ −∥δi ∥2 for constants a1 > 0 and a2 > 0. Consider the following Lyapunov function candidate Vi1′ (¯zi , ϵi ) = ℓ1 Vi0′ (δ) +

1 2

Bρϵ , Bρx¯ with z¯ = (¯z1 , . . . , z¯N )T , ϵ = (ϵ1 , . . . , ϵN )T , x¯ = (¯x1 , . . . , x¯ N )T , there is a distributed control (4) such that, for any (¯z (0), ϵ(0), ς x¯ (0), ς (0)) ∈ Bρz¯ × Bρϵ × Bρx¯ × Bρ ′ , the solution of the closedloop system composed of (16) and (4) is bounded over [0, ∞) and limt →∞ ei (t ) = 0, i = 1, . . . , N. Then the adaptive containment problem of the system (1) with leaders (2) can be solved.

Up to now, the containment problem has been converted into the stabilization problem of (16). For the distributed stabilization, based on the idea from Marino and Tomei (1995), we introduce an input-driven filter for each agent

ξ˙i = Af ξi + Bf u¯ i , i = 1, . . . , N with ξi (0) = 0 and     Ir − 2 0(r −2)×1 , Bf = Af = −λ[r −1] 0

1

where λ[r −1] = (λ1 , . . . , λr −1 ) is chosen to make Af Hurwitz. Denote ξ¯i = (ξ¯i1 , . . . , ξ¯i,r −1 )T with T

ξ¯i,j−1 = x¯ ij − ξi,j−1 − λj−1 ei ,

1

Then, from (16), we have

2

z˙¯ i = A0i z¯i + D0i ei ,

ζ¯is2i + ζ¯isi β¯ i δi + α¯ i⊤ Γi−1 α¯ i

whose time-derivative satisfies V˙ i1′ ≤ −(ℓ1 − ∥βiT β¯ i ∥)∥δi ∥2 − ¯li ζ¯is2i + ζ¯isi (βi + β¯ i A¯ i − ¯li β¯ i )δi .

(23)

j = 2, . . . , r .

ϵ˙i = gi (¯zi , ϵi , ei ) ˙ξ¯ = A ξ¯ + g ′ (¯z , ϵ , e ) f i i i i i i e˙ i = ξi1 + gi′′ (ξ¯i , ei ),

ξ˙i = Af ξi + Bf u¯ i

(24)

X. Wang et al. / Automatica (

 (λ2 − λ21 + D1i2 − λ1 D1i1 )ei  gi′ (¯zi , ϵi , ei ) =  ··· (D1ir − λr −1 λ1 − λr −1 D1i1 )ei + ϕi 

Therefore, the stabilization problem of (16) becomes a stabilization problem of the system (24). To present a recursive distributed control for the stabilization of the system (24), we introduce notations: 1

ξij ,

k ≥ 1,

πi1 = −2eiv

(25a)

and for j = 2, . . . , r

+

∂πi,j−1 1  (ξˆi1 − ξˆn1 ) ∂ eiv |Ni | n∈Ni

j −2   ∂πi,j−1 m=1

∂ξim

ξˆi,m+1 +

  ∂πi,j−1 ˆξn,m+1 ∂ξnm n∈Ni

(25b)

(26)

solves the adaptive containment problem of the multi-agent system (1) with the leaders (2). Proof. Four steps are given for the proof. Step 1: Take ε = (¯z1 , ϵ1 , . . . , z¯N , ϵN )T and V1 (t , ε) = i=1 Vi1 (t , z¯i , ϵi ). From Lemma 4.1, it is easy to see there is a smooth function V1 (t , ε) such that V 1 (ε) ≤ V1 (t , ε) ≤ V 1 (ε), V˙ 1 |(17) ≤ −∥ε∥2

