Semi-global containment control of discrete-time linear systems with actuator position and rate saturation

Communicated by Prof. Zidong Wang

Accepted Manuscript

Semi-global containment control of discrete-time linear systems with actuator position and rate saturation Zhiyun Zhao, Wen Yang, Hongbo Shi PII: DOI: Reference:

S0925-2312(18)31428-0 https://doi.org/10.1016/j.neucom.2018.12.001 NEUCOM 20218

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

4 July 2018 29 October 2018 1 December 2018

Please cite this article as: Zhiyun Zhao, Wen Yang, Hongbo Shi, Semi-global containment control of discrete-time linear systems with actuator position and rate saturation, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.12.001

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ACCEPTED MANUSCRIPT

Semi-global containment control of discrete-time linear systems with actuator position and rate saturation

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Zhiyun Zhao, Wen Yang, Hongbo Shi Key Laboratory of Advanced Control and Optimization for Chemical Processes, East China University of Science and Technology, Shanghai 200237, China.

Abstract

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In this paper, we study semi-global containment control problem for a multi-

agent system in the presence of multiple leader agents. We describe the dynamics of each follower agent in multi-agent system by a general discrete-time linear system subject to both actuator position and rate saturation. By using low gain approach, we construct both a linear state feedback control law and an observer-based output feedback control law for each follower agent. Under

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some standard assumptions, we provide a sufficient condition to guarantee that the states of follower agents converge to the convex hull formed by the leader

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agents asymptotically. Finally, we illustrate the theoretical results by numerical simulation results.

Keywords: Containment control; multi-agent systems; actuator saturation;

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rate saturation; discrete-time system.

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1. INTRODUCTION Consensus, as a basic approach of distributed cooperated control for multi-

agent systems, has drawn significant attention over past decades [1, 2, 3, 4].

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Consensus for multi-agent systems with multiple leaders, also called containment 1 This work supported in part by the National Natural Science Foundation of China under Grants 61703162 and 61573143, in part by the Shanghai Sailing Program under Grants 17YF1403400, in part by the Shanghai Natural Science Foundation under Grants 18ZR1409700, in part by Project funded by China Postdoctoral Science Foundation under Grant 2017M610233, and in part by the Fundamental Research Funds for the Central Universities under Grant 222201714030.

Preprint submitted to Neurocomputing

December 4, 2018

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control problem, requires all the states of follower agents converge to the convex hull formed by the leader agents. In early literatures, the dynamics of each agent is often simplified to the kinematics of a single-integrator [5, 6, 7] or the

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dynamics of a double-integrator [8, 9, 10, 11, 12]. In [7], the authors investigate finite-time containment control problem for a 10

group of single integrator agents in the presence of static and dynamic leaders.

It is shown that the multi-agent systems solving containment control at any

preset time if the communication graph has a spanning forest. The finite-time

containment control problem for a group of double-integrator agents was studied

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in [10]. In [11], the authors study mean-squared containment problem for firstorder and second-order integral multi-agent systems with communication noises. Necessary and sufficient conditions are proposed for ensuring the mean-squared containment, meanwhile, two effective distributed protocols are proposed for each follower agent with the help of a time-varying gain. In [12], the authors investigate containment control problem for a group of double-integrators in the presence of bounded unknown nonlinearity. A containment control law based

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on high-frequency feedback robust control is constructed for each follower agent

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such that all the followers will converge to the dynamic convex hull formed by dynamic leaders.

Reference [13, 14, 15, 16, 17, 18] deal with containment control for highorder linear systems. In [14], the authors discuss adaptive containment control

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for multi-agent systems with multiple leaders containing parametric uncertain-

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ties. Based on an adaptive internal model, the authors propose a distributed containment control for each follower agent such that the outputs of all agents will enter the moving convex set spanned by the leaders when the interconnection topology is a general directed graph. In particular, the authors of [15]

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discuss output containment control problem for heterogeneous multi-agent systems. Different with state containment control problem, output containment control problem entails all the outputs of follower agents converge to the convex hull formed by the leader agents. In [16], the authors propose sufficient con-

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ditions and effective control laws to ensure the containment control for linear 2

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multi-agent systems with exogenous disturbances. In the existing literatures, only a few results have been obtained on the containment problem for multi-agent systems subject to actuator saturation

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[19, 20]. In reference [20], the authors consider semi-global containment control problem for a group of general linear systems subjected to input saturation under switching topology. Both state feedback laws and output feedback laws

are constructed for each follower agent by using local information, thus the semiglobal containment control is solved when the communication topology among

agents is a jointly connected graph. Besides position saturation of actuator, rate saturation of actuator is also ubiquitous in practice applications and the

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phase-lag associated with rate saturation has a destabilizing effect, which could degrade the performance of the closed-loop system or even leads to instability in some extreme case. Rate saturation has been identified as an important factor to the mishaps of YF-22 [21] and the first production of Gripen [22]. Thus, it 50

is indispensable to take into account actuator position and rate saturation in

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the study of containment control for multi-agent systems. However, to the best of our knowledge, there is no existing result on containment control taking into

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consideration position and rate saturation. On the other hand, stabilization of an individual linear system subject to actuator position and rate saturation has 55

been obtained [23, 24, 25]. For a general linear system subject to both actuator

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position and rate saturation, whose open-loop poles are on the closed left-half plane and open-loop system is stabilizable, it is shown in [23] that the linear

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system is semi-globally stabilizable by both linear state feedback control laws and linear output feedback control laws. In particular, in [26], the authors show

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that semi-global leader-following consensus for a group of general continuous-

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time linear systems with actuator position and rate saturation can be achieved both by state feedback control laws and output feedback control laws when the communication topology among follower agents is a connected graph and the leader is a neighbor of at least of one follower. Reference [27] deals with the

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discrete-time counterparts of [26]. In this paper, we consider semi-global containment control problem for a 3

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group of general discrete-time linear systems in the presence of actuator position and rate saturation. For each follower agent, we propose both a linear state feedback containment control algorithm and an observe-based output feedback containment control algorithm by using low gain approach. We prove that

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these control algorithms together solve semi-global containment control prob-

lem, where the states of all follower agents converge to the convex hull formed by the leader agents asymptotically when the communication topology among fol-

lower agents is a connected undirected graph and each leader agent is a neighbor 75

of at least one follower agent.

