Output feedback-based consensus control for nonlinear time delay multiagent systems

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Brief paper

Output feedback-based consensus control for nonlinear time delay multiagent systems✩ ∗

Kuo Li a , Chang-Chun Hua a , , Xiu You b , Xin-Ping Guan c a

Institute of Electrical Engineering, Yanshan University, Qinhuangdao, 066004, China Institute of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China c Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, China b

article

info

Article history: Received 17 June 2018 Received in revised form 23 August 2019 Accepted 12 October 2019 Available online xxxx Keywords: Time delays Output feedback control Dynamic compensator Nonlinear multiagent systems

a b s t r a c t This paper focuses on the leader-following consensus problem for a class of identical nonlinear time delay multiagent systems. Different from the existing results, it is the first time that an output feedback-based consensus control algorithm is proposed for the multiagent systems based on a fixed directed topology. A novel dynamic compensator is constructed for each follower by only using the follower output and the relative output information of its neighbor agents. The distributed controller is designed for each follower based on the compensator, which is independent of time delays in the agent state. It is proved strictly that each state variable of followers can asymptotically track that of the leader based on the constructed controller. The conditions imposed on nonlinear terms are relaxed and the proposed algorithm has a simple design procedure. Moreover, the proposed control design method is applied to the consensus problem of chemical reactor systems. The simulation results illustrate the effectiveness of the proposed consensus protocols. © 2019 Published by Elsevier Ltd.

1. Introduction Over the past several years, the leader-following consensus control of multiagent systems has drawn broad attention due to its wide application in many industrial systems, including sensor networks, industrial processes, unmanned aerial vehicles, multiple robot systems, see Mu, Chen, Shi and Chang (2017), Mu, Zhang and Shi (2017) and Shi, Qin, and Ahn (2017). This issue aims to design the distributed consensus protocol for each agent based on the local information obtained from its neighbor agents, such that the agents can reach an agreement on a state of interest. The consensus algorithms have been developed for single integrator multiagent systems in Fan, Feng, Wang, and Song (2013), double integrator multiagent systems in Li, Du, and Lin (2011), highorder integrator multiagent systems in Zuo, Tian, Defoort, and Ding (2018), and linear multiagent systems in Li, Ren, Liu, and Fu (2013). ✩ This work was partially supported by National Key R&D Program of China (2018YFB1308300), National Natural Science Foundation of China (618255304, 61751309, 61673335, 61803243), Postgraduate’s Innovation Fund Project of Hebei Province (CXZZBS2019056). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Christos G. Cassandras. ∗ Corresponding author. E-mail addresses: [email protected] (K. Li), [email protected] (C.-C. Hua), [email protected] (X. You), [email protected] (X.-P. Guan).

Recently, the consensus problem was studied for various nonlinear multiagent systems from two aspects: the output consensus and full state variables consensus. For the output consensus problem, many results have been made based on the state feedback control or the output feedback control, see Hua, Li, and Guan (2019), Hua, You, and Guan (2016) and Wang, Chen, Lin, and Li (2017) and the references therein. For full state variables consensus problem, the work in Wang (2017) proposed the distributed coordinated tracking control for one-order uncertain nonlinear multiagent systems by using neural networks; the adaptive consensus algorithm has been developed in Hua, You, and Guan (2017) for second-order uncertain time-varying nonlinear multiagent systems; the work in Zhang, Liu, and Feng (2015) proposed the output feedback-based distributed consensus protocols for high-order time-varying nonlinear multiagent systems by using the dynamic gain method. However, its condition on nonlinear functions seems conservative, and the controller of the ith agent not only requires the agent output and the outputs of its neighbor agents, but also the estimated state information of the neighbors. Furthermore, all of above works did not consider the influence of time delays on the system state. It is well known that time delays are inevitable in many physical systems, which is a source of the instability or performance degradation of the system in Shi, Huang, and Yu (2013). Many methods have been proposed to address the consensus problem of the linear multiagent systems with time delays, see

https://doi.org/10.1016/j.automatica.2019.108669 0005-1098/© 2019 Published by Elsevier Ltd.

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.

2

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Dong, Xi, Lu, and Zhong (2014), Lin and Jia (2010) and Mu, Liao, and Huang (2015) and the references therein. Based on the fact that almost all real physical systems are nonlinear in nature, see Zhang et al. (2015), some interesting results have been presented to study the consensus problem of nonlinear time delay multiagent systems recently. In Chen, Wen, Liu, and Wang (2014), the neural networks-based adaptive control method was proposed to deal with the leader-following consensus problem for a class of nonlinear multiagent with time delays in the state, and then the method was extended to the nonlinear time delay multiagent systems with external noises in Wang, Wang, Liu, Yan, and Wei (2016) and the nonlinear uncertain time delay multiagent systems in Wen, Chen, Liu, and Liu (2017). Although the works also took into account time delays in the system state, there is a common assumption that all state variables of the system are available for measurement. Moreover, their results are semi-global due to the use of neural networks approximation technique under an undirected communication topology. Motivated by the above-mentioned discussions, under a fixed directed topology, this paper studies the output feedback-based leader-following consensus problem for a class of high-order nonlinear time delay multiagent systems. The main contributions are as follows. (i) The condition on nonlinear functions is less conservative than the one in existing works (see Assumption 3). It is the first time that the output feedback-based consensus problem is addressed for nonlinear multiagent systems with time delays on the state under a fixed directed topology. (ii) For each follower, a novel compensator is constructed by using the follower output and the output information of its neighbor agents. The distributed linear-like controller is designed based on the compensator, and it is independent of time delays. (iii) The proposed consensus algorithm has a simple design procedure, which does not use any recursive design methods. Moreover, the algorithm can reduce the number of the communication variables between ith agent and its neighbor agents compared with the existing results (e.g., Zhang et al. (2015)). Notation 1. In this paper, A ∈ ℜn×n , b ∈ ℜn , c ∈ ℜn , k ∈ ℜn and q ∈ ℜn are defined as follows,

⎡

⎤

0

⎢. A = ⎣ ..

