IT2 fuzzy adaptive containment control for fractional-order heterogeneous multi-agent systems with input saturation

Xia, Zhile; Mou, Jinping

doi:10.3233/JIFS-231350

IT2 fuzzy adaptive containment control for fractional-order heterogeneous multi-agent systems with input saturation

Article type: Research Article

Authors: Xia, Zhile^{; *} | Mou, Jinping

Affiliations: Department of Mathematics, Taizhou University, Zhejiang Province, China

Correspondence: [*] Corresponding author. Zhile Xia, Department of Mathematics, Taizhou University, Zhejiang Province, China, 318000. E-mail: [email protected].

Keywords: Containment problem, fractional-order systems, heterogeneous multi-agent systems, distributed type-II fuzzy adaptive controller

DOI: 10.3233/JIFS-231350

Journal: Journal of Intelligent & Fuzzy Systems, vol. 45, no. 3, pp. 4361-4370, 2023

Published: 24 August 2023

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Abstract

In this paper, the containment control problem of second-order nonlinear heterogeneous multi-agent system is studied. In order to deal with complex uncertainties such as unknown parts, uncertainties, and input constraints in the system, we designed a distributed fuzzy adaptive controller. The interval type-II (IT2) fuzzy set is adopted to deal with the uncertainty of membership functions. We construct a matrix equality and a matrix inequality to deal with the asymmetric Laplace matrix. The controller designed is simple and the designed controller only uses the information of itself and its neighbors. Therefore, it is very easy to be compensated in practice. Finally, a simulation example is introduced to verify the effectiveness of the proposed methods.

1Introduction

The coordinated control of multi-agent systems is becoming increasingly prevalent in various fields such as life, industry, and aerospace. As a result, many experts and scholars are showing great interest in this area and have been conducting extensive research [1–7]. Containment control, which involves designing control protocols that urge the state or output of followers into the convex hull spanned by those of leaders, is a common problem in multi-agent system coordination control. Although the containment control problem of integer-order multi-agent systems has been widely studied [8–16], many physical systems exhibit fractional-order (non-integer) dynamic behaviors due to their unique materials and characteristics, such as microorganisms in underwater environments and unmanned aerial vehicles operating in complex space environments. Compared with integer-order differential equations [17–19], fractional-order differential equations have non-local and long-memory effects [20, 21], which makes them of great value in studying nonlinear systems, chaotic phenomena, and functional calculus. Therefore, it is important to study the dynamics of multi-agent systems in the sense of fractional-order and to investigate the containment control problem of fractional-order nonlinear systems, which has practical significance [22–26]. Ye et al. [22] have proposed two containment control protocols for networked fractional-order systems with sampled position data. In [23], sufficient conditions for asymptotic stability of a specific case of multi-order fractional system were derived. Moreover, [24] and [25] established necessary and sufficient conditions to ensure the achievement of containment control for fixed topology. Yang et al. [26] have investigated the distributed cooperation control of heterogeneous fractional-order multi-agent systems with time delays and obtained the consensus condition for compounded delayed fractional-order multi-agent systems.

In many real-world scenarios, it can be challenging to obtain an accurate mathematical model due to the inherent complexity of the system. To address this issue, researchers have investigated the control problem of multi-agent systems based on adaptive fuzzy logic systems, as documented in several studies [27–31]. However, these studies have primarily focused on using traditional type-I fuzzy logic systems to study controller problems related to multi-agent systems. Few studies have explored the use of interval type-II (IT2) fuzzy adaptive systems for complex nonlinear fractional-order multi-agent systems [32–35]. For instance, in [32], the authors investigated cooperative control for time-delay multi-agent systems and proposed a new robust adaptive control technique. Zhang et al. [33] presented sufficient criteria for achieving containment control, while [34] discussed a new methodology for building and evolving hierarchical fuzzy systems. Additionally, [35] explored a secure type-II fuzzy ontology-based multi-agent framework. However, despite these efforts, to the best of our knowledge, there is a lack of a systematic study of containment control for complex nonlinear fractional-order multi-agent systems characterized by unknown nonlinear functions and external disturbances. As such, there exists a need for further research to explore the use of IT2 fuzzy adaptive systems and develop effective containment control strategies for these complex systems.

