Prescribed-time decentralized regulation of uncertain nonlinear multi-agent systems via output feedback

Systems & Control Letters 137 (2020) 104640

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Prescribed-time decentralized regulation of uncertain nonlinear multi-agent systems via output feedback✩ ∗

Xiandong Chen a , Xianfu Zhang a , , Qingrong Liu b a b

School of Control Science and Engineering, Shandong University, Jinan, 250061, PR China School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, PR China

article

info

Article history: Received 23 August 2019 Received in revised form 1 December 2019 Accepted 26 January 2020 Available online xxxx Keywords: Multi-agent systems Strict feedback form Prescribed-time regulation Decentralized control Time-scaling technique

a b s t r a c t This paper addresses the global prescribed-time decentralized regulation problem for a family of uncertain nonlinear multi-agent systems. The dynamics of each agent is supposed to be in the strict feedback form, where the uncertain nonlinearities admit Lipschitz growth conditions with an unknown positive constant gain multiplying by a time-varying continuous function gain. It is firstly defined error systems, which are composed of the leader and their followers, and shown that the prescribedtime decentralized regulation problem of the original systems is equivalent to the prescribed-time regulation problem of error systems. Then, by introducing a novel state transformation involving a time-scaling function, the prescribed-time regulation problem is then converted into designing appropriate gain parameters. Finally, based on the Lyapunov stability theorem, the prescribed-time regulation of the closed-loop systems is proved and the prescribed-time decentralized regulation of the original systems is thus guaranteed. An example is presented to show the feasibility of the proposed prescribed-time decentralized protocols. © 2020 Elsevier B.V. All rights reserved.

1. Introduction The multi-agent system has attracted much attention in recent years due to its broad range of applications such as unmanned air vehicles, microgrids, mobile robots and multiple agents [1–4]. The regulation problem for multi-agent systems as a generalization of the leader-follower consensus problem [5] has received increasingly attention for linear multi-agent systems [6]. The asymptotic control problems have also been intensively investigated for nonlinear multi-agent systems as the nonlinearity is ubiquitous in physical phenomena [7–9]. Specifically speaking, some researchers have been focusing on the cases when the agents can be described as the strict feedback forms [10–12]. The asymptotic consensus problem was investigated in [11] for nonlinear multi-agent systems whose nonlinearities satisfy Lipschitz conditions with known constant gains. [12] presented the dynamic high-gain method to study the asymptotic consensus of nonlinear multi-agent systems, where the nonlinearities are dominated by Lipschitz conditions with known time-varying function ✩ The work was supported by the National Natural Science Foundation of China (61973189, 61873334, 61503214 and 61573215), the Research Fund for the Taishan Scholar Project of Shandong Province of China (ts20190905), and the Foundation for Innovative Research Groups of National Natural Science Foundation of China (61821004). ∗ Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (X. Zhang), [email protected] (Q. Liu). https://doi.org/10.1016/j.sysconle.2020.104640 0167-6911/© 2020 Elsevier B.V. All rights reserved.

gains. [10] studied the asymptotic consensus problems for a class of nonlinear multi-agent systems, where the nonlinearities satisfy Lipschitz conditions with unknown constant gains multiplying by time-varying function gains. Sometimes, it may be desirable to achieve the control objective in a finite time and several kinds of finite-time consensus algorithms have been proposed for linear multi-agent systems [13,14]. For nonlinear multi-agent systems whose nonlinearities satisfy strict feedback forms, [15] studied the finite-time consensus problem for such systems whose nonlinearities satisfy Lipschitz conditions with time-varying function gains. [16] investigated the finite-time consensus problem of such systems whose nonlinearities satisfy Lipschitz conditions with unknown positive constant gains multiplying by time-varying function gains. However, the finite-time control methods proposed in [15,16] cannot guarantee the convergence within prescribed finite time, since the finite-time interval is depended on both initial conditions and some parameters. Motivated by this observation, the fixedtime stabilization is first proposed in [17] to make the finite-time interval be independent of initial states, and since then, the fixedtime method has also been widely used to investigate the control design for linear and nonlinear multi-agent systems [18–21] However, all the above results of finite-time and fixed-time control have one common feature, that is, the finite-time interval cannot be prescribed according to the task requirements. Moreover, as stated in [22], many practical engineering tasks need to be accomplished in a prescribed-time duration. For some

