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Formation-containment control for multi-agent systems with sampled data and time delays
Formation-containment control for multi-agent systems with sampled data and time delays
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Formation-containment control for multi-agent systems with sampled data and time delays Jinxin Zhang, Housheng Su PII: DOI: Reference:
S0925-2312(19)31628-5 https://doi.org/10.1016/j.neucom.2019.11.030 NEUCOM 21554
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
2 June 2019 20 September 2019 17 November 2019
Please cite this article as: Jinxin Zhang, Housheng Su, Formation-containment control for multi-agent systems with sampled data and time delays, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.11.030
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Formation-containment control for multi-agent systems with sampled data and time delays Jinxin Zhanga,b , Housheng Sub,∗ a School
of Science, Hunan University of Technology, Zhuzhou 412008, China of Artificial Intelligence and Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, China
b School
Abstract This paper investigates the formation containment control under sampling and with time delays for multi-agent systems (MASs). First, a novel sampling control protocol for the MASs to achieve formation containment is proposed, where the communication among agents is only occurred at the sampling instance. Then the communication load and the waste of energy can be decreased significantly. In addition, time delays are considered in this paper. Second, by establishing an appropriate Lyapunov function, sufficient conditions for MASs with sampled data and time delays reaching formation containment are obtained by handling the established Lyapunov function. Finally, the obtained results are demonstrated by a simulation example. Keywords: Multi-agent systems (MASs), formation containment, under sampling, time delays
1. Introduction Research on collaborative control of MASs has attracted more and more research workers and large numbers of research results have been obtained [1]-[3]. The collaborative control of MASs has been applied in robots, sensor networks, 5
unmanned aircrafts and many other fields [4]-[8]. In particular, formation con∗ Corresponding
author Email address: [email protected] (Housheng Su)
Preprint submitted to Journal of LATEX Templates
December 4, 2019
trol is an important aspect of the collaborative control of MASs. Formation control aims to design a control protocol which makes the agents achieve an expected shape. In [9] and [10], sufficient conditions were derived to make the MASs attain formation, in which only second-order dynamics were considered. 10
Formation control for high-order MASs were investigated in [11]-[13]. Group formation was studied in [14], where the agents were divided into some groups and each group achieved a specified formation. Many studies on MASs were with only one leader or without leaders. In [15][17], no leaders were considered and the MASs reached consensus. The problem
15
for collaborative control of MASs with one leader is regarded as coordinated tracking problem. The consensus problem with one leader was investigated in [18] where the dynamics of agents were second order. The high-order consensus with one leader was discussed in [19] and sufficient conditions ensuring consensus were obtained. However, some occasions require multiple leaders, thus,
20
containment problem arises. Containment control implies that followers are in the convex hull specified by the multiple leaders, which were discussed in [20][26]. Containment control problems for second-order MASs were investigated with finite-time observers [20] and sampled position data [21]. In [22]-[26], containment control problems were discussed for high-order and nonlinear MASs,
25
respectively. It should be pointed out that no communication was among leaders in [20]-[26]. Based on studies of formation and containment, formation containment problems arise which are more complex and require leaders achieve the designed formation and followers are stay in the formation. Formation containment has
30
been used in many engineering projects. For instance, the unmanned vehicles are denoted as leaders and the manned vehicles which are important are denoted as the followers. Leaders form a formation and followers stay in the formation. Thus, when facing foes, the followers can be protected by followers. In addition, formation problems and containment problems cannot directly combine into
35
formation containment problems, because the actions of leaders reaching formation effect followers. For formation containment problems, single-integrator 2
MASs were discussed in [27], double-integrator MASs were studied in [28], and high-order MASs were investigated in [29]. However, all the systems in [27]-[29] was continuous. For continuous-time systems, information transmission among 40
agents is continuous and the controller updates continuously, thus, much energy will be wasted. On some occasions, continuous information transmission cannot always be ensured. Therefore, sampling information for MASs is important and gains attention from researchers. Sampling strategy saves the energy and can be used in the situation where continuous information transmission cannot al-
45
ways be ensured. Sampling method for MASs to reach consensus was discussed in [30]-[34]. In [30], both position states and velocity states were obtained in sampling instances and conditions ensuring the MASs to achieve consensus were obtained. In [31] and [32], second-order consensus were achieved for MASs under sampling positions. In [33] and [34], the MASs with high-order dynamics
50
to reach consensus were investigated via sampling approach. To the best of our knowledge, sampling control problem for formation containment has not been fully discussed, which motivates us to do this work. This paper investigates formation containment of MASs based on sampling data. In addition, time delays are also considered. A sampling strategy for
55
formation containment is proposed. The controllers of both leaders and followers are under sampling. By establishing a suitable Lyapunov function, sufficient conditions for MASs with sampled data and time delays reaching formation containment are obtained. Main contributions are as follows. First, compared with the situation with one leader in [18] and [19], multiple leaders are considered in
60
this paper and the multiple leaders cause the analysis more complex. Second, in [20]-[26], there was no information transmission among leaders for containment problems, formation containment is investigated in this paper and communication among leaders exists. Different from containment control with no leader input, the controllers for leaders must be designed and formation functions are
65
also considered in the leader controllers. Thus, the analysis for the formation containment is more complicated. Third, the communication among agents of formation containment with sampling data and time delays is considered. In 3
[27]-[29], communication among agents for formation containment was continuous. Thus, the analysis methods in [27]-[29] cannot be used in this paper and the 70
Lyapunov function and handling methods in this paper are more complicated than those in [27]-[29]. In this paper, some preliminaries are shown in Section 2. Section 3 shows the problem transformation. Section 4 gives the results and the analysis. Section 5 shows the simulation example. Section 6 gives the conclusion. Notations: In this work, ⊗ is the Kronecker product. RP ×P means the order
75
of real matrices is P × P . The notation “∗” is regarded as the symmetric item in a symmetric matrix. The notation “T ” marks the transpose of a matrix. Other notations are routine.
2. Preliminaries 80
2.1. Graph theory The directed topology graph of P + Q agents can be depicted by G = ¯ Γ}, ˆ in which Γ = [κ1 , κ2 , ..., κP +Q ] denotes the group of nodes, Γ ¯ ⊂ Γ×Γ {Γ, Γ, ˆ = [aj ] ∈ R(P +Q)×(P +Q) is the weighted represents the group of edges and Γ
¯ adjacency matrix. $aj denotes an edge of G with a 6= j (a ∈ Γ, j ∈ Γ). $aj ∈ Γ
85
¯ means aj = 0. Any aj in Γ ˆ is nonnegative. means aj > 0 and $aj ∈ / Γ P +Q ¯ = [laj ] ∈ R(P +Q)×(P +Q) is the Laplacian matrix, where laa = P aj and L j=1,j6=a
laj = −aj (a 6= j).
2.2. Problem statement There are P + Q agents constituting of a MAS and the system is designed 90
as x ¯˙ v (t) = A¯ xv (t) + B u ¯v (t), v = 1, 2, ..., P + Q,
(1)
where x ¯v (t) ∈ Rn is the state. u ¯v (t) is the controller. Consider that P agents denote followers and the follower group is denoted as N = [1, 2, ..., P ]. Q agents denote leaders and the leader group is denoted as H = [P + 1, P + 2, ..., P + Q]. For the leader group, define the formation function as fH = 4
95
[fPT +1 , fPT +2 , ..., fPT +Q ]T ∈ RQn and for the bth agent with b ∈ H, fb is the formation vector. Definition 1. [29] For the MAS (1), the leaders achieve formation which is specified by fE¯ , when the following condition is attained, lim (¯ xb (t) − fb − g(t)) = 0,
t→∞
(2)
where b ∈ H and g(t) ∈ Rn is the formation reference function. 100
Remark 1. If fb = 0, we can see that the formation problem will be reduced to the consensus problem. Definition 2. [29] The MAS (1) achieves containment if the Eq.(3) is satisfied,
lim (¯ xa (t) −
t→∞
where a ∈ N , b ∈ H and 105
PP +Q
j=P +1
PX +Q
bj x ¯b (t)) = 0,
(3)
j=P +1
bj = 1 with the positive scalar bj (j ∈ H).
