Coordination of multi-agent systems with arbitrary convergence time

DOI: 10.1049/cth2.12086

B R I E F PA P E R

Coordination of multi-agent systems with arbitrary convergence time

Thanh Truong Nguyen

Minh Hoang Trinh

Chuong Van Nguyen

Nam Hoai Nguyen

M˜y V˘an Ðặng

1School of Electrical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam

2Department of Aerospace and Mechanical Engineering, University of Southern California, California, USA

(Email: [email protected])

Correspondence

Minh Hoang Trinh, School of Electrical Engineering, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi 11615, Vietnam.

Email:[email protected]

Funding information

Hanoi University of Science and Technology, Grant/Award Number: T2020-SAHEP-007

Abstract

This paper proposes fixed-time solutions to several problems in multi-agent systems. First, a fixed-time consensus protocol is investigated and under the assumption that the lower bound of smallest positive eigenvalue of the Laplacian matrix is known in advance, the consensus time of the protocol can be arbitrarily selected. Second, the consensus law to attenuate bounded disturbances acting on the system is modified. Third, the fixed-time design method is applied to the orientation alignment and network localization problems.

Finally, several simulations are provided to support the mathematical analysis.

1 INTRODUCTION

In recent years, there have been much research interests on con- trol and coordination of multi-agent systems due to its impacts on both civilian and military domains. Several applications of multi-agent systems include truck platooning in the automatic highway system, cooperative localization in sensor networks, distributed computation and optimization, formation control of unmanned vehicles, power generation and distribution, and modelling of social networks [1, 2].

There are many applications in which a multi-agent system needs to complete a task in a prescribed time. For example, the amount of energy generation and dispatch in smart-grid [3] are determined based on some decision variables which are solu- tions of a consensus protocol. Before making a target forma- tion, the agents need to complete the orientation estimation process [4]. Likewise, in pointing consensus problem [5], the agents should localize their positions and consent on the tar- get after a finite time before controlling their heading directions towards the target. These applications motivate the design of distributed finite-/fixed-time algorithm for multi-agent systems.

Finite-time controllers have been widely studied to stabilize the

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© 2021 The Authors. IET Control Theory & Applications published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology

states to the origin or the sliding surface [6–12]. The settling time in a finite-time controller is upper bounded by some value tf which is dependent on the initial conditions and system’s parameters. Thus, finite-time control laws cannot guarantee the states to converge to the origin after an arbitrarily selected time tf, or, i.e. they are inapplicable in problems with stringent requirements on the settling time. Several fixed-time controllers have been introduced recently, which allow the upper bound of the settling time t_f to be independent on the initial condition of the system [13, 14]. Finite- and fixed-time controller design methods have been applied to multi-agent systems, for example, finite-time average- and max-consensus protocols were intro- duced in [15, 16]. The authors in [17, 18] applied the fixed- time controller design methodology in [13] to consensus prob- lem with leaderless and leader–follower topologies, respectively.

Other fixed-time consensus protocols, designed for different assumptions on the agents’ dynamic and communication mod- els, can be found in [14, 19–22]. In [19], the authors pro- posed a robust fixed-time consensus algorithm for non-linear agents with disturbances. Fixed-time consensus tracking algo- rithm for second-order or higher-order integrator dynamics has been proposed in [20, 21]. The authors in [14] incorporates

900 wileyonlinelibrary.com/iet-cth IET Control Theory Appl.2021;15:900–909.

a time-dependent term into the consensus algorithm and proved fixed-time convergence of the system by using a time- transformation technique. In formation control and network localization, finite-time controllers were also studied, see [4, 23, 24] for examples. Note that finite-time convergence property of estimation/control law is beneficial in coordinating multiple tasks in multi-agent systems, as demonstrated in orientation- estimation based formation control [4] or pointing consensus [5]. Recently, fixed-time bearing-based network localization [25, 26] and bearing-based leader–follower formation control [27]

were also proposed.

The aim of this paper is designing fixed-time control laws for coordination of multi-agent systems based on the recently proposed method in [28–30]. First, a fixed-time consensus law for multi-agent systems interacting over an undirected graph is proposed. Although the control law in [28] is a global stabi- lization control law, for a pre-specified convergence time, the magnitude of the control law in [28] may increase greatly when the states are negative and sufficiently large [31]. To make the control input bounded, we propose to restrict the initial states in a box. It is shown that the states are maintained in that set for all time and the agents reach a consensus after an arbitrar- ily selected time. A method to expand the set is given. Second, in case there are bounded disturbances in the system, an addi- tional term is combined into the consensus law to attenuate the disturbances. Note that our disturbance rejection method is dif- ferent from [28], which is based on integral sliding mode con- trol. Third, we extend the control design for two other problems in multi-agent systems: orientation alignment and bearing-based network localization. It is worth noting that although the con- trol design for single-integrator system in [28] does not require any information on the systems’ parameters, when applying to the multi-agent systems, some information about the system’s topology has to be known in advance. In comparison with the fixed-time consensus and bearing-based network localization laws in [17, 25], the upper bound of the settling time t_f of the control laws in this paper can be easily selected. The structure of the proposed coordination laws in this paper is also simpler than the one studied in [27], where a dynamic auxiliary time-varying gain is introduced to the control law.