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The contribution of this paper are two-fold. First, this paper considers the containment control problem for a group of linear systems subject to not only actuator position saturation but also actuator rate saturation, which has not been investigated in existing results on the containment control [5, 8, 9, 11, 80

13, 14, 15, 19, 20]. Second, when compared with consensus subject to input saturation [19, 20], the results in this paper are more general, which have been

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extended to multiple leader case. The difference between the results in this paper and in [27] are two-fold. First, in this paper, we consider the case that

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there exist multiple leaders in the multi-agent system. The results in this paper can be degenerated to that in [27] if all leader agents share common states. Second, the actuator position and rate saturation are represented as standard

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saturation functions in [27], however, actuator position and rate saturation are represented as general saturation functions in this paper, which will bring many

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technical differences to be overcomed. The remainder of this paper is organized as follows. In Section 2, we intro-

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duce the problem of semi-global containment control. In Section 3, we construct

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both a linear state feedback containment control law and an observer-based output feedback control law for each follower agent in the system such that the containment control problem is solved when the communication topology among

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agents satisfy certain assumptions. Simulation examples are given in Section 4 to illustrate the theoretical results. Section 5 is a concluding remark of this paper. 4

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2. PROBLEM STATEMENT Consider a multi-agent system consisting of N follower agents, labelled as 1, 2, · · · , N . The dynamics of each follower agent i, i = 1, 2, · · · , N, is described

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by a discrete-time linear system in the presence of actuator position and rate saturation,

   x (t + 1) = Axi (t)+Bσp (vi (t)),   i vi (t + 1) = vi (t)+σr ((α − 1)vi (t) + ui (t)),     yi (t) = Cxi (t),

(1)

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where |α| < 1 is time constant, xi ∈ Rn , vi ∈ Rm , ui ∈ Rm and yi ∈ Rr are respectively the plant states, actuator state, control input and measurement

output of i-th agent. σp (s) is a saturation function representing the position saturation. σr (s) is a saturation function representing the rate saturation. Both σp (s) and σr (s) satisfies the following definition.

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Definition 1. [28] A function σ : Rm → Rm is said to be a saturation function if,

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• σ(s) is decentralized, i.e., σ(s) = [σ1 (s1 ), σ2 (s2 ), · · · , σm (sm )]; and for each i = 1 to m,

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• σi is continuous;

• σi is linear in a neighborhood of the origin and is bounded away from the

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vertical axis outside this neighborhood. Without loss of generality, assume that within this linear neighborhood the slope is unity, i.e.,   σ (s ) = s if |si | ≤ ∆, i i i  ∆ ≤ |σ (s )| ≤ b|s | if |si | > ∆, i i i

for some (arbitrarily small) ∆ > 0 and some (arbitrarily large) b ≥ 1.

There also exist M leader agents in the multi-agent system , labelled as

N + 1, N + 2, · · · , N + M . The dynamics of each leader agent is also described 5

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(2)

where xi ∈ Rn and yi ∈ Rr are the plant states and measurement output of

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by a discrete-time linear system,   x (t + 1) = Ax (t), i i  yi (t) = Cxi (t), leader agent i.

Make the following standard assumptions on the dynamics of agents.

Assumption 1. All eigenvalues of A are located inside or on the unit circle and the pair (A, B) is stabilizable. Assumption 2. The pair (A,C) is detectable.

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Lemma 1. [28] Under assumption 1, for each ε ∈ (0, 1], there exists a unique matrix P (ε) > 0 which solves the following algebraic Riccati equation (ARE): P (ε) = AT P (ε)A + εI − AT P (ε)B(B T P (ε)B + I)−1 B T P (ε)A,

(3)

1. limε→0 P (ε) = 0,

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such that A − B(B T P (ε)B + I)−1 B T P (ε)A is asymptotically stable. Moreover, 2. There exists an ε∗ ∈ (0, 1] such that for all ε ∈ (0, ε∗ ],

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1

1

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kP (ε) 2 AP (ε)− 2 k ≤ 1 2

− 21

kP (ε) AAP (ε)

√

k ≤ 2,

2,

(4) (5)

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Hereafter, we denote P = P (ε) for notation brevity. In this paper, we use an undirected graph GN = {VN , EN } to represent

the communication topology among the follower agents. In this graph, VN =

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{ν1 , ν2 , · · · , νN } is a finite, nonempty set of nodes, each denoting a follower

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agent, and EN ∈ VN × VN is a set of edges. An edge (νi , νj ) in an undirected graph means that νi and νj having access to the information form each other.

Let G = {V, E} be the graph generated by graph GN and the leaders, where V = {ν1 , ν2 , · · · , νN +M } is a finite, nonempty set of nodes, each denoting an

agent, and E = {(νi , νj ) : νi , νj ∈ V} is a set of edges. 6

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The neighborhood of follower agent i is defined as Ni = {j : (νj , νi ) ∈ E}. We assume that the leader agents have no neighbors. Let A = [aij ] be the adjacency matrix associated with G, where aij = 1 if (νj , νi ) ∈ E and aij = 0

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otherwise. Here we assume that aii = 0 for all i = 1, 2, · · · , N +M . Let L = [lij ] PN be the Laplacian matrix associated with A, where lii = j=1 aij and lij = −aij when i 6= j. Because the leaders have no neighbors, therefore, the Laplacian

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matrix L can be written as the following block matrix   LN LM , L= 0M ×N 0M ×M where LN ∈ RN ×N and LM ∈ RN ×M .

The communication topology among agents satisfies the following assumption.

Assumption 3. The undirected graph GN is connected and each leader agent

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is a neighbor of at least one follower agent.

Lemma 2. [8]Under Assumption 3, LN is positive definite, each entry of −L−1 N LM

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is nonnegative, and each row of −L−1 N LM has a sum equal to one.

Considering a multi-agent system consisting of the group of follower agents

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(1) and leader agents (2), we try to solve the following problems. Problem 1. For a priori given bounded sets X0 ⊂ Rn and V0 ⊂ Rm , construct

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a linear state feedback control law ui (t) = hi (xi (t), vi (t), {xj (t), j ∈ Ni })

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for each follower agent such that all these state feedback control laws together solve the semi-global containment control problem for the follower agents (1) and the leader agents (2). That is, for all i, i = 1, 2, · · · , N , there exists a set of nonnegative constants αij , j = N + 1, N + 2, · · · , N + M , satisfying NX +M

αij = 1,

j=N +1

7

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such that



lim xi (t) −

t→∞

NX +M

j=N +1



αij xj (t) = 0, i = 1, 2, · · · , N,

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holds for all xi (0) ∈ X0 , i = 1, 2, · · · , N + M , and vi (0) ∈ V0 , i = 1, 2, · · · , N .

Problem 2. Construct an observer-based output feedback law for each agent zi (t + 1) = Azi (t) − L(yi − Czi (t)) + Bvi (t), i = 1, 2, · · · , N,

zi (t) = Azi (t) − L(yi − Czi (t)), i = N + 1, N + 2, · · · , N + M,

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ui (t) = hi (zi (t), vi (t), {zj (t), j ∈ Ni })

such that all these observer-based output feedback control laws together solve the semi-global containment control problem for the follower agents (1) and the leader agents (2). That is, for all i, i = 1, 2, · · · , N , there exists a set of nonnegative constants αij , j = N + 1, N + 2, · · · , N + M , satisfying αij = 1,

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NX +M

j=N +1



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such that

lim xi (t) −

t→∞

NX +M

j=N +1



αij xj (t) = 0, i = 1, 2, · · · , N,

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holds for all xi (0), zi (0) ∈ X0 , i = 1, 2, · · · , N +M , and vi (0) ∈ V0 , i = 1, 2, · · · , N .