Definition 3. [29] If the requirements of Definition 1 and Definition 2 are both met, then MAS (1) achieves formation containment. ¯ for MAS (1) is defined as : The information topology G ¯1 ¯2 L L ¯= , L ¯3 0Q×P L
¯ is the Laplacian matrix, L ¯ 1 ∈ RP ×P , L ¯ 2 ∈ RP ×Q , L ¯ 3 ∈ RQ×Q . G ¯ H is where L
¯ 3 , and G ¯ N is the the information topology of the group H corresponding to L ¯1. information topology of the group F¯ corresponding to L 110
¯ H and G ¯ N are undirected and connected. Assumption 1. The topologies G Assumption 2. For every follower, there is at least one leader which has a directed path to it.
5
¯ 1 is positive, Lemma 1. [21] Based on Assumptions 1, 2, every eigenvalue of L −1 all entries of −L−1 1 L2 are nonnegative, and the sum of every row of −L1 L2 115
equals to one. Define tq as the sampling time where q is the non-negative integer. Define that τ¯ > 0 is the time delay and β = tq+1 − tq is the sampling period. Then, the controller of the group F¯ is designed as ¯1 u ¯a (t) =K
P X j=1
PX +Q
+
aj (¯ xa (tq − τ¯) − x ¯j (tq − τ¯))
b=P +1
(4)
ab (¯ xa (tq − τ¯) − x ¯b (tq − τ¯)) ,
and the controllers of the group H is designed as PX +Q
¯2 u ¯b (t) = K
j=P +1 120
bj ((¯ xb (tq − τ¯) − fb ) − (xj (tq − τ¯) − fj )) ,
(5)
¯ 1 represents the gain matrix of u ¯ 2 represents where a ∈ N , b ∈ H, K ¯a (t) and K the gain matrix of u ¯b (t). ¯ = t − tq + τ¯, where t ∈ [tq , tq+1 ). Then, the controllers (4) and Define d(t) (5) can be written as ¯1 u ¯a (t) =K
P X j=1
+
PX +Q
b=P +1
¯2 u ¯b (t) =K
¯ ¯ aj (¯ xa (t − d(t)) −x ¯j (t − d(t)) !
(6)
¯ ¯ ab (¯ xa (t − d(t)) −x ¯k (t − d(t))) (a ∈ N ),
PX +Q
j=P +1
¯ ¯ bj (¯ xk (t − d(t)) − fb ) − (¯ xj (t − d(t))
(7)
− fj ) (b ∈ H).
125
3. Problem Transformation
¯T2 (t), ..., x ¯TP (t)]T , x ¯H (t) = [¯ xTP +1 (t), x ¯TP +2 (t), ...¯ xTP +Q (t)]T , Let x ¯N (t) = [¯ xT1 (t), x one has ¯ ¯3 ⊗ BK ¯ 2 )(¯ x ¯˙ H (t) = (IQ ⊗ A)¯ xH (t) + (L xH (t − d(t)) − fH ), 6
(8)
¯ ¯1 ⊗ BK ¯ 1 )¯ x ¯˙ N (t) =(IP ⊗ A)¯ xN (t) + (L xN (t − d(t)) ¯ ¯2 ⊗ BK ¯ 1 )¯ + (L xH (t − d(t)) ,
(9)
¯ 3 M and the matrix M is nonsingular. Define the Jordan matrix JH = M −1 L ¯ 3 , where b ∈ H. Thus, 0 = cP +1 ≤ cP +2 ≤ Denote cb as the eigenvalues of L 130
· · · ≤ cP +Q .
Define ξb (t) = x ¯b (t) − fb and ξH (t) = [ξPT +1 (t), ξPT +2 (t), ..., ξPT +Q (t)]T . Then,
we get
and
ξ˙H (t) = (IQ ⊗ A)ξH (t) + (IQ ⊗ A)fH ¯ ¯3 ⊗ BK ¯ 2 )ξH (t − d(t)) + (L ¯ 3 M = JH = diag 0, J¯H , M −1 L
(10)
(11)
where J¯H =diag{cP +2 , cN +3 , ..., cP +Q }, M = [mTP +1 , mTP +2 , ..., mTP +Q ]T , M −1 =
135
¯ TP +Q ]T . ¯ TP +2 , ..., m [m ¯ TP +1 , m
T ¯ p+1 ⊗ In )ξE (t), and χH (t) = (t), χTH (t)]T , ηH (t) = (m Define ςH (t) = [ηH
ˆ ⊗ In )ξH (t) with M ˆ = [m (M ¯ TP +2 , m ¯ TP +3 , ..., m ¯ TP +Q ]T . We obtain ς˙H (t) = (IQ ⊗ A)ςH (t) + (M −1 ⊗ A)fH ¯ ¯ 2 )ςH (t − d(t)), + (JH ⊗ B K
η˙ H (t) = AηH (t) + (m ¯ 1 ⊗ A)fH ,
(12)
(13)
and ˆ −1 ⊗ A)fH χ˙ H (t) = (IQ−1 ⊗ A)χH (t) + (M ¯ ¯ 2 )χH (t − d(t)). + (JˆH ⊗ B K
(14)
Theorem 1. If limt→∞ χH (t) = 0, the leaders in group H reach formation. 140
Proof : Let
ηH (t) , ηˆH (t) = (M ⊗ In ) 0 0 . η¯H (t) = (M ⊗ In ) χH (t) 7
(15)
(16)
Thus, we get ηH (t) = mP +1 ⊗ ηH (t) = 1 ⊗ ηH (t). ηˆH (t) = (M ⊗ In ) 0
(17)
Because M ⊗In is nonsingular, then if limt→∞ χH (t) = 0, we get limt→∞ η¯H (t) = 0. Because ξH (t) = ηˆH (t) + η¯H (t), then if limt→∞ χH (t) = 0, we obtain lim (ξH (t) − ηˆH (t)) = lim (xH (t) − fH − 1 ⊗ ηH (t)) = 0.
t→∞
t→∞
(18)
145
Define ζa (t) =
P X j=1
+
¯ ¯ aj (¯ xa (t − d(t)) −x ¯j (t − d(t)))
PX +Q
b=P +1
¯ ¯ ab (¯ xa (t − d(t)) −x ¯b (t − d(t))),
and ζN (t) = [ζ1T (t), ζ2T (t), ..., ζPT (t)]T . we get ¯ ¯ ¯ 1 ⊗ In )¯ ¯ 2 ⊗ In )¯ ζN (t) = (L xF (t − d(t)) + (L xH (t − d(t)).
(19)
¯ ¯2 ⊗ ¯ −1 L If limt→∞ ζN (t) = 0, then we obtain limt→∞ (¯ xN (t − d(t)) − (L 1
¯ ¯ −1 L ¯ 2 ⊗ In )¯ In )¯ xH (t − d(t))) = 0, which implies limt→∞ (¯ xN (t) − (L xH (t)) = 0. 1 Thus, we have the following Lemma 3. 150
Lemma 2. If the requirements of Assumptions 1 and 2 are met, and if both limt→∞ χE (t) = 0 and limt→∞ ζF (t) = 0 are satisfied, then the system (1) reaches formation containment.