The remainder of this paper is organized in the following manner. In Section2, preliminaries on graph theory and fixed- time stabilization controllers are provided. Section3proposes the fixed-time consensus law. Further applications of fixed-time control laws for orientation alignment and bearing-based net- work localization problems are considered in Section 4. Sec- tion 5 contains some simulation results to demonstrate the fixed-time convergence property of the proposed consensus and network localization laws in Sections3and 4. A compar- ative study with other fixed-time consensus laws are also given.

Finally, Section6concludes the paper.

Notations. In this paper, ℝⁿ denotes the n-dimensional Euclidean space, ℝ^m×n is the set of m× n real matrices and ℝ_≥0 is the set of non-negative real numbers. The n× n iden- tity matrix is denoted by In, and 0 denotes the zero matrix of a suitable dimension. The range and null space ofA is denoted by Range(A) and  (A), respectively. We refer to [A]qand [A]i j

as the qth row of matrixA and the element in the ith row and

the j th column ofA, correspondingly. We use diag(A1,… , Am) to denote the block diagonal matrix formed by m matrices A1,… , Am. Fora ∈ ℝ^d, we denote |a| = [|a1|, … , |ad|]^⊤ and sgn(a) = [sgn(a1),… , sgn(ad)]^⊤, where sgn(⋅) is the signum function. The vector 1_ndenotes the n-dimension vector with all elements 1. The notations‖ ⋅ ‖1and‖ ⋅ ‖ represent the 1-norm and 2-norm, respectively.

2 BACKGROUND

2.1 Graph theory

Let  = (,  ) be a digraph with a vertex set  = {1, … , n}, and an edge set  ⊆ {(i, j )|i, j ∈ , i ≠ j} with |()| = m edges. For each edge (i, j ), i is the starting vertex and j is the end vertex. If (i, j )∈  implies ( j, i) ∈ , then  is an undi- rected graph. The neighbour set of a vertex i∈  is defined as

i = { j ∈ |(i, j ) ∈ }. The degree of a vertex i is given by di= |i|.

A path is a sequence of vertices i1i2… ip, where (ik, ik+1)∈

, ∀k = 1, … , p − 1, and the vertices ik(possibly except for the first and the last vertices) are all distinct. A cycle of length p is a path with the same end vertices, i.e. i₁≡ ip. A graph is acyclic if it does not contain any cycle. A graph is connected if and only if there exists a path connecting any pair of vertices in.

We define the Laplacian matrix of a graph by  = D − A, whereD = diag(di) is the degree matrix andA = [ai j]∈ ℝ^n×n is the adjacency matrix defined as

aij=

{1, ( j, i )∈ ,

0, otherwise.

If  is undirected and connected,  is symmetric, positive semi-definite and has only one zero eigenvalue 𝜆1= 0. The other eigenvalues of are given by 0 < 𝜆2≤ 𝜆3≤ … ≤ 𝜆n[2].

Labelling m edges in as e1,… , em and choose a direction for each edge. The incidence matrixH = [hki]∈ ℝ^m×n relates the edges and vertices in, and is defined as

h_ki=

⎧⎪

⎨⎪

⎩

−1, i is the starting vertex of ek

1, iis the end vertex of e_k 0, otherwise.

2.2 Fixed-time stability

We consider the following non-linear system:

̇x = f (t, x), x(t0)= x0, (1) wherex ∈  ⊆ ℝⁿ represents the state of the system and f : ℝ_≥0× ℝⁿis a non-linear function. We assume that the system (1) has the originx = 0 as its equilibrium.

Definition 1. [6] The origin of the system (1) is finite time stable if it is asymptotically stable and each solution starting

in reaches the origin at some finite time, i.e. x(t, t0,x0)= 0

∀t ≥ t0+ T (t0,x0), where T :ℝ_≥0×  ⟶ ℝ_≥0 is the set- tling time function.

Definition 2. ([13]) The origin of the system (1) is fixed-time stable if it is finite time stable and the settling time function is upper bounded by a positive number T_maxwhich is independent onx0. In other words,∃Tmax> 0 : T (t0,x0)≤ Tmax, ∀t0≥ 0,

∀x0∈ ℝⁿ.