2

X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

mechanical systems, controller switch is always involved so that the system states could change from the current one to the desired one at some prescribed time. For the practical purposes, the maneuver termination time has to be chosen to drive the states to satisfy a given accuracy. These requirements make the prescribed-time control law play a key role, and hence it is very significant and challenging to design the finite-time control strategy whose finite-time interval can be prescribed in advance. Therefore, many works have considered the prescribed-time control problems for normal-form nonlinear systems, control-affine systems and nonlinear nonholonomic systems [22–24] and the prescribed-time observer design of linear systems [25]. However, few works considered the prescribed-time control problem of multi-agent systems, and the existing prescribed-time control protocols were just proposed for linear multi-agent systems [26,27]. To the authors’ best knowledge, no work has been done for the prescribed-time control of nonlinear multi-agent systems, and motivated by the above results and analysis, designing prescribed-time decentralized regulation protocols for strict feedback nonlinear multi-agent systems on nonlinear manifold is a meaningful and challenging work and thus is considered in this paper. The main contributions of this paper are as follows: (i) Novel control protocols are proposed to achieve the global prescribedtime decentralized regulation of a class of uncertain nonlinear multi-agent systems by the high-gain method for the first time, where the nonlinearities of the original systems admit Lipschitz growth conditions with an unknown constant gain multiplying by a time-varying function gain. Specifically speaking, through introducing a state transformation involving a time-scaling function, the original systems can be transformed into a class of systems easier to construct prescribed-time protocols from the existing asymptotic control strategies under some modifications, which is obviously different from the existing control protocols in [15,18,27]. (ii) To propose the prescribed-time decentralized protocols, the prescribed-time high-gain observers are first proposed for the uncertain nonlinear multi-agent systems. In detail, the existed high-gain observers proposed in [12] focused on the asymptotic consensus of strict feedback nonlinear multi-agent systems, which cannot be applied to the prescribed-time control design. Moreover, the prescribed-time observer in [25] is designed for a class of linear systems and its high-gain parameters are difficult to obtain. Compared with these observers, the highgain observers proposed here can effectively compensate for the uncertainties and time variations in the considered systems and have a simpler form, and its high-gain parameters can be given in advance. The remainder of this paper is organized as follows. In Section 2, some mathematical preliminaries and the problem formulation are introduced. The global prescribed-time decentralized regulation problem is investigated in Section 3. Section 4 shows a simulation example to illustrate the proposed theoretical results. Finally, we give a conclusion in Section 5. 2. Preliminaries and problem formulation In this section, we present some preliminaries and the problem formulation. Denote by R the field of real numbers and Rm×n the set of m × n real matrices. Let ∥ · ∥ denote the Euclidean norm for vectors, or the induced Euclidean norm for matrices. Im is used to represent an identity matrix of m dimension. A1 ⊗ A2 denotes the Kronecker product of matrices A1 and A2 . For a symmetric matrix Q , λmax (Q ) and λmin (Q ) denote the largest and smallest eigenvalue of Q , respectively. The argument of functions will be omitted or simplified whenever no confusion can arise from the context. For example, we may denote xi (t) by xi .

2.1. Graph theory For later development, several basic concepts of graph theory are introduced for the uncertain nonlinear multi-agent systems. Let G = (V , E , A) be a weighted directed graph with the set of nods V = {1, . . . , N }, the set of directed edges E ⊆ V × V and a weighted adjacency matrix A = (aij ) ∈ RN ×N . An edge of E from node i to node j is denoted by (i, j), where the nodes i and j are called the parent and child nodes, respectively. A path on G between i1 and il is a sequence of edges of the form (ik , ik+1 ), k = 1, . . . , l − 1. The weighted adjacency matrix A = (aij ) is nonnegative, where aii = 0 indicates the graph G has not any self-loop, (j, i) ∈ E indicates aij > 0, otherwise aij = 0. A graph is simple if there is no self-loops or repeated edges. A directed tree is a directed graph where every nod i, i = 2, . . . , N, except the root nod 1, has exactly one parent, and there exists a unique directed path from 1 to i. A directed spanning tree of a network G is a directed tree, which contains all the nods of G . Furthermore, the Laplacian matrix L = (lij )N ×N of G is defined ∑N as: lij = −aij , i ̸ = j and lij = k=1,k̸ =i aik , i = j where i, j = 1, 2, . . . , N. Since the leader has no neighbors, the Laplacian matrix L associated with the interaction graph G can be partitioned as

( L=

0

0TN −1

⋆

Lˆ

) ,

where ⋆ ∈ RN −1 and Lˆ ∈ R(N −1)×(N −1) . 2.2. Problem formulation Consider the following uncertain nonlinear multi-agent systems described by x˙ k = Axk + Buk + f (t , xk ), yk = Cxk , k = 1, . . . , N ,