4. Main Results Theorem 2. Based on Assumptions 1, 2 and (4), (5), the MAS (1) reaches 155
formation-containment if Afb = 0 with b ∈ H and for given constants β > 0,
8
¯ 1, K ¯ 2 , there exist real matrices U ¯ > 0, W ¯ 1 > 0, τ¯ > 0, and given matrices K ¯ 2 > 0, X ¯ 1 > 0, X ¯ 2 > 0, U ˆ > 0, W ˆ 1 > 0, W ˆ 2 > 0, X ˆ 1 > 0, X ˆ 2 > 0, such that: W
where
Ω 11 ∗ Ω= ∗ ∗ Ξ 11 ∗ Ξ= ∗ ∗
Ω12
Ω13
Ω22
Ω23
∗
Ω33
∗
∗
Ξ12
Ξ13
Ξ22
Ξ23
∗
Ξ33
∗
∗
0
0 < 0, Ω34 Ω44 0 0 < 0, Ξ34 Ξ44
¯ (IQ−1 ⊗ A) + (IQ−1 ⊗ A)T U ¯ +W ¯1 Ω11 = U ¯1, + (IQ−1 ⊗ A)T F1 (IQ−1 ⊗ A) − X ¯1 Ω12 = X ¯ 2 )(U ¯ + (IQ−1 ⊗ A)T F1 ), Ω13 = (JˆH ⊗ B K ¯2 − W ¯1 − X ¯1 − X ¯2, Ω22 = W ¯2, Ω23 = X ¯ 2 + (JˆH ⊗ BK2 )T F1 (JˆH ⊗ B K ¯ 2 ), Ω33 = −2X ¯2, Ω34 = X ¯2 − X ¯2 Ω44 = −W ¯1 + β2X ¯2, F1 = τ¯2 X ˆ (IP ⊗ A) + (IP ⊗ A)T U ˆ +W ˆ1 Ξ11 = U ˆ1, + (IP ⊗ A)T F2 (IP ⊗ A) − X ˆ1, Ξ12 = X
9
(20)
(21)
¯1 ⊗ BK ¯ 1 )(U ˆ + (IP ⊗ A)T F2 ), Ξ13 = (L ˆ2 − W ˆ1 − X ˆ1 − X ˆ2, Ξ22 = W ˆ2, Ξ23 = X ¯1 ⊗ BK ¯ 1 )T F2 (L ¯1 ⊗ BK ¯ 1 ) − 2X ˆ2, Ξ33 = (L ˆ2, Ξ34 = X ˆ2 − X ˆ2, Ξ44 = −W ˆ1 + β2X ˆ2. F2 = τ¯2 X
Proof : If Afb = 0, we obtain
160
ˆ −1 ⊗ A)fE = 0. (M
(22)
¯ ¯ 2 )χH (t − d(t)). χ˙ H (t) = (IQ−1 ⊗ A)χH (t) + (JˆH ⊗ B K
(23)
The system (14) can be
Then, the Lyapunov function can be constructed as VH (t, χHt ) = VH1 (t, χHt ) + VH2 (t, χHt ) + VH3 (t, χHt ), where ¯ χH (t), VH1 (t, χHt ) = χTH (t)U
VH2 (t, χHt ) = +
Z
t
¯ 1 χH (s)ds χTH (s)W
t−¯ τ Z t−¯τ
t−β−¯ τ
VH3 (t, χHt ) =¯ τ
Z
0
−¯ τ
+β
Z
Z
t
t+α −¯ τ
−β−¯ τ
¯ 2 χH (s)ds, χTH (s)W
T ¯ χ˙ H (s)X1 χ˙ H (s) dsdα Z
t
t+α
10
¯ 2 χ˙ H (s) dsdα. χ˙ TH (s)X
(24)
Then, we obtain V˙ H (t, χHt ) = V˙ H1 (t, χHt ) + V˙ H2 (t, χHt ) + V˙ H3 (t, χHt ),
(25)
¯ χ˙ H (t), V˙ H1 (t, χHt ) = 2χTH (t)U
(26)
¯ 1 χH (t) + χT (t − τ¯) × (W ¯2 − W ¯ 1 )χH (t − τ¯) V˙ H2 (t, χHt ) =χTH (t)W H ¯ 2 χH (t − β − τ¯), − χTH (t − β − τ¯)W ¯ 1 χ˙ H (t) − τ¯ V˙ H3 (t, χHt ) =¯ τ 2 χ˙ TH (t)X ¯ 2 χ˙ H (t) β 2 χ˙ TH (t)X 165
−β
Z
t
(27)
¯ 1 χH (s)ds χTH (s)X
t−τ Z t−¯τ
t−β−¯ τ
(28) ¯ 2 χH (s)ds. χTH (s)X
From [Lemma 1 in [35]], we obtain Z t ¯ 1 χH (s)ds ≤ −¯ τ χTH (s)X
(29)
t−¯ τ
¯ 1 (χH (t) − χH (t − τ¯)), − (χH (t) − χH (t − τ¯)) X T
and −β
Z
t−¯ τ
t−β−¯ τ
−β
Z
¯ 2 χH (s)ds = χTH (s)X
t−¯ τ
¯ t−d(t)
¯ 2 χ˙ H (s)ds χ˙ TH (s)X
−β
Z
¯ t−d(t)
t−β−¯ τ
¯ 2 χ˙ H (s)ds χ˙ TH (s)X
T ¯ ¯ ¯ ≤ −(χH (t − τ¯) − χH (t − d(t))) X2 (χH (t − τ¯) − χH (t − d(t)))
(30)
¯ ¯2 − (χH (t − d(t)) − χH (t − β − τ¯))T X ¯ × (χH (t − d(t)) − χH (t − β − τ¯)). ¯ Let %(t) = [χTH (t), χTH (t − τ¯), χTH (t − d(t)), χTH (t − β − τ¯)]T . From (25)-(30), we get V˙ H (t, χHt ) < %T (t)Ω%(t).
(31)
If Ω < 0, then V˙ H (t, χHt ) < 0. Thus, the leaders achieve formation and it means 170
lim
t→∞
¯2L ¯ 3 ) ⊗ (B K ¯ 2 )(¯ (L xH − fH ) = 0. 11
(32)
From (19), we get ¯ ¯ ¯ 1 ⊗ In )x˙ N (t − d(t)) ¯ 2 ⊗ In )x˙ H (t − d(t)). ζ˙N (t) = (L + (L
(33)
From (8), (9), (19), (33), we obtain ¯ ¯1 ⊗ BK ¯ 1 )ζN (t − d(t)). ζ˙N (t) =(IN ⊗ A)ζN (t) + (L
(34)
For (34), the Lyapunov function is VN (t, ζN t ) = VN 1 (t, ζN t ) + VN 2 (t, ζN t ) + VN 3 (t, ζN t ),
(35)
where T ˆ ζN (t), VN 1 (t, ζN t ) = ζN (t)U
VN 2 (t, ζN t ) =
Z
t
t−¯ τ
+
Z
T ˆ 1 ζN (s)ds ζN (s)W t−¯ τ
t−β−¯ τ
VN 3 (t, ζN t ) =¯ τ
Z
0
−¯ τ
+β
Z
Z
t
t+α −¯ τ
−β−¯ τ
T ˆ 2 ζN (s)ds, ζN (s)W
˙ζ T (s)X ˙ ˆ 1 ζN (s) dsdα N
Z
t
t+α
˙ζ T (s)X ˙ ˆ 2 ζN (s) dsdα. N
With the similar analysis for (23), we can obtain if Ξ < 0, then ζN (t) is 175
stable. We see the MAS (1) achieves formation-containment from Lemma 3.