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3. MAIN RESULTS 3.1. The State Feedback Result

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We propose a linear state feedback containment control law for each follower

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agent (1), ui (t) = −γHA

NX +M j=1



1 aij (xi (t) − xj (t)) + 1 − b



γH

NX +M j=1

aij (xi (t) − xj (t))

  N X 1 − α + − 1 vi (t) − γHB (vi (t) − vj (t)), i = 1, 2, · · · , N, b j=1 8

(6)

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where γ ≤

1 λmax (LN )

is a constant and H = (I + B T P B)−1 B T P A,

(7)

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where P is the solution of the ARE (3). Now we are ready to establish the following result on semi-global containment control for multi-agent systems (1)-(2).

Theorem 1. Under assumptions 1 and 3, the state feedback containment con-

trol laws (6) solve semi-global containment control problem for multi-agent sys-

tems consisting of follower agents (1) and leader agents (2). That is, for any

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given bounded sets X0 ⊂ Rn and V0 ⊂ Rm , there is an ε∗ > 0 such that,

for any given ε ∈ (0, ε∗ ] and for all xi (0) ∈ X0 , i = 1, 2, · · · , N + M , and vi (0) ∈ V0 , i = 1, 2, · · · , N , 

lim xi (t) −

t→∞

NX +M

j=N +1



αij xj (t) = 0, i = 1, 2, · · · , N,

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where αij , i = 1, 2, · · · , N, j = N + 1, N + 2, · · · , N + M , are nonnegative con-

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stants satisfying

NX +M

αij = 1.

j=N +1

Proof: Denote xF (t) = [x1 (t)T , x2 (t)T , · · · , xN (t)T ]T , vF (t) = [v1 (t)T , v2 (t)T ,

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· · · , vN (t)T ]T , xR (t) = [xN +1 (t)T , xN +2 (t)T , · · · , xN +M (t)T ]T and u(t) = [u1 (t), u2 (t), · · · , uN (t)]T . Then, the dynamics of follower agents and leader agents can

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be written in the following compact form,    x (t+1) = (I ⊗ A)xF (t) + (I ⊗ B)σp (vF (t)),   F vF (t+1) = vF (t) + σr ((α − 1)vF (t) + u(t)),    x (t+1) = (I ⊗ A)x (t).

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Denote ψ(t) = xF (t) +

R

L−1 N LM


⊗ I xR (t), then


ψ(t + 1) = xF (t + 1) + L−1 N LM ⊗ I xR (t + 1)


= (I ⊗ A)xF (t) + (I ⊗ B)σp (vF (t)) + L−1 N LM ⊗ A xR (t)

= (I ⊗ A)ψ(t) + (I ⊗ B)σp (vF (t)), 9

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and 



Consider the following Lyapunov function candidate



γ(LN ⊗ H)ψ(t)

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1 vF (t + 1) = vF (t) + σr − γ(LN ⊗ HA)ψ(t) + 1 − b    1 − I + γ(LN ⊗ HB) vF (t) . b

V = ψ T (LN ⊗ P )ψ + κ(vF + γ(LN ⊗ H)ψ)T (vF + γ(LN ⊗ H)ψ), 2

(8)

b where κ ≥ max{3b, 2b−1 ( 32 + γθ )} is a positive constant. It is trivial to show that

V is positive definite due to the fact that both LN and P are positive definite.

c≥ 155

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Let c > 0 be a constant scalar such that sup

V.

(9)

ε∈(0,1],xi ∈X0 ,i=1,2,··· ,N +M,vi ∈V0 ,i=1,2,··· ,N

Such a c exists beacuse X0 and V0 are both bounded sets and limε→0 P = 0 according to Lemma 1.

Let LV (c) := {(ψ, vF ) ∈ RN (n+m) : V (ψ, vF ) ≤ c}. Let ε∗1 ∈ (0, 1] be such

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that for all ε ∈ (0, ε∗1 ], (ψ, vF ) ∈ LV (c) implies that

∆2 I, 2(b c + ∆) + ∆2 ∆ kγ(LN ⊗ H)ψk ≤ , 32b ∆ kγ(LN ⊗ HA)ψk ≤ , 32b r  

((LN ⊗ B T P A) − γ(L2N ⊗ B T P BH))ψ ≤ min ∆ , ∆ γ , 32b 2 θ   2 ∆ b ∆ kγ(LN ⊗ HB)vF k ≤ min , , 32b 4(κb2 + κ2 (b − 1)2 )   ∆ b2 ∆ kγ(LN ⊗ HB)σp (vF )k ≤ min , , 32b 4(κb2 + κ2 (b − 1)2 ) √

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LN ⊗ B T P B ≤

where θ =

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λ1 γ 2 λ2 ,

(10) (11) (12) (13) (14) (15)

λ1 = λmin (L2N ⊗H T H) and λ2 = λmax (((LN ⊗B T P A)−γ(LN ⊗

B T P BH))T ((LN ⊗ B T P A) − γ(LN ⊗ B T P BH))). The existence of such an ε∗1 is again due to Lemma 1 and the fact that |σp (s)| ≤ b|s|. Notice that, it follows from (3) and (7), we have (A − BH)T P (A − BH) − P = −εI − H T H. 10

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Let V 1 = ψ(t)T (LN ⊗ P )ψ(t), then we have

= ((I ⊗ A)ψ(t) + (I ⊗ B)σp (vF (t)))T (LN ⊗ P )

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V 1(t + 1) − V 1(t) ×((I ⊗ A)ψ(t) + (I ⊗ B)σp (vF (t))) − ψ(t)T (LN ⊗ P )ψ(t) = ψ T (LN ⊗ AT P A)ψ + σp (vF )T (LN ⊗ B T P B)σp (vF ) +2ψ T (LN ⊗ AT P B)σp (vF ) − ψ T (LN ⊗ P )ψ,

= −εψ T (LN ⊗ I)ψ − ψ T (LN ⊗ H T H)ψ + 2ψ T (LN ⊗ AT P BH)ψ +2ψ T (LN ⊗ AT P B)σp (vF )

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−ψ T (LN ⊗ H T B T P BH)ψ + σp (vF )T (LN ⊗ B T P B)σp (vF )

= −εψ T (LN ⊗ I)ψ − γψ T (L2N ⊗ H T H)ψ + (σp (vF ) + γ(LN ⊗ H)ψ)T ×(LN ⊗ B T P B)(σp (vF ) + γ(LN ⊗ H)ψ)

+2ψ T ((I ⊗ AT ) − γ(LN ⊗ H T B T ))(LN ⊗ P B)(σp (vF ) + γ(LN ⊗ H)ψ)

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Here, and hereafter in a similar situation, we have suppressed the dependence on t of the state variables. Therefore

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∆V = V (t + 1) − V (t)

= −εψ T (LN ⊗ I)ψ − γψ T (L2N ⊗ H T H)ψ

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+(σp (vF ) + γ(LN ⊗ H)ψ)(LN ⊗ B T P B)(σp (vF ) + γ(LN ⊗ H)ψ)