The dynamic behavior of multi-agent systems is typically described using single integrator, double integrator, and general linear systems. Among these, the double integrator system considers both position and velocity information of the agents and has gained widespread use in the field of robot cooperative control [15]. As a result, it is highly favored in this area. For the containment control problem of fractional-order multi-agent systems, reference [24] studied general linear systems, but the multi-agents studied were homogeneous. References [23, 25, 26] studied the dynamic behavior of agents represented by single-integrator systems. The system studied in reference [22] was represented by a double-integrator system, but the double-integrator system studied was relatively simple and did not consider practical situations such as system uncertainty, unknown functions, and controller saturation. Inspired by the above-mentioned arguments, In this paper, we study the design of distributed control protocol for a class of complex fractional-order multi-agent systems with saturated inputs, unknown nonlinear functions, and external disturbances, so that all followers can converge to the polyhedron formed by multiple pilots. the main work and contributions of this article are described as follows.

1) An adaptive IT2 fuzzy containment control method is firstly designed for the nonlinear fractional-order multi-agent systems with unknown nonlinear function, external interference, and input saturation.
2) The containment control problem is solved merely utilizing information of itself and its neighbors under a directed topology.
3) The adaptive controller designed in this paper is completely distributed, that is, each agent only uses the information of itself and its neighbors.
4) The controller designed does not need to know the specific information of the leaders. As long as the states of the leaders are bounded, it can solve the containment control problem.

The remainder of this article is organized as follows. We briefly introduce the nonlinear fractional-order multi-agent systems model and formulate the problem in Section II and III, respectively. In Section IV, both the proposed IT2 fuzzy adaptive method and effectiveness analysis are shown. In Section V, simulation examples are given to prove the effectiveness of this new method. Finally, Section VI presents the conclusion.

1.1Notations

This paper considers a multi-agent systems consisting of M leaders and N followers. Its topology is described by a directed graph G=(V,E,A) . Here V={v1,v2,…,vN+M} represents a node set, and node v_i stands for the ith agent. Without losing generality, we assume that the set F={1,2,…,N} composed of the first N nodes represents the followers. The set P={N+1,N+2,…,N+M} composed of the last M nodes represents the leaders. Edge set E⊆V×V , when eij=(vj,vi)∈E means agent v_i can receive information of agent v_j, that is, there is a directed path from agent v_j to agent v_i. Adjacency matrix A=(aij)∈R(N+M)×(N+M) . When edge eij∈E , element a_ij > 0, otherwise a_ij = 0.

Denote by dist( x,C ) the distance from x∈ℝn to the set C⊆Rn in the sense of Euclidean norm, that is dist(x,C)=infy∈C||x-y||2.

2Preliminaries

Let us first review the definition of fractional-order calculus, IT2 fuzzy systems and several useful Lemmas.

2.1Fractional-order calculus

Definition 1 [36, 37]. For an integrable function f(t):[0,∞)→ℝ , the Riemann-Liouville fractional integral of order α is defined as

Iαf(t)=1Γ(α)∫0tf(τ)(t-τ)1-αdτ,0<α<1,

where 0 < α < 1. the Gamma function Γ(α)=∫0∞e-ttα-1dt .

Definition 2 [36, 37]. The Caputo derivative of fractional order α of a function f∈ C1([t0,+∞),ℝ) is defined by

t0Dtαf(t)=1Γ(1-α)∫t0tf′(s)(t-s)αds,

where 0 < α < 1. 0 ≤ t₀ ≤ t. When t0=0,t0Dtαf(t) can be simplified as D^αf (t).

Lemma 1 [20, 38]. If x(t)∈C1([t0,+∞),ℝ) and 0 < α < 1, we have

Iα(Dαx(t))=x(t)-x(0).

Lemma 2 [21]. If f (t) is a continuous function, then we have

Iα1(Iα2f(t))=Iα2(Iα1f(t))=Iα1+α2(f(t)),

where 0 < α_i ≤ 1, i = 1, 2 .