(1)

where xk = (xk,1 , . . . , xk,n )T ∈ Rn , uk ∈ R and yk ∈ R are the state, input and output of agent k, respectively. The matrices (A, B) are a Brunovsky canonical pair(A = (aˆ ij )n×n , aˆ ij = 1, i = 1, . . . , n − 1, j = i + 1 and aˆ ij = 0, j ̸ = i + 1, and B = (0 0 · · · 0 1)T ), C = (1 0 · · · 0) and the nonlinearity is a n × 1 vector such that f (t , xk ) = (f1 (t , xk ) · · · fn (t , xk ))T with fi (t , xk ) being unknown functions, continuous and locally Lipschitz here. In our formulation, the agent indexed by 1 is referred as the leader and agents indexed by 2, . . . , N are called the followers. Since the control of nonlinear systems in general case is an open problem and there is not a systematic approach to design control protocols for these systems. Therefore, in this section, we consider the global prescribed-time decentralized regulation problem of system (1) under the following assumption on the unknown system nonlinearities f (t , xk ), k = 1, . . . , N. Assumption 1. There exist an unknown constant c and a known continuous function g(t), such that, for any (t , xk ) ∈ R+ × Rn and (t , xl ) ∈ R+ × Rn , k, l = 1, . . . , N,

|fi (t , xk ) − fi (t , xl )| ≤ cg(t)

i ∑

|xk,j − xl,j |, i = 1, . . . , n.

(2)

j=1

Assumption 1 can be viewed as a generalized form of the wellknown Lipschitz conditions [10]. Detailedly, due to the unknown constant c, Assumption 1 means that the sizes of nonlinearities f (t , xk ) are unknown. In this sense, we say that the system nonlinearities f (t , xk ) allow large uncertainties. More importantly, system (1) satisfying Assumption 1 is an essential time-varying system, in the sense that g(t) can be any continuous function

X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

well-defined on [t0 , +∞). This work also generalizes many research results of multi-agent systems with each agent being in the strict feedback form.

• The asymptotic consensus problem was considered in [11] for system (1) under Assumption 1 with g(t) ≡ 1 and c being a known positive constant.

• Distributed asymptotic consensus protocols were proposed in [12] for system (1) under Assumption 1 with c being a known positive constant. • The asymptotic consensus problem was investigated in [10] for system (1) under Assumption 1. • The finite-time consensus problem was considered in [15] for system (1) under Assumption 1 with c being a known positive constant. The objective of this paper is to design the protocols uk , k = 2, . . . , N for each follower based on its neighbors’ information, such that the global prescribed-time decentralized regulation of system (1) with the communication topology G is achieved, that is, through constructing output feedback controllers uk , k = 2, . . . , N in combination with time-varying high-gain observers zk , k = 1, . . . , N, all errors ∥xk (t) − x1 (t)∥, ∥zk (t) − z1 (t)∥, k = 2, . . . , N are bounded on [t0 , t0 +t ∗ ) and satisfy limt →t0 +t ∗ ∥xk (t)− x1 (t)∥ = 0, limt →t0 +t ∗ ∥zk (t) − z1 (t)∥ = 0, k = 2, . . . , N. Remark that it is very challenging to propose prescribedtime decentralized regulation protocols for system (1) under Assumption 1. Specifically speaking, a rather unified framework of the prescribed-time consensus for linear multi-agent systems was proposed in [26,28,29]. However, this framework hardly deals with the existence of the nonlinear terms f (t , xk ), especially for large uncertainties and essential time-varying features in (2). For the desired protocols, it is critical to introduce a certain mechanism which not only can effectively deal with system nonlinearities but also easily counteract system uncertainties and tackle essential time-varying features. To achieve this, motivated by [23–25,30,31], the following monotonically increasing function is introduced for our control design

µ(t − t0 ) =

t∗ t ∗ + t0 − t

, t ∈ [t0 , t0 + t ), ∗

(3)

where t ∗ > 0 is freely prescribed and independent of initial conditions with the properties that µ(0) = 1 and limt →t0 +t ∗ µ(t − t0 ) = +∞. The following definitions and lemmas are necessary to serve as the basis for the development of our desired time-varying protocols. Definition 2 ([23,24]). The system x˙ = f (t , x), (t , x) ∈ R+ × Rn with f (t , 0) ≡ 0 is said to be globally prescribed-time convergent to zero in time t ∗ , if for all x(t0 ) ∈ Rn , system states x(t) are well-defined on [t0 , t0 + t ∗ ) and satisfy limt →t0 +t ∗ ∥x(t)∥ = 0. Definition 3 ([23,24]). The system x˙ = f (t , x), (t , x) ∈ R+ × Rn with f (t , 0) ≡ 0 is said to be globally prescribed-time asymptotically stable in time t ∗ , if for all x(t0 ) ∈ Rn , there exists a class KL function ζ such that system states x(t) are well-defined on [t0 , t0 + t ∗ ) and satisfy

∥x(t)∥ ≤ ζ (∥x(t0 )∥, µ(t − t0 ) − 1)

(4)

with µ(t − t0 ) determined by (3). Notice that the function µ(t − t0 ) − 1 starts from zero at t = t0 and increases monotonically to infinity as t → t0 + t ∗ . Therefore, the function ζ (∥x(t0 )∥, µ(t − t0 ) − 1) decays to zero as t → t0 + t ∗ , i.e., at a time that is prescribed by t ∗ .

