Leader-following attitude consensus of multiple rigid body systems subject to jointly connected switching networks

Automatica 92 (2018) 63–71

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Leader-following attitude consensus of multiple rigid body systems subject to jointly connected switching networks✩ Tao Liu, Jie Huang * Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

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Article history: Received 8 May 2017 Received in revised form 8 September 2017 Accepted 28 January 2018

Keywords: Leader-following consensus Multi-agent systems Switched systems Nonlinear distributed observer

a b s t r a c t The leader-following attitude consensus problem of multiple rigid body systems has been studied by the distributed observer approach. The key assumption in the existing results is that the communication network among the rigid body systems is static and connected. Nevertheless, this assumption is undesirable since, typically, the communication network is time-varying and disconnected from time to time due to changes of the environment or failures of some subsystems. In this paper, we will further study the leader-following attitude consensus problem of multiple rigid body systems subject to a jointly connected switching communication network. This new problem is more challenging than the existing one since a jointly connected switching communication network can be disconnected at every time instant. To overcome the difficulty, we first show that the distributed observer for a nonlinear target system subject to a jointly connected switching communication network exists. Then, we further synthesize a distributed control law utilizing this distributed observer for the multiple rigid body systems. Finally, we show that this distributed control law solves our problem through the argument of the certainty equivalence principle. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The attitude control of rigid body systems has received constant attention not only because many space missions and robot applications demand precise attitude control, but also because the problem poses some specific challenges to control theory and technology. The problem has been studied under various scenarios with a variety of techniques in, say, Chen & Huang (2009, 2015), Luo, Chu, & Ling (2005), Sidi (1997), Tayebi (2008), and Yuan (1988). Recently, as more and more space missions are performed through coordinated operations of multiple spacecraft systems, the attitude consensus problem of multiple rigid body systems is getting more and more attentions from the control community. There are two types of attitude consensus problems for multiple rigid body systems: the leaderless attitude consensus problem and the leader-following attitude consensus problem. The leaderless ✩ This work has been supported by the Research Grants Council of Hong Kong, China, under Grant No. 14219516 and by Projects of Major International (Regional) Joint Research Program NSFC under Grant No. 61720106011. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michael M. Zavlanos under the direction of Editor Christos G. Cassandras. Corresponding author. E-mail addresses: [email protected] (T. Liu), [email protected] (J. Huang).

*

https://doi.org/10.1016/j.automatica.2018.02.012 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

attitude consensus problem is also called the attitude synchronization problem. It aims to synchronize the attitudes of all rigid body systems to a common trajectory, which is determined by the initial states of all systems (Abdessameud & Tayebi, 2009; Lawton & Beard, 2002). On the other hand, the leader-following attitude consensus problem aims to drive the attitudes of all rigid body systems to a desirable trajectory generated by a target system (Bai, Arcak, & Wen, 2008; Cai & Huang, 2014, 2016; Ren, 2007). In particular, Cai & Huang (2014, 2016) studied the leader-following attitude consensus problem of multiple rigid body systems by employing a socalled distributed observer, which is a dynamic compensator that can provide for each rigid body system the estimates of the angular velocity and the attitude of the target system. The results in Cai and Huang (2014, 2016) were obtained under the condition that the communication network of the multiple rigid body systems is static and connected, which is the least restrictive condition in the existing literature. Nevertheless, in some real applications, the assumption that the communication network is static and connected may be undesirable since, typically, the communication network is time-varying and disconnected from time to time due to changes of the environment, link failures, or network reconfigurations. In this paper, we will further study the more practical and more desirable scenario where the communication network is a jointly connected switching network (Jadbabaie, Lin, & Morse, 2003). The jointly connected assumption is the mildest assumption among all existing assumptions on the communication network because,

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T. Liu, J. Huang / Automatica 92 (2018) 63–71

Nomenclature

Rn 1N

⊗ ∥·∥ col(·)

n-dimensional Euclidean space An N dimensional column vector whose components are all 1 Kronecker product Euclidean norm of a vector or induced Euclidean norm of a matrix For Xi ∈ Rni ×p , i = 1, . . . , m

]T

col(X1 , . . . , Xm ) = X1T · · · XmT For x ∈ R3 , q(x) = col (x, 0) ∈ R4 Set of all quaternions Q = {q | q = col(qˆ , q¯ ), qˆ ∈ R3 , q¯ ∈ R} Set of all unit quaternions Qu = {q | q ∈ Q, ∥q∥ = 1} Quaternion identity, qI = col(0, 0, 0, 1) ∈ Qu Quaternion conjugate, for q ∈ Q, q∗ = col(−ˆq, q¯ ) Quaternion inverse, for q ∈ Qu , q−1 = q∗ Quaternion [ product, for q]i , qj ∈ Q

[

q(·) Q

Qu qI (·)∗ (·)−1

⊙

qi ⊙ qj = C (·) (·)×

For q ∈ Q, C (q) = (q¯ 2 − qˆ T qˆ )I3 + 2qˆ qˆ T − 2q¯ qˆ × If q ∈ Qu , C (q) is the direction cosine matrix 3 For x = [ col(x1 , x2 , x3 ) ∈ ]R x× =

I B0 Bi q0 qi

ω0 ωi Ji ui S E

v ϵi ϵˆi ωˇ i ξi ζi ηi ei eˆ i

ω˜ i

q¯ i qˆ j + q¯ j qˆ i + qˆ × qˆ i j q¯ i q¯ j − qˆ Ti qˆ j

0

−x3

x2

x3

0

−x1

−x2

x1

0

∈ R3×3

Inertial frame Body frame of the target system Body frame of the ith rigid body The attitude of B0 relative to I The attitude of Bi relative to I The angular velocity of B0 relative to I The angular velocity of Bi relative to I Inertia matrix of the ith rigid body Control torque of the ith rigid body System matrix of the angular velocity target system Output matrix of the angular velocity target system State of the angular velocity target system Relative attitude between Bi and B0 1 ϵi = q− 0 ⊙ qi The vector part of ϵi , ϵi = col(ϵˆi , ϵ¯i ) Angular velocity of Bi relative to B0 ωˇ i = ωi − C (ϵi )ω0 Estimate of v Estimate of ω0 Estimate of q0 Estimate of ϵi , ei = ηi∗ ⊙ qi The vector part of ei , ei = col(eˆ i , e¯ i ) Estimate of ω ˇ i , ω˜ i = ωi − C (ei )ζi