Lemma 3 [39]. Let x(t)∈ℝn be a differentiable vector-time function. Then, for any time instant t ≥ 0, we have

DαxT(t)Px(t)≤2xT(t)P(Dαx(t)),

where P∈ℝn×n is a symmetric positive definite matrix, The fractional order α satisfies the condition 0 < α < 1 .

2.2Interval Type-II fuzzy systems

Considering the interval type-II fuzzy logic system, the i₁i₂ ⋯ i_nth IF-THEN fuzzy rule can be expressed as:

R^{i₁i₂⋯i_n} : IF x₁ is M˜1i1 , ⋯, x_n is M˜nin ,

THEN y is [θ_{i₁i₂⋯i_n}, ϑ_{i₁i₂⋯i_n}] ,

where x = [x₁, x₂, …, x_n] ^T and y are the inputs and outputs of the fuzzy logic system respectively. M˜jij is an IT2 fuzzy set of the antecedent part, [θ_{i₁i₂⋯i_n}, ϑ_{i₁i₂⋯i_n}] is a weighting interval set of the consequent part.

In order to obtain the final output of the system, many scholars have devoted themselves to the research of model reduction, such as, in [40–43]. In this paper, the improved Biglarbegian-Melek-Mendel (BMM) direct defuzzification method [49, 50] is adopted, and the final output of the system is:

(1)

y=η∑θi1i2⋯inΠj=1nμ_Mjij(xj)∑Πj=1nμ_Mjij(xj)+(1-η)∑ϑi1i2⋯inΠj=1nμ¯Mjij(xj)∑Πj=1nμ¯Mjij(xj),

where μ_Mjij(xj) and μ¯Mjij(xj) represent the upper membership function (UMF) and the lower membership function(LMF), respectively. Regulation factor η meets the following condition 0 ≤ η ≤ 1 .

Define fuzzy basis function

(2)

Li(x)=μ_Mjij(xj)∑Πj=1nμ_Mjij(xj),Ri(x)=μ¯Mjij(xj)∑Πj=1nμ¯Mjij(xj).

Then the fuzzy logic system can be expressed as

(3)

y=θTL(x)+ϑTR(x),

where θ = η [θ₁, θ₂, …, θ_{i₁i₂⋯i_n}] ^T, L (x) = [L₁ (x) ,

L₂ (x) , …, L_{i₁i₂⋯i_n} (x)] ^T, ϑ = (1 - η) [ϑ₁, ϑ₂, …, ϑ_{i₁i₂⋯i_n}] ^T, R (x) = [R₁ (x) , R₂ (x) , …, R_{i₁i₂⋯i_n} (x)] ^T.

Remark 1. When applying the theory of IT2 fuzzy systems to solve practical problems, an important step is model reduction and defuzzification. The classic algorithm used for this task is the Karnik-Mendel (KM) algorithm [45]. However, in practical applications, the KM algorithm has several limitations, such as requiring iteration and not being convenient for stability analysis of controllers. Biglarbegian et al. [49, 50] proposed the BMM direct defuzzification method, which is more suitable for controller design and stability analysis. In this paper, the tuning factor is included with the unknown parameters, eliminating the special restrictions on the tuning factor.

Remark 2. Many research results show that fuzzy systems are universal approximators [44, 45].

2.3Other Lemmas

Lemma 4 [46]. For any ν > 0 and any ω, the following inequality holds

0≤|ω|-ωtanh(ων)≤ϱν,

where ϱ is a constant and meets condition ϱ = e^-(ϱ+1), namely ϱ = 0.2785 .

Lemma 5 [47]. (Barbalat) φ:ℝ→ℝ is a uniformly continuous function on the interval [0, ∞). If limt→∞∫0tφ(s)ds exists and is bounded, then limt→∞φ(t)=0 .