3

Lemma 4 ([32]). The following propositions are equivalent for a matrix Lˆ = (ˆlij )(N −1)×(N −1) ∈ R(N −1)×(N −1) . (a) Lˆ is an M-matrix. (b) ˆlij ≤ 0, i ̸ = j and there exists a positive definite diagonal matrix R = diag{r1 , . . . , rN −1 } such that LˆR + RLˆT > 0. (c) ˆlij ≤ 0, i ̸ = j and each of its eigenvalues has the positive real part. Remark 5. From (b) of Lemma 4, there exists a positive definite matrix R = diag{r1 , . . . , rN −1 } such that LˆR + RLˆT is a positive definite matrix, that is, a positive constant η can be found to satisfy LˆR + RLˆT > ηI. Lemma 6 ([33]). With the definition of A and B in (1), there exists a positive definite matrix P such that PA + AT P − PBBT P = −I . Lemma 7 ([33]). Suppose the matrix D ∈ Rn×n is defined as D = diag{1, 2, . . . , n} and P ∈ Rn×n is a positive defined matrix. Then, a positive constant α can be found such that PD + DP ≥ α I . Lemma 8 ([34]). With the definition of A and C in (1), there exists a column vector K2 = (g2,1 , . . . , g2,n )T ∈ Rn×1 such that A − K2 C is a Hurwitz matrix. 3. Main results This section is devoted to proposing a constructive procedure to design the global prescribed-time decentralized regulation protocols for the uncertain nonlinear multi-agent system (1) via output feedback control. Firstly, for k = 1, . . . , N, a high-gain observer for agent k is designed as z˙k = Azk + Buk + Γ −1 K2 (yk − zk,1 ),

(5)

where zk = (zk,1 , . . . , zk,n{) is the observer state, } K2 is given in Lemma 8, and Γ = diag 1/µ1+δ , . . . , 1/µn(1+δ ) with δ being a positive constant is a scaling matrix. Then, let T

x˜ k = xk − x1 , z˜k = zk − z1 , k = 2, . . . , N ,

(6)

and one can easily get the following error systems x˙˜ k = Ax˜ k + B(uk − u1 ) + f (t , xk ) − f (t , x1 )

(7)

and z˙˜k = Az˜k + B(uk − u1 ) + Γ −1 K2 (x˜ k,1 − z˜k,1 ).

(8)

Thus, under the condition that the communication topology G is connected, the global prescribed-time decentralized regulation of system (1) is solved, if the global prescribed-time regulation of the error systems (7) and (8) is solved via constructing output feedback protocols uk , k = 2, . . . , N. Let e˜ k = x˜ k − z˜k , k = 2, . . . , N ,

(9)

and in order to proceed the global prescribed-time regulation controllers for system (7) and (8), the following state transformations are introduced zˆk = Γ z˜k , eˆ k = Γ e˜ k , k = 2, . . . , N ,

(10)

which indicates the following matrix forms µ ˙

z˙ˆk = µ1+δ Azˆk − (1 + δ ) µ Dzˆk + µ1+δ K2 C eˆ k

+

1 B(uk µn(1+δ)

− u1 )

(11)

4

X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

and

Meanwhile, constructing the controllers

µ ˙ e˙ˆ k = µ1+δ (A − K2 C )eˆ k − (1 + δ ) µ Deˆ k + Γ (f (t , xk ) − f (t , x1 ))

(12)

with D given in Lemma 7. With the description before, we are ready to present our main result below. Theorem 9. Under Assumption 1 and the communication topology G , positive constants δ and γ , and positive definite matrix P can be chosen such that, the global prescribed-time decentralized regulation of system (1) is achieved by

∑N

uk = γ µ(1+δ )(n+1) BT P j=1 akj Γ (zj − zk ) + u1 , k = 2 , . . . , N

(13)

with µ determined by (3). Proof. From the description before, the main result can be deduced from the prescribed-time convergence of systems (11) and (12) under the designed controllers (13), and now we will choose an appropriate Lyapunov function to analyze the boundedness and convergence of system (11) and (12) in any prescribed finite time t ∗ . Since A−K2 C is a Hurwitz matrix, there always exists a positive definite matrix Q such that (A − K2 C )T Q + Q (A − K2 C ) = −I

j=1

akj (zˆj − zˆk ) + u1 , k = 2, . . . N ,

(18)

µ ˙

z˙ˆ = µ1+δ I ⊗ Azˆ − γ µ1+δ Lˆ ⊗ BBT P zˆ − (1 + δ ) µ I ⊗ Dzˆ + µ1+δ I ⊗ K2 C eˆ