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+2ψ T ((I ⊗ AT )−γ(LN ⊗ H T B T ))(LN ⊗ P B)(σp (vF )+γ(LN ⊗ H)ψ)     1 γ(LN ⊗ H)ψ +κ vF + σr −γ(LN ⊗ HA)ψ + 1 − b    T 1 − I +γ(LN ⊗ HB) vF +γ(LN ⊗ HA)ψ+γ(LN ⊗ HB)σr (vF ) b     1 × vF + σr −γ(LN ⊗ HA)ψ + 1 − γ(LN ⊗ H)ψ b    T 1 − I + γ(LN ⊗ HB) vF +γ(LN ⊗ HA)ψ+γ(LN ⊗ HB)σr (vF ) b −κ(vF + γ(LN ⊗ H)ψ)T (vF + γ(LN ⊗ H)ψ)

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≤ −ελmin (LN )ψ T ψ + 2

Nm X

k=1



 − 1 (hk )2 + 2pk (σ (v k ) + hk ) p γ

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∆ √ (σ (v k ) + hk )2 − κ(v k + hk )2 2(b( c + ∆) + ∆)2 p !2    1 +κ v k + σr − (v k + hk ) + hk − q k − rk + q k + tk  , b

+

where v k is the kth element of vF , hk is the kth element of γ(LN ⊗ H)ψ, q k is

the kth element of matrix γ(LN ⊗ HA)ψ, pk is the kth element of matrix (LN ⊗

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B T P )((I ⊗ A) − γ(L2N ⊗ BH))ψ, rk is the kth element of matrix γ(LN ⊗ HB)vF

and tk is the kth element of matrix γ(LN ⊗ HB)σr (vF ). We first consider the case that |v k | ≥

7∆ 8

for all k, k = 1, 2, · · · , N m, in this

case, it follows from (8) and (9) that

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√ (σr (v k ) + hk )2 ≤ ((b c + ∆) + ∆)2 , hence,

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1 ∆2 √ − (hk )2 + 2pk (σp (v k ) + hk ) + (σ (v k ) + hk )2 γ 2((b c + ∆) + ∆)2 p !2   1 k k k k k k k k +κ v + σr − (v + h ) + h − q − r + q + t − κ(v k + hk )2 b  2    2 2∆2 ∆2 ∆ k ∆ 1 7∆ ∆ ∆ k k ≤ + + |v |−κ |v | − +κ |v | − − + 322 2 16 32b b 8 8 16b 2 2 2 2∆ ∆ ∆ 21κ∆ k 483κ∆ ≤ + + |v k | − |v | + 322 2 16 16b 1024b2 2 ≤ −∆ .

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Therefore, we conclude that ∆V ≤ −ελmin (LN )ψ T ψ − ∆2 < 0 if |v k | ≥ all k = 1, 2, · · · , N n.

We next consider the case that |v k | <

7∆ 8

7∆ 8

for

for all k, k = 1, 2, · · · , N m, in this

case, according to (11), (12), (14) and (15) we have σp (v k ) = v k and   1 1 σr − (v k + hk ) + hk − q k − rk ) + q k + tk = − (v k + hk ) + hk − rk + tk , b b 12

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hence ∆2 (σ (v k ) + hk )2 2((b c + ∆) + ∆)2 p !2   1 k k k k k k k k +κ v + σr − (v + h ) + h − q − r + q + t b   θ k 2 1 γ ≤ (p ) + + (v k + hk )2 − κ(v k + hk )2 γ 2 θ  2 b−1 k +κ (v + hk ) + (tk − rk ) b   θ k 2 (2b − 1)κ 3 γ κb2 + κ2 (b − 1)2 k k k 2 − − (v (t − rk )2 , ≤ (p ) − + h ) + γ b2 2 θ b2 2pk (σp (v k ) + hk ) − κ(v k + hk )2 +

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√

7∆ 8

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Notice that γ(LN ⊗ HB)σr (vF ) = γ(LN ⊗ HB)vF if |v k | < 1, 2, · · · , N n. Then we have

for all k =

∆V ≤ −ελmin (LN )ψ T ψ − γψ T (L2N ⊗ H T H)ψ + ψ T ((LN ⊗ B T P A)

M

−γ(L2N ⊗ B T P BH))T ((LN ⊗ B T P A) − γ(L2N ⊗ B T P BH))ψ θλ2 T ψ ψ ≤ −ελmin (LN )ψ T ψ − γλ1 ψ T ψ + γ ≤ 0. Therefore, we conclude that ∆V < 0 if |v k | < k

180

ED

Finally, we consider the case that |v | ≥ in this case

PT

∆V ≤ − − ελmin (LN )ψ T ψ − ∆2 +

AC

(2b − 1)κ 3 γ − − b2 2 θ



for all k, k = 1, 2, · · · , N m.

for at least one k, k = 1, 2, · · · , N m,

X  1 θ − (hk )2 + (pk )2 γ γ 7∆ k

|v |<

8

 κb2 + κ2 (b − 1)2 k k 2 − (v + h ) + (t − r ) b2 θ ≤ −ελmin (LN )ψ T ψ − ∆2 + |((LN ⊗ B T P A) − γ(L2N ⊗ B T P BH))ψ|2 γ κb2 + κ2 (b − 1)2 + |γ(LN ⊗ HB)(vF ) − (LN ⊗ HB)σr (vF )|2 . b2

CE



7∆ 8

7∆ 8

k

k 2

According to (14) and (15), we have γ|(LN ⊗ HB)(vF ) − (LN ⊗ HB)σr (vF )| ≤ γ|(LN ⊗ HB)(vF )| + γ|(LN ⊗ HB)σr (vF )| b2 ∆ ≤ . 2 2(κb +κ2 (b−1)2 ) 13

ACCEPTED MANUSCRIPT

Therefore, we conclude that ∆V ≤ −ελmin (LN )ψ T ψ − ∆2 + k

|v | ≥

7∆ 8

∆2 2

+

∆2 2

≤ 0 if

for at least one k, k = 1, 2, · · · , N m.

∆V < 0, ∀(ψ, vF ) ∈ LV (c) \ {0}.

CR IP T

In conclusion, we have shown that, for all ε ∈ (0, ε∗ ],

Which implies that closed-loop system is asymptotically stable at (ψ, vF ) = (0, 0) with LV (c) included in the domain of attraction, and therefore, lim ψ(t) = 0,

lim vF (t) = 0,

t→∞

t→∞

follows that,

AN US

hold for all xi (0) ∈ X0 , i = 1, 2, · · · , N, and vi (0) ∈ V0 , i = 1, 2, · · · , N . It
lim xF (t) + L−1 N LM ⊗ In xR (t) = 0,

t→∞

hold for all xi (0) ∈ X0 , i = 1, 2, · · · , N, and vi (0) ∈ V0 , i = 1, 2, · · · , N . Ac-

cording to Lemma 2, each entry of −L−1 N LM is nonnegative and each row of

t→∞

M

−1 −L−1 N LM has a sum equal to one, choose αij = −[LN LM ]ij , thus we have   NX +M lim xi (t) − αij xj (t) = 0, i = 1, 2, · · · , N, j=N +1

hold for all xi (0) ∈ X0 , i = 1, 2, · · · , N + M , and vi (0) ∈ V0 , i = 1, 2, · · · , N .