under this assumption, the network can be disconnected at every time instant, and it includes the static and connected network as a special case. Nevertheless, the jointly connected switching network together with the nonlinearity of the target system poses two specific challenges that cannot be handled by the approach in Cai and Huang (2014, 2016). First, the establishment of the distributed observer in Cai and Huang (2014, 2016) explicitly relies on a Lyapunov function candidate for the distributed observer which exists only if the network is static and connected. Here, we will connect the stability of our distributed observer to

a newly established stability result for a perturbed linear switched system given recently in Liu and Huang (2017). Second, in order to overcome the nonlinearity of the target system, we will make use of a pseudo linear representation of the attitude kinematics of the target system. This new representation allows us to apply the result in Liu and Huang (2017) to our case. In particular, for the special case studied in Cai and Huang (2014, 2016), where the network is static and connected, our new approach will simplify the convergence analysis of the distributed observer in Cai and Huang (2014, 2016). It is noted that the leaderless consensus problem of rotation groups in SO(3) and SO(n) were studied in Matni and Horowitz (2014) and Tron, Afsari, and Vidal (2012), respectively. Instead of proposing distributed control laws to control the dynamics of each system, distributed algorithms for reaching consensus were proposed in Matni and Horowitz (2014) and Tron et al. (2012) by solving optimization problems. The remainder of the paper is organized as follows. In Section 2, we give a formulation of our problem and list two assumptions for the solvability of the problem. In Section 3, we focus on establishing the existence of the distributed observer subject to jointly connected switching networks. In Section 4, we further synthesize a distributed control law utilizing this distributed observer, and show that this distributed control law solves our problem through the argument of the certainty equivalence principle. In Section 5, an example is presented to illustrate the effectiveness of our approach. Finally, we conclude this paper in Section 6 with some remarks. 2. Problem formulation and preliminaries In this paper, we use unit quaternion to represent the attitude of a rigid body with respect to the inertial frame. As in Cai and Huang (2014), we consider a group of N rigid bodies, whose attitude kinematics and dynamics are governed by the following equations:

q˙ i =

1 2

qi ⊙ q(ωi )

Ji ω ˙ i = −ωi Ji ωi + ui , ×

(1a) i = 1, . . . , N

(1b)

where qi ∈ Qu is the unit quaternion representation of the attitude of the frame Bi relative to the inertial frame I ; ωi ∈ R3 is the angular velocity of the frame Bi relative to the inertial frame I ; Ji ∈ R3×3 is the positive definite inertia matrix and ui ∈ R3 is the control torque of the ith rigid body. Note that ωi , Ji , and ui are all expressed in Bi . Like in Cai and Huang (2016), we assume that the desired angular velocity ω0 ∈ R3 and attitude q0 ∈ Qu of the target system’s fixed body frame B0 relative to the inertial frame I are governed by the following equations:

v˙ = S v, q˙ 0 =

1 2

ω0 = E v

q0 ⊙ q(ω0 )

(2a) (2b)

where v ∈ Rm , and S ∈ Rm×m , E ∈ R3×m are constant matrices. Also, as in Cai and Huang (2014), we view the system composed of (1) and (2) as a multi-agent system of (N + 1) agents with (2) as the leader and the N subsystems of (1) as followers. However, here the communication network of the multi-agent ( ) system is described by a switching digraph1 G¯σ (t) = V¯ , E¯σ (t) with σ (t) being a piecewise constant switching signal, V¯ = {0, 1, . . . , N }, and E¯σ (t) ⊆ V¯ × V¯ for all t ≥ 0. Node 0 is associated with the leader system (2) and node i, i = 1, . . . , N, is associated with 1 See Appendix A for a summary of notation on digraph.

T. Liu, J. Huang / Automatica 92 (2018) 63–71

the ith follower system of (1). For i = 1, . . . , N , j = 0, 1, . . . , N, (j, i) ∈ E¯σ (t) if and only if agent i can use the information of agent j for control at time instant t. As a result, our control law has to satisfy the communication constraints described by the digraph G¯σ (t) . Such a control law is called a distributed control law. To describe such a control law, for i = 1, . . . , N, let N¯ i (t) denote the neighbor set of the ith follower at time t. Then, we will consider the distributed control law of the following form: ui = fi (qi , ωi , φi )

( ) φ˙ i = gi φi , φj − φi , j ∈ N¯ i (t) ,

i = 1, . . . , N

(3)

where φ0 = col(v, q0 ), and, for i = 1, . . . , N, fi (·), gi (·) are some nonlinear functions. The specific form of fi (·) and gi (·) are given in the right-hand side of (34) and (8), respectively. Thus, for i = 1, . . . , N, φi = col(ξi , ηi ). Remark 1. Like in Chen and Huang (2009), Sidi (1997), and Yuan (1988), the attitude and angular velocity errors, between each follower and the leader are defined as follows: 1 ϵi = q− 0 ⊙ qi ωˇ i = ωi − C (ϵi )ω0 ,

(4a) i = 1, . . . , N

(4b)

with ϵi = col(ϵˆi , ϵ¯i ) ∈ Qu and ω ˇ i ∈ R3 , whose kinematic and dynamic equations are described by

ϵ˙i =

1 2

ϵi ⊙ q(ωˇ i )

(5a)

˙ˇ i = −ω× Ji ωi + Ji ωˇ × C (ϵi )ω0 − C (ϵi )ω˙ 0 + ui . Ji ω i i

(

)

(5b)

Moreover, it can be deduced from Proposition 1 of Yuan (1988) that Bi , i = 1, . . . , N, and B0 coincide if and only if ϵˆi = 0, i = 1, . . . , N. Now, we are ready to describe our problem as follows. Problem 1. Given systems (1), (2), and a switching digraph G¯σ (t) , design a distributed control law of the form (3), such that for any initial conditions v (0), φi (0), ωi (0), i = 1, . . . , N, and qi (0) satisfying ∥qi (0)∥ = 1, i = 0, 1, . . . , N, the solution of the closedloop system exists and satisfies lim ϵˆi (t) = 0,

t →∞

lim ω ˇ i (t) = 0,

t →∞

i = 1, . . ., N .