(19)

with zˆ = (zˆ2T , . . . , zˆNT )T and eˆ = (eˆ T2 , . . . , eˆ TN )T . Let Vz = zˆ T R ⊗ P zˆ , where R and P defined in Lemmas 4 and 6, respectively, satisfy PA + AT P − PBBT P = −I , PD + DP ≥ α I , LˆR + RLˆT ≥ ηI

(20)

with η being a positive constant. Now, setting γ = λmax (R)/η, and from (20), the derivative of Vz along (19) can be calculated as V˙ z |(19) = µ1+δ zˆ T R ⊗ (AT P + PA − PBBT P)zˆ

− (1 + δ ) µµ˙ zˆ T R ⊗ (DT P + PD)zˆ + 2µ1+δ zˆ T R ⊗ PK2 C eˆ

(21)

≤ − λmin2 (R) µ1+δ ∥ˆz ∥2 − αλmin (R)(1 + δ ) µµ˙ ∥ˆz ∥2 + βµ1+δ ∥ˆe∥2 2

λmin (R)

∥R ⊗ PK2 C ∥2 .

∑N

Now, choosing the Lyapunov function V = Vz + 2β k=2 Vk , µ µ ˙ and by means of (17), (21) and µ = t ∗ , its derivative can be calculated as

DQ + QD ≥ α I with α being a positive constant. Firstly, choose the Lyapunov function candidates Vk = eˆ Tk Q eˆ k , k = 2, . . . , N and the derivative of Vk along (12) can be calculated as

− (1 + δ ) µµ˙ eˆ Tk (DQ + QD)eˆ k + 2eˆ Tk Q Γ (f (t , xk ) − f (t , x1 )) ∥ˆek ∥ − α (1 + 2

(14)

≤ cg(t)

∑i

{

∑i

j=1

⏐ ⏐ ⏐xk,j − x1,j ⏐

1

2

( µ1+δ +

2α (1+δ ) t∗

2α (1+δ ) t∗

2β ncg(t)∥Q ∥ λmin (R)

α (1+δ )

µ−

α (1+δ )

µ − 3ncg(t)∥Q ∥ ≥ 0.

t∗ t∗

Under Assumption 1, the scaling function (3), and by utilizing of Young’s inequality, the following estimate can be easily established 1 cg(t) µi(1+δ )

λmin (R)

( − β µ1+δ +

δ ) µµ˙ ∥ˆek ∥2

+ 2eˆ Tk Q Γ (f (t , xk ) − f (t , x1 )).

⏐ ⏐ ⏐ fi (t ,xk )−fi (t ,x1 ) ⏐ ⏐ µi(1+δ) ⏐ ≤

V˙ |(12)(19) ≤ −

µ−

j=1 µ(i−j)(1+δ ) µj(1+δ )

) µ − 6ncg(t)∥Q ∥ ∥ˆe∥2 .

≥ 0,

Firstly, we analyze the boundedness of all states of the closedloop system (12), (18) and (19) on [t0 , t1 ). For all t ∈ [t0 , t1 ) and by virtue of (22) and (23), we have V˙ |(12)(19) ≤ 2ν1 cˆ V (t)

(24)

(25)

∥ˆz (t)∥ + ∥ˆe(t)∥ ≤

2ν2 /ν3 eν1 cˆ (t1 −t0 )

√

(15)

and then using (15), the estimate 2eˆ Tk Q Γ (f (t , xk ) − f (t , x1 )) can be summarized compactly as (16)

µ

+ 3ncg(t)∥Q ∥∥ˆek ∥2 + ncg(t)∥Q ∥∥ˆzk ∥2 .

2

V˙ |(12)(19) ≤ −2δν4 µ(t − t0 )1+δ V (t) with ν4 =

Thus, substituting (16) into (14), we can easily have

(26)

with ν2 = λmax (R ⊗ P)∥ˆz (t0 )∥ + 2βλmax (Q )∥ˆe(t0 )∥ and ν3 = min {λmin (R ⊗ P), 2βλmin (Q )}. Combining with (25) and (26), we obtain that all states zˆ (t) and eˆ (t) do not escape on [t0 , t1 ) yielding the boundedness of all states zˆ (t) and eˆ (t) on [t0 , t1 ). Then, for all t ∈ [t1 , t0 + t ∗ ), and by means of (23), the inequality (22) satisfies 2

µ ˙ V˙ k |(12) ≤ −µ1+δ ∥ˆek ∥2 − α (1 + δ ) ∥ˆek ∥2

=

and

which indicates

≤ 3ncg(t)∥Q ∥∥ˆek ∥2 + ncg(t)∥Q ∥∥ˆzk ∥2 .