That is the containment control problem solved. This completes the proof. 

ED

185

3.2. The Output Feedback Result

PT

Construct an observer for each agent zi (t + 1) = Azi (t) − L(yi − Czi (t)) + Bvi (t), i = 1, 2, · · · , N,

CE

zi (t) = Azi (t) − L(yi − Czi (t)), i = N + 1, N + 2, · · · , N + M,

where L is any matrix such that A + LC is asymptotically stable. The existence

AC

of such an L is due to Assumption 2. We then propose an observer based output

190

feedback containment control law for each follower agent (1),   NX +M NX +M 1 γH aij (zi (t) − zj (t)) ui (t) = −γHA aij (zi (t) − zj (t)) + 1 − b j=1 j=1

  N X 1 − α+ −1 vi (t)−γHB aij (vi (t) − vj (t)), i = 1, 2, · · · , N, (16) b j=1 14

ACCEPTED MANUSCRIPT

where γ ≤

1 λmax (LN )

is a constant and H = (I + B T P B)−1 B T P A, where P is

the solution of the ARE (3). Theorem 2. Under assumptions 1, 2 and 3, the state feedback containment

CR IP T

control laws (16) solve semi-global containment control problem for multi-agent

systems consisting of follower agents (1) and leader agents (2). That is, for any given bounded sets X0 ⊂ Rn and V0 ⊂ Rm , there is an ε∗ > 0 such that,

for any given ε ∈ (0, ε∗ ] and for all xi (0), zi (0) ∈ X0 , i = 1, 2, · · · , N + M , and

lim xi (t) −

t→∞

NX +M

j=N +1



αij xj (t) = 0, i = 1, 2, · · · , N,

AN US

vi (0) ∈ V0 , i = 1, 2, · · · , N , 

where αij , i = 1, 2, · · · , N, j = N + 1, N + 2, · · · , N + M , are nonnegative constants satisfying

NX +M

αij = 1.

j=N +1

M

Proof: Denote xF (t) = [x1 (t)T , x2 (t)T , · · · , xN (t)T ]T , vF (t) = [v1 (t)T , v2 (t)T , · · · , vN (t)T ]T , xR (t) = [xN +1 (t)T , xN +2 (t)T , · · · , xN +M (t)T ]T and u(t) = [u1 (t),

ED

u2 (t), · · · , uN (t)]T . Then, the dynamics of follower agents and leader agents can

PT

be written in the following compact form,    x (t+1) = (I ⊗ A)xF (t) + (I ⊗ B)σp (vF (t)),   F vF (t+1) = vF (t) + σr ((α − 1)vF (t) + u(t)),    x (t+1) = (I ⊗ A)x (t). R

R

CE


Denote ψ(t) = xF (t) + L−1 N LM ⊗ I xR (t), then

AC


ψ(t + 1) = xF (t + 1) + L−1 N LM ⊗ I xR (t + 1)

195


= (I ⊗ A)xF (t) + (I ⊗ B)σp (vF (t)) + L−1 N LM ⊗ A xR (t)

= (I ⊗ A)ψ(t) + (I ⊗ B)σp (vF (t)).

Denote ei = xi − zi , i = 1, 2, · · · , N + M , then we have ei (t + 1) = (A + LC)ei (t) + B(σp (vi ) − vi ), i = 1, 2, · · · , N, ei (t + 1) = (A + LC)ei (t), i = N + 1, N + 2, · · · , N + M. 15

ACCEPTED MANUSCRIPT

Let eF (t) = [e1 (t)T , e2 (t)T , · · · , eN (t)T ]T and eR (t) = [eN +1 (t)T , eN +2 (t)T , · · · , eN +M (t)T ]T , then, the dynamics of eF and eR can be written in the following compact form,

eR (t + 1) = (I ⊗ (A + LC))eR (t).

CR IP T

eF (t + 1) = (I ⊗ (A + LC))eF (t) + (I ⊗ B)(σp (vF )(t) − vF (t)),
Denote ω(t) = eF (t) + L−1 N LM ⊗ I eR (t), then it follows

ω(t + 1) = (I ⊗ (A + LC))ω(t) + (I ⊗ B)(σp (vF )(t) − vF (t)),

and

AN US

200

(17)

 vF (t + 1) = vF (t) + σr − γ(LN ⊗ HA)(ψ(t) − ω(t))      1 1 + 1− γ(LN ⊗ H)(ψ(t) − ω(t)) − I + γ(LN ⊗ HB) vF . b b

M

Consider the Lyapunov function candidate

V = ψ T (LN ⊗ P )ψ + κ1 ω T (I ⊗ Q)ω

ED

+κ2 (vF + γ(LN ⊗ H)ψ)T (vF + γ(LN ⊗ H)ψ),

(18)

2

b where κ1 = λmax (P ), κ2 ≥ max{3b, 2b−1 ( 12 + γθ )} is a constant and Q > 0 is the

PT

unique solution to the following Lyapunov equation (A + LC)T Q(A + LC) − Q = −I.

CE

It is trivial to show that V is positive definite due to the fact that both P and

Q are positive definite.

AC

Let c > 0 be a constant scalar such that c≥

sup

V.

(19)

ε∈(0,1],xi ,zi ∈X0 ,i=1,2,··· ,N +M,vi ∈V0 ,i=1,2,··· ,N

Such a c exists beacuse X0 and V0 are both bounded sets and limε→0 P = 0

205

according to Lemma 1.

16

ACCEPTED MANUSCRIPT

Let LV (c) := {(ψ, ω, vF ) ∈ RN (n+m) : V (ψ, ω, vF ) ≤ c}. Let ε∗1 ∈ (0, 1] be

such that for all ε ∈ (0, ε∗1 ], (ψ, vF ) ∈ LV (c) implies that

∆2 √ I, 2(b c + ∆) + ∆2 ∆ kγ(LN ⊗ H)ψk ≤ , 64b ∆ kγ(LN ⊗ HA)ψk ≤ , 64b ∆ , kγ(LN ⊗ HA)ωk ≤ 64b

 

γ 1 − 1 (LN ⊗ H)ω ≤ ∆ ,

64b b 2

κ1 (σp (vF ) − vF )T (I ⊗ (B T QB))(σp (vF ) − vF ) ≤ ∆ , 32 ∆2 T T |2κ1 ω (I ⊗ (A + LC) QB)(σp (vF ) − vF )| ≤ 32 r  

((LN ⊗B T P A)−γ(L2N ⊗B T P BH))ψ ≤ min ∆ , ∆ γ , 64b 2 θ   2 b ∆ ∆ , , kγ(LN ⊗ HB)vF k ≤ min 64b 4(κb2 +κ2 (b−1)2 )   b2 ∆ ∆ , , kγ(LN ⊗ HB)σp (vF )k ≤ min 64b 4(κb2 +κ2 (b−1)2 )     b−1 T b−1 2 T A − I (L ⊗ H H) A − I N b b b2 κ1 ≤ 2 2 2γ (2κ2 b + κ22 (b − 1)2 ) 2 γ (vF − σ(vF ))T (L2N ⊗ B T H T HB)(vF − σ(vF ))