0

⎥ ⎦,

In−1 0

···

0

b = [0, 0, . . . , 1] c = [1, 0, . . . , 0]T k = [k1 , k2 , . . . , kn ]T q = [q1 , q2 , . . . , qn ]T , T

where ki and qi are positive constants. The arguments of the functions will be omitted or simplified whenever no confusion can arise from the context. For example, we may denote xik (t) and L(t) by xik and L, respectively. ℜv denotes the real v -dimensional space. ℜm×n and Cm×n denote the space of m × n matrices with real and complex entries, respectively. A ⊗ B denotes the Kronecker product of matrices A and B. λmin (·) is the minimum eigenvalue of a matrix. diag {·, ·} is a diagonal matrix. Ij is a j-dimensional identity matrix. For a matrix G = (gij ) ∈ ℜm×n , its Euclidean norm can be expressed as ∥G∥ =

(∑n ∑m j=1

i=1

|gij |2

) 21

.

2. Preliminaries and problem formulation 2.1. Topology theory (Hua et al., 2017) A topology is described by G = (V , E ), where E = {ν1 , . . . , νN } is a finite nonempty set of nodes, and E = V × V is a set of edges of the topology. (νi , νj ) ∈ E means there is an edge from nodes i to j. Denote the adjacency or connectivity matrix as A = [aij ] ∈ ℜN ×N , where aij > 0 if (νj , νi ) ∈ E and aij = 0 otherwise. The set of neighbors of a node νi is Ni =

{νj |(νj , νi ) ∈ E }. Define the in-degree matrix ∑ as a diagonal matrix D = diag {d1 , d2 , . . . , dN } with di = j∈Ni aij . The Laplacian matrix is expressed as L = D − A. Define the pinning matrix of topology G as B = diag {b1 , b2 , . . . , bN }, where bi > 0 if there exists an edge from the leader to the ith follower and bi = 0 otherwise. If a leader indexed by 0, we can get an augmented topology G¯, which consists of the graph G , the node 0 and edges between the leader and its neighbors. Then the matrix H can be defined as H = L + B. A lemma which plays a key role in the controller design is presented as follows. Lemma 1 (Li, Wen, Duan, & Ren, 2015). There exist functions x ≥ 0, y ≥ 0 and constants p > 0, q > 0 with 1p + 1q = 1, such that xy ≤

xp p

+

yq q

.

(1)

2.2. System formulation Consider a group of N + 1 agents consisting of a leader indexed by 0 and N followers indexed by 1, 2, . . . , N, which may be regarded as a class of nonlinear systems with time delays in the state. The dynamics of ith agent can be described by

⎧˙ xik (t) =xi(k+1) (t) + fk (t , x¯ ik (t), x¯ ik (t − d(t))) ⎪ ⎪ ⎨ 1≤k≤n−1 ⎪ x˙ (t) =ui (t) + fn (t , x¯ in (t), x¯ in (t − d(t))) ⎪ ⎩ in yi (t) =xi1 (t); 0 ≤ i ≤ N ,

(2)

where xij (t) ∈ ℜ, ui (t) ∈ ℜ and yi (t) ∈ ℜ are state variables, control input and output of the ith agent, respectively; and only the output is available for measurement; u0 (t) = 0; x¯ ij (t) = [xi1 (t), . . . , xij (t)]T ; x¯ ij (t − d(t)) = [xi1 (t − d(t)), . . . , xij (t − d(t))]T , ˙ the time delay d(t) and its derivative d(t) satisfy 0 ≤ d(t) ≤ ˙ d∗ and d(t) ≤ d¯ < 1, respectively, in which d∗ and d¯ are known constants; fj (·): ℜ+ × ℜj × ℜj → ℜ denotes an uncertain continuous function. Remark 1. Many real physics systems (e.g., chemical reactors, inverted pendulums, mobile manipulators, etc.) not only have the nonlinear complex feature, but also various delays features. However, many existing works on consensus problem assumed that there exist no delays features especially on the system state. On the other hand, not all state variables of the system can be available for measurement in practice, e.g., for a manipulator, its velocity signal is harder to be available for measurement than the position signal. Therefore, those motivate us to study the output feedback-based consensus control problem for nonlinear multiagent systems with time delays on the state. To the best of our knowledge, this problem is still open and awaits breakthrough. For designing distributed controller for each follower, some assumptions on system (2) are given as follows. Assumption 1. For the function fj (t , x¯ ij , x¯ ijd ), there exist known nonnegative constants γk1 and γk2 satisfying

|fj (t , x¯ ij , x¯ ijd ) − fj (t , x¯ 0j , x¯ 0jd )| ≤

j ∑ k=1

γk1 |xik − x0k | +

j ∑

γk2 |xikd − x0kd |,

(3)

k=1

where x¯ ijd = x¯ ij (t − d(t)) and xijd = xij (t − d(t)). Assumption 2. The topology G is fixed and directed; its augmented topology G¯ contains a spanning tree with the leader as the root.

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.

