Cooperative semi-global robust output regulation for a class of nonlinear uncertain multi-agent systems

Automatica 50 (2014) 1053–1065

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Cooperative semi-global robust output regulation for a class of nonlinear uncertain multi-agent systems✩ Youfeng Su a , Jie Huang b,1 a

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, China

b

Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

article

info

Article history: Received 15 October 2012 Received in revised form 6 August 2013 Accepted 7 January 2014 Available online 26 February 2014 Keywords: Cooperative control Multi-agent systems Output regulation Internal model Nonlinear uncertain systems

abstract In this paper, we study the cooperative semi-global robust output regulation problem for a class of minimum phase nonlinear uncertain multi-agent systems. This problem is a generalization of the leaderfollowing tracking problem in the sense that it further addresses such issues as disturbance rejection, robustness with respect to parameter uncertainties. To solve this problem, we first introduce a type of distributed internal model that converts the cooperative semi-global robust output regulation problem into a cooperative semi-global robust stabilization problem of the so-called augmented system. We then solve the semi-global stabilization problem via distributed dynamic output control law by utilizing and combining a block semi-global backstepping technique, a simultaneous high gain feedback control technique, and a distributed high gain observer technique. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction In the past decade, the leader-following tracking problem for various multi-agent systems such as, single-integrator systems (Jadbabaie, Lin, & Morse, 2003; Ren, 2007), double-integrator systems (Hu & Hong, 2007; Ren, 2008), general linear multi-agent systems (Li, Liu, Lin, & Ren, 2011), second order nonlinear systems with global Lipschitz assumption (Meng & Lin, 2012; Song, Cao, & Yu, 2010), thrust-propelled vehicles (Lee, 2012), and multiple mechanical systems (Dong, 2011), has been extensively studied, see also the recent books Qu (2009); Ren and Beard (2008). In particular, Dong (2011) employed the adaptive control technique to deal with some type of model uncertainty. Recently, the cooperative output regulation problem for multiagent systems has received more and more attention (Hong, Wang, & Jiang, 2013; Su & Huang, 2012, 2013; Wang, Hong, Huang, &

✩ This work has been supported in part by the Research Grants Council of the Hong Kong Special Administration Region under grant No. 412810, and in part by the National Natural Science Foundation of China under grant No. 61174049. The material in this paper was partially presented at the 2013 American Control Conference (ACC 2013), June 17–19, 2013, Washington, DC, USA. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Frank Allgöwer. E-mail addresses: [email protected] (Y. Su), [email protected] (J. Huang). 1 Tel.: +852 39438473; fax: +852 26036002.

http://dx.doi.org/10.1016/j.automatica.2014.02.010 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

Jiang, 2010; Yang, Stoorvogel, Grip, & Saberi, 2012). Instead of tracking a specific reference signal, this problem aims to design a distributed control law such that the output of each subsystem can asymptotically track a class of reference inputs in the presence of a class of disturbances and plant parameter uncertainties. Like the classical output regulation problem, here the class of reference inputs and the class of disturbances are both generated by a differential equation called the exosystem. Therefore, this problem can be viewed as a generalization of the leader-following tracking problem by treating the plant as the follower system and the exosystem as the leader system. So far, the problem has been widely studied for linear uncertain multi-agent systems in, say, Hong et al. (2013), Su and Huang (2013), Wang et al. (2010) and Yang et al. (2012). More recently, the cooperative robust output regulation problem for a class of nonlinear multi-agent systems in lower triangular form was further formulated and a global solution was obtained by a distributed state feedback control law in Su and Huang (2012). In the special case where the number of subsystems is equal to one, the problem in Su and Huang (2012) reduces to the conventional global robust output regulation problem as studied in Huang and Chen (2004). However, like the conventional robust output regulation problem for a single nonlinear system in lower triangular form, the global solution cannot be obtained via an output feedback control law. Therefore, in this paper, we will further study a so-called cooperative semi-global robust output regulation problem for the class of uncertain nonlinear multi-agent systems to be described

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Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

in (1). For the special case where the number of the subsystems of system (1) is equal to one, our problem reduces to the semi-global robust output regulation problem for a single nonlinear system (Khalil, 1994; Lan, Chen, & Huang, 2005; Serrani, Isidori, & Marconi, 2001). In comparison with the semi-global robust output regulation problem for a single nonlinear system, our problem is technically more challenging in at least two ways. First, the system in Khalil (1994), Lan et al. (2005) and Serrani et al. (2001) is a singleinput, single-output system. It can be converted into a semi-global stabilization problem of an augmented system through the employment of an internal model. The augmented system is still a single-input, single-output system whose stabilization problem can be handled by established techniques for the semi-global stabilization (Teel & Praly, 1995). In contrast, for a multi-agent system, the augmented system is a multi-input, multi-output nonlinear system and we have to develop specific techniques of block semi-global backstepping method that apply to multi-input, multioutput nonlinear systems. Second, due to the communication constraint which is described by a communication graph to be introduced in Section 2, we cannot use the full information of the system for feedback control, and we have to develop a distributed control law to stabilize the augmented system. The rest of this paper is organized as follows: In Section 2, we give a precise description of cooperative semi-global robust output regulation problem. In Section 3, we introduce a type of distributed internal model that converts the cooperative semi-global robust output regulation problem into a cooperative semi-global robust stabilization problem of the augmented system. In Section 4, we present some technical lemmas that are applicable to the block lower triangular nonlinear systems. These lemmas can be viewed as block techniques of semi-global backstepping method. In Section 5, by utilizing these technical lemmas, a simultaneous high gain feedback control technique and a distributed version of high gain observer technique, we solve the semi-global stabilization problem of the augmented system via distributed dynamic output feedback control. Thus, our problem is solvable. In Section 5, we provide two examples to illustrate our design. Finally, in Section 6 we present our conclusions. Notation: Given the column vectors ai , i = 1, . . . , s, we denote col(a1 , . . . , as ) = [aT1 , . . . , aTs ]T . The compact set Q¯ Rs , {x = col(x1 , . . . , xs ) ∈ Rs : |xi | ≤ R, i = 1, . . . , s}. Given a positive ¯ c (V (x)) definite and proper function V : Rs → R, the symbol Ω denotes the compact set {x ∈ Rs : V (x) ≤ c }, while the symbol Ωc (V (x)) denotes the open set {x ∈ Rs : V (x) < c }. Given two sets X1 ∈ Rn1 and X2 ∈ Rn2 , let X1 × X2 , {col(x1 , x2 ) : x ∈ X1 , x2 ∈ X2 }.

We assume the functions fki (·), bi (·), k = 0, 1, i = 1, . . . , N, and q0 (·) are sufficiently smooth functions with fki (0, . . . , 0, w) = 0, k = 0, 1, and q0 (0, w) = 0. The plant (1) and exosystem (2) together can be viewed as a multi-agent system of N + 1 agents with the exosystem as the leader and all the subsystems of (1) as the followers. With respect to (1) and (2), we can define a digraph2 G¯ = {V¯ , E¯ } where V¯ = {0, 1, . . . , N } with the node 0 associated with the exosystem and the other N nodes associated with the N followers, respectively, and (j, i) ∈ E¯ , j = 0, 1, . . . , N and i = 1, . . . , N, if and only if the control ui can make use of yi − yj for feedback control. Thus our control law is of the following form

2. Problem statement

Remark 1. If the control ui of each subsystem can access ei for feedback control, then we can design N individual controllers of the form ui = ui (ηi , ei ), η˙ i = gi (ηi , ei ) to solve the problem using the approach in, say, Lan et al. (2005) and Serrani et al. (2001). Such a control scheme is called (purely) decentralized control. However, due to the communication constraint among the agents, it is unrealistic to assume that all the followers know the information of the leader, i.e., the information of ei = yi − y0 . Thus, what makes our control law (4) interesting is that only those followers which are the neighbors of the leader need to know ei . Other followers can only indirectly access ei by sharing information with their neighbors. That is why we call our problem as cooperative semi-global robust output regulation problem for the multi-agent system (1).

In this paper, we consider the following nonlinear uncertain multi-agent systems z˙i = f0i (zi , x1i , v, w), x˙ si = x(s+1)i ,

s = 1, . . . , r − 1,

x˙ ri = f1i (zi , x1i , . . . , xri , v, w) + bi (w)ui , yi = x1i ,

i = 1, . . . , N ,

(1)

where zi ∈ R , xi , col(x1i , . . . , xri ) ∈ R , yi , ui , ∈ R, v ∈ Rq , w ∈ Rnw represents the parameter uncertainty. The exosystem is given by nzi

v˙ = S v,

y0 = q0 (v, w),

r

(2)

ui = ui (ξi , yi − yj , j ∈ Ni ),

ξ˙i = gi (ξi , yi − yj , j ∈ Ni ),

where Ni = {j : (j, i) ∈ E¯ }, and ui and gi are sufficiently smooth functions vanishing at the origin, and ξi ∈ Rni with ni to be defined later. A control law of the form (4) is called a distributed dynamic output feedback control law because the control of each subsystem can only take the output information of its neighbors and itself for feedback control. We call the composition of (1) and (4) as the overall closed-loop system which can be put in the following form x˙ c = fc (xc , v, w),

(3)

(5)

where xc = col(z1 , x1 , ξ1 , . . . , zN , xN , ξN ) ∈ Rnc for some integer nc and fc is sufficiently smooth satisfying fc (0, 0, w) = 0 for all w ∈ Rnw . Then our problem can be described as follows: Given systems (1) and (2), the digraph G¯ , a real number R > 0, and compact subsets V0 ⊆ Rq and W ⊆ Rnw which contain the origins of the respective Euclidean spaces, find a control law of the n form (4) such that for any v(0) ∈ V0 , w ∈ W, and xc (0) ∈ Q¯ R c , the trajectory of the closed-loop system (5) starting from xc (0) and v(0) exists and is bounded for all t ≥ 0, and limt →∞ e(t ) = 0, where e = col(e1 , . . . , eN ). The above problem will be called cooperative regional robust output regulation problem for the nonlinear multi-agent system (1) n on the compact subset Q¯ R c × V0 × W. If for any R > 0, and any q compact subsets V0 ⊆ R and W ⊆ Rnw which contain the origins of the respective Euclidean spaces, the cooperative regional robust output regulation problem for the nonlinear multi-agent system n (1) on the compact subset Q¯ R c × V0 × W is solvable, then we say that the cooperative semi-global robust output regulation problem for the nonlinear multi-agent system (1) is solvable.

where y0 ∈ R is the output of the exosystem. Then, for i = 1, . . . , N, the regulated errors for the subsystems are defined as ei = yi − y0 .

(4)

2 See Appendix A for a self-contained summary of digraph.

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

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For i = 1, . . . , N, let θi (v, w) = Ti col(ui (v, w),

3. Problem conversion In this section, we show that, like a single nonlinear system, the cooperative semi-global robust output regulation problem for system (1) can be converted into cooperative semi-global robust stabilization problem of a well defined augmented system under the following standard assumptions.