The proof is completed.
Remark 2. Sufficient conditions for MASs reaching formation containment with sampled data and time delays are given in this theorem. LMIs (20) and ¯ 1 , K¯2 , β, τ¯ are cho(21) cannot always ensure they are solvable. However, if K 180
sen properly, we can find LMIs (20) and (21) are solvable by using LMI tool in Matlab. Remark 3. Compared with the analysis of formation containment in [27]-[29] where communication among agents was continuous, communication among 12
agents in this paper was under sampling. Thus, the Lyapunov function and 185
the handling method are much different and complicated than those in [27]-[29]. Compared with the analysis of sampling problem for MASs to reach consensus in [30]-[34], the analysis of formation containment problem in this paper is more complicated. Remark 4. If we set the time delay τ¯ = 0, then results can be applied in
190
formation containment problem with sampled data and without time delays. If no communication among leaders exists, then the problem can be reduced to the containment problem. If Q = 1 which means that there is only one leader and the formation function does not exists, then the formation containment problem can be reduced to the consensus problem with one leader. Thus, this theorem is
195
more general.
Remark 5. In this paper, Theorem 2 establishes the relationship among network topologies, controller gains, time-delay and sampling period. If parameters except time-delay are determined, then the allowable maximum of time-delay can be obtained by solving LMIs (20) and (21). If parameters except sampling 200
period are determined, then the allowable maximum of sampling period can be obtained by solving LMIs (20) and (21). Remark 6. Many works about coordinated control problems investigated the case of nonlinear MAS [1], [24]-[26]. If the nonlinear function satisfies the Lipschitz condition, then by handling the Lyapunov function, we can use the
205
inequality (3) in [1] to linearize it. If the nonlinear function does not satisfy the Lipschitz condition, then the biggest difficulty is how to find an effective method to linearize it by handling the Lyapunov function.
5. Simulation example Consider that there are nine agents where five agents are followers and four agents are leaders. the system and the control can be described by (1), (4), and 13
7
3
6
2
1
4
8
5 9
Fig. 1: The directed interaction topology G.
(5) with 0 0 A= 0 0
1 0 0
0 1 B= 0 0
0 0 0 , 0 0 1 0 0 0
0 0 . 0 1
The communication topology is shown in Fig.1 and the Laplacian matrix is ¯1 ¯2 L L ¯= , L ¯3 0Q×P L where
4
−1 ¯1 = L −1 −1 −1
0
−1
−1
−1
4
−1
0
−1
4
−1
0
−1
4
−1
0
−1
−1 ¯2 = L 0 0 0
0
0
0
0
−1
0
0
−1
0
0
14
−1 −1 0 , −1 4 0
0 0 , 0 −1
2
−1 ¯ L3 = 0 −1
−1
0
2
−1
−1
2
0
−1
−1
0 . −1 2
In the simulation, the formation vectors are f6 = [−20, 0, −20, 0]T , f7 =
[20, 0, −20, 0]T , f8 = [20, 0, 20, 0]T , f9 = [−20, 0, 20, 0]T . Then, we see that Afb = 0. The controller gain matrices are K1 = I2 ⊗ [−0.06, −0.3], K2 = I2 ⊗ [−0.12, −0.33]. We set the sampling period β = 0.05s and choose the time delay τ¯ = 0.1s. By solving (20) and (21) using LMI Tool in Matlab, we find that LMIs (20)and (21) are solvable. In addition, the following parameters are obtained by solving (20) and (21) using LMI Tool in Matlab. 17.1560 −4.9312 7.4404 −2.1715 ¯ = I6 ⊗ ¯ 1 = I6 ⊗ ,W , U −4.9312 3.7035 −2.1715 1.8253
5.7991
¯ 2 = I6 ⊗ W −1.6269
−1.6269 5.3080 1.3693 ¯ 1 = I6 ⊗ ,X , 1.3970 1.3693 9.2038
5.3826 1.6036 49.8923 ¯ 2 = I6 ⊗ ˆ = I10 ⊗ ,U X 1.6036 8.7484 −7.2965
11.2427
ˆ 1 = I10 ⊗ W −2.5339
−2.5339 1.3368
9.1552
−7.2965 , 3.5496
ˆ 2 = I10 ⊗ ,W −2.0944
−2.0944 , 1.0115
6.8281 2.0355 7.8181 2.1582 ˆ 1 = I10 ⊗ ˆ 2 = I10 ⊗ ,X . X 2.0355 14.9412 2.1582 13.8879
210
The snapshots of positions of the agents are shown in Fig.2, where ”∗” means the follower and ”◦” means the leader. The curves of control inputs are shown in Fig.3, where red lines represent the the curves of follower control input and blue lines represent the curves of leader control input. We see that the leaders reach formation and the followers are in the formation. Thus, we see that the
215
MAS achieves formation-containment. 15
50
240
40
230
xi3
250
xi3
60
30
220
20
210
10 -30
-20
-10
0
10
200 -60
20
-50
-40
xi1
-30
-20
-10
-50
-40
-30
xi1
(a)
(b)
310
390
300
380
xi3
400
xi3
320
290
370
280
360
270 -70
-60
-50
-40
-30
350 -80
-20
-70
-60
xi1
xi1
(c)
(d)
Fig. 2: State snapshots of the leaders and the followers. (a) t=5s; (b) t=30s; (c)
30
30
20
20
10
10
u i4 (t)
u i2 (t)
t=40s; (d) t=50s.
0
0
-10
-10
-20
-20
-30
-30 0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
t
t
(a)
(b)
Fig. 3: The curves of control inputs.
16
30
35
40
45
50
6. Conclusion The current paper studies a novel sampling strategy for MASs with time delays to reach formation containment. A novel sampling strategy for the MASs to achieve formation containment is proposed. Only at the sampling instance, com220
munication among agents is occurred and the controller state updates. Compared with the strategy where communication among agents was continuous, this strategy can reduce the communication energy and the updating controller energy. By handling the suitable Lyapunov function, LMIs that ensuring the MAS to reach formation containment are obtained. Finally, a simulation is dis-
225
played. In the future, we will investigate more complex formation containment problems, such as the nonlinear MAS, the output feedback, and the event-based control mechanism based on sampled data.
Declaration of interests The authors declare that they have no known competing financial interests or 230
personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant No. 61873318, the Frontier Research Funds of Applied 235
Foundation of Wuhan under Grant No. 2019010701011421, the Natural Science Foundation of Hubei Province of China under Grant No. 2018CFA058, the Fundamental Research Funds for the Central Universities [grant number HUST: 2019kfyXKJC022], and the Program for HUST Academic Frontier Youth Team under Grant No. 2018QYTD07.
17
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21
Housheng Su received his B.S. degree in automatic control 350
and his M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and his Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a Postdoctoral researcher with the Department of Electronic Engineer-
355
ing, City University of Hong Kong, Hong Kong. Since November 2014, he has been a full professor with the School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.
360
Jinxin Zhang received the B.S. degree in automation from Hunan University, Changsha, China, in 2017. He is currently pursuing the M.S. degree in control science and engineering with Huazhong University of Science and Technology, Wuhan, China. His current research interests include time-delay systems and multi-agent systems.