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The control objective of this paper is to design distributed linear-like controller for each follower, such that each state variable of followers can asymptotically track that of the leader (i.e., limt →∞ |xij − x0j | → 0). Remark 2. If time delays on the state are not considered (i.e., d(t) = 0), the condition on nonlinear terms in Assumption 1 can be found in many existing works. There exist time delays on the state in many real physical systems (e.g., chemical reactors with delayed recycle streams) satisfying the reasonable condition on nonlinear functions in Assumption 1. Assumption 2 is a standard condition, and it can be found in Hua et al. (2019) and Hua et al. (2017). It can be known from the work in Li et al. (2015) that H is a M-matrix and its all eigenvalues λi have positive real parts (i.e., Re(λi ) > 0; i = 1, 2, . . . , N) under Assumption 2. In addition, it should be noted that the condition in Assumption 1 can be relaxed as the one in Assumption 3 (see Section 4). 3. Distributed controller design In order to achieve the output feedback consensus result and reduce the communication burden, we will propose a new output feedback consensus algorithm. Before proposing the algorithm, we first construct a novel dynamic gain compensator for each follower. Define the output consensus error for ith follower as ei =

∑

aij (yi − yj ) + bi (yi − y0 ),

(4)

where Fi = [f¯1i , f¯2i , . . . , f¯ni ]T , f¯ji = fj (t , x¯ ij , x¯ ijd ) − fj (t , x¯ 0j , x¯ 0jd ); A¯ 1 = LC2 (A − bkT )C2−1 , A¯ 2 = LC2 (A − qc T )C2−1 , k and q are defined in Notation 1; C2 = Lι0 C1 , in which C1 = diag {1, L, . . . , Ln−1 }, ι0 is a positive real number. Make a transformation as follows,

ξ¯i = C2−1 ξi ,

{

ξ˙ik =ξi(k+1) + qk Lk (ei − ξi1 ); ξ˙in =ui + qn Ln (ei − ξi1 ),

1≤k≤n−1

Theorem 1. Under a fixed directed topology, for the N + 1 agents satisfying Assumptions 1–2, each state variable of followers can asymptotically track that of the leader, by designing the compensator (5)-based distributed linear-like controller n ∑

Ln−j+1 kj ξij ,

(a) (6)

j=1

˙ = max{−β L2 + ρ L; L(t)

0},

L(0) ≥ 1,

(b)

where kj is a design parameter; β and ρ are positive constants chosen in (28) such that L(t) ≥ 1.

⎧ L˙ ⎪ ⎪ ⎪ ξ˙¯i =L(A − bkT − qc T )ξ¯i − (ι0 In + D)ξ¯i ⎪ ⎪ L ⎪ ⎪ ⎨ + qL1−ι0 ei ⎪ L˙ ⎪ ⎪ η˙¯ i =LAη¯ i + Lqc T ξ¯i − (ι0 In + D)η¯ i ⎪ ⎪ L ⎪ ⎪ ⎩ − qL1−ι0 ei + C2−1 Fi ,

ξi =[ξi1 , ξi2 , . . . , ξin ]T , (7)

=[xi1 − x01 − ξi1 , . . . , xin − x0n − ξin ]T .

where D = diag {0, 1, . . . , n − 1}. Then (10) can be rewritten as L˙ Ψ˙ i = LA¯ Ψi − D¯ Ψi − QL1−ι0 ei + F¯i , L

(11)

[ ] [ ] A − bkT − qc T 0n×n ξ¯i ¯ ,A= , T η¯ i qc A ] [ ] ] [ [ 0n×1 −q 0n×n ¯ = ι 0 In + D ,Q = . , F¯ = D q 0n×n ι0 In + D i C2−1 Fi where Ψi =

From (1), (4), (7) and (9), one has ei =

∑

aij (ηi1 + ξi1 − (ηj1 + ξj1 )) + bi (ηi1 + ξi1 )

j∈Ni

= Lι 0 C T

(∑

)

(12)

aij (Ψi − Ψj ) + bi Ψi ,

where C = [c T , c T ]T . Define Ψ = [Ψ1T , . . . , ΨNT ]T . It follows from (11) and (12) that L˙ Ψ˙ = LAˆ¯ Ψ − IN ⊗ D¯ Ψ + F¯ , L

(13)

where Aˆ¯ = IN ⊗ A¯ − H ⊗ (QC T ), F¯ = [F¯1T , F¯2T , . . . , F¯NT ]T . The following technical lemmas are given for proving stability of the system (13). Lemma 2. The matrix Aˆ¯ is Hurwitz if Aˆ¯ i = A¯ − λi QC T is Hurwitz, where λi (i = 1, 2, . . . , N) is the eigenvalue of the matrix H. Proof. Since H is a M-matrix and its all eigenvalues λi have positive real parts, we can take a unitary matrix U ∈ CN ×N satisfying U ∗ HU = Λ, where Λ is an upper-triangular matrix with λi (i = 1, 2, . . . , N) being its diagonal entries and U ∗ denotes conjugate transpose for the matrix U. A similar transformation shows

=(U ∗ ⊗ I2n )(IN ⊗ A¯ − H ⊗ (QC T ))(U ⊗ I2n ) =IN ⊗ A¯ − Λ ⊗ (QC T ),

(14)

which is a lower triangular block matrix with each diagonal block being Aˆ¯ i . Note that the eigenvalues of Aˆ¯ can be determined by

It follows from (2), (5)–(7) that

⎧ 1−ι0 1−ι0 ˙ ¯ ⎪ ⎨ ξi =A1 ξi + L C2 qei − L C2 qξi1 η˙ i =A¯ 2 ηi + L1−ι0 C2 q(ηi1 + ξi1 ) ⎪ ⎩ − L1−ι0 C2 qei + Fi ,

(10)

ˆ¯ ⊗ I ) (U ∗ ⊗ I2n )A(U 2n

Proof. Define the vectors ξi and ηi as follows,

ηi =[ηi1 , ηi2 , . . . , ηin ]T

(9)

j∈Ni

(5)

where ξij represents the state variable of the compensator, Lk denotes the kth power of L, and L is a dynamic gain designed in (6b), qj is defined in Notation 1. The main result of this section can be summarized as the following theorem.

ui = −

η¯ i = C2−1 ηi .