(20)

PT

ED

M

AN US

CR IP T

LN ⊗ B T P B ≤

≤

λ1 γ 2 λ2 ,

CE where θ =

b2 ∆ 2 2(2κ2 b2 + κ22 (b − 1)2 )

(21) (22)

(23)

(24)

(25) (26) (27) (28) (29)

(30)

(31)

λ1 = λmin (L2N ⊗H T H) and λ2 = λmax (((LN ⊗B T P A)−γ(LN ⊗

B T P BH))T ((LN ⊗ B T P A) − γ(LN ⊗ B T P BH))). The existence of such an ε∗1 is again due to Lemma 1 and the fact that |σp (s)| ≤ b|s|. Therefore, we have

AC

210

∆V = V (t + 1) − V (t) = −εψ T (LN ⊗ I)ψ − γψ T (L2N ⊗ H T H)ψ +(σp (vF ) + γ(LN ⊗ H)ψ)(LN ⊗ B T P B)(σp (vF ) + γ(LN ⊗ H)ψ) +2ψ T ((I ⊗AT )−γ(LN ⊗H T B T ))(LN ⊗P B)(σp (vF )+γ(LN ⊗H)ψ) 17

ACCEPTED MANUSCRIPT

−κ1 ω T ω + 2κ1 ω T (I ⊗ (A + LC)T QB)(σp (vF ) − vF )

AN US

CR IP T

+κ1 (σp (vF ) − vF )T (I ⊗ (B T QB))(σp (vF ) − vF )   +κ2 vF + σr − γ(LN ⊗ HA)(ψ − ω)      1 1 γ(LN ⊗ H)(ψ − ω) − I + γ(LN ⊗ HB) vF + 1− b b  + γ(LN ⊗ HA)ψ + γ(LN ⊗ HB)σr (vF )   × vF + σr − γ(LN ⊗ HA)(ψ − ω)      1 1 + 1− γ(LN ⊗ H)(ψ − ω) − I + γ(LN ⊗ HB) vF b b  + γ(LN ⊗ HA)ψ + γ(LN ⊗ HB)σr (vF ) −κ2 (vF + γ(LN ⊗ H)ψ)T (vF + γ(LN ⊗ H)ψ)

≤ −ελmin (LN )ψ T ψ + κ1 (σp (vF ) − vF )T (I ⊗ (B T QB))(σp (vF ) − vF )

M

−κ1 ω T ω + 2κ1 ω T (I ⊗ (A + LC)T QB)(σp (vF ) − vF )  Nm X ∆2 (σp (v k ) + hk )2  − 1 (hk )2 + 2pk (σ (v k ) + hk ) + √ + p γ 2(b( c + ∆) + ∆)2 +κ2

ED

k=1



1 v + σr − (v k + hk ) + pk + hk − sk − q k − rk b ! k

PT

−κ2 (v k + hk )2



k

k

+q +t

!2

,

CE

where v k is the kth element of vF , hk is the kth element of γ(LN ⊗ H)ψ, q k is

the kth element of matrix γ(LN ⊗HA)ψ, pk is the kth element of γ(LN ⊗HA)ω,

sk is the kth element of matrix γ(1 − 1b )(LN ⊗ H)ω, pk is the kth element of

matrix (LN ⊗ B T P )((I ⊗ A) − γ(LN ⊗ BH))ψ, rk is the kth element of matrix

AC

215

γ(LN ⊗ HB)vF and tk is the kth element of matrix γ(LN ⊗ HB)σr (vF ). We first consider the case that |v k | ≥

7∆ 8

for any k, k = 1, 2, · · · , N m, in this

case, it follows from (18) and (19) that √ (σr (v k ) + hk )2 ≤ ((b c + ∆) + ∆)2 , 18

ACCEPTED MANUSCRIPT

hence,

CR IP T

1 ∆2 √ − (hk )2 + 2pk (σp (v k ) + hk ) + (σ (v k ) + hk )2 γ 2((b c + ∆) + ∆)2 p !2   1 k k k k k k k k k k +κ2 v + σr − (v + h ) + p + h − s − q − r + q + t b −κ2 (v k + hk )2

   2  2 2∆2 ∆2 ∆ k 1 7∆ ∆ ∆ ∆ k k + + |v |+κ |v |− − + −κ |v | − 2 2 642 2 32 b 8 8 32b 64b 2∆2 ∆2 ∆ k 45κ2 ∆ k 2115κ2 ∆2 + + |v |− |v |+ ≤ 642 2 32 32b 4096b ≤ −2∆2 .

AN US

≤

We next consider the case that |v k | <

7∆ 8

for any k, k = 1, 2, · · · , N m, in

this case, according to (21), (22), (23), (25), (28) and (29) we have σp (v k ) = v k 220

and

M

  1 k k k k k k k σr − (v + h ) + p + h − s − q − r b 1 k = − (v + hk ) + pk + hk − sk − q k − rk , b

ED

hence

PT

1 ∆2 √ − (hk )2 + 2pk (σp (v k ) + hk ) + (σ (v k ) + hk )2 γ 2((b c + ∆) + ∆)2 p !2   1 +κ2 v k + σr − (v k + hk ) + pk + hk − sk − q k − rk + q k + tk b

AC

CE

−κ2 (v k + hk )2   γ 1 1 k 2 θ k 2 + ≤ − (h ) + (p ) + (v k + hk )2 γ γ θ 2 !2   1 k k k k k k +κ2 1− (v + h ) + (t − r ) + (p − s ) − κ2 (v k + hk )2 , b   1 θ κ2 (2b − 1) γ 1 ≤ − (hk )2 + (pk )2 − − − (v k + hk )2 γ γ b2 θ 2     κ2 (b − 1)2 κ2 (b − 1)2 (tk − rk )2 + 2κ2 + 2 2 (pk − sk )2 + 2κ2 + 2 2 b b

Notice that γ(LN ⊗ HB)σr (vF ) = γ(LN ⊗ HB)vF if |v k | < 19

7∆ 8

for all k =

ACCEPTED MANUSCRIPT

1, 2, · · · , N n. In this case, we have Nn X



 − 1 (hk )2 + θ (pk )2 γ γ k=1      κ2 (2b − 1) γ 1 κ22 (b − 1)2 k k 2 k k 2 − − (v (p − s ) + h ) + 2κ + − 2 b2 θ 2 b2 λ2 θ T ≤ −ελmin (LN )ψ T ψ − κ1 ω T ω − λ1 γψ T ψ + ψ ψ γ   1 κ2 (2b − 1) γ − − (vF + γ(LN ⊗ H)ψ)T (vF + γ(LN ⊗ H)ψ) − b2 θ 2       κ2 (b−1)2 2 T b−1 b−1 2 T + 2κ2 + γ ω A−I (L ⊗ H H) A−I ω N b2 b b κ1 ≤ −ελmin (LN )ψ T ψ − ω T ω. 2

AN US

CR IP T

∆V ≤ −ελmin (LN )ψ T ψ − κ1 ω T ω +

Therefore, we conclude that ∆V < 0 if |v k | <

7∆ 8

for all k, k = 1, 2, · · · , N m.