(6)

Remark 3. As pointed out in Remark 2.2 of Cai and Huang (2016)

dt

(7)

Thus, ∥qi (0)∥ = 1 implies that ∥qi (t)∥ = 1 for all t ≥ 0. For the solvability of Problem 1, we need the following two assumptions. Assumption 1. All the eigenvalues of the matrix S are semi-simple with zero real parts. Assumption 2. There exists a subsequence {jk : k = 0, 1, . . .} of {j : j = 0, 1, . . .} with tjk+1 − tjk < T for some T > 0, such that every node i, i = 1, . . . , N, is reachable from node 0 in the union ⋃j −1 digraph rk=+j1 G¯σ (tr ) . k

By Remark 1, if the control ui of every follower can directly access q0 and ω0 , then for each i = 1, 2, . . . , N, one can design a control ui depending on ϵi and ω ˇ i using the approach of Chen and Huang (2009) to solve Problem 1. Such a control scheme is called the purely decentralized control scheme in Cai and Huang (2014) and will be described in Remark 10 later. However, if the leader’s signal cannot be accessed by every follower, then the purely decentralized control law is not in the form of (3). To overcome this difficulty, under Assumptions 1 and 2 with the digraph G¯σ (t) being static, a distributed observer of the form (8) was introduced to estimate the leader’s state q0 and ω0 in Cai and Huang (2014), hence leading to a so-called distributed observer based control law to solve Problem 1 in Cai and Huang (2014). Nevertheless, in reality, the communication network is usually time-varying, and may be disconnected from time to time due to changes of the environment, or failures of sensors and actuators in some of the spacecraft systems. Thus, it is interesting and desirable to further study the leader-following attitude consensus problem of multiple rigid body systems without assuming that the digraph G¯σ (t) is static. 3. The distributed observer subject to jointly connected switching networks The main objective of this section is to extend the distributed observer for the target system introduced in Cai and Huang (2014) from static networks to jointly connected switching networks. Let (N +1)×(N +1) denote the weighted adjacency A¯σ (t) = [aij (t)]N i,j=0 ∈ R matrix of the digraph G¯σ (t) . Then, for the ith follower, i = 1, . . . , N, of (1), we define a dynamic compensator as follows:

ξ˙i = S ξi + µ1

∑

aij (t)(ξj − ξi )

(8a)

η˙ i =

1 2

∑

ηi ⊙ q(ζi ) + µ2

aij (t)(ηj − ηi )

(8b)

¯ i (t) j∈N

where µ1 , µ2 > 0, ξi ∈ Rm , ηi ∈ Q, ζi = E ξi ∈ R3 , ξ0 = v , and η0 = q0 . The dynamic compensator (8) is called a distributed observer of the leader system (2) if, for any initial conditions ξi (0), ηi (0), i = 1, . . . , N, and any trajectory generated by the leader system (2), the solution of (8a) and (8b) is such that

= 2qˆ Ti q˙ˆ i + 2q¯ i q˙¯ i ¯ ˆT = qˆ Ti (q¯ i I3 + qˆ × i )ωi − qi qi ωi = 0, i = 0, 1, . . . , N .

Remark 4. Under Assumption 1, the angular velocity ω0 of the leader system (2) is bounded but can be step function of any magnitude, sinusoidal function of various magnitudes and frequencies, and their finitely many combinations. Assumption 2 is called the jointly connected condition (Jadbabaie et al., 2003) and is perhaps the mildest condition on a switching network as it allows the network to be disconnected at any time instant. It includes the static and connected network as a special case if σ (t) remains constant for all t ≥ 0.

¯ i (t) j∈N

Remark 2. By Lemma 10.1 and Remark 10.2 of Chen and Huang (2015), if ϵˆi = 0, then qi = ±q0 and C (ϵi ) = I3 , i = 1, . . . , N, where q0 and −q0 correspond to the same attitude.

d∥qi ∥2

65

lim (ξi (t) − v (t)) = 0,

t →∞

lim (ηi (t) − q0 (t)) = 0.

t →∞

(9)

Under Assumption 1 and the assumption that the digraph G¯σ (t) is static and connected, it is shown in Cai and Huang (2016) that (8) is indeed a distributed observer of the leader system (2). In this section, we will further show that, under Assumption 1, (8) is still a distributed observer of the leader system (2) even if the digraph G¯σ (t) is switching as long as it satisfies Assumption 2. For this purpose, let L¯σ (t) ∈ R(N +1)×(N +1) be the Laplacian of G¯σ (t) and let ξ = col(ξ1 , . . . , ξN ), ξˆ = ξ − 1N ⊗ v . Then, (8a) can be put into the following compact form:

( ) ξ˙ˆ = IN ⊗ S − µ1 (Hσ (t) ⊗ Im ) ξˆ

(10)

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T. Liu, J. Huang / Automatica 92 (2018) 63–71

where Hσ (t) ∈ RN ×N is the matrix consisting of the last N rows and the last N columns of L¯σ (t) . It is shown in Lemma 2 of Su and Huang (2012) that, under Assumptions 1 and 2, for any µ1 > 0 and for any initial condition ξˆ (0), the solution of system (10) is such that, for any v (t) generated by the leader system (2), limt →∞ ξˆ (t) = 0 exponentially. Thus, we have, for i = 1, . . . , N lim (ξi (t) − v (t)) = 0,

lim (ζi (t) − ω0 (t)) = 0

t →∞

t →∞

(11)

x˙ = Aσ (t) + Md (t) x + F (t)

2 1

block diag {M(ζ1 ), . . . , M(ζN )}

col (q0 ⊙ q(ζ1 − ω0 ), . . . , q0 ⊙ q(ζN − ω0 )) . 2 To ascertain the exponential convergence property of system (18), we further put it into the following form: F (t) =

x˙ =

1 2

)

IN ⊗ M(ω0 ) − µ2 (Hσ (t) ⊗ I4 ) x

( +

M(ζ ) −

1 2

)

IN ⊗ M(ω0 ) x + F (t)

(19)

which is in the form of (12) with

Lemma 1. Consider the following time-varying system:

)

1

M (ζ ) =

(

exponentially. Nevertheless, since (8b) is nonlinear, Lemma 2 of Su and Huang (2012) does not apply to (8b). We need to first introduce the following result.