(22)

(23)

V (t) ≤ ν2 e2ν1 cˆ (t1 −t0 )

∑ |x˜ −˜z ,j +˜zk,j | ≤ cg(t) ij=1 k,jµj(1k+δ ) ) ∑ ( ≤ cg(t) ( ij=1 |ˆek,j | +) |ˆzk,j | ≤ cg(t) ∥ˆek ∥ + ∥ˆzk ∥ ,

2eˆ Tk Q Γ (f (t , xk ) − f (t , x1 ))

) ∥ˆz ∥2

with ν1 = max {β/λmin (R ⊗ P), 3/(2λmin (Q ))} and cˆ nc ∥Q ∥ maxt ∈[t0 ,t1 ) {g(t)}, and then we can easily obtain

|x˜ k,j |

∥Γ (f (t , xk ) − f (t , x1 ))∥ ≤ ncg(t)(∥ˆek ∥ + ∥ˆzk ∥),

4β ncg(t)∥Q ∥ λmin (R)

Since c is an unknown positive constant and g(t) is a continuous function well-defined on [t0 , +∞), thus there must exist a time t1 ∈ [t0 , t0 + t ∗ ), such that, for all t ∈ [t1 , t0 + t ∗ ),

V˙ k |(12) = µ1+δ eˆ Tk ((A − K2 C )T Q + Q (A − K2 C ))eˆ k

≤ −µ

∑N

system (11) can be further rewritten as the following compact form

with β =

and

1+δ

uk = γ µ(n+1)(1+δ ) BT P

1 4δ

min

{

λmin (R) , 1 λmax (R⊗P) λmax (Q )

(27)

}

and by means of (25), which

indicates (17)

V (t) ≤ ϱ1 ν2 e−2ν4 t

∗ µ(t −t )δ 0

(28)

X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640 ∗ δ e2ν1 cˆ (t1 −t0 )+2ν4 t µ(t1 −t0 ) .

with ϱ1 = Moreover, by means of (26) and (28), we can easily have

∥ˆz (t)∥ + ∥ˆe(t)∥ ≤

√

2ϱ1 ν2 /ν3 e−ν4 t

∗ µ(t −t )δ 0

.

(29)

Combining with (26) and (29), we obtain that all states zˆ (t) and eˆ (t) are bounded on [t0 , t0 + t ∗ ) and convergent to zero as t → t0 + t ∗ . Thus, the global prescribed-time regulation of system (11) and (12) is achieved under controllers (18). Next, we give the analysis of the global prescribed-time decentralized regulation of system (1) under controllers (13). Firstly, through the state transformations (10) and inequalities (26) and (29), we can easily obtain that ∥˜z (t)∥ + ∥˜e(t)∥ is bounded on [t0 , t1 ). Then, for all t ∈ [t1 , t0 + t ∗ ), we have

∥˜z (t)∥ + ∥˜e(t)∥ ≤

√

2nϱ1 ν2 (N − 1)/ν3 χ1 (t)e− n(N −1)(1+n)(1+δ )

ν4 t ∗

ν4 t ∗ 2

µ(t −t0 )δ

(30)

δ

2 with χ1 (t) = µ(t − t0 ) e− 2 µ(t −t0 ) . To obtain the boundedness of the time-varying function χ1 (t) on [t1 , t0 + t ∗ ), the derivatives of the function χ1 (t) is firstly obtained

χ˙ 1 (t) =

δν4

(

n(N − 1)(1 + δ )(1 + n)

2

δν4 t ∗

− µ(t − t0 )δ

× µ(t − t0 )χ1 (t).

(31)

> µ(t1 − t0 )δ , let χ˙ 1 (t2 ) = 0, and then we ) 1δ ( δν t ∗ t ∗ and its upper bound have t2 = t ∗ + t0 − n(N −1)(14+δ )(1+n) is χ1 (t2 ). Otherwise the upper bound of χ1 (t) is χ1 (t1 ). Now we have obtained the boundedness of χ1 (t) on [t1 , t0 + t ∗ ). Thus, combining with the boundedness of χ1 (t) and from (30), we have limt →t0 +t ∗ (∥˜z (t)∥ + ∥˜e(t)∥) = 0, and by virtue of (6) and (9), we finally obtain limt →t0 +t ∗ ∥xk (t) − x1 (t)∥ = 0, limt →t0 +t ∗ ∥zk (t) − z1 (t)∥ = 0, k = 2, . . . , N, which indiIf

n(N −1)(1+δ )(1+n) δν4 t ∗

cates the achievement of the global prescribed-time decentralized regulation of system (1). Moreover, the protocols (13) are bounded on [t0 , t1 ), and from (29), the protocols (13) are also well-defined on [t1 , t0 + t ∗ ) and satisfy