N

∂ V1 (t ,ε) ∥ ∂ε

≤ ν¯ 0 (∥ε∥ + ∥ε∥3 ) for smooth positive definite and N radially unbounded functions V 1 (ε) = zi , ϵi ), V 1 (ε) = i=1 V i1 (¯ N V (¯ z , ϵ ) and a positive constant ν ¯ . i 0 i=1 i1 i Step 2: Because Af is Hurwitz, there is a smooth function Vi2 (ξ¯i ) such that d1 ∥ξ¯i ∥2 ≤ Vi2 (ξ¯i ) ≤ d2 ∥ξ¯i ∥2 and V˙ i2 ≤ −∥ξ¯i ∥2 + d3 (e2i + ϕi2 (¯zi , ϵi )) with positive constants d1 , d2 and d3 . Define V2 (ξ¯ ) = N ¯ ¯ ¯ ¯ T ¯ 2 ≤ i=1 Vi2 (ξi ) with ξ = (ξ1 , . . . , ξN ) and we obtain ν1 ∥ξ ∥ N 2 2 2 2 ˙ ¯ ¯ ¯ V2 (ξ ) ≤ ν2 ∥ξ ∥ , V2 ≤ −∥ξ ∥ + ν3 i=1 (ei + ϕi (¯zi , ϵi )) for some positive constants ν1 , ν2 and ν3 . Step 3: Define the following Lyapunov functions N r −1 λ¯ 0 1  2 W0 (e) + ξ˜ij 2 ∥Q ∥ 2 i=1 j=1

λ¯ 20 k ∥e∥2 − k∥ξ˜ ∥2 + Θ1 2∥Q ∥2

(27)

with an appropriately defined function Θ1 satisfying Θ1 ≤ c0 (∥ξ¯ ∥2 + ∥e∥2 + ∥ξ˜ ∥2 ) for a positive constant c0 independent of parameter k.

N 

ϕi2 (¯zi , ϵi ) ≤ c2 ∥ε∥2

i =1

∥gi (¯zi , ϵi , ei ) − gi (¯zi , ϵi , 0)∥ ≤ c2 ∥e∥.

Consider the Lyapunov function candidate as follows: W2 (t , ε, ξ¯ , e, ξ˜ ) = V1 (t , ε) + ϱ1 V2 (ξ¯ ) + ϱ1 ϱ2 W1 (e, ξ˜ ) where ϱ1 = min 1, 4ν1 c }, ϱ2 = min 1, 2c1 . Clearly, letting 3 2 0 W 2 = V 1 + ϱ1 V2 + ϱ1 ϱ2 W1 , W 2 = V 1 + ϱ1 V2 + ϱ1 ϱ2 W1 yields W 2 ≤ W2 ≤ W 2 and Ω3ϱ (W 2 ) ⊂ Ω3ϱ (W 2 ) ⊂ Ω3ϱ+c1 (W 2 ). Also, Ω3ϱ+c1 (W 2 ) is a compact set satisfying





Ω3ϱ+c1 (W 2 ) ⊂ Ω3ϱ+c1 (V 1 ) × Ωϱ¯ 1 (V2 ) × Ωϱ¯ 2 (W1 ) with ϱ¯ 1 =

3ϱ+c1

3ϱ+c

and ϱ¯ 2 = ϱ ϱ 1 . 1 2 It can be verified that, on Ω3ρ+ ¯ c1 (W 2 ), ϱ1

˙ 2 ≤ − 1 ∥ε∥2 − 1 ϱ1 ∥ξ¯ ∥2 − ϱ1 ϱ2 (k − c0 )∥ξ˜ ∥2 W 4 2   ϱ1 ϱ2 λ¯ 20 k 1 4 − − c2 − ϱ1 ν3 − ϱ1 ϱ2 c0 ∥e∥2 . 2∥Q ∥2 2 Choosing k ≥ max{

2∥Q ∥2 ( 12 c24 +ϱ1 ν3 +ϱ1 ϱ2 c0 +1)