From (8) and (9), we have

j∈Ni

where aij and bi are defined in the topology theory. For ith follower, the dynamic gain compensator can be constructed as follows,

3

those of Aˆ¯ i . The proof is completed. (8)

Lemma 3. There exist the vectors k, q such that the matrix Aˆ¯ i is Hurwitz.

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.

4

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[ Proof. Define E =

In 0

−In

] . Then a similar transformation is

In

shown as follows, E −1 Aˆ¯ i E = E −1 (A¯ − λi QC T )E = Aˆ¯ i0 , where Aˆ¯ i0 =

[

A − bkT 0

(15)

]

[

0 bkT +(1−λi ) qc T A − λi qc T

0

−qc T

∥C2−1 Fi ∥ ≤

. Based

choosing the vectors k and q, which indicates that the matrix Aˆ¯ i is Hurwitz under the appropriate vectors k and q. The proof is completed. It follows from Lemmas 2–3 that the matrix Aˆ¯ is Hurwitz by choosing the appropriate vectors k and q. Therefore, there must exist the vectors k, q, a positive constant µ and a positive definite matrix P = diag {P1 , P2 , . . . , PN } satisfying

where ξ¯id = ξ¯i (t − d(t)), η¯ id = η¯ i (t − d(t)), Ψid = Ψi (t − d(t)); γ¯1 and γ¯2 are positive constants. It follows from (18) and (21) that 2

≤2

−2

get

N L˙ ∑

L

(18)

ΨiT Pi D¯ Ψi .

˙ ≥ 0, then we can It follows from (6b) that L(t) ≥ 1 and L(t) L˙ ≥ 0. This together with (16)–(18) that L

LΨ T (P Aˆ¯ + Aˆ¯ T P)Ψ ≤ − µLΨ T P Ψ

= − µL

N ∑

−2

L

ΨiT Pi D¯ Ψi = −

i=1

N L˙ ∑

L

N L˙ ∑

L

|f¯ji | ≤

≤

∑

∑

γk1 |xik − x0k | +

k=1

k=1

j ∑

j ∑

γk1 |ηik + ξik | +

k=1

N ∑

2γ¯2 ∥Pi ∥∥Ψi ∥∥Ψid ∥

i=1

≤

N ∑ γ¯ 2 2

δ1

∥Pi ∥∥Ψi ∥2 +

≤

ΨiT Pi Ψi .

∗ ∫ t N ∑ eϱ d e−ϱ(t −s) δ1 ∥Pi ∥∥Ψi (s)∥2 ds, ¯ t −d(t) 1 − d i=1

(24)

where ϱ is a positive constant. The derivative of V2 satisfies

N ∑

∗ N ∑ eϱ d δ ∥P ∥∥Ψi ∥2 ¯ 1 i 1 − d i=1

(25)

δ1 ∥Pi ∥∥Ψid ∥ . 2

i=1

N ∑

ΨiT Pi Ψi − ι0

i=1

− ϱ V2 +

N ∑

N L˙ ∑

L

ΨiT Pi Ψi

i=1

(26)

γ¯ ∥Pi ∥∥Ψi ∥2 ,

i=1

where γ¯ =

γk2 |ηikd + ξikd | (20)

γk1 Lk−1+ι0 |η¯ ik + ξ¯ik |

k=1 k−1+ι0

δ1 ∥Pi ∥∥Ψid ∥2 ,

where δ1 is a positive constant. Choose a candidate Lyapunov function as

V˙ ≤ − µL

γk2 |xikd − x0kd |

k=1

+ γk2 Ld

(23)

N ∑ i=1

) |η¯ ikd + ξ¯ikd | ,

where Ld = L(t − d(t)), ξikd = ξik (t − d(t)), ηikd = ηik (t − d(t)), ξ¯ikd = ξ¯ik (t − d(t)), η¯ ikd = η¯ ik (t − d(t)).

∗ eϱd 1−d¯ 1

δ +

γ¯22 δ1

+ 2γ¯1 .

From (6), (26) and ∥Ψi ∥2 ≤

j

∑(

2γ¯2 ∥Pi ∥∥Ψi ∥∥Ψid ∥.

Take the Lyapunov function for the resulting closed-loop system: V = V1 + V2 . Based on (18)–(25), we can get the derivative of V satisfying

Based on Assumption 1, (7) and (9), we can obtain j

N ∑

From (22) and Lemma 1, we have

(19)

i=1

j

(22)

i=1

−

i=1

≤ − ι0

2γ¯1 ∥Pi ∥∥Ψi ∥2 +

ΨiT Pi Ψi ,

¯ i ) Ψi ΨiT (Pi D¯ + DP

∥Ψi ∥∥Pi ∥∥C2−1 Fi ∥

V˙ 2 ≤ − ϱV2 +

i=1 N L˙ ∑

N ∑

V2 =

i=1

∥Ψi ∥∥Pi ∥∥F¯i ∥

i=1

N ∑

i=1

i=1

N ∑

i=1

(17)

ΨiT Pi F¯i

ΨiT Pi F¯i ≤ 2

i=1

Based on the work in Praly (2003), there exist the strictly positive real numbers ι0 , ¯ι0 satisfying −ι0 Pi ≤ Pi D1 + D1 Pi ≤ ¯ι0 Pi , where D1 = diag{D, D}. Then this inequality can be transformed into

N ∑

N ∑ i=1

≤

where ι1 = 2ι0 + ¯ι0 . In what follows, we choose a candidate Lyapunov function as V1 = Ψ T P Ψ , its derivative along (13) is

(21)

≤γ1 (∥η¯ i ∥ + ∥ξ¯i ∥) + γ2 (∥η¯ id ∥ + ∥ξ¯id ∥) ≤γ¯1 ∥Ψi ∥ + γ¯2 ∥Ψid ∥,

(16)

V˙ 1 =LΨ T (P Aˆ¯ + Aˆ¯ T P)Ψ + 2

L−(j−1+ι0 ) |f¯ji |

j=1

know that the system ζ˙ (t) = A¯ˆ i0 ζ (t) is asymptotically stable by

¯ i ≤ ι1 Pi , ι0 Pi ≤ Pi D¯ + DP

n ∑

]

on Theorems 1-3 in Wen, Zhao, Duan, Yu, and Chen (2016), we

P Aˆ¯ + Aˆ¯ T P ≤ −µP .