√ |uk (t)| ≤ γ 2ϱ1 ν2 /ν3 ∥P ∥∥Lk ⊗ I1×n ∥ ∗ δ × µ(t − t0 )(n+1)(1+δ) e−ν4 t µ(t −t0 ) + |u1 (t)|,

Theorem 12. Under Assumption 1 with c = 1 and the communication topology G , positive constants δ and γ , and positive definite matrix P can be chosen such that, the global prescribed-time decentralized stabilization of system (1) is solved by (13). Remark 13. In fact, the time-varying gain (3) tending to infinite as time goes to t0 + t ∗ , which is difficult to implement in practical control. To overcome this technical obstacle, a ‘‘turning off’’ mechanism, that is, setting some time tstop < t0 + t ∗ to achieve convergence of the estimation error to within some neighborhood of the origin that could be tuned by the user if desired, is introduced. One can, in principle, determine such a priori constant tstop as follows. (a∗ ) set the time-varying gain µ(t − t0 ) in (3) to some maximum value selected by the user, say µmax = 1010 ; (b∗ ) solve the expression µ(tstop − t0 ) = µmax and obtain tstop ; (c ∗ ) set the step size and the number of iterations as t0 + t ∗ − tstop t −t0 ⌋, respectively. and ⌊ t +stop t ∗ −t stop

0

)

(32)

where Lk is the kth row of Laplacian matrix L. Similar to the boundedness and prescribed-time convergence analysis of (30), we can get that all states of protocols (13) are also bounded on [t0 , t0 + t ∗ ) and satisfy limt →t0 +t ∗ uk (t) = u1 (t0 + t ∗ ), k = 2, . . . , N. ■ Remark 10. It should be noted that the proposed protocols (13) for system (1) are easy to understand and implement. Obviously, for system (1) satisfying Assumption 1 with the communication topology G , its construction consists of simply choosing positive constants δ and γ , and positive definite matrix P, independently selecting the prescribed time t ∗ . Remark 11. The choosing of parameters δ and γ affects the control effect of the closed-loop system. Specifically speaking, from (30) and (32), we can obtain that the larger the parameter δ , the faster the convergence speed of closed-loop system, but also accompanied by a little larger oscillation. Moreover, the parameter δ just need to satisfy δ > 0 and we often choose δ ≤ 1. From (13), we also have that the parameter γ depending on Lˆ and R directly affects the state response of the control protocols. Obviously, under Assumption 1 with c being a known positive constant, and combined with the idea of Theorem 9, one can easily obtain the global prescribed-time decentralized stabilization of system (1), whose proof is omitted here.

5

Moreover, once the ‘‘turning off’’ mechanism is implemented, the tracking errors between followers and leader may grow again. But if desired, treating the current time tstop as a new t0 , and then following (a∗ )-(c ∗ ), the new step size and the number of iterations can be chosen. Similarly detailed discussions can be also seen in [25]. 4. Example In this section, a simulation example is given to illustrate our main theoretical results. Consider the following nonlinear multi-agent system

{

x˙ k,1 = xk,2 + f1 (t , xk ), x˙ k,2 = uk + f2 (t , xk ), yk = xk,1 , k = 1, . . . , 5,

(33)

where xk = (xk,1 , xk,2 )T ∈ R2 , uk ∈ R and yk ∈ R are system state, input and output of agent k, respectively. The agent indexed by 1 is referred as the leader and agents indexed by 2, 3, 4 and 5 are called the followers. The weighted adjacent matrix and the Laplacian matrix, denoted respectively by A and L, are given as follows

⎛ ⎜ ⎜ ⎝

A = (ai,j )5×5 = ⎜

0 1 1 0 0

0 0 0 0 0

0 0 0 1 1

0 0 0 0 1

0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎠

and

⎛ ⎜ ⎜ ⎝

L = (li,j )5×5 = ⎜

0 −1 −1 0 0

0 1 0 0 0

0 0 1 −1 −1

0 0 0 1 −1

0 0 0 0 2

⎞ ⎟ ⎟ ⎟. ⎠

Let f1 (· ) = 0, f2 (· ) = −

sinxk,1 l(t)

− q(t)xk,2 ,

(34)

which means fi (· ), i = 1, 2 satisfy Assumption 1. Under this case, the five agents in system (33) can be viewed as five pendulums [35,36], whose length l(t) plays the role of an unknown time-varying parameter satisfying l(t) ≥ 5, t ≥ 0, and whose resistance due to pivot and surrounding air is governed by the law q(t) = t.

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X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

Fig. 1. The state responses of the closed-loop control system consisting of (33), (35) and (36) as t ∗ = 5.

X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

Fig. 2. The state responses of the closed-loop control system consisting of (33), (35) and (36) as t ∗ = 9.5.

7

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X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

Fig. 3. The state responses of control protocols (35) as the prescribed time choosing as t ∗ = 5 and t ∗ = 9.5, respectively.