Lemma 1. [28][Thm. 1] Consider the system (1) and let ⊂ ℝⁿbe a domain containing the equilibrium pointx = 0. Let 𝛼1(x) and 𝛼2(x) be two continuous positive definite functions on. Assume that there exist a real-valued continuously differentiable function V : I ×  ↦ ℝ_≥0(I is a finite time interval, i.e. I = [t0, tf]) and a real number𝜂 ≥ 1 such that

∙ 𝛼1(x) ≤ V (t, x) ≤ 𝛼2(x), ∀t ∈ I, ∀x ∈  ⧵ {0}

∙ V (t, 0) = 0, ∀t ∈ I

∙ ̇V ≤ ^−𝜂

tf−t(1− e^−V), ∀V ≥ 0, ∀t ∈ I

then the equilibrium pointx = 0 is fixed-time stable and Ta= tf − t0≥ Tr f (Tr f is the true fixed time in which the system is stabilized and tf is the settling time moment which is independent of the initial conditions).

3 ARBITRARY-TIME CONVERGENT CONSENSUS PROTOCOL

Consider an n-agent system characterized by an undirected, con- nected graph. The state of each agent is given by a state vec- tor xi(t )= [xi1,… , xid]^⊤, where xi(t0)∈  = [−0.5, 0.5]^d ⊂ ℝ^d, i = 1, … , n. Suppose that each agent i can obtain the relative variables (xi− xj),∀ j ∈ i, and update its state according to a single-integrator dynamics

̇xi= ui, i= 1, … , n, (2) whereuiis designed based on the relative variables available to agent i to drive their states to reach a same value.

The following fixed-time consensus protocol is proposed for single integrator agents:

u_i(t )=

⎧⎪

⎪⎨

⎪⎪

⎩ 𝜂 tf − t

∑

j∈Ni

diag(sgn(x_j− xi))œ(

|xj − xi|) , for t₀ ≤ t < tf,

0, otherwise.

(3)

Here, we define function𝜎(x) ≜ 1 − e^−xand𝝈(|xj − xi|) ≜ [1− e^{−|xj1−xi1|},… , 1 − e^{−|xjd−xid|}]^⊤. From Equation (3), we can express the n-agent system in matrix form as follows:

̇x = − 𝜂

t_f − t ̄H^⊤diag(sgn( ̄Hx))𝝈(| ̄Hx|), (4)

wherex = [x^⊤₁,… , x^⊤n]^⊤and ̄H = H ⊗ Id. Next, we prove the following lemmas:

Lemma 2. Under the consensus protocol (3), ifx(t0)∈  then x(t ) ∈

 for all to≤ t < tf.

Proof. Without loss of generality, we can assume that at some time t₀< t1< tf, x_ik(t₁)= 0.5 and |xjk(t )| ≤ 0.5, j = 1,… , n, j ≠ i. Then, it follows that

̇xik(t₁)= − 𝜂 t_f − t1

∑

j∈i

(1− e^{−|xjk−0.5|})

≤ 0,

and thus x_ik(t ) cannot grow over 0.5. Other cases can be treated

similarly. □

Lemma 3. Under the consensus protocol (3), the centroid is time- invariant. That is, ̄x(t ) = ¹

n(1^⊤_n ⊗ Id)x(t ) = ¹

n

∑n i=1xi(0).

Proof. SinceH1n= 0, we have

(1^⊤_n ⊗ Id)̇x = − 𝜂

t_f − t(1^⊤_n ⊗ Id) ̄H^⊤diag(sgn( ̄Hx))𝝈(| ̄Hx|)

= − 𝜂

t_f − t(1^⊤nH^⊤⊗ Id)diag(sgn( ̄Hx))𝝈(| ̄Hx|)

= 0.

It follows that ̄x(t ) = ̄x(0) = ¹

n

∑n

i=1xi(0), and thus the cen-

troid is time-invariant. □

The following lemma is important for further analysis.

Lemma 4. For z∈ ℝ, the function g : ℝ ↦ ℝ, where g(z) = z(1 − e^−z) satisfies:

∙ g(z) is positive definite.

∙ ̇g(z) > 0 if and only if z > 0, ̇g(z) < 0 if and only if z < 0, and

̇g(z) = 0 if and only if z = 0.

∙ g(z) ≥ g(|z|)

∙ g(z) is a convex function for z ∈ [0, 2].

Proof. The proof is trivial and will be omitted. □ Theorem 1. Under the consensus law (3), if𝜂 ≥ _𝜆^dm

2(), thenx(t ) converges to 1_n⊗ ̄x in fixed time. Moreover, x = 1n⊗ ̄x is a weakly stable equilibrium point with the settling time T_a= tf − t0 ≥ Tr f. Proof. Let𝜹i= xi− ̄x ∈ ℝ^d and let𝜹 = [𝜹^⊤1,… , 𝜹^⊤n]^⊤∈ ℝ^dn. Then, ̄H𝜹 = ̄Hx, ∑n

i=1𝜹i= 0. Moreover, it follows from Lemma2that𝛿ik∈ [−1, 1], ∀i = 1, … , n, k = 1, … , d.