ϱ1 ϱ2 λ¯0

Ω3ϱ+c1 (W 2 ),

2

, c0 + 1} yields that, on

˙ 2 ≤ − 1 ∥ε∥2 − 1 ϱ1 ∥ξ¯ ∥2 − ϱ1 ϱ2 ∥ξ˜ ∥2 − ∥e∥2 . W 4 2 ξ¯

ξ˜

Because Bρε¯ × Bρ¯ × Bρe¯ × Bρ¯ ⊂ Ω3ϱ+c1 (W 2 ), we have that, for each ξ¯

ξ˜

initial condition (ε(0), ξ¯ (0), e(0), ξ˜ (0)) ∈ Bρε¯ × Bρ¯ × Bρe¯ × Bρ¯ , the trajectory of the closed-loop system composed of (24) and (26) remains in Ω3ϱ+c1 (W 2 ) and also converges to the origin asymptotically. Thus, the conclusion follows by Lemma 4.2.  Remark 4.2. In a special case when the induced subgraph Gs associated with the N followers is undirected, the matrix H1 in Remark 2.2 is positive definite. In the construction of the Lyapunov function, W0 in Step 3 can be replaced by W0 = eT H1 e/2.

with Q specified in Remark 2.2, e = (e1 , . . . , eN )T , ξ˜ = (ξ˜1 , . . . , ξ˜N )T , ξ˜i = [ξ˜i1 · · · ξ˜i,r −1 ]T . Recalling (25) and by a direct computation, it can be verified that the time derivative of W1 satisfies

˙1 ≤ − W

gives Ω3ϱ (V 1 ) ⊂ Ω3ϱ (V 1 ) ⊂ Ω3ϱ+c1 (V 1 ). Clearly, Ω3ϱ+c1 (V 1 ) is a compact set. Consequently, there is a constant c2 > 0 such that, for all ε ∈ Ω3ϱ+c1 (V 1 )



u¯ i = kr πir

W1 (e, ξ˜ ) =

ξ˜

Bρe¯ × Bρ¯ ⊂ Ωϱ (W1 ).

i =1

Theorem 4.1. Under Assumptions 2.1–2.3 and 3.1, the distributed control composed of the internal model (14), the filter (23) and the stabilization control of the form

W0 (e) = eT Qe,

Bρ¯ ⊂ Ωϱ (V2 ),

Moreover, defining c1 = max{1, maxε∈Ω3ϱ (V 1 ) |V 1 (ε) − V 1 (ε)|}

N 

where ξˆN +l = 0, l = 1, . . . , K . We are ready to present our main result.

and ∥

ξ¯

Bρε¯ ⊂ Ωϱ (V 1 ),

   ∂ V1 (t , ε)     ∂ε  ≤ c2 ∥ε∥,

ξ˜i,j−1 = ξˆi,j−1 − πi,j−1 πij = −2ξ˜i,j−1 +

5

W1 (e, ξ˜ ) are positive definite and radially unbounded, there exists a positive constant ϱ such that

gi′′ (ξ¯i , ei ) = ξ¯i1 + (λ1 + D1i1 )ei .

kj

–

ξ¯ ξ˜ Step 4: Note that, there exist Bρε¯ , Bρ¯ , Bρe¯ , Bρ¯ such that (ε(0), ξ¯ (0), ξ¯ ξ˜ e(0), ξ˜ (0)) ∈ Bρε¯ × Bρ¯ × Bρe¯ × Bρ¯ for any (¯z (0), ϵ(0), x¯ (0)) ∈ Bρz¯ × Bρϵ × Bρx¯ and ξi (0) = 0. Because smooth functions V 1 (ε), V2 (ξ¯ ),

where

ξˆij =

)

In a sum, the design procedure can be given as follows: (i) after the verification of Assumptions 2.1 to 2.3, convert the containment problem of (1) and (2) into a virtual leaderfollowing problem of the agents (9) with (7); (ii) solve the regulator equations, compute (12), and verify Assumption 3.1; (iii) construct the distributed internal model (14) and the coordinate transformation (15) to obtain the distributed stabilization problem; (iv) solve the distributed stabilization problem by constructing the filter (23) and the feedback (26).

6

X. Wang et al. / Automatica (

)

–

polytope spanned by two leaders’ outputs despite the parametric changes of the leaders. Besides, for each i, αˆ i approaches its true value −π 2 /4 in the time interval [0, 40] s and −π 2 /16 after 40 s. 6. Conclusions Fig. 1. Interconnection graph G for the example.