(

where

( (12)

Aσ (t) =

1 2

)

IN ⊗ M(ω0 ) − µ2 (Hσ (t) ⊗ I4 )

(20)

where σ (t) is a piecewise constant switching signal; Aσ (t) is a switching matrix such that the origin of the following system

Md (t) =

x˙ = Aσ (t) x

by viewing ω0 as a bounded time function. Let us first note that, from (11)

(13)

is exponentially stable; Md (t) and F (t) are piecewise continuous and bounded over [0, ∞), and both Md (t) and F (t) converge to zero exponentially as t tends to infinity. Then, for any initial condition x(0), the solution of system (12) converges to zero exponentially. Remark 5. Lemma 1 can be extracted from the main result of Lemma 3.1 in Liu and Huang (2017). For convenience of the readers, we provide a self-contained proof in Appendix B.

(

Lemma 2. Under Assumptions 1 and 2, for any µ2 > 0 and for any initial conditions ηi (0), i = 1, . . . , N, the solution of (8b) is such that, for any q0 (t) generated by the leader system (2) lim (ηi (t) − q0 (t)) = 0

(14)

t →∞

exponentially.

=

1 2 1 2

1

ηi ⊙ q(ζi ) − q0 ⊙ q(ω0 ) + µ2 2

xi ⊙ q(ζi ) +

+ µ2

∑

1 2

∑

lim (ζi (t) − ω0 (t)) = 0,

⎢−y3 M(y) = ⎢ ⎣ y2 −y1 1 2

i = 1, . . ., N

(22)

exponentially. Also, we note that M(y) is skew symmetric and linear in y. Thus, both Md (t) and F (t) converge to zero exponentially as t tends to infinity. To make use of Lemma 1, it suffices to establish that, for any ω0 generated by (2a), the origin of the following system: x˙ =

1 2

)

IN ⊗ M(ω0 ) − µ2 (Hσ (t) ⊗ I4 ) x

(23)

is exponential stable. For this purpose, let Φ (τ , t) ∈ R4×4 be the state transition matrix of the system

ϕ˙ =

1 2

M(ω0 )ϕ

(24)

where ϕ ∈ R4 , and define z(t) = (IN ⊗ Φ (0, t))x(t). Then, we have z(0) = x(0), and along the trajectory of system (23)

1

¯ i (t) j∈ N

2

q0 ⊙ q(ζi − ω0 )

aij (t)(xj − xi ).

(15)

y3

−y2

0

y1

−y1 −y2

0

−y3

y1

⎤

y2 ⎥ ⎥ y3 ⎦

.

(16)

M(ζi )xi + µ2

∑

aij (t)(xj − xi ) +

¯ i (t) j∈N

(25)

By Corollary 4 of Su and Huang (2012), under Assumption 2, for any µ2 > 0, the origin of the linear switched system (25) is exponentially stable, i.e., there exist positive constants α and λ such that

∥z(t)∥ ≤ α∥z(0)∥e−λt = α∥x(0)∥e−λt ,

t ≥ 0.

(26)

d∥ϕ (t)∥2

1 2

q0 ⊙ q(ζi − ω0 ).

(17)

Letting x = col(x1 , . . . , xN ) and ζ = col(ζ1 , . . . , ζN ) gives x˙ = M(ζ )x − µ2 (Hσ (t) ⊗ I4 )x + F (t)

= −(IN ⊗ Φ (0, t))µ2 (Hσ (t) ⊗ I4 )x = −µ2 (Hσ (t) ⊗ I4 )(IN ⊗ Φ (0, t))x = −µ2 (Hσ (t) ⊗ I4 )z .

Then, a simple calculation shows that

0

Then, system (15) is equivalent to x˙ i =

(21)

= − (IN ⊗ Φ (0, t)M(ω0 )) x 2 (1 ) + (IN ⊗ Φ (0, t)) IN ⊗ M(ω0 ) − µ2 (Hσ (t) ⊗ I4 ) x

aij (t)(ηj − ηi )

To obtain a compact form of system (15), define a matrix function M(·) : R3 ↦ → R4×4 , such that for each y = col(y1 , y2 , y3 ) ∈ R3 0

IN ⊗ M(ω0 )

) ∂ z˙ = IN ⊗ Φ (0, t) x + (IN ⊗ Φ (0, t))x˙ |(23) ∂t

¯ i (t) j∈N

⎡

2

(

Proof. Let xi = ηi − q0 , i = 1, . . . , N. Then x˙ i =

1

t →∞

( Now, we are ready to complete the proof that the dynamic compensator (8) is indeed a distributed observer of the leader system (2) by the following lemma.

M(ζ ) −

)

= 2ϕ (t)ϕ˙ (t) = ϕ (t)M(ω0 )ϕ (t) = 0. (27) dt Thus, for any ϕ (0), ∥ϕ (t)∥ = ∥Φ (t , 0)ϕ (0)∥ = ∥ϕ (0)∥ for all t ≥ 0, and hence the matrix Φ (t , 0) is bounded for all t ≥ 0. As a result ∥x(t)∥ = ∥(IN ⊗ Φ (t , 0))z(t)∥ ≤ ∥(IN ⊗ Φ (t , 0))∥∥z(t)∥

(18)

≤ α∥ ˆ x(0)∥e−λt ,

t≥0

(28)