Consider the Lyapunov function V =¹

2‖𝜹‖², which is pos- itive definite, continuously differentiable for t₀≤ t < tf. The

derivative of V along a trajectory of (4) is given by

̇V = − 𝜂

tf − t 𝜹^⊤̄H^⊤diag(sgn( ̄Hx))𝝈(| ̄Hx|)

= − 𝜂

tf − t 𝜹^⊤̄H^⊤diag(sgn( ̄H𝜹))𝝈(| ̄H𝜹|)

= − 𝜂

t_f − t |̄H𝜹|^⊤𝝈(| ̄H𝜹|)

= − 𝜂

t_f − t

∑

(i, j )∈

∑d k=1

|𝛿jk− 𝛿ik|(

1− e^{−|𝛿jk−𝛿ik|})

= − 𝜂

t_f − t

∑

(i, j )∈

∑d k=1

g(|𝛿jk− 𝛿ik|). (5)

It follows that ̇V ≤ 0, ∀t0≤ t < tf. Since g(⋅) is a convex func- tion on [0,1] and|𝛿jk− 𝛿ik| = |xjk− xik| ≤ 1, it follows from Jensen’s inequality that

∑d k=1

g(

|𝛿jk− 𝛿ik|)

≥ dg (∑^d

k=1|𝛿jk− 𝛿ik| d

)

= dg(‖𝜹j − 𝜹i‖1

d )

. (6)

Furthermore, since g(⋅) is non-decreasing in [0,1], it follows from the norm inequality‖x‖1≥ ‖x‖ that

∑

(i, j )∈

∑d k=1

g(

|𝛿jk− 𝛿ik|)

≥ d ∑

(i, j )∈

g(‖𝜹j − 𝜹i‖ d

)

≥ (dm)g (∑

(i, j )∈‖𝜹j− 𝜹i‖ dm

)

≥ (dm)g

(‖ ̄H𝜹‖

dm )

. (7)

On the other hand, since is connected,  ( ̄H) = Range(1n⊗ Id). Moreover, since (1^⊤_n ⊗ Id)𝜹 = 0, we have 𝜹 ⟂  ( ̄H). This implies

𝜆2()𝜹^⊤𝜹 ≤ 𝜹^⊤̄H^⊤̄H𝜹 = 𝜹^⊤ ̄𝜹 ≤ 𝜆n()𝜹^⊤𝜹. (8) Combining (5), (7), and (8), we have

̇V ≤ − 𝜂 t_f − t(dm)g

(√𝜆2()‖𝜹‖

dm )

= − 𝜂

tf − t

√𝜆2()‖𝜹‖

( 1− e⁻

√𝜆2()‖𝜹‖

dm

)

. (9)

Let𝛾 =^{√𝜆2()}

dm

√2V = ^{√𝜆2()}

dm ‖𝜹‖, it follows from (9) that

̇𝛾 =

√𝜆2() dm

√̇V

2V ≤ − 𝜂 t_f − t

𝜆2()

dm (1− e^−𝛾). (10)

For 𝜂1= 𝜂^𝜆2()

dm ≥ 1, based on Lemma 1, there holds𝛾(t ) = 0,∀t ≥ tf. Thus, V = 0 or 𝜹 = 0, for t ≥ tf. □ Remark1. We can extend the region of fixed-time convergence by modifying the consensus law (3) as follows:

u_i(t )=

⎧⎪

⎪⎨

⎪⎪

⎩ 𝜂 t_f − t

∑

j∈Ni

diag(sgn(x_j− xi))œ(

𝜅|xj − xi|) for t0≤ t < tf,

0, otherwise.

(11)

where 0< 𝜅 < 1 and 𝜂 is increased, respectively. Correspond- ingly, fixed-time convergence of (11) is guaranteed forxi(0)∈

^′= [−_𝜅¹,¹

𝜅]^d,∀i = 1, … , n.

Next, we consider the effects of disturbances in the consen- sus protocol. For simplicity, it is assumed that the disturbances appear when the agents use the relative information from other agents to calculate the update law

u_i(t )=

⎧⎪

⎪⎨

⎪⎪

⎩ 𝜂 t_f − t

∑

j∈Ni

diag(sgn(x_j− xi+ ıij))œ(

|xj− xi+ ıij|) ,

for t₀≤ t < tf,

0, otherwise.

(12) Suppose that (xj− xi)can still be measured but it is affected by the disturbances𝜻_{i j}∈ ℝ^d, which satisfy the following assump- tion:

Assumption 1. The disturbances𝜻_{i j},∀(i, j ) ∈  are (i) continu- ous and upper bounded by a known constant𝜁max(or‖𝜻_{i j}‖_∞<

𝜁max), (ii) [𝜻_{i j}]_k= −[𝜻_ji]_k,∀k = 1, … , d, and (iii) If xik= xjk, then [𝜻_{i j}]_k= 0.