3Problem description

Consider the second-order uncertain fractional-order nonlinear multi-agent systems composed of N followers and M leaders. The system model of follower i(i∈F={1,2,…,N}) is described as

(4)

{Dαx1i(t)=x2i(t),Dαx2i(t)=fi(xi(t))+sat(ui(t))+ρi(t),

where x1i(t),x2i(t)∈ℝ and sat(ui(t))∈ℝ represent the position status, speed status and saturation control input of follower i, respectively; x_i = [x_1i, x_2i] ^T; fi(xi(t)):ℝ2→ℝ are uncertain nonlinear dynamics. ρi(t):[0,∞)→ℝ is a bounded input disturbance, that is, there is a normal number ρ¯i , such that |ρi(t)|≤ρ¯i , where ρ¯i an unknown constant. Saturation function

(5)

sat(ui)={u_i,ui<u_i<0,ui,u_i≤ui≤u¯i,u¯i,ui>u¯i>0,

where u_i and u¯i are the known lower and upper bounds of u_i (t) (i = 1, 2, …, N).

The system model of leader k(k∈P={N+1,N+2,…,N+M}) is described as

(6)

{Dαω1k(t)=ω2k(t),Dαω2k(t)=φk(ωk,t),

where ω1k,ω2k∈ℝ indicate the position and speed status of the leader k, respectively; ω_k = [ω_1k, ω_2k] ^T; φk(ωk,t):ℝ2×[0,+∞) is an unknown time-varying nonlinear function for any follower i.

Assumption 1. For each follower, there exists at least one directed path from the leader to the follower. For each leader, there is no directed path to the leader from any other agent.

Lemma 6. Under Assumption 1, the Laplace matrix L has the following structure:

(7)

L=[L1L20M×N0M×M].

where L1=D+B-A∈RN×N , D=diag{d1,d2, …, d_N}, B=diag{b1,b2,…,bN} , A=(aij)N×N, bi=∑m=N+1N+Maim , L2=-(ai,N+k)N×M∈RN×M , di=∑l=1Nail , i, j = 1, 2, …, N, k = 1, 2, …, M . The set Ni={vj|eij∈E} denotes a collection of neighbors agent v_i. Moreover, matrix L1 is a nonsingular M-matrix, each entry of -L1-1L2 is nonnegative, and each row of -L1-1L2 has a sum equal to 1.

Remark 3. Define

(8)

-L1-1L2=[ξ1,N+1ξ1,N+2…ξ1,N+Mξ2,N+1ξ2,N+2…ξ2,N+M⋮⋮⋮⋮ξN,N+1ξN,N+2…ξN,N+M].

Based on Lemma 6, one has

(9)

∑k=N+1N+Mξik=1,0≤ξik≤1,i=1,2,…,N.

Assumption 2. There is a closed convex set Ω⊂ℝ2 , so that for any ω_k ∈ Ω, there is φ_k (ω_k, t) ≤ M_φ (k∈P) , where M_φ is an unknown constant.

The objective of this paper is to design controllers for the follower agents to achieve containment control subject to input constraints: For i∈F , for any initial state x_i (0),

1. x_i (t) converges to the convex hull formed by the leaders, that is,
(10)
limt→∞dist(xi(t),Co(ωk(t),k∈P))=0.
2. u_i (t) stays within the desired range u_i≤ui(t)≤u¯i , where u_i<0 and u¯i>0 are the predetermined constants.

4Main results

The communication between agents in multi-agent systems is local, so only the error between neighboring agents can be used to design the distributed controller ui(i∈F) . For agent i, the consensus error signal between neighboring nodes is described as

(11)

{e1i=∑j=1Naij(x1i-x1j)+∑k=N+1N+Maik(x1i-ω1k),e2i=∑j=1Naij(x2i-x2j)+∑k=N+1N+Maik(x2i-ω2k).

Then the vector form of consensus tracking error:

(12)

{e1=L1x1+L2ω1,e2=L1x2+L2ω2.

By taking the derivative of both sides of equation (12), we can get:

(13)

{Dαe1=e2,Dαe2=L1(f(x)+sat(u)+ρ(t)+L1-1L2φ(ω,t)),

where f (x) = [f₁ (x₁) , f₂ (x₂) , …, f_N (x_N)] ^T, x = [x₁, x₂, …, x_N] ^T, sat (u) = [sat (u₁) , sat (u₂) , …, sat (u_N)] ^T, φ (ω, t) = [φ_N+1 (ω_N+1, t) , φ_N+2 (ω_N+2, t) , …, φ_N+M (ω_N+M, t)] ^T, ρ (t) = [ρ₁ (t) , ρ₂ (t) , …, ρ_N (t)] ^T.