Based on (6b) and d(t) ≥ 0, it can be verified that L(t) ≥ L(t − L(t −d(t)) d(t)) ≥ 1, namely L(t) ≤ 1. Then it follows from (10), (20) that

V˙ ≤ − µ ¯L

N ∑

1

λmin (Pi )

ΨiT Pi Ψi − ϱV2

i=1

( −

ΨiT Pi Ψi , we get

) N γ¯ ∥P ∥ ∑ T ρι0 − Ψi P i Ψi , λmin (P)

(27)

i=1

where µ ¯ = µ − ι0 β .

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.

K. Li, C.-C. Hua, X. You et al. / Automatica xxx (xxxx) xxx

4. The extension of the proposed algorithm

For (27), we can take

β<

µ , ι0

ρ>

γ¯ ω , ι0

where ω = max{ λ

(28)

∥P ∥ min (P)

|H ∈ Ω (H)}, Ω (H) is all the cases of

topology matrix H for given N + 1 agents. Then we can ensure ∥ > 0. that µ ¯ > 0 and ρι0 − λγ¯ ∥P(P) min

Substituting (28) into (27) gives V˙ ≤ − µ ¯ V1 − ϱV2 ≤ −χ V ,

(29)

where χ = min{µ, ¯ ϱ}. Due to V = V1 + V2 , we can obtain from (29) that V1 = Ψ T P Ψ ≤ V (t) ≤V (0)e−χ t ;

t ∈ [0, +∞).

V (0)

λmin (P)

e−χ t ,

∥ξ¯ ∥2 ≤

V (0)

λmin (P)

e−χ t .

ξ ≤

V (0)Lm

λmin (P)

e−χ t (32)

2(j+ι0 −1)

2 ij

λmin (P)

e

−χ t

|fj (t , x¯ ij , x¯ ijd ) − fj (t , x¯ 0j , x¯ 0jd )| j ∑

γk1 (t)|xik − x0k | +

j ∑

(33)

γk2 (t)|xikd − x0kd |,

k=1

υ1 t

where γk1 (t) ≤ ϵ1 e and γk2 (t) ≤ ϵ2 eυ2 t , in which ϵ1 and ϵ2 are unknown nonnegative constants, υ1 and υ2 are known nonnegative constants.

(31)

2(j+ι0 −1)

V (0)Lm

Assumption 3. For the function fj (t , x¯ ij , x¯ ijd ), there exist unknown nonnegative time-varying functions γk1 (t) and γk2 (t) satisfying

k=1

(30)

It follows from (6b) that L(0) ≤ L(t) ≤ Lm , where Lm is a ρ constant with Lm = max{ β ; L(0)}. This together with (9) and (31) that

ηij2 ≤

In this section, the condition in Assumption 1 is relaxed as the one in Assumption 3. Then we will propose the output feedbackbased distributed consensus algorithm for the system (2) with the condition in Assumption 3.

≤

Based on (30), one has

∥η∥ ¯ 2≤

5

.

Due to |xij − x0j | = |xij − x0j − ξij + ξij | ≤ |ηij | + |ξij |, we can get from (32) that limt →∞ |xij − x0j | → 0. The proof of Theorem 1 is completed. Remark 3. In Zhang et al. (2015), by designing the common dynamic gain observer for the agents including all followers and leader, for the ith follower, they proposed the distributed controller constituted by the relative outputs and the estimated state information of its neighbor agents. Different from their work, we design the new dynamic gain compensator including the output consensus error ei for all followers. By means of the compensator, our designed controller (6a) for the ith follower only requires its own output and the output information of its neighbor agents, which avoids the communication of the estimated state information of its neighbor agents, such that the number of the communication variable are reduced greatly. Therefore, our proposed compensator (5) has more advantages. Moreover, our proposed output feedback consensus algorithm based on the compensator can address the consensus problem for nonlinear multiagent systems with time delays on the state, while their algorithm is ineffective for such multiagent systems under directed topology. Remark 4. By designing the new dynamic gain L(t) in (6b), the consensus problem of nonlinear multiagent systems with time delays on the state can be addressed. By choosing β and ρ in (25), we can conclude that L(t) is a nondecreasing time-varying function, and it is bounded with 1 ≤ L(0) ≤ L(t) ≤ Lm . Moreover, since the time delay d(t) is a nonnegative bounded time-varying function, we can verify easily that L(t) ≥ L(t − d(t)) ≥ 1. The result in Theorem 1 is obtained based on a conservative condition on nonlinear terms (namely, the Lipschitz growth rates are known nonnegative constants γk1 , γk2 ). It should be noted that by modifying slightly the dynamic gain L(t) in (6b), the condition on nonlinear terms in Assumption can be relaxed greatly. The corresponding algorithm is proposed in the next section.