Examples of [𝜻_{i j}]k that satisfy these properties include odd functions of the relative states such as sin(xik− xjk) or tanh(xik− xjk). The matrix form of the system at the time t ∈ [t0, tf):

̇x = − 𝜂

t_f − t ̄H^⊤diag(sgn( ̄Hx + 𝜻 ))𝝈(

| ̄Hx + 𝜻 |)

, (13)

where𝜻 = [… , 𝜻^⊤_{i j},…]^⊤= [𝜁1,… , 𝜁dm]^⊤is a dm-dimension dis- turbance vector corresponding to the edges in.

In order to compensate the disturbance𝜻_{i j}acting on the edge (i, j ), each agent i includes the term ̂𝜻_{i j} = 𝜁maxsgn(xi− xj) into the control law. The stacked vector of the compensating component

̂𝜻 = −𝜁maxsgn( ̄Hx) (14)

is then applied to the consensus law in the following form:

u = − 𝜂

tf − t ̄H^⊤diag(sgn( ̄Hx + 𝜻 − ̂𝜻 ))𝝈(

| ̄Hx + 𝜻 − ̂𝜻 |) . (15) Theorem 2. Let Assumption1hold. Under the control law (15), the system achieves fixed-time consensus under the presence of the disturbances.

Proof. Consider the Lyapunov function V = ¹

2‖𝜹‖², the deriva- tive of V along a trajectory of (15) is given by

̇V = − 𝜂

t_f − t( ̄H𝜹)^⊤diag(sgn( ̄H𝜹 + 𝜻 − ̂𝜻 ))𝝈(| ̄H𝜹 + 𝜻 − ̂𝜻 |).

(16) For each k= 1, … , dm, it follows from our assumption on 𝜻 and the definition of ̂𝜻 that

𝜁k− ̂𝜁k= 𝜁max

( 𝜁k

𝜁max

+ sgn([ ̄H𝜹]k) )

.

This implies

⎡⎢

⎢⎣ sgn

(𝜁k− ˆ𝜁k

)= sgn( [ ̄Hffi]_k)

, if [ ̄Hffi]_k≠ 0 sgn

(𝜁k− ˆ𝜁k

)= 0, if [ ̄Hffi]_k= 0 (17)

and both cases are equivalent to sgn(𝜁k− ̂𝜁k)= sgn([ ̄H𝜹]_k), or i.e. sgn(𝜻 − ̂𝜻 ) = sgn( ̄H𝜹). It follows that (16) can be rewritten as follows:

̇V = − 𝜂

t_f − t( ̄H𝜹)^⊤diag(sgn( ̄H𝜹))𝝈(| ̄H𝜹 + 𝜻 − ̂𝜻 |)

= − 𝜂

t_f − t

∑dm k=1

|[ ̄H𝜹]k|𝜎(|[ ̄H𝜹 + 𝜻 − ̂𝜻 ]k|)

≤ − 𝜂

t_f − t

∑dm k=1

|[ ̄H𝜹]k|𝜎(|[ ̄H𝜹]k|)

≤ − 𝜂

tf − t |̄H𝜹|^⊤𝝈(| ̄H𝜹|), (18) where, in the first inequality, we have used the fact that 𝜎(|[ ̄H𝜹 + 𝜻 − ̂𝜻 ]k|) = 𝜎(|[ ̄H𝜹]k| + |[𝜻 − ̂𝜻 ]k|) ≥

𝜎(|[ ̄H𝜹]k|) ≥ 0. The rest of this proof is similar to the proof of Theorem1and will be omitted. □ It is worth remarking that in practice, due to the time- dependent term ¹

t_f−t, as ̄H𝜹 + 𝜻 − ̂𝜻 ≠ 0 when t is close to tf, the fixed-time control law (15) would make the control input very large. In order to implement the control law (15), we can terminate the control input after a time t₁= tf − 𝜖, where 𝜖 > 0 is a small positive number.

4 EXTENSIONS

Due to the boundedness condition imposing on the states of each agent, the consensus law (3) and its variations are applicable to problems where the states of the agents are always bounded in time. Two problems satisfying such the boundedness properties, namely, orientation alignment and bearing-based network localization will be introduced in this section.

4.1 Fixed-time orientation alignment

Consider a group of n agents in the two-dimensional space inter- acting over an undirected graph. Each agent has a local coor- dinate systemⁱΣ (i = 1, 2, … , n) rotated from a global coordi- nate system^gΣ (unknown) by an angle xi(unknown and assume that xi(t0)∈ D = [−𝜋, 𝜋]). Suppose that each agent i can sense the local bearing vectorbⁱ_{i j}in its local coordinate systemⁱΣ and receivedb^j_jifrom its neighbour agents. Note that the local bear- ing vectors can be represented as

bⁱ_{i j} = R^⊤_i p_j − p_i

‖p_j − p_i‖ = R^⊤_i bi j,

b^j_ji = R^⊤_j p_i− p_j

‖p_i− p_j‖ = −R^⊤_jbi j,

whereRi=

[cos(x_i)− sin(xi) sin(x_i) cos(x_i)

]

is the rotation matrix that aligns the axes of^gΣ andⁱΣ. By receiving b^j_jifrom agent j , agent i can solve for the matrixRiR^⊤_j based on the equation:

bⁱ_{i j} = R^⊤_ibi j = −R^⊤_iRjb^j_ji. (19)

FromR^⊤_i Rj, agent i can find the relative orientation (x_i− xj) [4, 32, 33].