(n −1) ui (v,w) d ηi (n −1) dt ηi

dui (v,w) dt

,..., ˙ ). Then it can be verified that θi (v, w) = Ti Φi Ti−1 θi

(v, w). By performing the coordinate and input transformation z¯i = zi − zi (v, w),

x¯ si = xsi − xsi (v, w),

η¯ i = ηi − θi (v, w),

u¯ i = ui − Γi Ti

−1

s = 1, . . . , r ,

ηi ,

we obtain that the augmented system takes the following form: Assumption 1. The exosystem is neutrally stable, i.e., all the eigenvalues of S are semi-simple with zero real parts. Assumption 2. bi (w) > 0, i = 1, . . . , N, for all w ∈ R

nw

.

Assumption 3. There exist sufficiently smooth functions zi (v, w) with zi (0, 0) = 0 such that for any v ∈ Rq and w ∈ Rnw ,

∂ zi (v, w) S v = f0i (zi (v, w), q0 (v, w), v, w). ∂v

(6)

z˙¯ i = f¯0i (¯zi , x¯ 1i , v, w) x˙¯ si = x¯ (s+1)i ,

s = 1, . . . , r − 1 ˙x¯ ri = f¯1i (¯zi , x¯ 1i , . . . , x¯ ri , v, w) + bi (w)Γi T −1 η¯ i + bi (w)¯ui i ˙η¯ i = (Mi + Qi Γi Ti−1 )η¯ i + Qi u¯ i ei = x¯ 1i

(10)

where f¯0i (¯zi , x¯ 1i , v, w) = f0i (¯zi + zi (v, w), x¯ 1i + x1i (v, w), v, w)

− f0i (zi (v, w), x1i (v, w), v, w), Remark 2. Under Assumption 1, given any compact subset V0 , there exists a compact subset V such that, for any v(0) ∈ V0 , the trajectory v(t ) of the exosystem remains in V for all t ≥ 0. Under Assumptions 2 and 3, for i = 1, . . . , N, let x1i (v, w) = q0 (v, w), xsi (v, w) = xri (v,w)

∂ x(s−1)i (v,w) S v, ∂v

s = 2, . . . , r, ui (v, w) =

bi (w)( ∂v S v − f1i (zi (v, w), x1i (v, w), . . . , xri (v, w), v, w)), and xi (v, w) = col(x1i (v, w), . . . , xri (v, w)). Then it can be verified that x(v, w) = col(z1 (v, w), x1 (v, w), . . . , zN (v, w), xN (v, w)), and u(v, w) = col(u1 (v, w), . . . , uN (v, w)) constitute the global solution of the regulator equations associated with (1) and (2) (Isidori & Byrnes, 1990). −1

f¯1i (¯zi , x¯ 1i , . . . , x¯ ri , v, w)

= f1i (¯zi + zi (v, w), x¯ 1i + x1i (v, w), . . . , x¯ ri + xri (v, w), v, w) − f1i (zi (v, w), x1i (v, w), . . . , xri (v, w), v, w). The augmented system (10) has an important property that the origin is an equilibrium point for all v and w , and e is identically zero at the origin. As a result, it is possible to solve the cooperative semi-global robust output regulation problem for system (1) by stabilizing the augmented system (10) semi-globally via a distributed dynamic output feedback control law. To elaborate this point further, let A¯ = [aij ]Ni,j=0 be any weighted adjacency matrix of G¯ . For i = 1, . . . , N, let ev i = j=0 aij (yi − yj ). Then we will consider a class of dynamic output feedback controller as follows

N

Assumption 4. The function ui (v, w) is a polynomial in v with coefficients depending on w . Remark 3. It is known from Huang (2004) that, under Assumptions 1–4, there exist integers nηi , i = 1, . . . , N, and real coefficients polynomials Pi (λ) = λnηi − ϱ1i − ϱ2i λ1 − · · · − ϱnηi i λnηi −1 whose roots are all imaginary, such that, for all trajectories v(t ) of the exosystem and all w ∈ W, ui (v, w) satisfy dui (v,w) 2i dt

ϱ1i ui (v, w) + ϱ

+ ··· + ϱ

n −1 d ηi ui (v,w) . nηi i n −1 dt ηi

n d ηi ui (v,w) n dt ηi

=

Now we define

the dynamic compensator (Nikiforov, 1998):

η˙ i = Mi ηi + Qi ui ,

i = 1, . . . , N

(7)

Rnηi ×nηi

Rnηi

where Mi ∈ are any Hurwitz matrices, and Qi ∈ are any column vectors such that the pairs (Mi , Qi ) are controllable. By Remark 3.7 in Huang and Chen (2004), under Assumptions 1–4, the dynamic compensator (7) is an internal model of system (1). The system composed of the plant (1) and the internal model (7) is called the augmented system (Huang & Chen, 2004). For i = 1, . . . , N, let

0  .. . Φi =  0

ϱ1i

1

.. .

0

ϱ2i

··· .. . ··· ···

0



..  . ,  1

ϱnηi i

 T 1

0  Γi =   ..  . .

ζ˙i = ϕi (ζi , evi ),

(11)

where ζi ∈ R ζi for some integer nζi , and ki , ϕi are globally defined sufficiently smooth functions that vanish at the origin. It can be seen that u¯ i depends on (yi − yj ) if and only if the node j is a neighbor of the node i. Thus a control law of the form (11) is indeed a distributed control law in the sense of Remark 1. We now further define the cooperative regional robust stabilization problem of system (10) as follows: Given a real number R¯ > 0, and some compact subsets V ⊆ Rq and W ⊆ Rnw , which contain the origins of the respective Euclidean spaces, find a control law of the form (11) such that, for any v ∈ V and w ∈ W, the equilibrium point at the origin of the closed-loop system composed of (10) and (11) is locally n asymptotically stable with its domain of attraction containing Q¯ R¯ c . The above problem is called cooperative regional robust stabilizan tion problem for system (10) on the compact subset Q¯ R¯ c × V × W. n

If for any R¯ > 0, and any compact subsets V ⊆ Rq and W ⊆ Rnw which contain the origins of the respective Euclidean spaces, the cooperative regional robust stabilization problem for system (10) n on the compact subset Q¯ R¯ c × V × W is solvable, then we say that the cooperative semi-global robust stabilization problem for system (10) is solvable.

(8)

0

Since (Γi , Φi ) is observable, (Mi , Qi ) is controllable, and the eigenvalues of Mi and Φi do not coincide, there exists a nonsingular matrix Ti that satisfies the Sylvester equation (Nikiforov, 1998) Ti Φi − Mi Ti = Qi Γi .

u¯ i = ki (ζi , ev i ),

(9)

Lemma 1. Under Assumptions 1–4, given any R > 0, and any compact subsets V0 ⊆ Rq and W ⊆ Rnw , there exist a number R¯ > 0 and a compact subset V ⊆ Rq , such that if the cooperative regional robust n stabilization problem of (10) on the compact subset Q¯ R¯ c × V × W is solvable by the distributed dynamic output feedback controller of the form (11), then the cooperative regional robust output regulation problem of the multi-agent system (1) on the compact subset

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Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

n

Q¯ R c × V0 × W is solvable by a distributed dynamic output feedback controller of the following form ui = ki (ζi , ev i ) + Γi Ti−1 ηi ,

ζ˙i = ϕi (ζi , evi ), η˙ i = Mi ηi + Qi ui .

(12)

Assume that for any given compact set Ξ1 ⊆ Rn1 , there exist positive real numbers c1 > 0 and ε1 ≥ 1, and a C 2 positive definite ¯ c1 (U0 (χ1 )), and for all and proper function U0 (χ1 ) such that, Ξ1 ⊆ Ω ¯ c1 +ε1 (U0 (χ1 )), col(v, w) ∈ V × W, and all χ1 ∈ Ω U˙ 0 (χ1 )(13) ≤ −α∥χ1 ∥2 ,



(15)

where α is a positive real number. Define the C 2 function

Proof. For any R > 0, and any xc (0) ∈ Q¯ R c , there exists R¯ > 0 n such that x¯ c (0) ∈ Q¯ R¯ c , where x¯ c = col(¯z1 , x¯ 11 , . . . , x¯ r1 , η¯ 1 , ζ1 . . . , z¯N , x¯ 1N , . . . , x¯ rN , η¯ N , ζN ). By Remark 2, for any given V0 ⊆ Rq , we can always find the compact subset V ⊆ Rq such that v(t ) ∈ V for any t ≥ 0. Then there exists a distributed dynamic output feedback controller of the form (11), such that the state x¯ c (t ) of the n closed-loop system consisting of (10) and (11) with x¯ c (0) ∈ Q¯ R¯ c is bounded for any t ≥ 0 and approaches zero asymptotically. It is noted that xc = col(¯z1 + z1 (v, w), x¯ 11 + x11 (v, w), x¯ 21 + x21 (v, w), . . . , x¯ r1 + xr1 (v, w), η1 + θ1 (v, w), ζ1 , . . . , z¯N + zN (v, w), x¯ 1N + x1N (v, w), x¯ 2N + x2N (v, w), . . . , x¯ rN + xrN (v, w), ηN + θN (v, w), ζN ). And for any v ∈ V and w ∈ W, xsi (v, w), θi (v, w) are bounded for any t ≥ 0. Then xc (t ) is also bounded n for any xc (0) ∈ Q¯ R c and any t ≥ 0. On the other hand,

where P is a positive definite solution to the inequality M T P + PM ≤ −In2 . Then V (χ1 , χ2 ) is positive definite defined on Ωc1 +ε1 (U0 (χ1 ))× Rn2 . Furthermore, for any other given compact set Ξ2 ⊆ Rn2 , there ¯ c ∗ (V (χ1 , χ2 )), and exist c ∗ > 0 and ν > 0, such that, Ξ1 × Ξ2 ⊆ Ω ¯ c ∗ +ε∗ (V (χ1 , χ2 )), for all col(v, w) ∈ V × W and all col(χ1 , χ2 ) ∈ Ω

lim ei (t ) = lim (x1i (t ) − x1i (v, w)) = lim x¯ 1i (t ) = 0.

Lemma 4. Consider the nonlinear system.

n

t →∞

t →∞

t →∞

Thus the cooperative regional robust output regulation problem of (1) is solvable by the distributed dynamic output feedback controller (12). The proof is thus completed.  Remark 4. As indicated in Lemma 1, Assumptions 1–4 guarantee that the cooperative semi-global robust output regulation problem for system (1) can be converted into the cooperative semi-global robust stabilization problem of the augmented system (10). It is noted that these four assumptions are needed even for the semiglobal robust output regulation problem of a single nonlinear system as studied in Khalil (1994), Lan et al. (2005) and Serrani et al. (2001). 4. Technical lemmas Having shown that the cooperative semi-global robust output regulation problem for system (1) can be converted into the cooperative semi-global robust stabilization problem of the augmented system (10), we will further tackle the semi-global stabilization problem of the augmented system (10). For this purpose, we need to establish three technical lemmas on the block semi-global backstepping that are applicable to block lower triangular nonlinear systems. The proofs of Lemmas 3 and 4 will be put in Appendix B. Lemma 2. Given a C 1 mapping F (x, y) : Rm × Rn → Rp satisfying F (0, y) = 0 for any y ∈ Rn , and a compact set C1 × C2 which contains x = 0, there exists a positive number µ such that for any (x, y) ∈ C1 × C2 , ∥F (x, y)∥ ≤ µ∥x∥. Proof. Since F (x, y) is C 1 ,

   ∂ F (x,y) 

∂ F (x,y) ∂x

is continuous. Let µ

max(x,y)∈C1 ×C2  ∂ x  . Then µ is finite and ∥F (x, y)∥ µ∥x∥. 