Approximation of unknown smooth function f_i (x_i) (i = 1, 2, …, N) with fuzzy logic system of type-II (1)-(3)

(14)

fi(xi)=θi*TLi(xi)+ϑi*TRi(xi)+ɛi(t).

where the ideal parameters θi* and ϑi* are given by

(15)

[θi*,ϑi*]=argminθi,ϑi[supxi|fi(xi)-θi*TLi(xi)- ϑi*TRi(xi)|],

The fuzzy logic system has been proved to have universal approximation function. Therefore, it is reasonable to assume that the approximation is bounded, namely

(16)

|ɛi(t)|≤ɛ¯i,

where ɛ¯i is an unknown positive constant.

To deal with the saturation function, we make the following assumption [48].

(17)

|Δ(ui)|≤Mui,i=1,2,…,N,

where Δ (u_i) = sat (u_i) - u_i, and M_{u_i} is an unknown constant.

Theorem 1. For fractional-order multi-agent system (4)-(6), if there are positive numbers k, λ, τ, positive definite symmetric matrix P, the following conditions are satisfied:

(18)

PL1+L1TP=I,

(19)

Ω=[2kP-τI-k2P+I*-2kI]<-λI.

Then the following controller can be designed so that all the followers can converge to the polyhedron formed by multiple leaders.

(20)

ui=-τˆiδi-θˆiTLi(xi)-ϑˆiTRi(xi)-Mˆitanh(Mˆiδiνi(t)),i=1,2,…,N,

where δ_i = ke_1i + e_2i. The corresponding parameter adaptation rates are designed as

(21)

{Dαθˆfi=Proj(θˆi,δiLi(x)), i=1,2,…,N,Dαϑˆi=Proj(ϑˆi,δiRi(x)), i=1,2,…,N,Dατˆi=δi2, i=1,2,…,N,DαMˆi=|δi|, i=1,2,…,N,

where θˆi,ϑˆi,τˆi are the estimated value of θi* , ϑi* and τ, respectively. Mˆi is the estimated value of the lumped uncertain parameter, which will be explained later. ν_i (t) >0 satisfies ∫0+∞νi(t)<∞. The projection operator is defined as follows

(22)

Proj(z1,z2){=z2-∇g(z1)(∇g(z1))T||∇g(z1)||2z2g(z1),if g(z1)>0 and z2T∇g(z1)>0,=z2, if not,

with

(23)

g(z1)=||z1||2-zMɛzzM2,

where z_M is a specifies boundary, and ɛ_z is a specifies boundary tolerance.

Proof. Let δ = ke₁ + e₂, based on the definition of Δ (u_i) in (17), we have

(24)

Dαδ=ke2+L1(f(x)+u+Δ(u)+ρ(t)+L1-1L2φ(ω,t)).

where u = [u₁, u₂, …, u_N] ^T, Δ (u) = [Δ (u₁) , Δ (u₂) , …, Δ (u_N)] ^T.

For system (24), construct the following Lyapunov candidate function

(25)

V=V1+V2,V1=δTPδ+e1Te1,V2=12∑i=1N(θ˜iTθ˜i+ϑ˜iTϑ˜i+τ˜i2+M˜i2).

where θ˜i=θˆi-θi* , ϑ˜i=ϑˆi-ϑi* , τ˜i=τˆi-τ , Mi˜=Mˆi-Mi .

First, for V₁, derivation on both sides, according to Lemma 3 and condition (18), we can get

(26)

DαV1≤2kδTPe2+2δTPL1(f(x)+u+Δ(u)+ ρ(t)+L1-1L2φ(ω,t))+2e1Te2=2kδTPe2+δT(PL1+L1TP)(f(x)+u+ Δ(u)+ρ(t)+L1-1L2φ(ω,t))+2e1Te2,=2kδTP(δ-ke1)+δT(f(x)+u+Δ(u) +ρ(t)+L1-1L2φ(ω,t))+2e1T(δ-ke1),=δ¯TΩδ¯+∑i=1Nδi(τδi+θi*TLi(x)+ϑi*TRi(x) +Oi+ui),

where δ¯T=[δT,e1T] and Oi=ɛi(t)+ρi(t)-∑k=N+1N+Mξikφk(ωk,t)+Δ(ui) . According to the boundedness of ɛ_i (t), ρ_i (t), φ_k (ω_k, t) and Δ (u_i), it is known that exists M_i > 0 so that

(27)

δiOi≤|δi|Mi.