On the other hand, we consider the case that |v k | ≥

225

k, k = 1, 2, · · · , N m, in this case

7∆ 8

for at least one

PT

ED

M

∆2 ∆2 ∆V ≤ −εψ T (LN ⊗ I)ψ − κ1 ω T ω + + − 2∆2 32 32  X   κ2 (2b − 1) γ 1 − − − (v k + hk )2 + 2 b θ 2 |v k |< 7∆ 8      κ2 (b − 1)2 κ2 (b − 1)2 + 2κ2 + 2 2 (tk − rk )2 + 2κ2 + 2 2 (pk − sk )2 b b   31∆2 κ22 (b − 1)2 ≤ − + 2κ2 + γ 2 (vF − σ(vF ))T 16 b2

AC

CE

×(L2N ⊗ B T H T HB)(vF − σ(vF ))       κ2 (b−1)2 2 T b−1 T b−1 + 2κ2 + 2 2 γ ω A −I (L2N ⊗ H T H) A−I ω b b b 2 κ1 23∆ ≤ − − ωT ω 16 2 < 0,

Therefore, we conclude that ∆V ≤ 0 if |v k | ≥

7∆ 8

for at least one k, k =

1, 2, · · · , N m.

In conclusion, we have shown that, for all ε ∈ (0, ε∗ ], ∆V < 0, ∀(ψ, vF ) ∈ LV (c) \ {0}. 20

ACCEPTED MANUSCRIPT

Which implies that closed-loop system is asymptotically stable at (ψ, vF ) = (0, 0) with LV (c) included in the domain of attraction, and therefore, lim ψ(t) = 0,

lim vF (t) = 0,

t→∞

CR IP T

t→∞

hold for all xi (0) ∈ X0 , i = 1, 2, · · · , N, and vi (0) ∈ V0 , i = 1, 2, · · · , N . It follows that,


lim xF (t) + L−1 N LM ⊗ In xR (t) = 0,

t→∞

hold for all xi (0) ∈ X0 , i = 1, 2, · · · , N, and vi (0) ∈ V0 , i = 1, 2, · · · , N . Ac-

cording to Lemma 2, each entry of −L−1 N LM is nonnegative and each row of

t→∞

j=N +1

AN US

−1 −L−1 N LM has a sum equal to one, choose αij = −[LN LM ]ij , thus we have   NX +M lim xi (t) − αij xj (t) = 0, i = 1, 2, · · · , N,

hold for all xi (0) ∈ X0 , i = 1, 2, · · · , N + M , and vi (0) ∈ V0 , i = 1, 2, · · · , N .

That is the containment control problem solved. This completes the proof.  4. Simulation Results

M

230

ED

Consider a multi-agent system consisting of 4 follower agents and 2 leader

PT

agents, described by (1) and (2), respectively, with     1 1 0 , B =  , A= 0 1 1 α = 0.5, σp = min{1, |s|}, σr = min{1, |s|}.

The communication topology among agents is shown in Fig. 1, where nodes

CE

235

1, 2, 3 and 4 are follower agents and nodes 5 and 6 are leader agents. It is easy

AC

to verify that the communication topology among agents satisfies Assumption 3. According to the communication topology, we can obtain    2 −1 −1 0 0 0        −1  0 2 0 −1  0  , LM =  LN =      −1  −1 0 2 0  0    0 −1 0 2 0 −1 21



   .   

CR IP T

ACCEPTED MANUSCRIPT

Figure 1: The undirected graph representing the communication topology.

The maximum eigenvalue of LN is λmax (LN ) = 3.618. We then choose γ = 240

0.25 <

1 λmax (LN ) .



= =

h

h

x1 (0) x2 (0) x3 (0) x4 (0) x5 (0) x6 (0)  2 −90 14 21 61 95 , 9 −2 −2 14 10 −2 i v1 (0) v2 (0) v3 (0) v4 (0) i 0.1 0.2 −0.3 0.4 .

i

M

h

AN US

Choose the initial values of the agents randomly as

ED

Here, we consider two different values of ε, ε = 1 and ε = 0.1. Then we can

PT

obtain the corresponding solutions of the ARE (3) respectively as     2.9471 2.3692 0.3815 0.4869  , P (0.1) =  . P (1) =  2.3692 4.6131 0.4869 1.3704

Figs. 2 and 3 respectively show the trajectories of states of follower agents

and leader agents, the input of follower agents and the rates of the actuators of

CE 245

follower agents, for ε = 1 and ε = 0.1. It is shown in Fig. 2 that the containment

AC

control is not solved under control law (6), which indicates that not all the initial conditions of the agents are in the domain of attraction resulting from the value of ε = 1. It is shown in Fig. 3 that the states of all follower agents

250

converge to the convex hull formed by the lader agents under control law (6), which indicates that all the initial conditions of the agents are in the domain of attraction resulting from the value of ε = 0.1. That is these regions of the 22

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100

4000 x11

x21

x31

x41

x51

x61

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x22

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x62

2000 50 0

-4000

-50

-6000 -8000

-100 -10000 -12000

0

500

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1500

2000

2500

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3000

0

500

1000

t(s)

u2

u3

1500

2000

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3000

2500

3000

t(s)

2500 u1

CR IP T

0

xi2

xi1

-2000

u4

σ(v1)

1

1500

σ(v2)

σ(v3)

σ(v4)

AN US

2000

0.5

σ(v i)

ui

1000 500 0

-0.5

-500 -1000 -1500

0

-1

0

500

1000

1500

2000

2500

0

3000

500

1000

1500

2000

t (s)

M

t (s)

Figure 2: The evolutions of the agents with ε = 1.

ED

initial conditions can be increased by deceasing the value of ε. This verifies that

255

PT

the semi-global containment control problem is solved.

5. Conclusions

CE

In this paper, we have studied semi-global containment control problem of a group of discrete-time linear systems in the presence of actuator position and rate saturation. By using low gain approach, we have constructed both a linear

AC

state feedback control law and an observer based output feedback control law

260

for each follower agent such that, the states of follower agents converge to the convex hull formed by the leader agents asymptotically. Numerical simulation results show that, for arbitrarily large initial conditions, semi-global containment control problem can always be solved under the proposed control law by sufficiently decreasing the value of ε. 23

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30

4000 x21

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xi2

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-1000

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3000

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500

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CR IP T

x11

1500

2000

2500

3000

t(s)

t(s) 1

15 u1

u2

u3

σ(v1)

u4

10

ui

σ(v i)

5

AN US

0.5

0

σ(v2)

σ(v3)

σ(v4)

0

-0.5

-5

-10

0

500

1000

1500

2000

2500

-1

3000

0

500

1000

1500

2000

2500

3000

t (s)

M

t (s)

Figure 3: The evolutions of the agents with ε = 0.1.