= ≤

V (χ1 , χ2 ) =

c1 U0 (χ1 ) c1 + ε1 − U0 (χ1 )

+ νχ2T P χ2 ,

V˙ (χ1 , χ2 )(13)+(14) ≤ −β∥(χ1 , χ2 )∥2 ,



where β is a positive real number and ε ∗ ≥ 1 can be arbitrarily chosen.

χ˙ 1 = Z (χ1 , v, w) + G1 (χ1 , τ (χ2 , ρ), v, w), χ˙ 2 = ρ Y (w)χ2 + G2 (χ1 , τ (χ2 , ρ), v, w),

(16)

where χ1 ∈ Rn1 , χ2 ∈ Rn2 , v ∈ V and w ∈ W. Z (χ1 , v, w), G1 (χ1 , τ (χ2 , ρ), v, w), and G2 (χ1 , τ (χ2 , ρ), v, w) are sufficiently smooth functions satisfying Z (0, v, w) = 0, G1 (χ1 , 0, v, w) = 0 and G2 (0, 0, v, w) = 0. Y (w) is Hurwitz for any w ∈ W. τ (χ2 , ρ) is a sufficiently smooth function satisfying that τ (0, ρ) = 0 and ∥τ (χ2 , ρ)∥ ≤ ∥χ2 ∥ for any ρ ∈ [ρ0 , +∞) with some ρ0 > 0. Suppose that given any compact set Ξ1 ⊆ Rn1 , there exists a C 2 function V (χ1 , w), positive definite and proper functions V (χ1 ) and V (χ1 ), and a positive real number α , such that for all col(v, w) ∈ ¯ c1 +ε1 (V (χ1 )), V × W, and all χ1 ∈ Ω V (χ1 ) ≤ V (χ1 , w) ≤ V (χ1 ),

(17)

∂ V (χ1 , w) Z (χ1 , v, w) ≤ −α∥χ1 ∥2 , ∂χ1

(18)

¯ c1 (V (χ1 )), and ε1 ≥ 1. Then for where c1 > 0 is such that Ξ1 ⊆ Ω any other compact set Ξ2 ⊆ Rn2 , there exist a sufficient large positive number ρ , a positive real number c ∗ , a C 2 function W (χ1 , χ2 , w), and positive definite functions W (χ1 , χ2 ) and W (χ1 , χ2 ), such that ¯ c ∗ (W (χ1 , χ2 )), and for all col(v, w) ∈ V × W, and all Ξ1 × Ξ2 ⊆ Ω ¯ c ∗ +ε∗ (W (χ1 , χ2 )), col(χ1 , χ2 ) ∈ Ω W (χ1 , χ2 ) ≤ W (χ1 , χ2 , w) ≤ W (χ1 , χ2 ),

(19)

 ˙ (χ1 , χ2 , w) ≤ −β∥(χ1 , χ2 )∥2 , W (16)

(20)

where β is a positive real number, and ε ∗ ≥ 1 can be chosen arbitrarily. Specially, if ε ∗ is chosen so that ε ∗ ≥ max(χ1 ,χ2 )∈Ω¯ c ∗ (W (χ1 ,χ2 ))

{|W (χ1 , χ2 ) − W (χ1 , χ2 )|, 1}, the equilibrium (χ1 , χ2 ) = (0, 0) of system (16) is locally asymptotically stable with its domain of attraction containing Ξ1 × Ξ2 .

Lemma 3. Consider the nonlinear system

χ˙ 1 = ϕ1 (χ1 , v, w), χ˙ 2 = M χ2 + ϕ2 (χ1 , v, w),

(13) (14)

where χ1 ∈ R , χ2 ∈ R , v ∈ V and w ∈ W. M is Hurwitz, ϕ1 (χ1 , v, w) and ϕ2 (χ1 , v, w) are sufficiently smooth with ϕ1 (0, v, w) = 0 and ϕ2 (0, v, w) = 0 for all col(v, w) ∈ V × W. n1

n2

Remark 5. When V (χ1 , w) = V (χ1 ) and Y (w) = Y are independent of w , we can obtain that W (χ1 , χ2 , w) = W (χ1 , χ2 ) is also independent of w , and ε1 , ε2 and ε ∗ can be chosen all equaling to 1. Such a case is essentially studied as Lemma 2.2 in Teel and Praly (1995) or Corollary 12.1.2 in Isidori (1999). Therefore, Lemma 4 somewhat extends these results.

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

5. Main result

Then

By Lemma 1, we only need to focus on solving the cooperative semi-global stabilization problem for the augmented system (10). For this purpose, we further need two more assumptions. Assumption 5. For the ith subsystem of the augmented system (10), there exists a C 2 positive definite and proper function V0i : Rnzi → R, such that for all col(v, w) ∈ V × W, V0i (¯zi ) along the trajectory of the zero dynamics, that is, z˙¯ i = f¯0i (¯zi , 0, v, w), satisfies

∂ V0i ¯ f0i (¯zi , 0, v, w) ≤ −α0i ∥¯zi ∥2 , ∂ z¯i

(21)

where α0i , i = 1, . . . , N, are some known positive real numbers. Assumption 6. Every node i = 1, . . . , N is reachable from the node 0 in the digraph G¯ . Remark 6. Under Assumption 5, the zero dynamics of the ith subsystem in the augmented system (10) is global asymptotically stable. Assumptions 1–6 are quite standard in the sense that the Assumptions 1–5 are needed even for the solvability of the semiglobal robust output regulation problem for the single nonlinear system (Lan et al., 2005). Many practical systems such as Vol der Pol system, Duffing system, Chua’s circuit system, and Lorenz system, satisfy these assumptions. Assumption 6 has been shown to be necessary for the solvability of the cooperative robust output regulation problem for multi-agent systems (Su & Huang, 2013). n×n

Lemma 5 (p. 131 in Horn & Johnson, 1991). Let A ∈ R be an M-matrix3 and D ∈ Rn×n be a diagonal matrix with positive main diagonal entries d1 , . . . , dn . Then τ (DA) ≥ τ (A) min1≤i≤n di , where τ (A) = min{Re λ, λ ∈ σ (A)} and σ (A) denotes the spectral set of A.

N

Remark 7. Let H = [hij ]Ni,j=1 with hii = j=0 aij and hij = −aij , for any i ̸= j, i, j = 1, . . . , N. By Lemma 4 in Hu and Hong (2007), all of the nonzero eigenvalues of the matrix H have positive real parts. Furthermore, H has all eigenvalues with positive real parts, i.e., H is an M -matrix, if and only if Assumption 6 is satisfied. Let H (w) = diag{b1 (w), . . . , bN (w)}H. Under Assumption 2, H (w) is nonsingular if and only if H is nonsingular. Furthermore, by Lemma 5, τ (H (w)) ≥ τ (H ) min1≤i≤n {bi (w)} > 0 if H is an M matrix. Then again by Lemma 4 in Hu and Hong (2007), we can conclude that H (w) has all the eigenvalues with positive real parts for all w ∈ W if and only if Assumption 6 is satisfied. We further define the following coordinate transformations for each subsystem of (10), xˆ si =

x¯ si

,

1057

s = 1, . . . , r − 1

x˙ˆ si =

x˙¯ si g s−1

x˙ˆ (r −1)i =

=

x¯ (s+1)i

1 g r −2

g s−1

= g xˆ (s+1)i ,

s = 1, . . . , r − 2

ϑi − g γr −1 xˆ (r −1)i − · · · − g γ2 xˆ 2i − g γ1 xˆ 1i

ϑ˙ i = f˜1i (¯zi , xˆ 1i , . . . , xˆ (r −1)i , ϑi , v, w, g ) + bi (w)¯ui + bi (w)Γi Ti−1 η˜ i + Γi Ti−1 Qi ϑi where f˜1i (¯zi , xˆ 1i , . . . , xˆ (r −1)i , ϑi , v, w, g ) = f¯1i (¯zi , xˆ 1i , . . . , g r −2 xˆ (r −1)i ,

ϑi − g r −1 (γr −1 xˆ (r −1)i + · · · + γ2 xˆ 2i + γ1 xˆ 1i ), v, w) + g r (γr −2 xˆ (r −1)i + · · · + γ2 xˆ 3i + γ1 xˆ 2i ) + g γr −1 (ϑi − g r −1 (γr −1 xˆ (r −1)i + · · · + γ2 xˆ 2i + γ1 xˆ 1i )) and 1 ˜ zi , xˆ 1i , . . . , xˆ (r −1)i , ϑi , v, w, g ) η˙˜ i = Mi η˜ i − b− i (w)Qi f1i (¯ 1 + b− i (w)Mi Qi ϑi .

Let xˆ ai = col(ˆx1i , . . . , xˆ (r −1)i ), i = 1, . . . , N. Then under the coordinate transformations (22)–(24), the augmented system (10) takes the following form: z˙¯ i = fˆ0i (¯zi , xˆ ai , v, w) x˙ˆ ai = gAc xˆ ai + Bc (g )ϑi

η˙˜ i = Mi η˜ i + fˆ1i (¯zi , xˆ ai , ϑi , v, w, g ) ϑ˙ i = fˆ2i (¯zi , xˆ ai , η˜ i , ϑi , v, w, g ) + bi (w)¯ui

(25)

where fˆ0i (¯zi , xˆ ai , v, w) = f¯0i (¯zi , x¯ 1i , v, w), 1 fˆ1i (¯zi , xˆ ai , ϑi , v, w, g ) = b− i (w)Mi Qi ϑi 1 ˜ zi , xˆ 1i , . . . , xˆ (r −1)i , ϑi , v, w, g ), − b− i (w)Qi f1i (¯ fˆ2i (¯zi , xˆ ai , η˜ i , ϑi , v, w, g ) = f˜1i (¯zi , xˆ 1i , . . . , xˆ (r −1)i , ϑi , v, w, g )

+ Γi Ti−1 Qi ϑi + bi (w)Γi Ti−1 η˜ i ,  0 1 0 ··· 0 1 ···  0  . .. .. .. . Ac =   . . . .  0 0 0 ··· −γ1 −γ2  0    Bc (g ) =   

−γ3

···

0 0

.. .

1

   ,  

−γr −1

0 

..  .  . 0  1 g r −2

(22)

5.1. Stabilization of the augmented system

ϑi = x¯ ri + g γr −1 x¯ (r −1)i + · · · + g r −2 γ2 x¯ 2i + g r −1 γ1 x¯ 1i

(23)

1 η˜ i = η¯ i − b− i (w)Qi ϑi

(24)

Let us first consider the distributed static state feedback control law as follows.

g s −1

where the number g and the coefficients γs , s = 1, . . . , r − 1 are to be determined later.

3 A matrix M ∈ Rn×n is called a M -matrix if all the non-diagonal elements are all non-positive and M has all the eigenvalues with positive real parts.

u¯ i = −K

N 

aij (ϑi − ϑj ) , −K ϑv i

(26)

j =0

where ϑ0 , 0 and ϑi , i = 1, . . . , N, is defined by (23) with the coefficients γs , s = 1, . . . , r − 1, being chosen so that the polynomial f (s) = sr −1 + γr −1 sr −2 + · · · + γ2 s + γ1 is stable, and g and K are positive numbers to be determined. Then the

1058

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

closed-loop system consisting of the augmented system (25) and the distributed control law (26) takes the following form: z˙¯ i = fˆ0i (¯zi , xˆ ai , v, w) x˙ˆ ai = gAc xˆ ai + Bc (g )ϑi

η˙˜ = M η˜ + F1 (¯z , xˆ a , 0, v, w, g ).