22
Journal Pre-proof
Formation-containment control for multi-agent systems with sampled data and time delays Jinxin Zhang, Housheng Su PII: DOI: Reference:
S0925-2312(19)31628-5 https://doi.org/10.1016/j.neucom.2019.11.030 NEUCOM 21554
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
2 June 2019 20 September 2019 17 November 2019
Please cite this article as: Jinxin Zhang, Housheng Su, Formation-containment control for multi-agent systems with sampled data and time delays, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.11.030
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Formation-containment control for multi-agent systems with sampled data and time delays Jinxin Zhanga,b , Housheng Sub,∗ a School
of Science, Hunan University of Technology, Zhuzhou 412008, China of Artificial Intelligence and Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, China
b School
Abstract This paper investigates the formation containment control under sampling and with time delays for multi-agent systems (MASs). First, a novel sampling control protocol for the MASs to achieve formation containment is proposed, where the communication among agents is only occurred at the sampling instance. Then the communication load and the waste of energy can be decreased significantly. In addition, time delays are considered in this paper. Second, by establishing an appropriate Lyapunov function, sufficient conditions for MASs with sampled data and time delays reaching formation containment are obtained by handling the established Lyapunov function. Finally, the obtained results are demonstrated by a simulation example. Keywords: Multi-agent systems (MASs), formation containment, under sampling, time delays
1. Introduction Research on collaborative control of MASs has attracted more and more research workers and large numbers of research results have been obtained [1]-[3]. The collaborative control of MASs has been applied in robots, sensor networks, 5
unmanned aircrafts and many other fields [4]-[8]. In particular, formation con∗ Corresponding
author Email address: [email protected] (Housheng Su)
Preprint submitted to Journal of LATEX Templates
December 4, 2019
trol is an important aspect of the collaborative control of MASs. Formation control aims to design a control protocol which makes the agents achieve an expected shape. In [9] and [10], sufficient conditions were derived to make the MASs attain formation, in which only second-order dynamics were considered. 10
Formation control for high-order MASs were investigated in [11]-[13]. Group formation was studied in [14], where the agents were divided into some groups and each group achieved a specified formation. Many studies on MASs were with only one leader or without leaders. In [15][17], no leaders were considered and the MASs reached consensus. The problem
15
for collaborative control of MASs with one leader is regarded as coordinated tracking problem. The consensus problem with one leader was investigated in [18] where the dynamics of agents were second order. The high-order consensus with one leader was discussed in [19] and sufficient conditions ensuring consensus were obtained. However, some occasions require multiple leaders, thus,
20
containment problem arises. Containment control implies that followers are in the convex hull specified by the multiple leaders, which were discussed in [20][26]. Containment control problems for second-order MASs were investigated with finite-time observers [20] and sampled position data [21]. In [22]-[26], containment control problems were discussed for high-order and nonlinear MASs,
25
respectively. It should be pointed out that no communication was among leaders in [20]-[26]. Based on studies of formation and containment, formation containment problems arise which are more complex and require leaders achieve the designed formation and followers are stay in the formation. Formation containment has
30
been used in many engineering projects. For instance, the unmanned vehicles are denoted as leaders and the manned vehicles which are important are denoted as the followers. Leaders form a formation and followers stay in the formation. Thus, when facing foes, the followers can be protected by followers. In addition, formation problems and containment problems cannot directly combine into
35
formation containment problems, because the actions of leaders reaching formation effect followers. For formation containment problems, single-integrator 2
MASs were discussed in [27], double-integrator MASs were studied in [28], and high-order MASs were investigated in [29]. However, all the systems in [27]-[29] was continuous. For continuous-time systems, information transmission among 40
agents is continuous and the controller updates continuously, thus, much energy will be wasted. On some occasions, continuous information transmission cannot always be ensured. Therefore, sampling information for MASs is important and gains attention from researchers. Sampling strategy saves the energy and can be used in the situation where continuous information transmission cannot al-
45
ways be ensured. Sampling method for MASs to reach consensus was discussed in [30]-[34]. In [30], both position states and velocity states were obtained in sampling instances and conditions ensuring the MASs to achieve consensus were obtained. In [31] and [32], second-order consensus were achieved for MASs under sampling positions. In [33] and [34], the MASs with high-order dynamics
50
to reach consensus were investigated via sampling approach. To the best of our knowledge, sampling control problem for formation containment has not been fully discussed, which motivates us to do this work. This paper investigates formation containment of MASs based on sampling data. In addition, time delays are also considered. A sampling strategy for
55
formation containment is proposed. The controllers of both leaders and followers are under sampling. By establishing a suitable Lyapunov function, sufficient conditions for MASs with sampled data and time delays reaching formation containment are obtained. Main contributions are as follows. First, compared with the situation with one leader in [18] and [19], multiple leaders are considered in
60
this paper and the multiple leaders cause the analysis more complex. Second, in [20]-[26], there was no information transmission among leaders for containment problems, formation containment is investigated in this paper and communication among leaders exists. Different from containment control with no leader input, the controllers for leaders must be designed and formation functions are
65
also considered in the leader controllers. Thus, the analysis for the formation containment is more complicated. Third, the communication among agents of formation containment with sampling data and time delays is considered. In 3
[27]-[29], communication among agents for formation containment was continuous. Thus, the analysis methods in [27]-[29] cannot be used in this paper and the 70
Lyapunov function and handling methods in this paper are more complicated than those in [27]-[29]. In this paper, some preliminaries are shown in Section 2. Section 3 shows the problem transformation. Section 4 gives the results and the analysis. Section 5 shows the simulation example. Section 6 gives the conclusion. Notations: In this work, ⊗ is the Kronecker product. RP ×P means the order
75
of real matrices is P × P . The notation “∗” is regarded as the symmetric item in a symmetric matrix. The notation “T ” marks the transpose of a matrix. Other notations are routine.
2. Preliminaries 80
2.1. Graph theory The directed topology graph of P + Q agents can be depicted by G = ¯ Γ}, ˆ in which Γ = [κ1 , κ2 , ..., κP +Q ] denotes the group of nodes, Γ ¯ ⊂ Γ×Γ {Γ, Γ, ˆ = [aj ] ∈ R(P +Q)×(P +Q) is the weighted represents the group of edges and Γ
¯ adjacency matrix. $aj denotes an edge of G with a 6= j (a ∈ Γ, j ∈ Γ). $aj ∈ Γ
85
¯ means aj = 0. Any aj in Γ ˆ is nonnegative. means aj > 0 and $aj ∈ / Γ P +Q ¯ = [laj ] ∈ R(P +Q)×(P +Q) is the Laplacian matrix, where laa = P aj and L j=1,j6=a
laj = −aj (a 6= j).
2.2. Problem statement There are P + Q agents constituting of a MAS and the system is designed 90
as x ¯˙ v (t) = A¯ xv (t) + B u ¯v (t), v = 1, 2, ..., P + Q,
(1)
where x ¯v (t) ∈ Rn is the state. u ¯v (t) is the controller. Consider that P agents denote followers and the follower group is denoted as N = [1, 2, ..., P ]. Q agents denote leaders and the leader group is denoted as H = [P + 1, P + 2, ..., P + Q]. For the leader group, define the formation function as fH = 4
95
[fPT +1 , fPT +2 , ..., fPT +Q ]T ∈ RQn and for the bth agent with b ∈ H, fb is the formation vector. Definition 1. [29] For the MAS (1), the leaders achieve formation which is specified by fE¯ , when the following condition is attained, lim (¯ xb (t) − fb − g(t)) = 0,
t→∞
(2)
where b ∈ H and g(t) ∈ Rn is the formation reference function. 100
Remark 1. If fb = 0, we can see that the formation problem will be reduced to the consensus problem. Definition 2. [29] The MAS (1) achieves containment if the Eq.(3) is satisfied,
lim (¯ xa (t) −
t→∞
where a ∈ N , b ∈ H and 105
PP +Q
j=P +1
PX +Q
bj x ¯b (t)) = 0,
(3)
j=P +1
bj = 1 with the positive scalar bj (j ∈ H).