Remark 5. In this section, since the Lipschitz growth rates are unknown nonnegative time-varying functions γk1 (t), γk2 (t), the condition in Assumption 3 contains the one in Assumption 1 as a special case. Compared with the work in Zhang et al. (2015), our proposed Lipschitz growth rates can be unknown nonnegative time-varying functions, while their Lipschitz growth rates cannot, hence our condition on nonlinear terms is less conservative. Moreover, we not only consider time-varying delays on the state of the system, but also are based on directed communication topology to solve the consensus problem. Therefore our work is more challenging. In what follows, we give the main result of this section, which can be summarized as the following theorem. Theorem 2. Under a fixed directed topology, for the N + 1 agents satisfying Assumptions 2–3, each state variable of followers can asymptotically track that of the leader by designing the compensator (5)-based distributed linear-like controller in (6a) with

˙ = max{−β L2 + ρ (t)L; L(t)

(34)

0},

ν1 t

where ρ (t) = ν0 e , ν0 is an arbitrary positive constant, ν1 and β are designed in (36); L(0) should be chosen in (44), such that L(t) ≥ 1. Proof. The proof procedure is similar to that in the previous section, we only need to define γk1 := γk1 (t), γk2 := γk2 (t) in (20); γ¯1 := γ¯1 (t), γ¯2 := γ¯2 (t) in (21)–(22); γ¯ := γ¯ (t) in (26)–(28). Then we have V˙ ≤ − µ ¯L

N ∑

ΨiT Pi Ψi − ϱV2

i=1

( −

e

σ¯ 1 t

(35)

) N ∑ σ0 − ν0 ι0 eσ1 t ΨiT Pi Ψi , ν0 ι0 i=1

where µ ¯ = µ − ι0 β , σ¯ 1 = ν1 − σ1 , σ0 eσ1 t = λ (P) γ¯ (t), and σ0 min is an unknown nonnegative constant, σ1 is a known nonnegative constant; ϱ is chosen in (44). For (35), we can choose ∥P ∥

β<

µ ι0

and

ν1 > σ1 .

(36)

There must exist a time t1 < +∞ satisfying eσ¯1 t −

σ0 ≥ 0; ν0 ι0

t ∈ [t1 , +∞).

(37)

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.

6

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From (35)–(37), one has V˙ ≤ − χ1 V ;

t ∈ [t1 , +∞),

(38)

where χ1 = min{µ ¯ L(0), ϱ}. Based on (38), we can obtain t ∈ [t1 , +∞),

V (t) ≤V (t1 )e−χ1 (t −t1 ) ;

(39)

which means that V1 = Ψ P Ψ ≤ V (t1 )e Then one has T

∥η∥ ¯ 2 ≤ Vtp e−χ1 (t −t1 ) , ∥ξ¯ ∥2 ≤ Vtp e−χ1 (t −t1 ) ;

−χ1 (t −t1 )

on t ∈ [t1 , +∞).

(40)

t ∈ [t1 , +∞),

V (t )

where Vtp = λ 1(P) . min It follows from (9) and (40) that

ηij2 ≤ V¯ tp Ln¯ j (t)e−χ1 t , ξij2 ≤ V¯ tp Ln¯ j (t)e−χ1 t ;

(41)

t ∈ [t1 , +∞),

where V¯ tp = Vtp eχ1 t1 and n¯ j = 2(j + ι0 − 1). It can be verified that L(t) ≤ max{L(0), Then we can get

{

ρ (t) } β

based on (34).

}

n¯ j

Ln¯ j (t) ≤ max L0 ; ν¯ 0j eν¯ 1j t , ν

(42)

n¯ j

where ν¯ 0i = ( β0 )n¯ j , L0 = Ln¯ j (0) and ν¯ 1j = ν1 n¯ j . On t ∈ [t1 , +∞), it follows from (41)–(42) that

} { n¯ j ν¯ 1j t ¯ e−χ1 t , η ≤ Vtp max L0 ; ν¯ 0i e { } n¯ ξij2 ≤ V¯ tp max L0j ; ν¯ 0i eν¯1j t e−χ1 t . 2 ij

(43)

By taking

ϱ ≥¯ν1(n+1) + χ¯ 1 , (44) ν¯ 1(n+1) + χ¯ 1 L(0) ≥ max{1; }, µ ¯ where χ¯ 1 is an arbitrary positive constant, we can ensure that χ1 > ν¯ 1j (j = 1, 2, . . . , n) in (43). This together with (44) that lim ηij2 (t) → 0,

t →∞

lim ξij2 (t) → 0,

(45)

t →∞

which indicates that limt →∞ |xij (t) − x0j (t)| → 0. The proof is completed for Theorem 2. Remark 6. By taking ϱ and L(0) in (44), we can ensure that the proposed distributed linear-like controller (6a) with (34) is effective under the condition in Assumption 3, such that our control objective can be achieved. Moreover, on t ∈ [t1 , +∞), it follows from (6a), and (42)–(44) that

  ∑  n n¯ 2 2 ˆ −0.5χ¯ t 1 , |ui (t)|≤√n L j kj ξij ≤ V¯ tp e

(46)

j=1

√ where Vˆ¯ tp =

{

n¯ (n+1)

n∥k∥2 V¯ tp max L0

} ; ν¯ 0(n+1) . Then we can get

limt →∞ ui (t) → 0, which means that ui (t) is bounded on t ∈ [0, +∞). It should be noted that if d¯ and d∗ are unknown constants in Theorem 2, our proposed algorithm is still valid. Remark 7. In Zhang et al. (2015), they study the output feedback consensus control problem for nonlinear multiagent systems with a conservative condition on nonlinear functions under a fixed undirected communication topology. Different from their work,

Fig. 1. The communication topology.

we consider time delays on the state of the system, and our results are obtained based on a fixed directed topology under the less conservative condition on nonlinear terms. Moreover, our output feedback-based algorithm proposed by means of the compensator (5) can reduce communication burden between ith agent and its neighbor agents. Therefore our obtained results are more general. Remark 8. Although the works in Chen et al. (2014), Wang et al. (2016) and Wen et al. (2017) have addressed the leaderfollowing consensus problem for nonlinear multiagent systems with time delays on the state, their results are based on state feedback control under an undirected connected graph, and the resulting closed-loop system is stable in the sense of semi-global boundedness due to the use of neural networks. Their results seem conservative. Different from their works, we propose the new output feedback-based consensus algorithms under a fixed directed communication topology, which render that each state variable of followers can asymptotically track that of the leader in the global sense. Furthermore, the proposed consensus algorithm can be extended directly to deal with the containment control problem of nonlinear multiagent systems with time delays on the state. 5. Simulation example In this section, to verify the effectiveness of our proposed consensus protocols, we consider the following two-stage chemical reactors with delayed recycle streams as agents (e.g., the isothermal continuous stirred tanks reactors, see Jia, Chen, Xu, Zhang, and Zhang (2017)) based on the fixed directed communication topology shown in Fig. 1. The dynamics can be described by