η˙˜ i = Mi η˜ i + fˆ1i (¯zi , xˆ ai , ϑi , v, w, g ) ϑ˙ i = fˆ2i (¯zi , xˆ ai , η˜ i , ϑi , v, w, g ) − bi (w)K ϑvi . (27) Let z¯ = col(¯z1 , . . . , z¯N ), xˆ a = col(ˆxa1 , . . . , xˆ aN ), η˜ = col(η˜ 1 , . . . , η˜ N ), and ϑ = col(ϑ1 , . . . , ϑN ). Then system (27) is equivalent to the following block lower triangular form

η˜˙ = M η˜ + F1 (¯z , xˆ a , ϑ, v, w, g ) ϑ˙ = −KH (w)ϑ + F2 (¯z , xˆ a , η, ˜ ϑ, v, w, g )

(28)

where M = diag(M1 , . . . , MN ), F0 (¯z , xˆ a , v, w) = col(fˆ01 , . . . , fˆ0N ), F1 (¯z , xˆ a , ϑ, v, w, g ) = col(fˆ11 , . . . , fˆ1N ), and F2 (¯z , xˆ a , η, ˜ ϑ,  v, w, g ) = col(fˆ21 , . . . , fˆ2N ). Let nz = Ni=1 nzi , nxa = (r − 1)N, i=1

nηi , nϑ = N. We have the following theorem.

Theorem 1. Under Assumptions 2, 5 and 6, given any arbitrarily large number R¯ > 0, there exist sufficiently large positive real num¯ such that the equilibrium at the bers g and K , which depend on R, origin of the closed-loop system consisting of (10) and (26) is locally n asymptotically stable with its domain of attraction containing Q¯ R¯ c , where nc = rN +

N

i =1

(nzi + nηi ).

z˙¯ = F0 (¯z , xˆ a , v, w),

F0 (¯z , xˆ a , v, w) g (IN ⊗ Ac )ˆxa

, and ϕ2 (χ1 , v, w) = F1 (¯z , xˆ a ,

nη nz +nxa 0, v, w, g ). By (24), for any η¯ ∈ Q¯ R¯ and any col(¯z , xˆ a ) ∈ Q¯ R¯ , 1 nη ¯ ¯ there exists R2 > 0 such that η˜ ∈ Q ¯ . Then by (30), applying

1

(29)

¯ By (22), for any g ≥ 1, Then given any R¯ > 0, let R¯ 1 = R. n x¯ = col(¯x11 , . . . , x¯ r1 , . . . , x¯ 1N , . . . , x¯ rN ) ∈ Q¯ R¯rN and z¯ ∈ Q¯ R¯ z , n z +n xa

implies col(¯z , xˆ a ) ∈ Q¯ R¯ . Note that Ac is Hurwitz. There exists 1 a positive definite matrix Pc such that ATc Pc + Pc Ac ≤ −Ir −1 . Under Assumption 5, let V0 (¯z ) = Then

N

i =1

V0i (¯zi ) and α0 = mini=1,...,N {α0i }.

≤ −α0 ∥¯z ∥2 .

We then consider the C positive definite function V1 (¯z , xˆ a ) = V0 (¯z ) + xˆ Ta (IN ⊗ Pc )ˆxa . Let c1 > 0 be sufficiently large such that 2

nz +nxa

¯ c1 (V1 (¯z , xˆ a )). Note that Fˆ0 (¯z , xˆ a , v, w) , F0 (¯z , xˆ a , ⊆ Ω 1 v, w) − F0 (¯z , 0, v, w) is sufficiently smooth with Fˆ0 (¯z , 0, v, w) = ∂V ∂V 0, and ∂ z¯0 (¯z ) is C 1 with ∂ z¯0 (0) = 0. Then given any fixed ε1 ≥ 1, ¯ for all col(¯z , xˆ a ) ∈ Ωc1 +ε1 (V1 (¯z , xˆ a )) and all col(v, w) ∈ V × W, by Lemma 2, there exist L1 > 0 and L2 > 0 such that ∥Fˆ0 (¯z , xˆ a ,

Q¯ R¯

L2 L2

v, w)∥ ≤ L1 ∥ˆxa ∥ and ∥ ∂∂Vz¯0 ∥ ≤ L2 ∥¯z ∥. Let g ≥ 21α02 + 1. Then for all ¯ c1 +ε1 (V1 (¯z , xˆ a )) and all col(v, w) ∈ V × W, col(¯z , xˆ a ) ∈ Ω  ∂ V0 ˆ V˙ 1 (¯z , xˆ a )(29) ≤ −α0 ∥¯z ∥2 + F0 (¯z , xˆ a , v, w) − g ∥ˆxa ∥2 ∂ z¯ ≤ −α0 ∥¯z ∥2 + L1 L2 ∥¯z ∥ · ∥ˆxa ∥ − g ∥ˆxa ∥2   α0 L21 L22 2 ≤ − ∥¯z ∥ − g − ∥ˆxa ∥2 2 2α0 ≤ −α1 ∥(¯z , xˆ a )∥2   L2 L2 α where α1 = min 20 , g − 21α 2 > 0. 0

2

¯ c2 +ε2 (V2 (¯z , xˆ a , η)) ˜ , V × W and all col(¯z , xˆ a , η) ˜ ∈Ω V˙ 2 (¯z , xˆ a , η) ˜ (31) ≤ −α2 ∥(¯z , xˆ a , η)∥ ˜ 2



(30)

(32)

where α2 is a positive real number and ε2 ≥ 1 can be arbitrarily chosen. Step-3: Consider now system (28). Note that it has the form of (16) with χ1 = col(¯z , xˆ a , η) ˜ , χ2 = ϑ , ρ = K , and τ (χ2 , ρ) = χ2 = ϑ , Y (w) = −H (w). Note that here τ (χ2 , ρ) is independent of ρ . n n +n By (23), for any x¯ ∈ Q¯ R¯Nr and any col(¯z , xˆ a , η) ˜ ∈ Q¯ R¯ z xa × Q¯ R¯ η , 1

n

2

there exists R¯ 3 > 0 such that ϑ ∈ Q¯ R¯ ϑ . By (32), applying Lemma 4 3 nη

n

a with Ξ1 = Q¯ R¯ × Q¯ R¯ and Ξ2 = Q¯ R¯ ϑ , there exist a sufficient 1 2 3 large positive number K , a positive real number c3 , a C 2 function ˜ ϑ) and V3 (¯z , xˆ a , η, ˜ ϑ, w), positive definite functions V 3 (¯z , xˆ a , η, n +n V 3 (¯z , xˆ a , η, ˜ ϑ), and a positive real number α3 , such that Q¯ R¯ z xa ×

nη

x˙ˆ a = g (IN ⊗ Ac )ˆxa .

z¯ =F0 (¯z ,0,v,w)

η˜ , ϕ1 (χ1 , v, w) =

nz +nx

Proof. The proof is based on block semi-global backstepping technique introduced in Section 4. It contains the following 3 steps. Step-1: Consider the col(¯z , xˆ a ) subsystem with ϑ = 0, that is,

 V˙ 0 (¯z ) ˙

Note that it has the form  of (13) and  (14) with χ1 = col(¯z , xˆ a ), χ2 =

nη nz +nxa Lemma 3 with Ξ1 = Q¯ R¯ and Ξ2 = Q¯ R¯ , there exist a positive 1 2 real number c2 , and a C 2 positive definite function V2 (¯z , xˆ a , η) ˜ , such n nz +nxa ¯ c2 (V2 (¯z , xˆ a , η)) ˜ , and for all col(v, w) ∈ that, Q¯ R¯ × Q¯ R¯ η ⊆ Ω

x˙ˆ a = g (IN ⊗ Ac )ˆxa + (IN ⊗ Bc (g ))ϑ

N

(31)

R2

z˙¯ = F0 (¯z , xˆ a , v, w)

nη =

Step-2: Consider the col(¯z , xˆ a , η) ˜ subsystem with ϑ = 0, that is z˙¯ = F0 (¯z , xˆ a , v, w), x˙ˆ a = g (IN ⊗ Ac )ˆxa ,

1

n

¯ c3 (V 3 (¯z , xˆ a , η, ˜ ϑ)), and for all col(v, w) ∈ V × W, Q¯ R¯ × Q¯ R¯ ϑ ⊆ Ω 2

3

¯ c3 +ε3 (V 3 (¯z , xˆ a , η, and all col(¯z , xˆ a , η, ˜ ϑ) ∈ Ω ˜ ϑ)), ˜ ϑ) ≤ V3 (¯z , xˆ a , η, ˜ ϑ, w) ≤ V 3 (¯z , xˆ a , η, ˜ ϑ) V 3 (¯z , xˆ a , η,  V˙ 3 (¯z , xˆ a , η, ˜ ϑ, w)

(28)

≤ −α3 ∥(¯z , xˆ a , η, ˜ ϑ)∥

2

(33) (34)

where ε3 is chosen so that

ε3 ≥

max

¯ c (V 3 (z¯ ,ˆxa ,η,ϑ) (¯z ,ˆxa ,η,ϑ)∈ ˜ Ω ˜ ) 3

 V 3 (¯z , xˆ a , η, ˜ ϑ)

  − V 3 (¯z , xˆ a , η, ˜ ϑ) , 1 .

(35)

By Lemma 4, from (33) to (35), any trajectory of col(¯z , xˆ a , η, ˜ ϑ) n nz +nxa n × Q¯ R¯ η × Q¯ R¯ ϑ converges to the origin starting in compact set Q¯ R¯ 1

2

3

asymptotically as t → ∞. Then by (22)–(24), and noting that x¯ c ∈ n Q¯ R¯ c , where x¯ c = col(¯z1 , x¯ 11 , . . . , x¯ r1 , η¯ 1 , . . . , z¯N , x¯ 1N , . . . , x¯ rN , η¯ N ), nz +nx

nη

n

a implies col(¯z , xˆ a , η, ˜ ϑ) ∈ Q¯ R¯ × Q¯ R¯ × Q¯ R¯ ϑ , any trajectory of x¯ c 1 2 3 n starting in compact set Q¯ R¯ c converges to the origin asymptotically as t → ∞. Thus, the equilibrium at the origin of the closedloop system consisting of (10) and (26) is asymptotically stable n with its domain of attraction containing Q¯ R¯ c . The proof is thus completed. 