In this paper, the adaptive containment problem was studied for a class of multi-agent systems under a general digraph. An adaptive IM was constructed for the leaders with parametric uncertainties and a semi-global distributed control was proposed to make the outputs of all agents enter the polytope determined by the leaders. References

Fig. 2. The responses of the output trajectories of the leaders and the followers and the trajectories of estimations.

5. Numerical example In this section, we consider a containment problem for a multiagent system consisting of four followers and two leaders. The follower agents are the mass–damper–spring systems with unit mass described by: y¨ i + di y˙ i + fi yi = ui ,

i = 1, 2, 3, 4

(28)

where yi , ui ∈ R are the output and input, respectively. di and fi are the damping coefficient and spring coefficient of agent i, which are uncertain with 0 ≤ di ≤ 2 and 0 ≤ fi ≤ 2. Obviously, the widely studied double-integrator systems is a special case of the system (28). The leaders are specified by harmonic oscillators:

v˙ j1 = ωvj2 ,

v˙ j2 = −ωvj1 ,

j = 1, 2

with yN +j = vj1 = [1, 0]vj as their outputs. Clearly, Assumption 2.1 to 2.3 are satisfied with r = 2. The interconnection graph is shown in Fig. 1 with associated matrices H1 and H2 discussed in Remark 2.2. With xi1 = yi , xi2 = y˙ i + di xi1 , the system (28) is rewritten as x˙ i1 = xi2 − di xi1 ,

x˙ i2 = −fi xi1 + ui ,

xi = [xi1 , xi2 ]T .

The associated regulator equations can be solved with Ui (ω) = −(H1−1 H2 )(i) (I2 ⊗([fi −ω2 , di ω])) where ⊗ means Kronecker product. Thus, the system (12) can be obtained with si = 2 and αi1 = −ω2 and it can be verified that for vi (0) ̸= 0 and |fi −ω2 |+|di ω| ̸= 0, Assumption 3.1 is satisfied. Suppose that Bρx = {x ∈ R8 : ∥x∥ ≤

5}, V = {v ∈ R4 : ∥v∥ ≤ 10}, W = {ω ∈ R : |ω| ≤ 5}. Therefore, applying Theorem 4.1, the internal model (14) can be constructed with µi = [1, 1], L = [2, 1]T , ¯li = 3, γi = 30, and then the filter (23) and the stabilization control (26) can be designed directly with λ1 = 5 and k = 10. The simulation results are shown in Fig. 2, where ω = π /2, (v1 (0), v2 (0))T = [5, 0, 0, 5]T , (di , fi ) = (0.6 + 0.2i, 2.5 − 0.5i), xi (0) = (1.5 − 0.5i, 0) and the initial values of the remaining states are zero. Moreover, at time t = 40 s, the leaders resettle their states as (v1 (40), v2 (40))T = [3, 0, 0, 3]T , and dynamics with ω = π /4. As shown in Fig. 2, the followers’ outputs converge to the dynamic