6. References

PT

References

ED

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[1] R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with swiching topology and time-delays, IEEE Trans. on Automatic

CE

Control, 2004, 49(9): 1520-1533.

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[2] W. Ren. Distributed leaderless consensus algorithms for networked Euler-

AC

Lagrange systems. International Journal of Control, 2009, 82(11): 21372149.

[3] W. Yang, Z. Wang, Z. Zuo, C. Yang and H. Shi. Nodes selection strategy in cooperative tracking problem. Automatica, 2016, 74: 118-125.

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[4] Q. Li, B. Shen, Z. Wang and F.E. Alsaadi. Event-triggered H∞ state estimation for state-saturated complex networks subject to quantization effects and distributed delays. Journal of the Franklin Institute, 2018, 355(5):

CR IP T

2874-2891. [5] M. Ji, G. Ferrari-Trecate, M. Egerstedt and A. Buffa. Containment control mobile networks, IEEE Trans. on Automatic Control, 2008, 53(8): 1972–

280

1975.

[6] B. Li, Z. Chen, Z. Liu, C. Zhang and Q. Zhang. Containment control

AN US

of multi-agent systems with fixed time-delays in fixed directed networks. Neurocomputing, 2016, 173: 2069-2075. 285

[7] H. Wang, C. Wang and G. Xie. Finite-time containment control of multiagent systems with static or dynamic leaders. Neurocomputing, 2017, 226: 1-6.

M

[8] Z. Meng, W. Ren and Z. You. Distributed finite-time attitude containment control for multiple rigid bodies, Automatica, 2010, 46(12): 2092-2099. [9] J. Li, W. Ren and S. Xu. Distributed containment control with multiple

ED

290

dynamic leaders for double-integrator dynamics using only position mea-

PT

surements, IEEE Trans. on Automatic Control, 2012, 57(6): 1553-1559. [10] F. Wang, H. Yang, Z. Liu and Z. Chen. Containment control of leaderfollowing multi-agent systems with jointly-connected topologies and timevarying delays, Neurocomputing, 2017, 260: 341-348.

CE

295

AC

[11] Y. Wang, L Cheng, Z. Hou, M. Tan and M. Wang. Containment control of multi-agent systems in a noisy communication environment, Automatica, 2014, 50(7): 1922-1928.

[12] Z. Liu, Q. Jin and Z. Chen. Distributed containment control for bounded

300

unknown second-order nonlinear multi-agent systems with dynamic leaders. Neurocomputing, 2015, 168: 1138-1143.

25

ACCEPTED MANUSCRIPT

[13] Z. Li, W. Ren, X. Liu and M. Fu. Distributed containment control of multiagent systems with general linear dynamics in the presence of multiple leaders, International Journal of Robust and Nonlinear Control, 2013, 23(5):

CR IP T

534-547.

305

[14] X. Wang, Y. Hong and H. Ji. Adaptive multi-agent containment control

with multiple parametric uncertain leaders, Automatica, 2014, 50(9): 23662372.

[15] S. Zuo, Y. Song, F. Lewis and A. Davoudi. Adaptive output containment

matica, 2018, 92: 235-239.

AN US

control of heterogeneous multi-agent systems with unknown leaders, Auto-

310

[16] C. Xu , Y. Zheng, H. Su and H. Zeng. Containment for linear multi-agent systems with exogenous disturbances. Neurocomputing, 2015, 160: 206212.

[17] Q. Ma and G. Miao. Distributed containment control of linear multi-agent

M

315

systems. Neurocomputing, 2014, 133: 399-403.

ED

[18] X. Mu and Z. Yang. Containment control of discrete-time general linear multi-agent systems under dynamic digraph based on trajectory analysis.

320

PT

Neurocomputing, 2016, 171: 1655-1660. [19] Z. Li, Z. Duan, W. Ren and G. Feng. Containment control of linear multiagent systems with multiple leaders of bounded inputs using distributed

CE

continuous controllers, International Journal of Robust and Nonlinear Control, 2015, 25(13): 2101-2121.

AC

[20] H. Su and M.Z.Q. Chen. Multi-agent containment control with input satu-

325

ration on switching topologies, IET Control Theory & Applications, 2015, 9(3): 399-409.

[21] M.A. Dornheim. Report pinpoints factors leading to YF-22 crash, Aviation Week Space Technol 1992, 9: 53–54.

26

ACCEPTED MANUSCRIPT

[22] C.A. Shifrin,Gripen likely to fly again soon, Aviation Week Space Technol. (Aug. 23, 1993) 72.

330

[23] Z. Lin. Semi-global stabilization of discrete-time linear systems with po-

CR IP T

sition and rate-limited actuators, Systems & Control Letters, 1998, 34: 313–322.

[24] Y.H. Lim and H.S. Ahm. Decentralized control of nonlinear interconnected

systems under both amplitude and rate saturations, Automatica, 2013,

335

49(8): 2551-2555.

AN US

[25] A.A. Stoorvogel and A. Saberi. Output regulation of linear plants with actuators subject to amplitude and rate constraints, International Journal of Robust and Nonlinear Control, 1999, 9: 631-657. 340

[26] Z. Zhao and Z.Lin. Semi-global leader-following consensus of multiple linear systems with position and rate limited actuators, International Journal of

M

Robust and Nonlinear Control, 2015, 25(13): 2083-2100.

[27] H. Zhou, Y. Yang, H. Su and W. Zeng. Semi-global leader-following consen-

ED

sus of discrete-time linear multi-agent systems subject to actuator position and rate saturation, International Journal of Robust and Nonlinear Con-

345

PT

trol, 2017, 27(16): 2921-2936.

AC

CE

[28] Z. Lin, Low Gain Feedback, Springer, London, 1998.

27

ACCEPTED MANUSCRIPT

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Zhiyun Zhao is currently a Postdoctoral Researcher in the Department of Automation, East China University of Science and Technology, Shanghai, China. She received her MSc degree in Control Theory and Control Engineering from Jiangsu University of Science and Technology in 2010, Jiangsu, China, and Ph. D. degree in Control Theory and Control Engineering from Shanghai Jiao Tong University in 2016, Shanghai, China. She is a recipient of the Shanghai Sailing Program award. Her main research interest is in coordinated control of multi-agent systems.

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Wen Yang is a Professor at East China University of Science and Technology(ECUST). She received her BSc degree in Mineral Engineering in 2002 and MSc degree in Control Theory and Control Engineering from Central South University in 2005, Hunan, China, and PhD degree in Control Theory and Control Engineering from Shanghai Jiao Tong University, Shanghai, China, in 2009. She was a Visiting Student with the University of California, Los Angeles, from 2007 to 2008. Her research interests include distributed state estimation, network security, coordinated and cooperative control, and complex networks.

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Hongbo Shi is a Professor at East China University of Science and Technology. He received the M.S. degree and the Ph.D. degree in control theory and control engineering from East China University of Science and Technology, China in 1989 and 2000, respectively. He was the 2003 Shu Guang Scholar of Shanghai. His research interest covers modeling of industrial process and advanced control technology, theory and methods of integrated automation systems, condition monitoring and fault diagnosis of industrial process.