When r = 1, the coordinate transformations (22) and (23) reduce to ϑi = x¯ 1i . Then by Theorem 1, we can obtain the following corollary. Corollary 1. Suppose all the subsystems in (10) has uniformly relative degree r = 1. Then under Assumptions 2, 5 and 6, given any arbitrarily large number R¯ > 0, there exists a sufficiently large ¯ such that the equilibrium positive real number K , which depends on R,

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

at the origin of the closed-loop system consisting of system (10) and the distributed controller u¯ i = −K

N 

aij (¯x1i − x¯ 1j ) = −K

N 

j=0

aij (yi − yj )

j =0

1059

where the notation (a)i denotes the ith component of vector a, and [Ac ]r −1 denotes the (r − 1)th row of the matrix Ac . Let ψi = [ψ1i , . . . , ψri ]T , ψ = [ψ1 , . . . , ψN ]T , Dh = diag{hr −1 , . . . , h, 1}, and nψ = rN. Then system (37) is equivalent to the following block lower triangular form

is locally asymptotically stable with domain of attraction containing N n Q¯ R¯ c , where nc = N + i=1 (nzi + nηi ).

z˙¯ = F0 (¯z , xˆ a , v, w)

When r > 1, since the state of each subsystem in the augmented system (10) is not allowed for feedback control, the distributed static state feedback controller (26) is not applicable. Instead, by introducing a distributed version of high gain observer (Esfandiari & Khalil, 1992), and combining the distributed state feedback controller (26), we can show that the augmented system (25) is stabilizable by the following distributed dynamic output feedback control laws.

η˙˜ = M η˜ + F1 (¯z , xˆ a , ϑ, v, w, g ) ϑ˙ = −KH (w)ϑ + F¯2 (¯z , xˆ a , η, ˜ κ(ψ, h), v, w, g , K )

u¯ i = −K ϑˆ v i

ϑˆ vi = ζri + g γr −1 ζ(r −1)i + · · · + g r −2 γ2 ζ2i + g r −1 γ1 ζ1i ζ˙i = Ao (h)ζi + Bo (h)evi

(36)

where ζi , col(ζ1i , . . . , ζri ) ∈ R , r

 −h δ r −h2 δr −1  .. Ao (h) =   .  r −1 −h δ 2 −h r δ 1  hδ 

1 0

0 1

.. .

··· ··· .. .

0 0

0 0

0 0

··· ···

1 0

.. .



..   . , 

˙ i − g ((H ⊗ [Ac ]r −1 )x˙ˆ a )i ). col(0, . . . , 0, (H ϑ)

Theorem 2. Under Assumptions 2, 5 and 6, given any arbitrarily large number R¯ > 0, there exist sufficiently large positive real num¯ such that the equilibrium at the bers g, K and h, which depend on R, origin of the closed-loop system consisting of (10) and (36) is locally n asymptotically stable with its domain of attraction containing Q¯ R¯ c ,

Nr +nψ

Then by (38), for any h ≥ 1 and any col(¯x, ζ ) ∈ Q¯ R¯ , where nψ ¯ ζ = col(ζ1 , . . . , ζN ), implies ψ ∈ QR¯ . By (33) and (34), again 4

n nz +nxa n applying Lemma 4 with Ξ1 = Q¯ R¯ × Q¯ R¯ η × Q¯ R¯ ϑ and Ξ2 = 1 2 3

z˙¯ i = fˆ0i (¯zi , xˆ ai , v, w)

nψ

Q¯ R¯ , there exist a sufficient large positive number h, a positive 4

real number c4 , a C 2 function V4 (¯z , xˆ a , η, ˜ ϑ, ψ, w), positive defi˜ ϑ, ψ), and a pos˜ ϑ, ψ) and V 4 (¯z , xˆ a , η, nite functions V 4 (¯z , xˆ a , η, nz +nx

nη

n

nψ

a itive real number α4 , such that Q¯ R¯ × Q¯ R¯ × Q¯ R¯ ϑ × Q¯ R¯ ⊆ 1 2 3 4 ¯ c4 (V 4 (¯z , xˆ a , η, ˜ ϑ, ψ)), and for all col(v, w) ∈ V × W, and all Ω ¯ c4 +ε4 (V 4 (¯z , xˆ a , η, ˜ ϑ, ψ)), col(¯z , xˆ a , η, ˜ ϑ, ψ) ∈ Ω

V 4 (¯z , xˆ a , η, ˜ ϑ, ψ) ≤ V4 (¯z , xˆ a , η, ˜ ϑ, ψ, w)

x˙ˆ ai = gAc xˆ ai + Bc (g )ϑi

≤ V 4 (¯z , xˆ a , η, ˜ ϑ, ψ)  V˙ 4 (¯z , xˆ a , η, ˜ ϑ, ψ, w)(39) ≤ −α4 ∥(¯z , xˆ a , η, ˜ ϑ, ψ)∥2

η˙˜ i = Mi η˜ i + fˆ1i (¯zi , xˆ ai , ϑi , v, w, g ) ϑ˙ i = fˆ2i (¯zi , xˆ ai , η˜i , ϑi , v, w,g ) − bi (w)K ϑvi + bi (w)K ϑvi − ϑˆ vi ζ˙i = Ao (h)ζi + Bo (h)evi .

ε4 ≥ (37)

aij (¯x(s+1)i − x¯ (s+1)j ) + ai0 x¯ (s+1)i .

j =1

For i = 1, . . . , N, let s = 1, . . . , r .

Then it can be verified that for any i = 1, . . . , N,

ψ˙ si = h(ψ(s+1)i − δr −s+1 ψ1i ), s = 1, . . . , r − 1 ˙ i − g ((H ⊗ [Ac ]r −1 )x˙ˆ a )i ψ˙ ri = −hδ1 ψ1i + (H ϑ)

(40) (41)

where ε4 is chosen so that

Note that the sth derivatives of ev i (t ), s = 1, . . . , r − 1, for each subsystem i, i = 1, . . . , N, are as follows:

ψsi = hr −s (e(vsi−1) − ζsi ),

1 where κ(ψ, h) = (IN ⊗ D− z , xˆ a , v, w) and F1 (¯z , xˆ a , ϑ, v, h )ψ , F0 (¯ w, g ) are the same as those in (28), and F¯2 = F2 + col(b1 (w) 1 −1 KD− h ψ1 , . . . , bN (w)KDh ψN ), F3 = col(F31 , . . . , F3N ), where F3i =

Proof. It is noted that the system (39) has the form of (16) with χ1 = col(¯z , xˆ a , η, ˜ ϑ), χ2 = ψ , ρ = h, and τ (χ2 , ρ) = κ(ψ, h) = 1 (IN ⊗ D− h )ψ , Y (w) = IN ⊗ Ao (1). Note that for any h ≥ 1, 1 ¯ ¯ ∥κ(ψ, h)∥ = ∥(IN ⊗ D− h )ψ∥ ≤ ∥ψ∥. Let R4 = (N + 1)R.

and the coefficients g, K , and γj , j = 1, . . . , r − 1, are the same as those in (26), δk , k = 1, . . . , r, are chosen so that the polynomial g (s) = sr + δr sr −1 + · · · + δ2 s + δ1 is stable, and h is a positive number to be determined. The closed-loop system consisting of the augmented system (25) and the distributed dynamic output feedback control law (36) takes the following form:

(s)

(39)

N

r

ev i =

ψ˙ = h(IN ⊗ Ao (1))ψ + F3 (¯z , xˆ a , η, ˜ ϑ, κ(ψ, h), v, w, g , K )

where nc = 2rN + i=1 (nzi + nηi ). Thus, the cooperative semi-global stabilization problem of system (10) is solvable.

h2 δr −1   .   Bo (h) =   ..  ,  r −1  h δ2 hr δ1

N 

x˙ˆ a = g (IN ⊗ Ac )ˆxa + (IN ⊗ Bc (g ))ϑ

(38)

max

¯ c (V 4 (z¯ ,ˆxa ,η,ϑ,ψ) ˜ (¯z ,ˆxa ,η,ϑ,ψ)∈ ˜ Ω ) 4

  − V 4 (¯z , xˆ a , η, ˜ ϑ, ψ) , 1 .

 V 4 (¯z , xˆ a , η, ˜ ϑ, ψ) (42)

˜ By Lemma 4, from (40) to (42), any trajectory of col(¯z , xˆ a , η, n n n +n n ϑ, ψ) starting in compact set Q¯ R¯ z xa × Q¯ R¯ η × Q¯ R¯ ϑ × Q¯ R¯ ψ 1 2 3 4 converges to the origin asymptotically as t → ∞. Then by (22)–(24), and noting that x¯ c = col(¯z1 , x¯ 11 , . . . , x¯ r1 , η¯ 1 , ζ1 , . . . , n +n n z¯N , x¯ 1N , . . . , x¯ rN , η¯ N , ζN ) ∈ Q¯ R¯ c imply col(¯z , xˆ a , η, ˜ ϑ, ψ) ∈ Q¯ R¯ z xa 1 n n n × Q¯ ¯ η × Q¯ ¯ ϑ × Q¯ ¯ ψ , any trajectory of x¯ c starting in compact set Q¯ ¯nc R2

R3

R4

R

converges to the origin asymptotically as t → ∞. Thus, the equilibrium at the origin of the closed-loop system consisting of (10) and (36) is locally asymptotically stable with its domain of attracn tion containing Q¯ R¯ c . The proof is thus completed. 

1060

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

µ

µ

and w = col(w1 , w1b , . . . , wN , wNb ) being the uncertainty vector. We assume that w belongs to some known compact set W which satisfies that 010×1 ∈ W and W ⊆ {w ∈ R10 : b¯ i + wib > 0, i = 1, . . . , 5}. The exosystem is an unforced harmonic oscillator

v˙ 1 = v2 ,

Fig. 1. The network topology. (Agent 0 is the leader.)

5.2. Solvability of the problem By Lemma 1 and Theorem 2, we can obtain the following result. Theorem 3. Under Assumptions 1–6, given any R > 0, and any compact subsets V0 ⊆ Rq and W ⊆ Rnw , there exist sufficiently large numbers g > 0, K > 0, h > 0, which depend on R, such that the cooperative regional robust output regulation problem of N n system (1) on the compact set Q¯ R c , where nc = 2rN + i=1 (nzi + nηi ), is solved by a distributed dynamic output feedback controller: ui = −K ϑˆ v i + Γi Ti−1 ηi ,

ϑˆ vi = ζri + g γr −1 ζ(r −1)i + · · · + g r −2 γ2 ζ2i + g r −1 γ1 ζ1i , ζ˙i = Ao (h)ζi + Bo (h)evi , η˙ i = Mi ηi + Qi ui ,

i = 1, . . . , N .

(43)

Thus, the cooperative semi-global robust output regulation problem of system (1) is solvable. For the special case where r = 1, there is no need to introduce the distributed high gain observer. By Lemma 1 and Corollary 1, we have the following corollary. Corollary 2. Suppose all the subsystems in (1) has uniformly relative degree r = 1. Then under Assumptions 1–6, given any R > 0, and any compact subsets V0 ⊆ Rq and W ⊆ Rnw , there exists a sufficiently large number K > 0, which depend on R, such that the cooperative regional robust output regulation problem of N n system (1) on the compact set Q¯ R c , where nc = N + i=1 (nzi + nηi ), is solved by a distributed dynamic output feedback controller: ui = −Kev i + Γi Ti−1 ηi ,

η˙ i = Mi ηi + Qi ui ,

i = 1, . . . , N .

(44)

Thus, the cooperative semi-global robust output regulation problem of system (1) is solvable.

v˙ 2 = −v1 ,

y0 = v1 .