T. Liu, J. Huang / Automatica 92 (2018) 63–71

for some αˆ > 0. Thus, we have shown that the origin of the linear switched system (23) is exponentially stable. By Lemma 1, for any initial condition x(0), the solution of system (19) is such that limt →∞ x(t) = 0 exponentially. Consequently, for i = 1, . . . , N, limt →∞ (ηi (t) − q0 (t)) = 0 exponentially. □ Remark 6. Lemma 2 contains Lemma 3.2 of Cai and Huang (2016) as a special case where the communication network is static and connected. It is worth noting that, even for this special case, the proof here is much simpler than the proof of Lemma 3.2 of Cai and Huang (2016) since we have realized and made use of the relation xi ⊙ q(ζi ) = M(ζi )xi , i = 1, . . . , N. 4. Solvability of the problem The establishment of Lemma 2 allows us to make use of the certainty equivalence principle to synthesize our control law. Instead of using the error signals ϵi and ω ˇ i between each follower and the leader as defined in (4), we define the following estimates of the error signals as in Cai and Huang (2014): ei = ηi∗ ⊙ qi i = 1, . . . , N

(29b)

where ei = col(eˆ i , e¯ i ) ∈ Qu and ω ˜ i ∈ R3 . Next, for each follower, we will synthesize a control law ui that relies on ei and ω ˜ i . The collection of ui , i = 1, . . . , N , together with the distributed observer (8) will constitute the overall distributed control law. For this purpose, note that, as derived in detail in Appendix C, ei and ω ˜ i , i = 1, . . . , N , are governed by the following equations: e˙ i =

2

ei ⊙ q(ω ˜ i ) + edi

(30a)

˙˜ i = −ω× Ji ωi + Ji ω˜ × C (ei )ζi − C (ei )ES ξi Ji ω i i

(

)

− Jdi + ui

(30b)

where edi = col(eˆ di , e¯ di ) ∈ Q, Jdi ∈ R3 , and they are given by 1

− 1)q(ζi ) ⊙ ei + ηdi ⊙ qi 2 Jdi = Ji (Cdi ζi + C (ei )E ξdi )

edi =

(eTi ei

∗

(31b)

i

Remark 7. As stated in Remark 4.1 of Cai and Huang (2014), the estimated error signal ei , i = 1, . . . , N, defined in (29a) has the property that ∥ei (t)∥ = ∥ηi (t)∥ for all t ≥ 0. By Lemma 2, limt →∞ ∥ηi (t)∥ = ∥q0 (t)∥ = 1 exponentially. It follows that limt →∞ ∥ei (t)∥ = 1 exponentially, and hence it is bounded. Consequently, ηdi , edi , Cdi , ξdi , and Jdi all tend to zero exponentially. As in Cai and Huang (2014), Chen and Huang (2009), and Luo et al. (2005), we further perform on system (30) the following transformation: zi = ω ˜ i + ki1 eˆ i ,

i = 1, . . . , N

(32)

where ki1 is a positive constant. Then, system (30) becomes 2

¯ ˆ ˆ (eˆ × i + ei I3 )(zi − ki1 ei ) + edi

(33a)

1 e˙¯ i = − eˆ Ti (zi − ki1 eˆ i ) + e¯ di 2 ( Ji z˙i = −ωi× Ji ωi + Ji (zi − ki1 eˆ i )× C (ei )ζi 1

¯ ˆ − C (ei )ES ξi + ki1 (eˆ × i + ei I3 )(zi − ki1 ei )

(33b)

+

1 2

¯ ˆ ki1 (eˆ × i + ei I3 )(zi − ki1 ei ) − ki2 zi

)

(34)

where ki2 is a positive constant. The proof of our main result needs to use Lemma 4.1 of Cai and Huang (2014), which is rephrased as follows. Lemma 3. Consider systems (33a) and (33b). For any piecewise continuous time function zi (t), i = 1, . . . , N, defined for all t ≥ 0 satisfying limt →∞ zi (t) = 0, the solution of the ith subsystem is bounded for all t ≥ 0 and limt →∞ eˆ i (t) = 0, i = 1, . . . , N. Remark 8. It is noted that a special case of Lemma 3 where edi , i = 1, . . . , N, are identically zero and N = 1 was proved in Lemma 3.1 of Chen and Huang (2009).

− Jdi + ui

Theorem 1. Given systems (1), (2), and a switching digraph G¯σ (t) , under Assumptions 1 and 2, Problem 1 is solvable by a distributed control law composed of (8) and (34). Proof. By the definitions of ϵi and ei in (4a) and (29a), respectively, we have, for i = 1, . . . , N 1 ei − ϵi = ηi∗ ⊙ qi − q− 0 ⊙ qi

= (ηi − q0 )∗ ⊙ qi .

(35)

By Lemma 2, under Assumptions 1 and 2, limt →∞ (ηi (t) − q0 (t)) = 0 exponentially. Thus lim (ei (t) − ϵi (t)) = 0

t →∞

(36)

Ji z˙i = −ki2 zi − Jdi′ ,

i = 1, . . . , N .

(37)

Since Ji is positive definite, ki2 > 0, and limt →∞ Jdi′ = 0 exponentially, system (37) can be viewed as strictly stable linear systems with exponentially decaying disturbances. Thus, we have limt →∞ zi (t) = 0, i = 1, . . . , N. Then, as a result of Lemma 3 and (36) lim ϵˆi (t) = lim eˆ i (t) = 0.

t →∞

t →∞

(38)

Hence, the leader-following consensus of the attitudes of the multiple rigid body systems is achieved. On the other hand, by (32) lim ω ˜ i (t) = 0.

t →∞

(39)

From (4b) and (29b), we have

ωˇ i = ωi − C (ϵi )ω0 = ω˜ i + C (ei )ζi − C (ϵi )ω0 = ω˜ i + C (ei )(ζi − ω0 ) ( ) + C (ei ) − C (ϵi ) ω0 .