Then, following the design procedure proposed in Section 3, the global prescribed-time decentralized regulation protocols of system (33) can be constructed as follows uk = γ µ6 BT P

5 ∑

akj Γ (zj − zk ) + u1 , k = 2, . . . , 5,

(35)

j=1

where zk = (zk,1 , zk,2 )T is the state of the observer

{

z˙k,1 = zk,2 + µ2 (yk − zk,1 ), z˙k,2 = uk + µ4 (yk − zk,1 ), k = 1, . . . , 5.

(36)

Firstly, from Lemmas 4–7, the matrices P and R can be expressed as follows

( P =

1.7321 1

1 1.7321

)

and 0.232 ⎜ 0 R=⎝ 0 0

⎛

0 0.3457 0 0

0 0 0.5 0

⎞

0 0 ⎟ ⎠, 0 1.2557

and we can easily obtain λmin (LˆR + RLˆT ) = 0.4259. In order to verify the effectiveness of the proposed prescribedtime decentralized protocols, the initial conditions (x1,1 , x1,2 ) = (2, −4), (x2,1 , x2,2 ) = (3, −6), (x3,1 , x3,2 ) = (4, −8), (x4,1 , x4,2 ) = (5, −10), (x5,1 , x5,2 ) = (1.6, −5.2) and (z1,1 , z1,2 ) = (0, 0), k = 1, . . . , 5 are chosen firstly under controller u1 ≡ 0. Then, to analyze the influence of parameter δ and γ on the control effect of the closed-loop system (33), (35) and (36), δ = 0.5 and δ = 1 are chosen, respectively. Moreover, from choosing η = 0.4 and η = 0.2, γ = 3.13925 and γ = 6.2785 can be obtained, respectively. Figs. 1 and 2 show the state responses of the closed-loop control system consisting of (33), (35) and (36). In detail, Figs. 1 and

X. Chen, X. Zhang and Q. Liu / Systems & Control Letters 137 (2020) 104640

2 show that all states xk,i − x1,i , zk,i − z1,i , k = 2, . . . , 5, i = 1, 2 are bounded and convergent to zero at the prescribed time t ∗ = 5 and t ∗ = 9.5, respectively. Comparing with Figs. 1(a) and 1(b), we can see that the choosing of parameter δ affects the convergence rate of the closed-loop system, and generally speaking, the larger the parameter δ choosing, the faster the convergence speed of the closed-loop system. The same conclusion can also be found in Figs. 2(a) and 2(b). From 1(a) and Fig. 1(c), we can get that the choosing of parameter γ has little effect on the closed-loop system. The same conclusion can also be found in Figs. 2(a) and 2(c). Fig. 3 shows the state responses of control protocols (35) as the prescribed time choosing as t ∗ = 5 and t ∗ = 9.5, respectively. In detail, the control protocols uk , k = 2, . . . , 5 are bounded and convergent to zero as the prescribed time choosing as t = 5 and t = 9.5, respectively. Furthermore, comparing with Figs. 3(a) and 3(b), we can see that the choosing of parameter δ also affects the convergence rate of the closed-loop system, and the larger the parameter δ choosing, the faster the convergence speed of the closed-loop system, but also accompanied by a little larger oscillation. Comparing with Figs. 3(a) and 3(c), we can get that the choosing of parameter γ has little effect on the convergence speed of the closed-loop system, and just affects the state responses of control protocols at the initial time. 5. Conclusions In this paper, we have tackled the global prescribed-time decentralized regulation of uncertain nonlinear multi-agent systems under Assumption 1 via output feedback control. Remarkably, the systems in question allow unknown parameters and essential time-varying features in nonlinearities, and hence are considerably different from those in the closely related literatures. Firstly, define error systems, which consisted of the leader and their followers. Then, by introducing a time-scaling function, an appropriate state transformation was applied to make the prescribed-time decentralized regulation problem transformed into the prescribed-time regulation problem of the error systems. Apparently, the current schemes are invalid for heterogeneous multi-agent systems, or system with external disturbances, and unknown control coefficients for which the study will be attempted in the future. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Xiandong Chen: Conceptualization, Methodology, Software, Validation, Writing - review & editing. Xianfu Zhang: Conceptualization, Methodology, Supervision, Writing - review & editing. Qingrong Liu: Software, Writing - review & editing. References [1] Y.C. Cao, W.W. Yu, W. Ren, G.R. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Trans. Ind. Inf. 9 (1) (2013) 427–438. [2] C.C. Hua, K. Li, X.P. Guan, Semi-global/global output consensus for nonlinear multiagent systems with time delays, Automatica 103 (2019) 480–489. [3] J.P. Hua, G. Feng, Distributed tracking control of leader-follower multiagent systems under noisy measurement, Automatica 46 (8) (2010) 1382–1387.

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