(46)

We assume that the interconnection among the subsystems is determined by Fig. 1. Note that all the subsystems in Fig. 1 are reachable from the exosystem. Therefore, Assumption 6 is satisfied. Let a weighted adjacency matrix of G¯ be given by A¯ = [aij ]Ni,j=0 with a10 = a20 = a31 = a35 = a42 = a45 = a53 = a54 = 1 and zero for the others. It can be also verified that Assumptions 1 and 2 are satisfied. In this example, the solution to the regulator equation of each plant in (45) is given by x1i (v, w) = v1 , x2i (v, w) = v2 , ui (v, w) = bi (w)−1 µi (w)v2 (1 − v12 ). Then it can be verified that for i = 1, . . . , 5, d4 ui (v, w) dt 4

= −9ui (v, w) − 10

d2 ui (v, w) dt 2

.

(47)

Thus, Assumptions 3 and 4 are satisfied. Since the plant (45) contains no zero dynamics, Assumption 5 is satisfied automatically. It is noted that, even for the special case where N = 1, the output regulation problem for van del Pol system does not have a global solution with output feedback control. However, it has been verified that system (45) has the form of (1) with r = 2, and satisfies Assumptions 1–6. Therefore, applying Theorem 3, it is possible to solve the cooperative semi-global robust output regulation problem for this system via distributed output feedback control. Let Φi and Γi , i = 1, . . . , 5, in (8) be given by



0 0 Φi =  0 −9

1 0 0 0

 T



0 1 0 −10

0 0 , 1 0

1

0 Γi =   . 0

(48)

0

Let



0 0 Mi =  0 −4

1 0 0 −12

0 1 0 −13

 



0

0 0  , 1  −6

0 Qi =   . 0

(49)

1

Then for i = 1, . . . , N, the Sylvester equation (9) admits a unique solution Ti whose inverse is

−5.0000 −54.0000 = −27.0000 432.0000 

Ti−1

6. Two examples

12.0000 −5.0000 −54.0000 −27.0000

3.0000 −48.0000 −35.0000 426.0000

6.0000 3.0000  , −48.0000 −35.0000



and In this section, we will use two examples, namely, multiple Van der Pol oscillator systems and multiple generalized Lorenz systems, to illustrate our design. To our knowledge, neither of these two examples can be handled by existing methods in the literature since their nonlinear terms do not satisfy the global Lipschitz-like condition.

12.0000

3.0000

6.0000 .



Consider a group of Van der Pol oscillators as follows: x˙ 1i = x2i

(50)

Therefore, by Theorem 3, we can obtain the distributed dynamic output feedback controller: ui = −K ϑˆ v i + Γi Ti−1 ηi

ϑˆ vi = ζ2i + g ζ1i ζ˙i = Ao (h)ζi + Bo (h)evi η˙ i = Mi ηi + Qi ui , i = 1, . . . , 5,

6.1. Van der Pol oscillators

(51)

where

x˙ 2i = −x1i + µi (w)x2i (1 − x21i ) + bi (w)ui yi = x1i ,

 Γi Ti−1 = −5.0000

i = 1, . . . , 5,

(45) µ

where the coefficients µi (w) = µ ¯ i + wi and bi (w) = b¯ i + wib with ¯ µ ¯ i and bi , i = 1, . . . , 5, being their nominal values, respectively,

Ao (h) =



−10 ∗ h −30 ∗ h2



1 , 0

Bo (h) =





10 ∗ h . 30 ∗ h2

Assume that µ ¯ i = 5.5, b¯ i = 1, i = 1, . . . , 5, and W = {w ∈ µ R10 : wi ∈ [−2, 2], wib ∈ [−0.5, 0.5], i = 1, . . . , 5}. Then we

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

1061

is the nominal value of ai and wi , col(w1i , . . . , w5i , wbi ) is the uncertainty of ai . We assume that the uncertainty wi belongs to a known compact set W that satisfies W ⊆ {wi ∈ R6 , a¯ 1i + w1i < 0, a¯ 5i + w5i < 0, b¯ i + wbi > 0, i = 1, . . . , 5}. Performing the coordinate transformation (z1i , z2i , yi ) = (x1i , x3i , x2i ) on the subsystems of system (52) yields the following system z˙1i = a1i z1i + a2i yi z˙2i = a5i z2i + z1i yi y˙ i = a3i z1i − z1i z2i + a4i yi + bi ui .

(53)

Note that system (53) takes the form of (1) with zi = col(z1i , z2i ) and r = 1. The function zi (v, w) in (6) is given by zi (v, w) =

r1i v1 + r2i v2 z1i (v, w) = z2i (v, w) r3i v12 + r4i v22 + r5i v1 v2





where r1i = − r5i a5i

Fig. 2. Responses of the states x1i (t ) and x2i (t ).

, r5i = −

a1i a2i

, r2i = − 1+a2ia2 , r3i = −

1+a21i r2i a5i +2r1i a25i +4





(a25i +2)r1i −a5i r2i

1i

a5i (a25i +4)

, r4i =

. Then we have

ui (v, w) = c1i v1 + c2i v2 + c3i v13 + c4i v23 + c5i v12 v2 + c6i v1 v22 1 −1 where c1i = −b− i (a4i + a3i r1i ), c2i = bi (1 − a3i r3i ), c3i = −1 −1 −1 1 bi r1i r3i , c4i = bi r2i r4i , c5i = bi (r2i r3i + r1i r5i ), c6i = b− i (r1i r4i + r2i r5i ). It can be verified that ui (v, w) also satisfies (47). As a result, we can obtain that Φi , Γi , and Mi , Qi are the same as those in (48) and (49), respectively. Thus Γi Ti−1 can be chosen the same as (50). Now we have shown that Assumptions 1–4, and 6 are all satisfied. To verify Assumption 5, performing the coordinate transformation z¯1i = z1i − z1i (v, w) and z¯2i = z2i − z2i (v, w)  yields the 

function f¯0i (¯zi , 0, v, w) in (21) as f¯0i (¯zi , 0, v, w) = Let V0i (¯zi ) =

µi 2 ¯ 2

z1i +

1 2 z , 2 2i

¯

a1i z¯1i a5i z¯2i + v1 z¯1i

where µi > maxcol(v,w)∈V×W



v12 2a1i a5i



.

.

Then

Fig. 3. Responses of the regulated errors ei (t ).

2 V˙ 0i z˙ =f¯ (¯z ,0,v,w) = a1i µi z¯1i + a5i z¯2i2 + v1 z¯1i z¯2i i 0i i



design the control gains in (51) as g = 2, K = 60 and h = 30. The simulation result assuming that the exact uncertainty vector is w = [−1, −0.2, −0.5, −0.1, 0, 0, 0.5, 0.1, 1, 0.2]T and the initial conditions of the closed-loop system are v(0) = [5, −5]T , x1 (0) = [−5, 4]T , x2 (0) = [−1, 0]T , x3 (0) = [3, −4]T , x4 (0) = [7, −8]T , x5 (0) = [11, −12]T , and zero for the others, is reported in Figs. 2 and 3. From Fig. 2, we can see that the states of all subsystems converge to the same trajectory. And from Fig. 3, we can see that the tracking errors of all the followers converge to zero asymptotically. 6.2. Generalized Lorenz systems Consider a group of generalized Lorenz systems (Liang, Zhang, & Xia, 2008) as follows:

 ≤ − −a1i µi −

v12 −2a5i



2 z¯1i −

−a5i 2

2 z¯2i

≤ −α0i z¯i2 v2





where α0i = mincol(v,w)∈V×W −a1i µi − −2a1 , 25i . Thus, As5i sumption 5 is satisfied. Therefore, by Corollary 2, we can construct a distributed dynamic output feedback controller of the form (44) to solve this problem. Assume that a¯ i = [−10, 10, 28, −1, −8/3, 1]T and W = {wi ∈ 6 R : wsi ∈ [−2, 2], s = 1, 2, 3, w5i ∈ [− 23 , 23 ], w4i = wbi = 0, i = 1, . . . , 5}. By Corollary 2, we design the distributed controller as ui = −Kev i + Γi Ti−1 ηi ,

η˙ i = Mi ηi + Qi ui ,

−a

i = 1, . . . , 5,

where Mi , Qi are given by (49), Γi Ti is given by (50), and the control gain K = 50. The simulation result assuming that the exact uncertain parameter are w1 = [1, −1, −1, 0, − 13 ]T , w2 = −1

x˙ 1i = a1i x1i + a2i x2i x˙ 2i = a3i x1i + a4i x2i − x1i x3i + bi ui x˙ 3i = a5i x3i + x1i x2i yi = x2i ,

i = 1, . . . , 5,

(52)

where ai , col(a1i , . . . , a5i , bi ) is a constant parameter vector that satisfies a1i < 0, a5i < 0, and bi > 0. The exosystem is given as (46), and the regulated error is defined as ei = yi − y0 = x2i − v1 . The interconnection among the subsystems is also assumed to be determined by Fig. 1 which satisfies Assumption 6. Here we allow the parameter vector ai to undergo the perturbation, and is assumed to be ai = a¯ i + wi , where a¯ i , col(¯a1i , . . . , a¯ 5i , b¯ i )

[0, 0, −1, 0, 13 ]T , w3 = [−1, 1, 0, 0, 0]T , w4 = [−1, 1, −1, 0, 0]T , w5 = [0, 0, −1, 0, 13 ]T , and the initial conditions of the closedloop system are v(0) = [5, −5]T , z1 (0) = [4, 8]T , y1 (0) = −5, z2 (0) = [0, 9]T , y2 (0) = −1, z3 (0) = [−4, 10]T , y3 (0) = 3, z4 (0) = [−8, 11]T , y4 (0) = 7, z5 (0) = [−12, 12]T , y5 (0) = 11,

and zero for the others, is reported in Figs. 4–5. From Fig. 4, we can see that the outputs of all the subsystems converge to that of the leader. And from Fig. 5, we can see that the tracking errors of all the followers converge to zero asymptotically.

1062

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

Appendix B. Proofs of Lemmas 3 and 4 We further have the following lemmas and remarks. Remark 8. Given a positive definite function V : Rn → R, and cV (χ ) any c > 0 and ε ≥ 1, the function W (χ ) = c +ε−V (χ ) is a positive definite function defined on Ωc +ε (V (χ )). It is noted that ¯ c (V (χ )) ⊆ Ωc +ε (V (χ )). Furthermore, for all χ ∈ Ω ¯ c (V (χ )), i.e., Ω 2

0 ≤ V (χ ) ≤ c, W (χ ) ≤ cε ≤ c 2 . On the other hand, let the function U (χ ) be such that 0 ≤ U (χ ) ≤ V (χ ), for any χ ∈ Rn . Then for all χ ∈ Ωc +ε (V (χ )), we have 0 ≤ U (χ ) < c + ε , and c (c +ε) c . hence (c +ε−U (χ ))2 ≥ c +ε Fig. 4. Tracking performance.

Lemma 6. Let the functions Vi : Rni → R, i = 1, 2, be positive defc V1 (χ1 ) inite. Then for any c1 > 0 and ε1 ≥ 1, W (χ1 , χ2 ) = c +ε1 − + 1 1 V1 (χ1 ) n2 V2 (χ2 ) is a positive definite function defined on Ωc1 +ε1 (V1 (χ1 ))×R . Furthermore,

¯ c2 (V2 (χ2 )) ⊆ ¯ c1 (V1 (χ1 )) × Ω 1. For any c2 > 0 and ε ∗ ≥ 1, Ω ¯ c 2 +c (W (χ1 , χ2 )), and Ω ¯ c 2 +c +ε∗ (W (χ1 , χ2 )) ⊆ Ωc1 +ε1 Ω 2 2 1

1

(V1 (χ1 )) × Rn2 . 2. For any col(χ1 , χ2 ) ∈ Ωc 2 +c2 +ε∗ (W (χ1 , χ2 )), 1

c1

c1 (c1 +ε1 ) (c1 +ε1 −V1 (χ1 ))2

≥

. c1 +ε1 c V (χ )

Fig. 5. Responses of the regulated errors ei (t ).