Suppose that maxi, j∈(xi− xj)≤ ^𝜋

2. Let agent i update its orientation by a single integrator dynamics:

̇xi= ui, i= 1, … , n. (20)

The orientation alignment process is completed after xi (i= 1,… , n) achieved a consensus. Note that uiis the angular veloc- ity that rotates the axes ofⁱΣ, and thus it is independent of the coordinate system.

By adopting the control law (11) with 0< 𝜅 < ²

𝜋, based on Remark1, for a sufficiently large chosen𝜂, the orientation align- ment process is completed after a fixed time upper bounded by tf.

4.2 Bearing-based network localization

Consider a sensor network of n nodes (agents) (n≥ 3) in the two-dimensional space. Let each agent have a local reference frame ⁱΣ. By applying the result from previous subsection, we can assume that the local coordinate systems of n agents are aligned to each other and to a common global coordi- nate system. Let agent i maintain an estimate ̂p_i of p_i—the position of agent i in the global coordinate system. Agent i updates its estimated position according to the following equation:

̇̂p_i= ui, i= 1, … , n, (21) whereuiis the update law that will be designed.

The network is represented by (, p), where  is an undi- rected graph characterizing the sensing and communication among the agents. An agent i can sense the bearing vectorsbi j =

p_i−p_j

‖p_i−p_j‖ and share its own estimated position ̂p_i with several neighbouring agents j ∈ i. The network localization prob- lem aims at determining a configuration ̂p = [̂p^⊤₁,… , ̂p^⊤_n]^⊤∈ ℝ²ⁿwhich is bearing congruent [34] to the true configuration p = [p^⊤₁,… , p^⊤_n]^⊤of the network, i.e.

P_{bi j}(̂p_i− ̂p_j)= 0, ∀i, j ∈ , i ≠ j. (22)

Here, P_{bi j} = I2− bi jb^⊤_{i j} ∈ ℝ^2×2 is an orthogonal projection matrix that projects any vector in ℝ² onto the orthogo- nal complement of span(bi j). It is not hard to see that P_{bi j} is symmetric, positive semi-definite, and  (P_{bi j})= span(bi j).

We adopt the assumptions from [34, 35] that the network (, p) is infinitesimally bearing rigid, or i.e.  (diag(P_bk) ̄H) = Range(1n⊗ I2,p). An example of infinitesimally bearing rigid network is shown in Figure1.

The following update law is proposed to solve the fixed-time bearing-based network localization problem:

u_i(t )=

⎧⎪

⎪⎨

⎪⎪

⎩ 𝜂 t_f − t

∑

j∈i

Pb_ijdiag (

sgn(Pb_ijˆpji) )

œ

(|Pb_ijˆpji|) ,

if t ∈ [t0, t_f),

0, otherwise,

(23)

F I G U R E 1 The network (, p) is infinitesimally bearing rigid

where ̂p_ji= ̂p_j − ̂p_i. Moreover, in (23), the matricesP_{bi j}, j ∈

i can be calculated from the sensed bearing vectorsbi j, and

̂p_j can be obtained via communication with j∈ i.

We can express the n-agent system under the network local- ization law (23) as follows:

̇̂p = − 𝜂

t_f − t ̄H^⊤diag(P_bk)diag(sgn(y))𝝈(

|y|)

, (24)

wherey = diag(P_bk) ̄H ̂p, ̄H = H ⊗ I2,bk is the bearing vec- tor corresponding to the kth edge in , k ∈ {1, … , m}, and diag(P_bk) is the block diagonal matrix with main diagonal blocks P_bk.

Let the agents initiate ̂p_i(0), i= 1, … , n, satisfying ̂p_i(0)∈

 = [−0.5, 0.5]². Similar to the consensus problem, it can be shown that under the control law (24), the formation’s centroid

̄̂p = ¹

n(1^⊤_n ⊗ In)̂p(t ) is time invariant because ̇̄̂p = 0. Let ̂p^∗be the final estimated configuration, which satisfies the bearing congruency condition (22) and has the same centroid as ̂p_i(0), we prove the following theorem.