(40)

Thus, by (11) and (39) lim ω ˇ i (t) = 0

)

t →∞

2

′

(

(31a)

∑

1

ui = ωi× Ji ωi − Ji (zi − ki1 eˆ i )× C (ei )ζi − C (ei )ES ξi

exponentially. Under the control law (34), system (33c) becomes

with ηdi = µ2 j∈N¯ (t) aij (t)(ηj −ηi ), Cdi = 2(e¯ i e¯ di −ˆeTi eˆ di )I3 +2(eˆ i eˆ Tdi + i ∑ ¯ ˆ× eˆ di eˆ Ti − e¯ i eˆ × ¯ (t) aij (t)(ξj − ξi ). di − edi ei ), and ξdi = µ1 j∈N

e˙ˆ i =

where Jdi′ = Jdi − ki1 Ji eˆ di . By Remark 7, eˆ di , e¯ di , and Jdi′ all tend to zero exponentially. We are now ready to present our control law. For i = 1, . . . , N, let

(29a)

ω˜ i = ωi − C (ei )ζi ,

1

67

(33c)

which completes the proof. □

(41)

68

T. Liu, J. Huang / Automatica 92 (2018) 63–71

Remark 9. It can be seen that the overall distributed control law consists of (8) and (34), which is in the form of (3) with φi = col(ξi , ηi ), i = 1, . . . , N. Remark 10. If the control ui , i = 1, . . . , N, of every follower can directly access q0 , ω0 , and ω ˙ 0 , then we can obtain a purely decentralized control law as follows: ui = ωi× Ji ωi − Ji (zi − ki1 ϵˆi )× C (ϵi )ω0 − C (ϵi )ω ˙0

(

+

1 2

ki1 (ϵˆi× + ϵ¯i I3 )(zi − ki1 ϵˆi ) − ki2 zi

)

(42)

(a) G¯1 .

(b) G¯2 .

(c) G¯3 .

(d) G¯4 .

where zi = ω ˇ i + ki1 ϵˆi . It can be seen that the distributed control law (34) is obtained from the purely decentralized control law (42) by replacing ϵi , ω ˇ i in (42) with their estimates ei , ω˜ i , and ω0 , ω˙ 0 with their estimates ζi , ES ξi . It is in this sense that we say our control law is an application of the certainty equivalence principle.

5. An example We consider a group of four rigid body systems described by (1) with inertial matrices given by Ji = diag {i, i + 2, 2i} , i = 1, 2, 3, 4. Suppose the desired angular velocity takes the form

ω0 (t) = col (A1 cos(t + θ1 ), A2 cos(t + θ2 ), A3 )

Fig. 1. Switching topology G¯σ (t) with P = {1, 2, 3, 4}.

(43)

for arbitrary magnitudes A1 , A2 , A3 , and arbitrary initial phases θ1 , θ2 . Then, this class of signals can be generated by the leader system of the form (2) with

⎡

0

S = ⎣−1 0

1

0

⎤

0

0⎦ ,

0

0

E = I3

(44)

and an appropriate initial condition v (0). Clearly, the matrix S satisfies Assumption 1. We assume that the switching digraph G¯σ (t) is dictated by the following switching signal:

σ (t) =

⎧ ⎪ ⎪ ⎪1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2, ⎪ ⎪ 3, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩4,

if

sT0 ≤ t < (s +

if

(s +

if

(s +

if

(s +

1 4 1 2 3 4

1 4

)T0

)T0 ≤ t < (s + )T0 ≤ t < (s +

1 2 3 4

)T0 (45) )T0

)T0 ≤ t < (s + 1)T0

where T0 = 2 and s = 0, 1, 2, . . .. The four digraphs G¯i , i = 1, 2, 3, 4 are described in Fig. 1, where node 0 is associated with the leader system and the other four nodes are associated with the four followers. It can be seen that Assumption 2 is satisfied even though G¯σ (t) is disconnected for all t ≥ 0. By Theorem 1, we can design a distributed control law composed of (8) and (34) with the following design parameters: µ1 = µ2 = 10, ki1 = ki2 = 20, i = 1, 2, 3, 4. Simulation of the closed-loop system is performed with q0 (0) = [1, 0, 0, 0]T , q1 (0) = [0, 1, 0, 0]T , q2 (0) = [0, 0, 1, 0]T , q3 (0) = [ ] [0, 0, 0, 1]T , q4 (0) =

1/2,

√

3/2, 0, 0

T

, v (0) = 13 , and other

randomly generated initial conditions. We let aij (t) = 1 whenever (j, i) ∈ E¯σ (t) . The estimation errors of the leader’s angular velocity ω0 and attitude q0 by the distributed observer are shown in Figs. 2 and 3, respectively. Figs. 4 and 5 further show the consensus errors of attitudes and angular velocities, respectively. Satisfactory performance of the leader-following consensus is observed for both attitudes and angular velocities.

Fig. 2. Estimation errors of the leader’s angular velocity.

6. Conclusions The leader-following attitude consensus problem of multiple rigid body systems with the attitude being represented by the unit quaternion has been studied by a distributed observer approach in Cai and Huang (2014, 2016). The key assumption in the existing results is that the communication network among the rigid body systems is static and connected. Nevertheless, this assumption is undesirable since, typically, the communication network is timevarying and disconnected from time to time due to changes of the environment or failures of some subsystems. In this paper, we have further studied the same problem under the assumption that the communication network is jointly connected. We have first shown that the distributed observer for a nonlinear target system subject to a jointly connected switching communication network exists. Then, we have synthesized a distributed control law utilizing this

T. Liu, J. Huang / Automatica 92 (2018) 63–71

69

Fig. 5. Leader-following consensus of angular velocities. Fig. 3. Estimation errors of the leader’s attitude.

node i1 . Given a set of n0 digraphs ⋃{nG0 i = (V , Ei ), i = 1, . . . , n0 }, the digraph G = (V , E ) with E = i=1 Ei , is called the union of the ⋃ n digraphs Gi , and is denoted by G = i=0 1 Gi . The weighted adjacency matrix of a digraph G is a nonnegative matrix A = [aij ]Ni,j=1 ∈ RN ×N satisfying aii = 0 and aij > 0 if and only if (j, i) ∈ E , i, j = 1, . . . , N. The Laplacian of G is then defined as L = [lij ]Ni,j=1 ∈ RN ×N , where

∑N

Fig. 4. Leader-following consensus of attitudes.