7. Conclusion In this paper, we have studied the cooperative semi-global robust output regulation problem for a class of minimum phase nonlinear uncertain multi-agent systems. We have introduced a type of distributed internal model that converts the problem into a cooperative semi-global robust stabilization problem of the socalled augmented system. We solved the semi-global stabilization problem via distributed dynamic output control law by utilizing and combining a block semi-global backstepping technique, a simultaneous high gain feedback control technique, and a distributed high gain observer technique. Our result generalized the semi-global robust output regulation problem from a single nonlinear system to a multi-agent system. Our future work will focus on the same problem subject to the time-varying network topologies. It would also be interesting to further extend our result to the case that the leader system (2) also contains uncertainty.

1 1 is a Proof. By Remark 8, it is ready to see that c +ε1 − 1 1 V1 (χ1 ) positive definite function defined on Ωc1 +ε1 (V1 (χ1 )). So W (χ1 , χ2 ) is positive definite function defined on Ωc1 +ε1 (V1 (χ1 )) × Rn2 . Furthermore, we further have the following results. ¯ c1 (V1 (χ1 )) × (1) For any c2 > 0 and any col(χ1 , χ2 ) ∈ Ω ¯ Ωc2 (V2 (χ2 )), by Remark 8, we have

W (χ1 , χ2 ) ≤

c12

ε1

+ c2 ≤ c12 + c2 .

¯ c1 (V1 (χ1 )) × Ω ¯ c2 (V2 (χ2 )) ⊆ Ω ¯ c 2 +c (W (χ1 , χ2 )). On Therefore, Ω 2 1 ∗ the other hand, for any ε ≥ 1, if W (χ1 , χ2 ) ≤ c12 + c2 + ε ∗ , then c1 V1 (χ1 ) c1 +ε1 −V1 (χ1 )

≤ c12 + c2 + ε ∗ . Therefore, c12 + c2 + ε ∗

V1 (χ1 ) ≤ (c1 + ε1 )

c12

+ c2 + ε ∗ + c1

¯ c 2 +c +ε∗ (W (χ1 , χ2 )) ⊆ Ωc1 +ε1 So χ1 ∈ Ωc1 +ε1 (V1 (χ1 )). Thus, Ω 2 1 n2 (V1 (χ1 )) × R . (2) By the first part, for any col(χ1 , χ2 ) ∈ Ωc 2 +c2 +ε∗ (W (χ1 , χ2 )), 1

we have χ1 ∈ Ωc1 +ε1 (V1 (χ1 )). Then by Remark 8, we have c1 (c1 + ε1 )

(c1 + ε1 − V1 (χ1 ))

2

≥

c1 c1 + ε1

The proof is thus completed. Appendix A. Graph notation We summarize some graph notation which can be found in Godsil and Royle (2001). A digraph G = (V , E ) consists of a finite set of nodes V = {1, . . . , N } and an edge set E ⊆ V × V . For the edge (i, j) ∈ E , node i is called a neighbor of node j. If the digraph G contains a sequence of edges of the form (i1 , i2 ), (i2 , i3 ), . . . , (ik , ik+1 ), then the set {(i1 , i2 ), (i2 , i3 ), . . . , (ik , ik+1 )} is called a path of G from i1 to ik+1 , and node ik+1 is said to be reachable from node i1 . A matrix A = [aij ] ∈ RN ×N is said to be a weighted adjacency matrix of a digraph G if aii = 0, aij > 0 ⇔ (j, i) ∈ E , and aij = aji if (j, i) ∈ E implies (i, j) ∈ E .

< c1 + ε1 .

.



Lemma 7. Let the functions Vi : Rni → R, i = 1, 2, be positive definite. Then for any ci > 0 and εi ≥ 1, i = 1, 2, W (χ1 , χ2 ) =

c1 V1 (χ1 ) c1 + ε1 − V1 (χ1 )

+

c2 V2 (χ2 ) c2 + ε2 − V2 (χ2 )

(B.1)

is a positive definite function defined on Ωc1 +ε1 (V1 (χ1 )) × Ωc2 +ε2 (V2

(χ2 )). Furthermore,

¯ c1 (V1 (χ1 )) × Ω ¯ c2 (V2 (χ2 )) ⊆ Ω ¯ c 2 +c 2 (W (χ1 , 1. For any ε ∗ ≥ 1, Ω 1 2 ¯ c 2 +c 2 +ε∗ (W (χ1 , χ2 )) ⊆ Ωc1 +ε1 (V1 (χ1 )) × Ωc2 +ε2 χ2 )) and Ω (V2 (χ2 )).

1

2

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

¯ c 2 +c 2 +ε∗ (W (χ1 , χ2 )), 2. For any col(χ1 , χ2 ) ∈ Ω 2 1 ci

≤

ci +εi

ci (ci +εi ) , (ci +εi −Ui (χi ))2

and

ci (ci +εi ) (ci +εi −Vi (χi ))2

≤

of Lemma 6, for all col(v, w) ∈ V × W, and all col(χ1 , χ2 ) ∈

(c12 +c22 +ε ∗ +ci )2 , ci (ci +εi )

where

i = 1, 2, and Ui : R → R is positive definite satisfying Ui (χi ) ≤ Vi (χi ) for any χi ∈ Rni . ni

Proof. By Remark 8, it can be seen that W (χ1 , χ2 ) positive definite on Ωc1 +ε1 (V1 (χ1 ))× Ωc2 +ε2 (V2 (χ2 )). Furthermore, we further have the following results. ¯ c2 (V2 (χ2 )), by ¯ c1 (V1 (χ1 )) × Ω (1) For any col(χ1 , χ2 ) ∈ Ω Remark 8, W (χ1 , χ2 ) ≤

c12

ε1

c22

+

ε2

≤ c12 + c22 .

Vi (χi ) ≤ (ci + εi )

ci Vi (χi ) ci +εi −Vi (χi )

1

2

≤ c12 + c22 + ε ∗ , i = 1, 2. Then

c12 + c22 + ε ∗ c12

+ c22 + ε ∗ + ci

< ci + εi .

(B.2)

¯ c 2 +c 2 +ε∗ (W (χ1 , χ2 )) ⊆ Ωc1 +ε1 So χi ∈ Ωci +εi (Vi (χi )). Thus, Ω 1 2 (V1 (χ1 )) × Ωc2 +ε2 (V2 (χ2 )). ¯ c 2 +c 2 +ε∗ (W (χ1 , χ2 )), (2) By the first part, for any col(χ1 , χ2 ) ∈ Ω we have χi ci (ci +εi ) . (ci +εi −Ui (χi ))2

1

2

Ωci +εi (Vi (χi )). Then by Remark 8,

∈

ci ci +εi

≤

On the other hand, by (B.2), we have

ci (ci + εi ) ≤  (ci + εi − Vi (χi ))2

ci (ci + εi )

(

+ ε + ci ) . ci (ci + εi )

c12 +c22 +ε ∗ i 2 c +c 2 +ε ∗ +c

ci + εi − (ci + ε )

≤

c12

+

c22

The proof is thus completed.

∗

1

2

2



B.1. Proof of Lemma 3 Proof. Since M is Hurwitz, there indeed exists a positive definite matrix P such that M T P + PM ≤ −In2 . Then by Lemma 6, V (χ1 , χ2 ) is positive definite defined on Ωc1 +ε1 (U0 (χ1 )) × Rn2 . Note that ϕ2 (χ1 , v, w) is sufficiently smooth with ϕ2 (0, v, w) = 0. Then by Lemma 2, there exists L1 > 0 such that for any ¯ c1 +ε1 (U0 (χ1 )), ∥P ϕ2 (χ1 , v, w)∥ ≤ L1 ∥χ1 ∥. Let U1 (χ2 ) = χ1 ∈ Ω T χ2 P χ2 . Then for all col(v, w) ∈ V × W and all col(χ1 , χ2 ) ∈ ¯ c1 +ε1 (U0 (χ1 )) × Rn2 , Ω U˙ 1 (χ2 )(14) ≤ −∥χ2 ∥2 + 2∥χ2 ∥ · ∥P ϕ2 (χ1 , v, w)∥



≤ −∥χ2 ∥2 + 2L1 ∥χ2 ∥ · ∥χ1 ∥ ≤ − ∥χ2 ∥ + Let ν =

α c1 4L21 (c1 +ε1 )

W (χ1 , χ2 , w) =

W (χ1 , χ2 ) = W (χ1 , χ2 ) =

2L21

c1 V (χ1 , w) c1 + ε1 − V (χ1 , w)

+

c2 U (χ2 , w) c2 + ε2 − U (χ2 , w)

,

c1 V (χ1 ) c1 + ε1 − V (χ1 )

∥χ1 ∥ .

, c2 be a positive real number such that Ξ2

c1 + ε1 − V (χ1 )

+

c2 λmin ∥χ2 ∥2 c2 + ε2 − λmin ∥χ2 ∥2 c2 λmax ∥χ2 ∥2 c2 + ε2 − λmax ∥χ2 ∥2

, .

2

(0, s2 ). Therefore, we have for any (χ1 , χ2 ) ∈ Ωc1 +ε1 (V (χ1 )) × Ωc2 +ε2 (λmax ∥χ2 ∥2 ), W (χ1 , χ2 ) ≤ W (χ1 , χ2 , w) ≤ W (χ1 , χ2 ). c12

∗

(B.3)

c22 .