Definition 3. [29] If the requirements of Definition 1 and Definition 2 are both met, then MAS (1) achieves formation containment. ¯ for MAS (1) is defined as : The information topology G ¯1 ¯2 L L ¯= , L ¯3 0Q×P L
¯ is the Laplacian matrix, L ¯ 1 ∈ RP ×P , L ¯ 2 ∈ RP ×Q , L ¯ 3 ∈ RQ×Q . G ¯ H is where L
¯ 3 , and G ¯ N is the the information topology of the group H corresponding to L ¯1. information topology of the group F¯ corresponding to L 110
¯ H and G ¯ N are undirected and connected. Assumption 1. The topologies G Assumption 2. For every follower, there is at least one leader which has a directed path to it.
5
¯ 1 is positive, Lemma 1. [21] Based on Assumptions 1, 2, every eigenvalue of L −1 all entries of −L−1 1 L2 are nonnegative, and the sum of every row of −L1 L2 115
equals to one. Define tq as the sampling time where q is the non-negative integer. Define that τ¯ > 0 is the time delay and β = tq+1 − tq is the sampling period. Then, the controller of the group F¯ is designed as ¯1 u ¯a (t) =K
P X j=1
PX +Q
+
aj (¯ xa (tq − τ¯) − x ¯j (tq − τ¯))
b=P +1
(4)
ab (¯ xa (tq − τ¯) − x ¯b (tq − τ¯)) ,
and the controllers of the group H is designed as PX +Q
¯2 u ¯b (t) = K
j=P +1 120
bj ((¯ xb (tq − τ¯) − fb ) − (xj (tq − τ¯) − fj )) ,
(5)
¯ 1 represents the gain matrix of u ¯ 2 represents where a ∈ N , b ∈ H, K ¯a (t) and K the gain matrix of u ¯b (t). ¯ = t − tq + τ¯, where t ∈ [tq , tq+1 ). Then, the controllers (4) and Define d(t) (5) can be written as ¯1 u ¯a (t) =K
P X j=1
+
PX +Q
b=P +1
¯2 u ¯b (t) =K
¯ ¯ aj (¯ xa (t − d(t)) −x ¯j (t − d(t)) !
(6)
¯ ¯ ab (¯ xa (t − d(t)) −x ¯k (t − d(t))) (a ∈ N ),
PX +Q
j=P +1
¯ ¯ bj (¯ xk (t − d(t)) − fb ) − (¯ xj (t − d(t))
(7)
− fj ) (b ∈ H).
125
3. Problem Transformation
¯T2 (t), ..., x ¯TP (t)]T , x ¯H (t) = [¯ xTP +1 (t), x ¯TP +2 (t), ...¯ xTP +Q (t)]T , Let x ¯N (t) = [¯ xT1 (t), x one has ¯ ¯3 ⊗ BK ¯ 2 )(¯ x ¯˙ H (t) = (IQ ⊗ A)¯ xH (t) + (L xH (t − d(t)) − fH ), 6
(8)
¯ ¯1 ⊗ BK ¯ 1 )¯ x ¯˙ N (t) =(IP ⊗ A)¯ xN (t) + (L xN (t − d(t)) ¯ ¯2 ⊗ BK ¯ 1 )¯ + (L xH (t − d(t)) ,
(9)
¯ 3 M and the matrix M is nonsingular. Define the Jordan matrix JH = M −1 L ¯ 3 , where b ∈ H. Thus, 0 = cP +1 ≤ cP +2 ≤ Denote cb as the eigenvalues of L 130
· · · ≤ cP +Q .
Define ξb (t) = x ¯b (t) − fb and ξH (t) = [ξPT +1 (t), ξPT +2 (t), ..., ξPT +Q (t)]T . Then,
we get
and
ξ˙H (t) = (IQ ⊗ A)ξH (t) + (IQ ⊗ A)fH ¯ ¯3 ⊗ BK ¯ 2 )ξH (t − d(t)) + (L ¯ 3 M = JH = diag 0, J¯H , M −1 L
(10)
(11)
where J¯H =diag{cP +2 , cN +3 , ..., cP +Q }, M = [mTP +1 , mTP +2 , ..., mTP +Q ]T , M −1 =
135
¯ TP +Q ]T . ¯ TP +2 , ..., m [m ¯ TP +1 , m
T ¯ p+1 ⊗ In )ξE (t), and χH (t) = (t), χTH (t)]T , ηH (t) = (m Define ςH (t) = [ηH
ˆ ⊗ In )ξH (t) with M ˆ = [m (M ¯ TP +2 , m ¯ TP +3 , ..., m ¯ TP +Q ]T . We obtain ς˙H (t) = (IQ ⊗ A)ςH (t) + (M −1 ⊗ A)fH ¯ ¯ 2 )ςH (t − d(t)), + (JH ⊗ B K
η˙ H (t) = AηH (t) + (m ¯ 1 ⊗ A)fH ,
(12)
(13)
and ˆ −1 ⊗ A)fH χ˙ H (t) = (IQ−1 ⊗ A)χH (t) + (M ¯ ¯ 2 )χH (t − d(t)). + (JˆH ⊗ B K
(14)
Theorem 1. If limt→∞ χH (t) = 0, the leaders in group H reach formation. 140
Proof : Let
ηH (t) , ηˆH (t) = (M ⊗ In ) 0 0 . η¯H (t) = (M ⊗ In ) χH (t) 7
(15)
(16)
Thus, we get ηH (t) = mP +1 ⊗ ηH (t) = 1 ⊗ ηH (t). ηˆH (t) = (M ⊗ In ) 0
(17)
Because M ⊗In is nonsingular, then if limt→∞ χH (t) = 0, we get limt→∞ η¯H (t) = 0. Because ξH (t) = ηˆH (t) + η¯H (t), then if limt→∞ χH (t) = 0, we obtain lim (ξH (t) − ηˆH (t)) = lim (xH (t) − fH − 1 ⊗ ηH (t)) = 0.
t→∞
t→∞
(18)
145
Define ζa (t) =
P X j=1
+
¯ ¯ aj (¯ xa (t − d(t)) −x ¯j (t − d(t)))
PX +Q
b=P +1
¯ ¯ ab (¯ xa (t − d(t)) −x ¯b (t − d(t))),
and ζN (t) = [ζ1T (t), ζ2T (t), ..., ζPT (t)]T . we get ¯ ¯ ¯ 1 ⊗ In )¯ ¯ 2 ⊗ In )¯ ζN (t) = (L xF (t − d(t)) + (L xH (t − d(t)).
(19)
¯ ¯2 ⊗ ¯ −1 L If limt→∞ ζN (t) = 0, then we obtain limt→∞ (¯ xN (t − d(t)) − (L 1
¯ ¯ −1 L ¯ 2 ⊗ In )¯ In )¯ xH (t − d(t))) = 0, which implies limt→∞ (¯ xN (t) − (L xH (t)) = 0. 1 Thus, we have the following Lemma 3. 150
Lemma 2. If the requirements of Assumptions 1 and 2 are met, and if both limt→∞ χE (t) = 0 and limt→∞ ζF (t) = 0 are satisfied, then the system (1) reaches formation containment.