Theorem 3. Let𝜂 > ^dm

𝜆4()where𝜆4() is the smallest positive eigen- value of the matrix = ̄H^⊤diag(P_bk) ̄H. Under the network localiza- tion law (23), if̂p_i(t )≠ ̂p_j(t ) for all time 0≤ t ≤ tf,̂p converges to a final configuration ̂p^∗which is bearing congruent to the true configuration p.

Proof. Consider the Lyapunov function

V =1

2

(̂p − ̂p^∗)⊤(

̂p − ̂p^∗) ,

which is continuously differentiable and positive definite. The derivative of V along a trajectory of (24) is

̇V = − 𝜂 t_f − t

(̂p − ̂p^∗)⊤

̄H^⊤diag(P_bk)⋅

× diag(

sgn(diag(P_bk) ̄H ̂p)) 𝝈(

|diag(P_bk) ̄H(

̂p − ̂p^∗)

|) ,

= − 𝜂

t_f − t q^⊤diag(sgn(q))𝝈(|q|), (25) whereq ≜ diag(P_bk) ̄H( ̂p − ̂p^∗) and we have used the fact that diag(P_bk) ̄H ̂p^∗= 0. It follows from (25) that

̇V = − 𝜂

tf − t |q|^⊤𝝈(

|q|)

≤ − 𝜂

t_f − t(dm)g (‖‖q‖‖1

dm )

≤ − 𝜂

tf − t(dm)g(‖‖q‖‖

dm )

. (26)

Combine with

‖‖q‖‖ =√

q^⊤q =√(

̂p − ̂p^∗)⊤

(

̂p − ̂p^∗)

≥√

𝜆4()‖‖‖̂p − ̂p^∗‖‖‖, (27) where𝜆4() > 0 because (, p) is infinitesimally bearing rigid, it follows that

̇V ≤ − 𝜂 tf − t

√𝜆4()√ 2V

( 1− e⁻

√𝜆4()√ 2V dm

)

. (28)

The rest of the proof is similar to the proof of Theorem1and

will be omitted. □

Define the estimated network scale ̂s =

√

1 n

∑n

i=1‖̂p_i− ̄̂p‖². Since ̄̂p is time-invariant, we have ̄̂p^∗= ̄̂p(t ). Define ̂r = ̂p − 1n⊗ ̄̂p and ̂r^∗= ̂p^∗− 1n⊗ ̄̂p^∗, one haŝr − ̂r^∗= ̂p − ̂p^∗. Tak- ing the time derivative of̂s = ^‖̂r‖√

n along a trajectory of the sys- tem yields

̇̂s = 1√ n

(̂p − 1n⊗ ̄̂p)^⊤

‖‖̂p − 1ⁿ⊗ ̄̂p‖‖ ̇̂p.

Thus,

̂s ̇̂s = ̇V

n ≤ 0, (29)

which implies that ̂s is a non-increasing function. Integrating both sides of Equation (29), and keeping in mind that V (t_f)= 0 gives

̂s²(0)− ̂s²( tf

)= 2V (0)

n = ‖̂p(0) − ̂p^∗‖²

n . (30)

It follows from Equation (30) that

̂r(0)^⊤̂r(0) − (̂r^∗)^⊤̂r^∗=(

̂r(0) − ̂r^∗)⊤(

̂r(0) − ̂r^∗) or, 2(̂r^∗)^⊤(

̂r(0) − ̂r^∗)

= 0. (31)

It can be seen that ̂p(t ) → ̂p(0) is equivalent to ̂r(t ) → ̂r^∗. Equation (31) also implies that ̂r^∗is the orthogonal projection of̂r(0) onto the null space of the matrix .

5 SIMULATIONS

5.1 Fixed-time consensus over undirected graphs

In this subsection, we provide simulation results to illustrate the effectiveness of the fixed-time consensus protocol (3) and compare it with several existing consensus law in the litera- ture. Consider a 10-agent system interacting over an undirected cycle graph10. Each agent has a state vectorxi ∈ ℝ². In all simulations, we use a same set of initial statesxi(0) chosen in

 = [−0.5, 0.5]².

In the first simulation, the fixed-time consensus protocol (3) is employed. Since the smallest positive eigenvalue of the Lapla- cian matrix of the cycle graphn is𝜆2= 2(1 − cos^2𝜋

n ) [2], it is sufficient to choose𝜂 ≥ 𝜂0= ^dm_𝜆

2 =_2(1−cos^2×10𝜋

5) ≈ 52.3607 to ensure a fixed-time convergence according to1, with the con- vergent time is chosen to be tf = 0.25 s. Figure2depicts the simulation results with𝜂 = 80 and tf = 0.1 s. Observe that the states x1iand x2i(i= 1, … , 10) converge to the consensus value after a finite time which is earlier than tf = 0.1 s and the control u(t ) is also bounded.