⎧ 1 ⎪ ⎪ x˙ i1 = − xi1 − ωi1 xi1 + ⎪ ⎪ θ ⎪ i1 ⎪ ⎪ ⎪ 1 ⎨ x˙ i2 = − xi2 − ωi2 xi2 + θi2 ⎪ ⎪ mi2 ⎪ ⎪ + ui + δ¯ 2 (·) ⎪ ⎪ ϖ ⎪ i2 ⎪ ⎩ yi =xi1 ,

1 − ri2

ϖi1 ri1

ϖi2

xi2 + δ¯ 1 (·)

xi1 (t − d(t))

(47)

where xi1 (t) and xi2 (t) are compositions, ri1 and ri2 are recycle flow rates, θi1 and θi2 are reactor residence times, ωi1 and ωi1 are reaction constants, mi2 is the feed rate, ϖi1 and ϖi2 are reactor volumes, δ¯ 1 (·) and δ¯ 2 (·) are nonlinear functions for describing the system uncertainties and external disturbances. Take θi1 = θi2 = 10, ωi1 = 0.02, ωi2 = 0.05, ri1 = ri2 = 0.2, ϖi1 = ϖi2 = 0.8, mi2 = 0.8, δ¯1 (·) = 0.03x1 , δ¯2 (·) = 0.25xi2 (t − d). d(t) = 0.6 + 0.2 sin(t). It can be verified that Assumptions 1–3 hold in system (47). Then we take d∗ = 0.8, d¯ = 0.2, ϱ = 0.1, q1 = 2, q2 = 1, k1 = 1, k2 = 3, µ = 0.22, ι0 = 1, ι1 = 6, β = 0.21, ρ = 295.2. The initial values are chosen as x01 (0) = −0.5, x02 (0) = 1, x11 (0) = −0.3, x12 (0) = 12, x21 (0) = −0.6, x22 (0) = −15, x31 (0) = 0.4, x32 (0) = −5, x41 (0) = −0.2, x42 (0) = 10,

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.

K. Li, C.-C. Hua, X. You et al. / Automatica xxx (xxxx) xxx

Fig. 2. The responses of yi (t), i = 0, 1, 2, 3, 4.

7

Fig. 7. The responses of ui (t), i = 0, 1, 2, 3, 4.

From Figs. 2–5, we can see that each state variable of followers can asymptotically track that of the leader. The dynamic gain is shown in Fig. 6, from which one sees that it is bounded. The input signals are shown in Fig. 7. The simulation results illustrate that our proposed consensus protocols are effective. 6. Conclusion Fig. 3. The responses of yi (t) − y0 (t), i = 1, 2, 3, 4.

Fig. 4. The responses of xi2 (t), i = 0, 1, 2, 3, 4.

Fig. 5. The responses of xi2 (t) − x02 (t), i = 1, 2, 3, 4.

Fig. 6. The response of L(t).

ξ11 (0) = −0.4, ξ12 (0) = 0.1, ξ21 (0) = 0.1, ξ22 (0) = −0.2, ξ31 (0) = 0.1, ξ32 (0) = 0.3, ξ41 (0) = −0.2, ξ42 (0) = 0.1, L(0) = 10. The simulation results are shown in Figs. 2–7.

Under a fixed directed graph, this paper proposes the output feedback distributed consensus control algorithm for nonlinear time delay multiagent systems. The proposed algorithm can not only relax the condition on nonlinear terms, but also reduce communication burden between ith agent and its neighbor agents. The proposed consensus protocols are distributed, which are independent of the time delays in the system state. Moreover, the protocols can deal with the consensus problem of the chemical reactor systems. The future work is to study the output feedback-based distributed containment control problem for nonlinear multiagent systems with nonidentical time delays in the state under switching directed topologies. References Chen, C., Wen, G., Liu, Y., & Wang, F. (2014). Adaptive consensus control for a class of nonlinear multiagent time-delay systems using neural networks. IEEE Transactions on Neural Networks and Learning Systems, 25(6), 1217–1226. Dong, X., Xi, J., Lu, G., & Zhong, Y. (2014). Formation control for high-order linear time-invariant multiagent systems with time delays. IEEE Transactions on Control of Network Systems, 1(3), 232–240. Fan, Y., Feng, G., Wang, Y., & Song, C. (2013). Distributed event-triggered control of multi-agent systems with combinational measurements. Automatica, 49(2), 671–675. Hua, C., Li, K., & Guan, X. (2019). Leader-following output consensus for high order nonlinear multiagent systems. IEEE Transactions on Automatic Control, 64(3), 1156–1161. Hua, C., You, X., & Guan, X. (2016). Leader-following consensus for a class of high-order nonlinear multi-agent systems. Automatica, 73(11), 138–144. Hua, C., You, X., & Guan, X. (2017). Adaptive leader-following consensus for second-order time-varying nonlinear multiagent systems. IEEE Transactions on Cybernetics, 47(6), 1532–1539. Jia, X., Chen, X., Xu, S., Zhang, B., & Zhang, Z. (2017). Adaptive output feedback control of nonlinear time-delay systems with application to chemical reactor systems. IEEE Transactions on Industrial Electronics, 64(6), 4792–4799. Li, S., Du, H., & Lin, X. (2011). Finite-time consensus algorithm for multiagent systems with double-integrator dynamics. Automatica, 47(8), 1706–1712. Li, Z., Ren, W., Liu, X., & Fu, M. (2013). Distributed containment control of multiagent systems with general linear dynamics in the presence of multiple leaders. International Journal of Robust & Nonlinear Control, 23(5), 534–547. Li, Z., Wen, G., Duan, Z., & Ren, W. (2015). Designing fully distributed consensus protocols for linear multiagent systems with directed graphs. IEEE Transactions on Automatic Control, 60(4), 1152–1157. Lin, P., & Jia, Y. (2010). Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies. IEEE Transactions on Automatic Control, 55(3), 778–784. Mu, B., Chen, J., Shi, Y., & Chang, Y. (2017). Design and implementation of nonuniform sampling cooperative control on a group of two-wheeled mobile robots. IEEE Transactions on Industrial Electronics, 64(6), 5035–5044.