= + Then by the first part of Lemma 7, for any ¯ c ∗ +ε∗ (W (χ1 , χ2 )) ⊆ Ωc1 +ε1 (V (χ1 )) × Ωc2 +ε2 ε ∗ ≥ 1, we have Ω ¯ c1 (V (χ1 ))× (λmax ∥χ2 ∥2 ). Thus, (B.3) yields (19). Since Ξ1 ×Ξ2 ⊆ Ω ¯ c2 (λmax ∥χ2 ∥2 ), again by the first part of Lemma 7, we have Ξ1 × Ω ¯ c ∗ (W (χ1 , χ2 )). Ξ2 ⊆ Ω On the other hand, by the second part of Lemma 7, for all ¯ c ∗ +ε∗ (W (χ1 , χ2 )), the col(v, w) ∈ V × W, and all col(χ1 , χ2 ) ∈ Ω derivative of W (χ1 , χ2 , w) along the closed-loop system (16) satisfies

˙ (χ1 , χ2 , w) = W (16) 

c1 (c1 + ε1 ) ∂V [Z (χ1 , v, w) (c1 + ε1 − V (χ1 , w))2 ∂χ1

+ G1 (χ1 , τ (χ2 , ρ), v, w)]

2

¯ c2 (ν U1 (χ2 )), and c ∗ = c12 + c2 . Note that Ξ1 × Ξ2 ⊆ ⊆ Ω ¯ ¯ c2 (ν U1 (χ2 )). Then by the first part of Lemma 6, Ωc1 (U0 (χ1 )) × Ω ¯ c ∗ (V (χ1 , χ2 )) and Ω ¯ c ∗ +ε∗ (V (χ1 , χ2 )) ⊆ we have Ξ1 × Ξ2 ⊆ Ω ¯ c ∗ +ε∗ Ωc1 +ε1 (U0 (χ1 )) × Rn2 . Therefore, for any col(χ1 , χ2 ) ∈ Ω (V (χ1 , χ2 )), inequality (15) is still valid. Thus, by the second part

c1 V (χ1 )

+

Then W (χ1 , χ2 ) and W (χ1 , χ2 ) are both positive definite on the open set Ωc1 +ε1 (V (χ1 )) × Ωc2 +ε2 (λmax ∥χ2 ∥2 ). Note that for any s t s1 > 0 and s2 > 0, the function s 1−t is non-decreasing over

Let c

2



Proof. Since Y (w) is Hurwitz for any w ∈ W, there exists a positive definite matrix P (w) that depends on w , such that Y (w)T P (w) + P (w)Y (w) = −I. Let U (χ2 , w) = χ2T P (w)χ2 . Then λ(P (w)) ∥χ2 ∥2 ≤ U (χ2 , w) ≤ λ(P (w))∥χ2 ∥2 , where λ(P (w)) and λ(P (w)) denote the minimum and maximum eigenvalues of P (w), respectively. Since Y (w) and hence P (w) is continuous with respect to w over the compact set W, λmin ≤ λ(P (w)) ≤ λ(P (w)) ≤ λmax for some positive numbers λmin and λmax . Choose c2 large enough so ¯ c2 (λmax ∥χ2 ∥2 ). Let ε2 ≥ 1. Then we define the functhat Ξ2 ⊆ Ω tion

i

2

We are now ready to prove Lemmas 3 and 4.

2

(13)+(14)

ν α c1 (c1 + ε1 ) ∥χ1 ∥2 − ∥χ2 ∥2 + 2ν L21 ∥χ1 ∥2 ≤− 2 (c1 + ε1 − U0 (χ1 )) 2 α c1 ν 2 2 ≤− ∥χ1 ∥ − ∥χ2 ∥ ≤ −β∥(χ1 , χ2 )∥2 2(c1 + ε1 ) 2   α c1 ν where β = min 2(c +ε ) , 2 . The proof is thus completed. 1 1

on the open set Ωc1 +ε1 (V (χ1 )) × Ωc2 +ε2 (λmax ∥χ2 ∥2 ). Let

Remark 9. It is noted that when ε1 = ε2 = ε ∗ = 1 in (B.1), such a type of Lyapunov function was explored in Teel and Praly (1995), and is widely used in semi-global backstepping control design.

1

¯ c ∗ +ε∗ (V (χ1 , χ2 )), Ω  V˙ (χ1 , χ2 )

B.2. Proof of Lemma 4

¯ c 2 +c 2 (W (χ1 , χ2 )). On ¯ c2 (V2 (χ2 )) ⊆ Ω ¯ c1 (V1 (χ1 )) × Ω Therefore, Ω 1 2 ∗ ¯ c 2 +c 2 +ε∗ the other hand, for any ε ≥ 1 and any col(χ1 , χ2 ) ∈ Ω (W (χ1 , χ2 )), we have

1063

+

c2 (c2 + ε2 ) 2χ T P (w)[ρ Y (w)χ2 (c2 + ε2 − U (χ2 , w))2 2

+ G2 (χ1 , τ (χ2 , ρ), v, w)] ≤−

α c1 (c1 + ε1 ) ∥χ1 ∥2 (c1 + ε1 − V (χ1 ))2

−ρ

c2 (c2 + ε2 )

(c2 + ε2 − λmin ∥χ2 ∥2 )2

∥χ2 ∥2

1064

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065

    ∂V c1 (c1 + ε1 )  G1 (χ1 , τ (χ2 , ρ), v, w) +  2 ∂χ (c1 + ε1 − V (χ1 )) 1  T  2c2 (c2 + ε2 ) χ P (w)G2 (χ1 , τ (χ2 , ρ), v, w) + 2 2 2 (c2 + ε2 − λmax ∥χ2 ∥ ) α c1 ρ c2 ≤− ∥χ1 ∥2 − ∥χ2 ∥2 c1 + ε1 c2 + ε2    (c 2 + c22 + ε ∗ + c1 )2   ∂ V  · ∥G1 (χ1 , τ (χ2 , ρ), v, w)∥ + 1  c1 (c1 + ε1 ) ∂χ1  +

2(c12 + c22 + ε ∗ + c2 )2 c2 (c2 + ε2 )

totically stable with its domain of attraction containing Ξ1 × Ξ2 . The proof is thus completed. 

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∥χ2 ∥

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· ∥P (w)G2 (χ1 , τ (χ2 , ρ), v, w)∥ . ∂V (χ1 , w) Since V (χ1 , w) is a C 2 function, by (17) and (18), ∂χ 1 ∂V 1 is a C function satisfying ∂χ (0, w) = 0. Meanwhile, G1 (χ1 , 1 τ (χ2 , ρ), v, w), and G2 (χ1 , τ (χ2 , ρ), v, w) are sufficiently smooth functions satisfying G1 (χ1 , 0, v, w) = 0 and G2 (0, 0, v, w) = 0. By Lemma 2, for any col(v, w) ∈ V × W, any (χ1 , χ2 ) ∈ ¯ c ∗ +ε∗ (W (χ1 , χ2 )) and any Ω  ρ ≥ ρ0 , there exist positive numbers ∂V Lk , k = 1, 2, 3, such that  ∂χ  ≤ L1 ∥χ1 ∥ and





1

∥P (w)G2 (χ1 , τ (χ2 , ρ), v, w)∥ ≤ L3 (∥χ1 ∥ + ∥τ (χ2 , ρ)∥) ≤ L3 (∥χ1 ∥ + ∥χ2 ∥) . Therefore, for all col(v, w) ∈ V × W, and all col(χ1 , χ2 ) ∈ ¯ c ∗ +ε∗ (W (χ1 , χ2 )), Ω

 ˙ (χ1 , χ2 , w) W (16)

where

C3 =

c1 + ε1 2(

c12

+

> 0, c22

C2 =

+ ε + c2 ) ∗

c2 (c2 + ε2 )

2

c2 c2 + ε2

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(c12 + c22 + ε ∗ + c1 )2 (c 2 + c22 + ε ∗ + c2 )2 L1 L2 + 1 L3 C4 = 2c1 (c1 + ε1 ) c2 (c2 + ε2 ) > 0, and ν > 0 is small enough so that C1 − ν C4 > 0. Let ρ = ∗

Ren, W., & Beard, R. W. (2008). Communications and control engineering series, Distributed consensus in multi-vehicle cooperative control. London: SpringerVerlag. C C3 + ν4 C2 C4

.

For any ρ ≥ max{ρ ∗ , ρ0 }, let β = min{C1 − ν C4 , ρ C2 − C3 − ν }. Then β > 0, and



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> 0,

L3 > 0,

˙ (χ1 , χ2 , w) ≤ −β∥(χ1 , χ2 )∥2 . W (16)

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≤ −C1 ∥χ1 ∥2 − (ρ C2 − C3 )∥χ2 ∥2 + 2C4 ∥χ1 ∥ · ∥χ2 ∥   C4 2 ≤ −(C1 − ν C4 )∥χ1 ∥ − ρ C2 − C3 − ∥χ2 ∥2 ν c1 α

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∥G1 (χ1 , τ (χ2 , ρ), v, w)∥ ≤ L2 ∥τ (χ2 , ρ)∥ ≤ L2 ∥χ2 ∥

C1 =

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(B.4)

Since for any (χ1 , χ2 ) ∈ Ξ1 × Ξ2 , by the choice of c1 and c2 , we ¯ c2 (λmax ∥χ2 ∥2 ). It is noted that ¯ c1 (V (χ1 )) × Ω have (χ1 , χ2 ) ∈ Ω 2 ¯ c1 (V (χ1 )) × Ω ¯ c2 (λmax ∥χ2 ∥ ) ⊆ Ω ¯ c ∗ (W (χ1 , χ2 )) ⊆ Ω ¯ c ∗ (W (χ1 , Ω ¯ c ∗ (W (χ1 , χ2 )). Now if ε∗ ≥ max(χ ,χ )∈Ω¯ ∗ (W (χ ,χ )) χ2 , w)) ⊆ Ω 1 2 1 2 c

¯ c ∗ (W (χ1 , χ2 )) ⊆ {|W (χ1 , χ2 ) − W (χ1 , χ2 )|, 1}, we have Ω ¯ c ∗ +ε∗ (W (χ1 , χ2 )). Then by (B.4), for any initial condition in Ξ1 × Ω ¯ c ∗ +ε∗ (W (χ1 , χ2 )). Again by Ξ2 , the trajectory remains in the set Ω (B.3) and (B.4), this trajectory converges to (0, 0) asymptotically as t → ∞. Thus, the equilibrium (χ1 , χ2 ) = (0, 0) is locally asymp-

Serrani, A., Isidori, A., & Marconi, L. (2001). Semiglobal nonlinear output regulation with adaptive internal model. IEEE Transactions on Automatic Control, 46(8), 1178–1194. Song, Q., Cao, J., & Yu, W. (2010). Second-order leader-following consensus of nonlinear multi-agents via pinning control. Systems & Control Letters, 59(9), 553–562. Su, Y., & Huang, J. (2012). Global robust output regulation for nonlinear multi-agent systems in strict feedback form. In Proc. 12th international conference on control, automation, robotics and vision, ICARCV (pp. 436–441). Guangzhou, China, 5–7 December. Su, Y., & Huang, J. (2013). Cooperative robust output regulation of a class of heterogeneous linear uncertain multi-agent systems. International Journal of Robust and Nonlinear Control. Published online http://dx.doi.org/10.1002/rnc.3027. Teel, A., & Praly, L. (1995). Tools for semiglobal stabilization by partial state and output feedback. SIAM Journal on Control and Optimization, 33(5), 1443–1488. Wang, X., Hong, Y., Huang, J., & Jiang, Z. (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. Yang, T., Stoorvogel, A.A., Grip, H.F., & Saberi, A. (2012). Semi-global regulation of output synchronization for heterogeneous networks of non-introspective, invertible agents subject to actuator saturation. In Proc. 51st IEEE conference on decision and control (pp. 5298–5303). Maui, Hawaii, USA, 10–13 December.

Y. Su, J. Huang / Automatica 50 (2014) 1053–1065 Youfeng Su received the B.S. degree in 2005 and the M.S. degree in 2008, both from East China Normal University, Shanghai, China, and the Ph.D. degree in 2012 from The Chinese University of Hong Kong, Hong Kong, China. From May 2012 to June 2013, he was a Postdoctoral Fellow at The Chinese University of Hong Kong. He is currently with the College of Mathematics and Computer Science, Fuzhou University, Fuzhou, China. His research interests include cooperative control, multi-agent systems, output regulation, and switched systems.

1065 Jie Huang is a Professor and Chairman with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China. His research interests include nonlinear control theory and applications, multi-agent systems, and flight guidance and control. Dr. Huang is a Fellow of IEEE, and a Fellow of IFAC.