4. Main Results Theorem 2. Based on Assumptions 1, 2 and (4), (5), the MAS (1) reaches 155
formation-containment if Afb = 0 with b ∈ H and for given constants β > 0,
8
¯ 1, K ¯ 2 , there exist real matrices U ¯ > 0, W ¯ 1 > 0, τ¯ > 0, and given matrices K ¯ 2 > 0, X ¯ 1 > 0, X ¯ 2 > 0, U ˆ > 0, W ˆ 1 > 0, W ˆ 2 > 0, X ˆ 1 > 0, X ˆ 2 > 0, such that: W
where
Ω 11 ∗ Ω= ∗ ∗ Ξ 11 ∗ Ξ= ∗ ∗
Ω12
Ω13
Ω22
Ω23
∗
Ω33
∗
∗
Ξ12
Ξ13
Ξ22
Ξ23
∗
Ξ33
∗
∗
0
0 < 0, Ω34 Ω44 0 0 < 0, Ξ34 Ξ44
¯ (IQ−1 ⊗ A) + (IQ−1 ⊗ A)T U ¯ +W ¯1 Ω11 = U ¯1, + (IQ−1 ⊗ A)T F1 (IQ−1 ⊗ A) − X ¯1 Ω12 = X ¯ 2 )(U ¯ + (IQ−1 ⊗ A)T F1 ), Ω13 = (JˆH ⊗ B K ¯2 − W ¯1 − X ¯1 − X ¯2, Ω22 = W ¯2, Ω23 = X ¯ 2 + (JˆH ⊗ BK2 )T F1 (JˆH ⊗ B K ¯ 2 ), Ω33 = −2X ¯2, Ω34 = X ¯2 − X ¯2 Ω44 = −W ¯1 + β2X ¯2, F1 = τ¯2 X ˆ (IP ⊗ A) + (IP ⊗ A)T U ˆ +W ˆ1 Ξ11 = U ˆ1, + (IP ⊗ A)T F2 (IP ⊗ A) − X ˆ1, Ξ12 = X
9
(20)
(21)
¯1 ⊗ BK ¯ 1 )(U ˆ + (IP ⊗ A)T F2 ), Ξ13 = (L ˆ2 − W ˆ1 − X ˆ1 − X ˆ2, Ξ22 = W ˆ2, Ξ23 = X ¯1 ⊗ BK ¯ 1 )T F2 (L ¯1 ⊗ BK ¯ 1 ) − 2X ˆ2, Ξ33 = (L ˆ2, Ξ34 = X ˆ2 − X ˆ2, Ξ44 = −W ˆ1 + β2X ˆ2. F2 = τ¯2 X
Proof : If Afb = 0, we obtain
160
ˆ −1 ⊗ A)fE = 0. (M
(22)
¯ ¯ 2 )χH (t − d(t)). χ˙ H (t) = (IQ−1 ⊗ A)χH (t) + (JˆH ⊗ B K
(23)
The system (14) can be
Then, the Lyapunov function can be constructed as VH (t, χHt ) = VH1 (t, χHt ) + VH2 (t, χHt ) + VH3 (t, χHt ), where ¯ χH (t), VH1 (t, χHt ) = χTH (t)U
VH2 (t, χHt ) = +
Z
t
¯ 1 χH (s)ds χTH (s)W
t−¯ τ Z t−¯τ
t−β−¯ τ
VH3 (t, χHt ) =¯ τ
Z
0
−¯ τ
+β
Z
Z
t
t+α −¯ τ
−β−¯ τ
¯ 2 χH (s)ds, χTH (s)W
T ¯ χ˙ H (s)X1 χ˙ H (s) dsdα Z
t
t+α
10
¯ 2 χ˙ H (s) dsdα. χ˙ TH (s)X
(24)
Then, we obtain V˙ H (t, χHt ) = V˙ H1 (t, χHt ) + V˙ H2 (t, χHt ) + V˙ H3 (t, χHt ),
(25)
¯ χ˙ H (t), V˙ H1 (t, χHt ) = 2χTH (t)U
(26)
¯ 1 χH (t) + χT (t − τ¯) × (W ¯2 − W ¯ 1 )χH (t − τ¯) V˙ H2 (t, χHt ) =χTH (t)W H ¯ 2 χH (t − β − τ¯), − χTH (t − β − τ¯)W ¯ 1 χ˙ H (t) − τ¯ V˙ H3 (t, χHt ) =¯ τ 2 χ˙ TH (t)X ¯ 2 χ˙ H (t) β 2 χ˙ TH (t)X 165
−β
Z
t
(27)
¯ 1 χH (s)ds χTH (s)X
t−τ Z t−¯τ
t−β−¯ τ
(28) ¯ 2 χH (s)ds. χTH (s)X
From [Lemma 1 in [35]], we obtain Z t ¯ 1 χH (s)ds ≤ −¯ τ χTH (s)X
(29)
t−¯ τ
¯ 1 (χH (t) − χH (t − τ¯)), − (χH (t) − χH (t − τ¯)) X T
and −β
Z
t−¯ τ
t−β−¯ τ
−β
Z
¯ 2 χH (s)ds = χTH (s)X
t−¯ τ
¯ t−d(t)
¯ 2 χ˙ H (s)ds χ˙ TH (s)X
−β
Z
¯ t−d(t)
t−β−¯ τ
¯ 2 χ˙ H (s)ds χ˙ TH (s)X
T ¯ ¯ ¯ ≤ −(χH (t − τ¯) − χH (t − d(t))) X2 (χH (t − τ¯) − χH (t − d(t)))
(30)
¯ ¯2 − (χH (t − d(t)) − χH (t − β − τ¯))T X ¯ × (χH (t − d(t)) − χH (t − β − τ¯)). ¯ Let %(t) = [χTH (t), χTH (t − τ¯), χTH (t − d(t)), χTH (t − β − τ¯)]T . From (25)-(30), we get V˙ H (t, χHt ) < %T (t)Ω%(t).
(31)
If Ω < 0, then V˙ H (t, χHt ) < 0. Thus, the leaders achieve formation and it means 170
lim
t→∞
¯2L ¯ 3 ) ⊗ (B K ¯ 2 )(¯ (L xH − fH ) = 0. 11
(32)
From (19), we get ¯ ¯ ¯ 1 ⊗ In )x˙ N (t − d(t)) ¯ 2 ⊗ In )x˙ H (t − d(t)). ζ˙N (t) = (L + (L
(33)
From (8), (9), (19), (33), we obtain ¯ ¯1 ⊗ BK ¯ 1 )ζN (t − d(t)). ζ˙N (t) =(IN ⊗ A)ζN (t) + (L
(34)
For (34), the Lyapunov function is VN (t, ζN t ) = VN 1 (t, ζN t ) + VN 2 (t, ζN t ) + VN 3 (t, ζN t ),
(35)
where T ˆ ζN (t), VN 1 (t, ζN t ) = ζN (t)U
VN 2 (t, ζN t ) =
Z
t
t−¯ τ
+
Z
T ˆ 1 ζN (s)ds ζN (s)W t−¯ τ
t−β−¯ τ
VN 3 (t, ζN t ) =¯ τ
Z
0
−¯ τ
+β
Z
Z
t
t+α −¯ τ
−β−¯ τ
T ˆ 2 ζN (s)ds, ζN (s)W
˙ζ T (s)X ˙ ˆ 1 ζN (s) dsdα N
Z
t
t+α
˙ζ T (s)X ˙ ˆ 2 ζN (s) dsdα. N
With the similar analysis for (23), we can obtain if Ξ < 0, then ζN (t) is 175
stable. We see the MAS (1) achieves formation-containment from Lemma 3.