Further simulations are conducted to compare the proposed consensus protocol (3) with the conventional consensus proto- col [1] (Simulation 2), the fixed-time consensus protocol in [17]

(Simulation 3), and the fixed-time consensus protocol in [14]

(Simulation 4). The control gain𝜂 > 0 is included into all con- sidered consensus protocols so that we can have a relatively fair comparison of their convergence times and control magnitudes.

Simulation results are depicted in Figures2–5. Under all consen- sus laws, the agents asymptotically converge to their geometric centre. It is observed that the fixed-time consensus laws outper- form the usual consensus protocol in term of convergence rate.

Indeed, the usual consensus law cannot steer the agents to the consensus point in 0.1 s. On the other hand, the cost of fixed- time convergence is shown in the control magnitude‖u‖, which are very large at t = 0.

F I G U R E 2 Simulation 1: (a) Trajectories ofxi, i= 1, … , 10, under the consensus law (3); (b) Statesxi1andxi2, i= 1, … , 10 vs time; and (c) ‖u(t )‖ versus time

F I G U R E 3 Simulation 2: (a) Trajectories ofxi, i= 1, … , 10, under the normal consensus law ui= 𝜂∑

j∈i(xj− xi) (see [1]); (b) Statesxi1andxi2, i= 1,… , 10, versus time; and (c) ‖u(t )‖ versus time

It is remarked that when t_f > 1, for 0 < t < tf − 1, the time- varying gain ^𝜂

t_f−t < 𝜂, and thus, the magnitude of the proposed control law during that time interval is smaller than which of the usual consensus protocol (̇xi= −𝜂∑

j∈i(xi− xj)). Thus, fixed-time convergence property is mostly due to the control effort in the time span (t_f − 1, tf).

5.2 Fixed-time bearing-based network localization

In this subsection, we simulate a network of 20 agents with (, p) as shown in Figure 1. The initial estimates ̂p_i(0), i= 1,… , 20, have been randomly chosen inside  = [−0.5, 0.5]².

F I G U R E 4 Simulation 3: (a) Trajectories ofxi, i= 1, … , 10, under the fixed-time consensus law ui= 𝜂∑

j∈i(sig(xj− xi)^0.75+ sig(xj− xi)^1.25), where sig(x )^𝛼= sgn(x)|x|^𝛼(see [17]); (b) Statesxi1andxi2, i= 1, … , 10, versus time; and (c) ‖u(t )‖ versus time

F I G U R E 5 Simulation 4: (a) Trajectories ofxi, i= 1, … , 10, under the fixed-time consensus law ui=_t^𝜂

f−t

∑j∈i(xj− xi), 0≤ t < tf andui= 0, other- wise (see [14]); (b) Statesxi1andxi2, i= 1, … , 10, versus time; and (c) ‖u(t )‖ versus time

F I G U R E 6 Simulation 5: (a) Trajectories of̂p_i, i= 1, … , 10, under the network localization law (23); (b) Trajectories of̂p_ifor another set of initial condition, and (c) Values of V (t ) versus time with𝜂 = 50, 30, and 20

The bearing-based network localization algorithm (23) is adopted with parameters𝜂 = 50, 30, 20 and tf = 7. Simulation results depicted in Figure6show that the estimated configura- tion can be achieved within 7 s (= tf) and it is bearing congruent to the true configuration. For two different set of initial states, the estimated configuration ̂p may converge to a configuration satisfies ̂b^∗i j = −bi j,∀i, j ∈ , i ≠ j (Figure 6a) or a configu- ration satisfies ̂b^∗i j = bi j,∀i, j ∈ , i ≠ j (Figure6b). In both cases, the convergence times are both upper bounded by the pre-selected convergence time t_f, as shown in Figure6(c).

6 CONCLUSIONS

This paper proposed fixed-time convergence solutions to the consensus, orientation alignment, and bearing-based network localization problems. Besides having a simple structure, the convergence time of those fixed-time consensus/network local- ization laws can be easily selected. Furthermore, a disturbance rejection strategy was also presented. The proposed fixed-time control law design method in this paper can be also applied to other coordination problems in multi-agent systems. However,

it is worth noting that when the convergence time t_f is chosen to be small, the magnitude of the update laws will accordingly become very large. Thus, in designing fixed-time control laws using the proposed methodology in this paper, one needs to consider the trade-off between the magnitude of the control law and the convergence time of the system. Finally, the discretiza- tion and implementation of these proposed laws will be left for future studies.

A C K N OW L E D G E M E N T S

This research was funded by the Hanoi University of Sci- ence and Technology (HUST) under project number T2020- SAHEP-007.

DA TA AVA I L A B I L I T Y S TA T E M E N T

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

C O N F L I C T O F I N T E R E S T S

The authors declare that there are no conflicts of interest regard- ing this submission. There is no licensing material from other sources that was reproduced in this submission.

O RC I D

Minh Hoang Trinh https://orcid.org/0000-0001-5736-6693

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IET Control Theory Appl. 2021;15:900–909.

https://doi.org/10.1049/cth2.12086