lii = j=1 aij and lij = −aij for i ̸ = j. We call a time function σ : [0, ∞) ↦ → P = {1, 2, . . . , n0 }, where n0 is some positive integer, a piecewise constant switching signal if there exists a sequence {tj : j = 0, 1, 2, . . .} satisfying t0 = 0, tj+1 − tj ≥ τ for some positive constant τ , such that for all t ∈ [tj , tj+1 ), σ (t) = p for some p ∈ P . P is called the switching index set, tj is called the switching instant, and τ is called the dwell time. Given a piecewise constant switching signal σ (t) and a set of n0 digraphs Gi = (V , Ei ), i = 1, . . . , n0 , with the corresponding weighted adjacency matrices being Ai , we call the time-varying digraph Gσ (t) = (V , Eσ (t) ) a switching digraph, and denote the weighted adjacency matrix and the Laplacian of Gσ (t) by Aσ (t) and Lσ (t) , respectively. Appendix B. Proof of Lemma 1

distributed observer for the multiple rigid body systems. Finally, we have shown that this distributed control law solves our problem through the argument of the certainty equivalence principle. Acknowledgments We would like to thank the editors, and the anonymous reviewers for their careful reading of the manuscript and constructive comments.

Proof. Let Φ (τ , t) be the state transition matrix of system (13), i.e., it is the unique solution of the following matrix equation:

∂ Φ (τ , t) = −Φ (τ , t)Aσ (t) , ∂t

Φ (t , t) = I .

(B.1)

Let

∫ P(t) =

∞

Φ (τ , t)T Q Φ (τ , t)dτ

(B.2)

t

Appendix A. Notation on digraph A digraph G = (V , E ) consists of a finite set of nodes V = {1, . . . , N } and an edge set E ⊆ V × V . An edge of E from node i to node j is denoted by (i, j), and node i is called a neighbor of node j. Then, Ni = {j | (j, i) ∈ E } denotes the neighbor set of node i. If the digraph contains a sequence of edges of the form {(i1 , i2 ), (i2 , i3 ), . . . , (ik−1 , ik )}, then this set is called a directed path of G from node i1 to node ik , and node ik is said to be reachable from

where Q is some constant positive definite matrix. Clearly, P(t) is continuous for all t ≥ 0. Since the origin of the linear switched system (13) is exponentially stable, we have

∥Φ (τ , t)∥ ≤ α1 e−λ1 (τ −t) ,

τ ≥t≥0

(B.3)

for some α1 > 0 and λ1 > 0. Then, it can be verified that c1 ∥x∥2 ≤ xT P(t)x ≤ c2 ∥x∥2

(B.4)

70

T. Liu, J. Huang / Automatica 92 (2018) 63–71

for some positive constants c1 , c2 . Hence, P(t) is positive definite and bounded, and there exists some positive constant c3 such that ∥P(t)∥ ≤ c3 for all t ≥ 0. Then, for t ∈ [tj , tj+1 ), j = 0, 1, 2, . . .

∫

∞

T

In (8b), for i = 1, . . . , N, let

ηdi = µ2

=

+ 2xT P(t)Md (t)x + 2xT P(t)F (t) T

= −x Qx + 2x P(t)Md (t)x + 2x P(t)F (t) 2

=

2

≤ −λmin (Q )∥x∥ + 2c3 ∥Md (t)∥∥x∥ ∥P(t)∥2 + ∥x∥2 + ε∥F (t)∥2 ε ) (

≤ − λmin (Q ) − 2c3 ∥Md (t)∥ −

c32

ε

+ ε∥F (t)∥2

(B.6)

where λmin (Q ) denotes the smallest eigenvalue of the matrix Q . Let

ε > 0 be such that c4 := λmin (Q ) −

c32

ε

> 0. Then

˙ |(12) ≤ − (c4 − 2c3 ∥Md (t)∥) ∥x∥2 + ε∥F (t)∥2 . U(t)

(B.7)

Since Md (t) converges to zero exponentially, there exists some positive integer l, such that c5 := (c4 − 2c3 ∥Md (t)∥) > 0,

t ≥ tl .

(B.8)

Hence, we have

˙ |(12) ≤ −c5 ∥x∥2 + ε∥F (t)∥2 U(t) ≤ −λ2 U(t)|(12) + ε∥F (t)∥2 , c5 . c2

where λ2 =

t ≥ tl

(B.9)

e−λ2 (t −τ ) ε∥F (τ )∥2 dτ ,

t ≥ tl .

(B.10)

t ≥ tl .

(B.11)

Without loss of generality, we assume that 0 < λ3 < λ2 . Then t

−λ2 (t −τ )

e

ε∥F (τ )∥ dτ 2

∗

ei ⊙ q(ωi ) −

∗

2 −λ2 t λ3 tl

t

∫

e

1 2

(

1 2

e

(λ2 −λ3 )τ

ηi ⊙ qi ⊙ q(ωi ) ∗

1

e∗i ⊙ q(ζi ) ⊙ ei = q (C (ei )ζi ) .

(C.2)

(C.3)

C˙ (ei ) = 2e¯ i e˙¯ i I3 − 2eˆ Ti e˙ˆ i I3 + 2e˙ˆ i eˆ Ti ×

¯ ˙ˆ + 2eˆ i e˙ˆ i − 2e˙¯ i eˆ × i − 2ei ei = e¯ i (−ˆeTi ω˜ i + 2e¯ di )I3

− eˆ Ti (e¯ i ω˜ i + eˆ × ˜ i + 2eˆ di )I3 i ω + (e¯ i ω˜ i + eˆ × ˜ i + 2eˆ di )eˆ Ti i ω + eˆ i (e¯ i ω˜ i + eˆ × ˜ i + 2eˆ di )T i ω = −2e¯ i eˆ Ti ω˜ i I3 + e¯ i ω˜ i eˆ Ti + eˆ × ˜ i eˆ Ti i ω

dτ

ˆ T ˜ i eˆ × + e¯ i eˆ i ω˜ iT − eˆ i ω˜ iT eˆ × i + ei ω i

tl

) α3 ∥F (tl )∥2 ( −λ3 (t −tl ) e − e−λ2 (t −tl ) = λ2 − λ3 α3 ∥F (tl )∥2 −λ3 (t −tl ) ≤ e . λ2 − λ3