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K. Li, C.-C. Hua, X. You et al. / Automatica xxx (xxxx) xxx

Mu, N., Liao, X., & Huang, T. (2015). Event-based consensus control for a linear directed multiagent system with time delay. IEEE Transactions on Circuits and Systems II: Express Briefs, 62(3), 281–285. Mu, B., Zhang, K., & Shi, Y. (2017). Integral sliding mode flight controller design for a quadrotor and the application in a heterogeneous multi-agent system. IEEE Transactions on Industrial Electronics, 64(12), 9389–9398. Praly, L. (2003). Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate. IEEE Transactions on Automatic Control, 48(6), 1103–1108. Shi, Y., Huang, J., & Yu, B. (2013). Robust tracking control of networked control systems: application to a networked DC motor. IEEE Transactions on Industrial Electronics, 60(12), 5864–5874. Shi, Y., Qin, J., & Ahn, H. (2017). Distributed coordination control and industrial applications. IEEE Transactions on Industrial Electronics, 64(6), 4967–4971. Wang, J. (2017). Distributed coordinated tracking control for a class of uncertain multiagent systems. IEEE Transactions on Automatic Control, 62(7), 3423–3429. Wang, F., Chen, B., Lin, C., & Li, X. (2017). Distributed adaptive neural control for stochastic nonlinear multiagent systems. IEEE Transactions on Cybernetics, 47(7), 1795–1803. Wang, Z., Wang, D., Liu, D., Yan, P., & Wei, Q. (2016). Neural network-based distributed adaptive robust control for a class of nonlinear multiagent systems with time delays and external noises. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46(6), 750–758. Wen, G., Chen, C., Liu, Y., & Liu, Z. (2017). Neural network-based adaptive leaderfollowing consensus control for a class of nonlinear multiagent state-delay systems. IEEE Transactions on Cybernetics, 47(8), 2151–2160. Wen, G., Zhao, Y., Duan, Z., Yu, W., & Chen, G. (2016). Containment of higherorder multi-leader multi-agent systems: a dynamic output approach. IEEE Transactions on Automatic Control, 61(4), 1135–1140. Zhang, X., Liu, L., & Feng, G. (2015). Leader-follower consensus of time-varying nonlinear multi-agent systems. Automatica, 52(2), 8–14. Zuo, Z., Tian, B., Defoort, M., & Ding, Z. (2018). Fixed-time consensus tracking for multiagent systems with high-order integrator dynamics. IEEE Transactions on Automatic Control, 63(2), 563–570.

Kuo Li received the B.S. degree in the College of Automation and Electrical Engineering, Qingdao University, Qingdao, China, in 2015. He is currently pursuing the Ph.D. degree in the School of Electrical Engineering, Yanshan University, Qinhuangdao, China. His current research interests mainly include control of switched systems, control of nonlinear systems, and cooperative control of multiagent systems.

Chang-Chun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He was a research Fellow in National University of Singapore from 2006 to 2007. From 2007 to 2009, he worked in Carleton University, Canada, funded by Province of Ontario Ministry of Research and Innovation Program. From 2009 to 2010, he worked in University of Duisburg-Essen, Germany, funded by Alexander von Humboldt Foundation. Now he is a full Professor in Yanshan University, China. He is the author or coauthor of more than 120 papers in mathematical, technical journals, and conferences. He has been involved in more than 15 projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. He is Cheung Kong Scholars Programme Special appointment professor. His research interests are in nonlinear control systems, multiagent systems, control systems design over network, teleoperation systems and intelligent control. Xiu You received her B.S., M.S. and Ph.D. degrees in Automation from Yanshan University, Qinhuangdao, China, in 2012, 2014 and 2018 respectively. She is currently an associate professor at the School Mathematical Sciences, Shanxi University. She was awarded the nomination for excellent doctoral dissertation from Chinese Association of Automation in 2018. Her current research interests include nonlinear control, multiagent systems and networked control systems.

Xinping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering, both from Harbin Institute of Technology, in 1986, 1991, and 1999, respectively. He is with the Department of Automation, Shanghai Jiao Tong University. He is the (co)author of more than 200 papers in mathematical, technical journals, and conferences. As (a)an (co)-investigator, he has finished more than 20 projects supported by National Natural Science Foundation of China (NSFC), the National Education Committee Foundation of China, and other important foundations. He is Cheung Kong Scholars Programme Special appointment professor. His current research interests include networked control systems, robust control and intelligent control for complex systems and their applications. Dr. Guan is serving as a Reviewer of Mathematic Review of America, a Member of the Council of Chinese Artificial Intelligence Committee, and Vice-Chairman of Automation Society of Hebei Province, China.

Please cite this article as: K. Li, C.-C. Hua, X. You et al., Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica (2019) 108669, https://doi.org/10.1016/j.automatica.2019.108669.