The proof is completed.
Remark 2. Sufficient conditions for MASs reaching formation containment with sampled data and time delays are given in this theorem. LMIs (20) and ¯ 1 , K¯2 , β, τ¯ are cho(21) cannot always ensure they are solvable. However, if K 180
sen properly, we can find LMIs (20) and (21) are solvable by using LMI tool in Matlab. Remark 3. Compared with the analysis of formation containment in [27]-[29] where communication among agents was continuous, communication among 12
agents in this paper was under sampling. Thus, the Lyapunov function and 185
the handling method are much different and complicated than those in [27]-[29]. Compared with the analysis of sampling problem for MASs to reach consensus in [30]-[34], the analysis of formation containment problem in this paper is more complicated. Remark 4. If we set the time delay τ¯ = 0, then results can be applied in
190
formation containment problem with sampled data and without time delays. If no communication among leaders exists, then the problem can be reduced to the containment problem. If Q = 1 which means that there is only one leader and the formation function does not exists, then the formation containment problem can be reduced to the consensus problem with one leader. Thus, this theorem is
195
more general.
Remark 5. In this paper, Theorem 2 establishes the relationship among network topologies, controller gains, time-delay and sampling period. If parameters except time-delay are determined, then the allowable maximum of time-delay can be obtained by solving LMIs (20) and (21). If parameters except sampling 200
period are determined, then the allowable maximum of sampling period can be obtained by solving LMIs (20) and (21). Remark 6. Many works about coordinated control problems investigated the case of nonlinear MAS [1], [24]-[26]. If the nonlinear function satisfies the Lipschitz condition, then by handling the Lyapunov function, we can use the
205
inequality (3) in [1] to linearize it. If the nonlinear function does not satisfy the Lipschitz condition, then the biggest difficulty is how to find an effective method to linearize it by handling the Lyapunov function.
5. Simulation example Consider that there are nine agents where five agents are followers and four agents are leaders. the system and the control can be described by (1), (4), and 13
7
3
6
2
1
4
8
5 9
Fig. 1: The directed interaction topology G.
(5) with 0 0 A= 0 0
1 0 0
0 1 B= 0 0
0 0 0 , 0 0 1 0 0 0
0 0 . 0 1
The communication topology is shown in Fig.1 and the Laplacian matrix is ¯1 ¯2 L L ¯= , L ¯3 0Q×P L where
4
−1 ¯1 = L −1 −1 −1
0
−1
−1
−1
4
−1
0
−1
4
−1
0
−1
4
−1
0
−1
−1 ¯2 = L 0 0 0
0
0
0
0
−1
0
0
−1
0
0
14
−1 −1 0 , −1 4 0
0 0 , 0 −1
2
−1 ¯ L3 = 0 −1
−1
0
2
−1
−1
2
0
−1
−1
0 . −1 2
In the simulation, the formation vectors are f6 = [−20, 0, −20, 0]T , f7 =
[20, 0, −20, 0]T , f8 = [20, 0, 20, 0]T , f9 = [−20, 0, 20, 0]T . Then, we see that Afb = 0. The controller gain matrices are K1 = I2 ⊗ [−0.06, −0.3], K2 = I2 ⊗ [−0.12, −0.33]. We set the sampling period β = 0.05s and choose the time delay τ¯ = 0.1s. By solving (20) and (21) using LMI Tool in Matlab, we find that LMIs (20)and (21) are solvable. In addition, the following parameters are obtained by solving (20) and (21) using LMI Tool in Matlab. 17.1560 −4.9312 7.4404 −2.1715 ¯ = I6 ⊗ ¯ 1 = I6 ⊗ ,W , U −4.9312 3.7035 −2.1715 1.8253
5.7991
¯ 2 = I6 ⊗ W −1.6269
−1.6269 5.3080 1.3693 ¯ 1 = I6 ⊗ ,X , 1.3970 1.3693 9.2038
5.3826 1.6036 49.8923 ¯ 2 = I6 ⊗ ˆ = I10 ⊗ ,U X 1.6036 8.7484 −7.2965
11.2427
ˆ 1 = I10 ⊗ W −2.5339
−2.5339 1.3368
9.1552
−7.2965 , 3.5496
ˆ 2 = I10 ⊗ ,W −2.0944
−2.0944 , 1.0115
6.8281 2.0355 7.8181 2.1582 ˆ 1 = I10 ⊗ ˆ 2 = I10 ⊗ ,X . X 2.0355 14.9412 2.1582 13.8879
210
The snapshots of positions of the agents are shown in Fig.2, where ”∗” means the follower and ”◦” means the leader. The curves of control inputs are shown in Fig.3, where red lines represent the the curves of follower control input and blue lines represent the curves of leader control input. We see that the leaders reach formation and the followers are in the formation. Thus, we see that the
215
MAS achieves formation-containment. 15
50
240
40
230
xi3
250
xi3
60
30
220
20
210
10 -30
-20
-10
0
10
200 -60
20
-50
-40
xi1
-30
-20
-10
-50
-40
-30
xi1
(a)
(b)
310
390
300
380
xi3
400
xi3
320
290
370
280
360
270 -70
-60
-50
-40
-30
350 -80
-20
-70
-60
xi1
xi1
(c)
(d)
Fig. 2: State snapshots of the leaders and the followers. (a) t=5s; (b) t=30s; (c)
30
30
20
20
10
10
u i4 (t)
u i2 (t)
t=40s; (d) t=50s.
0
0
-10
-10
-20
-20
-30
-30 0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
t
t
(a)
(b)
Fig. 3: The curves of control inputs.
16
30
35
40
45
50
6. Conclusion The current paper studies a novel sampling strategy for MASs with time delays to reach formation containment. A novel sampling strategy for the MASs to achieve formation containment is proposed. Only at the sampling instance, com220
munication among agents is occurred and the controller state updates. Compared with the strategy where communication among agents was continuous, this strategy can reduce the communication energy and the updating controller energy. By handling the suitable Lyapunov function, LMIs that ensuring the MAS to reach formation containment are obtained. Finally, a simulation is dis-
225
played. In the future, we will investigate more complex formation containment problems, such as the nonlinear MAS, the output feedback, and the event-based control mechanism based on sampled data.
Declaration of interests The authors declare that they have no known competing financial interests or 230
personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant No. 61873318, the Frontier Research Funds of Applied 235
Foundation of Wuhan under Grant No. 2019010701011421, the Natural Science Foundation of Hubei Province of China under Grant No. 2018CFA058, the Fundamental Research Funds for the Central Universities [grant number HUST: 2019kfyXKJC022], and the Program for HUST Academic Frontier Youth Team under Grant No. 2018QYTD07.
17
240
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21
Housheng Su received his B.S. degree in automatic control 350
and his M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and his Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a Postdoctoral researcher with the Department of Electronic Engineer-
355
ing, City University of Hong Kong, Hong Kong. Since November 2014, he has been a full professor with the School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.
360
Jinxin Zhang received the B.S. degree in automation from Hunan University, Changsha, China, in 2017. He is currently pursuing the M.S. degree in control science and engineering with Huazhong University of Science and Technology, Wuhan, China. His current research interests include time-delay systems and multi-agent systems.
22