)

qi ⊙ q(ωi )

¯ ¯ ˜ i + eˆ × + (eˆ Ti ω˜ i − 2e¯ di )eˆ × ˜ i + 2eˆ di )× i − ei (ei ω i ω

tl

≤ α3 ∥F (tl )∥ e

q(ζi ) ⊙ ηi ⊙ qi + ηdi ⊙ qi + ∗

T

Since F (t) also converges to zero exponentially, there exist some positive constants α3 and λ3 such that

ε∥F (t)∥2 ≤ α3 e−λ3 (t −tl ) ∥F (tl )∥2 ,

⊙ qi + ηi∗ ⊙

we have

t tl

∫

2 1

)∗

C (ei ) = (e¯ 2i − eˆ Ti eˆ i )I3 + 2eˆ i eˆ Ti − 2e¯ i eˆ × i

U(t)|(12) ≤ e−λ2 (t −tl ) U(tl )|(12)

∫

2 1

ηi ⊙ q(ζi ) + ηdi

Next, since

Then, we have

+

1

∗ q(ζi ) ⊙ ei + ηdi ⊙ qi 2 2 ) 1( = ei ⊙ q(ωi ) − ei ⊙ e∗i ⊙ q(ζi ) ⊙ ei 2 ) 1( ∗ + ei ⊙ e∗i ⊙ q(ζi ) ⊙ ei − q(ζi ) ⊙ ei + ηdi ⊙ qi 2 ) ( 1 = ei ⊙ q(ωi ) − e∗i ⊙ q(ζi ) ⊙ ei 2 ) 1( ∗ + ⊙ qi ei ⊙ e∗i − qI ⊙ q(ζi ) ⊙ ei + ηdi 2 1 = ei ⊙ q(ωi − C (ei )ζi ) 2 1 ∗ + (eTi ei − 1)q(ζi ) ⊙ ei + ηdi ⊙ qi 2 1 = ei ⊙ q(ω˜ i ) + edi . 2 In the above derivation, we have used the identity

=

∥x∥2

(C.1)

e˙ i = η˙ i∗ ⊙ qi + ηi∗ ⊙ q˙ i

)

T

aij (t)(ηj − ηi ).

Then, we compute

(

˙ + ATσ (t) P(t) + P(t)Aσ (t) x P(t)

∑ ¯ i (t) j∈N

(B.5)

Next, let U(t) = xT P(t)x. Then, along the trajectory of system (12), for any t ∈ [tj , tj+1 ) with j = 0, 1, 2, . . .

T

(B.13)

Appendix C. Derivation of system (30)

= −P(t)Aσ (t) − ATσ (t) P(t) − Q .

˙ |(12) = x U(t)

t ≥ tl .

That is, limt →∞ U(t)|(12) = 0 exponentially. Since U(t)|(12) ≥ c1 ∥x(t)∥2 , limt →∞ x(t) = 0 exponentially. □

t

( T

and substituting (B.12) into (B.10) gives

U(t)|(12) ≤ e−λ2 (t −tl ) U(tl )|(12) + W (tl )e−λ3 (t −tl )

)

(

α3 ∥F (tl )∥2 λ2 −λ3

( ) ≤ U(tl )|(12) + W (tl ) e−λ3 (t −tl ) ,

∂ Φ (τ , t) dτ Φ (τ , t) Q ∂ t t ( ) ∫ ∞ T ∂ + Φ (τ , t) Q Φ (τ , t)dτ − Q ∂t ∫t ∞ =− Φ (τ , t)T Q Φ (τ , t)dτ Aσ (t) t ∫ ∞ Φ (τ , t)T Q Φ (τ , t)dτ − Q − ATσ (t)

˙ = P(t)

Letting W (tl ) =

− e¯ 2i ω˜ i× − e¯ i (eˆ × ˜ i )× + Cdi i ω = 2e¯ i (eˆ i ω˜ iT − eˆ Ti ω˜ i I3 ) − ω˜ i× eˆ i eˆ Ti − e¯ 2i ω˜ i× (B.12)

− (eˆ i ω˜ iT − eˆ Ti ω˜ i I3 )eˆ × i + Cdi ˆ× = 2e¯ i ω˜ i× eˆ × ˜ i× eˆ i eˆ Ti − e¯ 2i ω˜ i× − ω˜ i× eˆ × i −ω i ei + Cdi

(C.4)

T. Liu, J. Huang / Automatica 92 (2018) 63–71

= 2e¯ i ω˜ i× eˆ × ˜ i× eˆ i eˆ Ti − e¯ 2i ω˜ i× i −ω − ω˜ i (eˆ i eˆ Ti − eˆ Ti eˆ i I3 ) + Cdi ) ( + Cdi = −ω˜ i× (e¯ 2i − eˆ Ti eˆ i )I3 + 2eˆ i eˆ Ti − 2e¯ i eˆ × i ×

= −ω˜ i× C (ei ) + Cdi .

(C.5)

In (8a), for i = 1, . . . , N, let

ξdi = µ1

∑

aij (t)(ξj − ξi ).

(C.6)

¯ i (t) j∈N

Then

˙˜ i = Ji ω˙ i − Ji C˙ (ei )ζi − Ji C (ei )ζ˙i Ji ω = −ωi× Ji ωi + Ji ω˜ i× C (ei )ζi − Ji Cdi ζi − Ji C (ei )E(S ξi + ξdi ) + ui ( ) = −ωi× Ji ωi + Ji ω˜ i× C (ei )ζi − C (ei )ES ξi − Jdi + ui .

(C.7)

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Tao Liu received the B.Eng. degree in automation from the University of Science and Technology of China, Hefei, China, in 2014. He is currently pursuing the Ph.D. degree with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China. His research interests include output regulation, cooperative control, and their applications to robotic systems and spacecraft systems.

Jie Huang is Choh-Ming Li professor and chairman of 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. He is a Fellow of IEEE, a Fellow of IFAC, and a Fellow of CAA.