Berman, A., & Plemmons, R. (1979). Nonnegative matrices in the mathematical sciences. New York: Academic Press. Cao, Y., Stuart, D., Ren, W., & Meng, Z. (2011). Distributed containment control for multiple autonomous vehicles with double-integrator dynamics: algorithms and experiments. IEEE Transactions on Control Systems Technology, 19(4), 929–938. Godsil, C., & Royle, G. (2001). Algebraic graph theory. New York: Springer-Verlag. Hong, Y., Wang, X., & Jiang, Z. P. (2013). Distributed output regulation of leader–follower multi-agent systems. International Journal of Robust and Nonlinear Control, 23(1), 48–66. Huang, J. (2004). Nonlinear output regulation: theory and application. Philadelphia: SIAM. Isidori, A., Marconi, L., & Praly, L. (2012). Robust design of nonlinear internal models without adaptation. Automatica, 48(10), 2409–2419. Khalil, H. K. (1996). Nonlinear systems (2nd ed.). New Jersey: Prentice Hall. Li, Z., Liu, X., Ren, W., & Xie, L. (2013). Distributed tracking control for linear multiagent systems with a leader of bounded unknown input. IEEE Transactions on Automatic Control, 58(12), 518–523. Liu, L., Chen, Z., & Huang, J. (2009). Parameter convergence and minimal internal model with an adaptive output regulation problem. Automatica, 45(5), 1206–1311. Lou, Y., & Hong, Y. (2012). Target containment control of multi-agent systems with random switching interconnection topologies. Automatica, 48(5), 879–885. Marino, R., & Tomei, P. (2003). Output regulation for linear systems via adaptive internal model. IEEE Transactions on Automatic Control, 48(12), 2199–2202. Marino, R., & Tomei, P. (1995). Nonlinear control design: geometric, adaptive, and robust. London: Prentice-Hall. Mei, J., Ren, W., & Ma, G. (2012). Distributed containment control for Lagrangian networks with parametric uncertainties under a directed graph. Automatica, 48(4), 653–659. Meng, Z., Ren, W., & You, Z. (2010). Distributed finite-time attitude containment control for multiple rigid bodies. Automatica, 46(12), 2092–2099. Obregón-Pulido, G., Castillo-Toledo, B., & Loukianov, A. G. (2011). A structurally stable globally adaptive internal model regulator for MIMO linear systems. IEEE Transactions on Automatic Control, 56(1), 160–165. Rockafellar, R. T. (1972). Convex analysis. New Jersey: Princeton University Press. Serrani, A., Isidori, A., & Marconi, L. (2001). Semiglobal nonlinear output regulation with adaptive internal model. IEEE Transactions on Automatic Control, 46(8), 1178–1194. Shi, G., & Hong, Y. (2009). Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies. Automatica, 45(5), 1165–1175. Shi, G., Hong, Y., & Johansson, K. (2012). Connectivity and set tracking of multi-agent systems guided by multiple moving leaders. IEEE Transactions on Automatic Control, 57(3), 663–676. Su, Y., & Huang, J. (2013). Cooperative adaptive output regulation for a class of nonlinear uncertain multi-agent systems with unknown leader. Systems & Control Letters, 62(6), 461–467. Wang, X., Hong, Y., Huang, J., & Jiang, Z. P. (2010). A distributed control approach to a robust output regulation problem for multi-agent linear systems. IEEE Transactions on Automatic Control, 55(12), 2891–2895.

Xinghu Wang received his B.Sc. and Ph.D. degrees from Shandong University, Weihai, and University of Science and Technology of China in 2007 and 2012, respectively. He is currently a postdoctoral in Academy of Mathematics and Systems Science, Chinese Academy of Sciences. His research interests include nonlinear control and multi-agent systems.

X. Wang et al. / Automatica ( Yiguang Hong received his B.Sc. and M.Sc. degrees from Peking University, China, and his Ph.D. degree from Chinese Academy of Sciences (CAS). He is currently a professor in the Academy of Mathematics and Systems Science, CAS. His research interests include nonlinear control, multiagent systems, hybrid systems, and software systems. He is a recipient of Guang Zhaozhi Award of Chinese Control Conference, Young Author Prize of IFAC World Congress, Young Scientist Award of CAS, Youth Award for Science and Technology of China, and State Natural Science Prize of China. He is also the director of Key Lab of Systems and Control, CAS and the IEEE Control Systems Society chapter activities chair. Moreover, he is the Editor-in-Chief of Control Theory and Technology and deputy Editorin-Chief of Acta Automatica Sinica. In addition, he serves or served as the associate editors of some journals including IEEE Control Systems Magazine, IEEE Trans. Automatic Control, and Nonlinear Analysis: Hybrid Systems.

)

–

7 Haibo Ji was born in Anhui, China, in 1964. He received the B.Eng. degree and Ph.D. degree in Mechanical Engineering from Zhejiang University and Beijing University, in 1984 and 1990, respectively. He is currently a Professor in the Department of Automation, University of Science and Technology of China, Hefei, China. His research interests include nonlinear control and its applications.