Substitute controller (20) into (26) and consider inequality (27), we can get

(28)

DαV1≤δ¯TΩδ¯ +∑i=1Nδi(-τ˜iδi-θ˜i*TLi(x)-ϑ˜i*TRi(x)) +∑i=1Nδi(sgn(δi)Mi-Mˆitanh(Mˆiδiνi(t))).

For V₂, taking its fractional-order differential, we can get

(29)

DαV2≤∑i=1N(θ˜iTDαθˆi+ϑ˜iTDαϑˆi +τ˜iTDατˆi+M˜iTDαMˆi),

Combine (29) (30) and consider the adaptive rate (21), we have

(30)

DαV1+DαV2≤δ¯TΩδ¯+∑i=1Nδi(-τ˜iδi-θ˜iTLi(x)-ϑ˜iTRi(x)) +∑i=1Nδi(-sgn(δi)M˜i) +∑i=1Nδi(sgn(δi)Mˆi-Mˆitanh(Mˆiδiνi(t))) +∑i=1N(θ˜iTD(α)θˆi+ϑ˜iTD(α)ϑˆi+τ˜iTD(α)τˆi) +∑i=1NM˜iTD(α)Mˆi=δ¯TΩδ¯+∑i=1N(θ˜iT(proj(θˆi,δiLi(x))-δiLi(x))) +∑i=1N(ϑ˜iT(proj(ϑˆi,δiRi(x))-δiRi(x))) +∑i=1N(Mˆi|δi|-Mˆiδitanh(Mˆiδiνi(t))),

Combined with the definition of projection operator (22)-(23), and condition (19),

(31)

DαV1+DαV2≤-λ(δTδ+e1Te1)+∑i=1Nϱνi(t).

Based on Lemma1, integrating on both sides of equation (31), we can get

(32)

I1-αIαDαV+λ∫0T(δTδ+e1Te1)dt ≤∑i=1N∫0Tϱνi(t)dt.

According to Definition 1, the following inequality holds

(33)

I1-αIαDαV=I1-αV=1Γ(1-α)∫0TV(s)(t-s)αds≥0.

Thus

(34)

∫0T(δTδ+e1Te1)dt≤1λ∑i=1N∫0Tϱνi(t)dt.

Moreover

(35)

limT→∞∫0Tϱνi(t)dt<∞,i=1,2,…,N,

Therefore,

(36)

limT→∞∫0T(δTδ+e1Te1)dt

is bounded.

According to Lemma 5, there

(37)

limt→∞(δTδ+e1Te1)=0

(38)

limt→∞δi=0,limt→∞e1i=0,i=1,2,…,N.

Based on the definition of δ_i, there are

(39)

limt→∞e1i=0,limt→∞e2i=0,i=1,2,…,N.

i.e.,

(40)

limt→∞e1=0,limt→∞e2=0.

According to (12), it is known that all the followers of the system converge to the inside of the polyhedron formed by the leaders.

5Simulation

This section gives five followers and two leaders to verify the effectiveness of the containment control algorithm designed in this paper. The network topology diagram is shown in the Fig. 1. Among them, indexes 1,2,3,4,5 refer to followers, and indexes 6,7 refer to leaders.

Fig. 1

Network communication topology diagram.

According to Fig. 1, we can get the Laplace matrix (7), where

(41)

L1=[1-1000020-100-110000-120000-11]],L2=[00-10000-100].

The dynamic systems of five followers and two leaders are described as follow

D0.96x11(t)=x12(t),D0.96x21(t)=x22(t),D0.96x31(t)=x32(t),D0.96x41(t)=x42(t),D0.96x51(t)=x52(t),D0.96x12(t)=-0.25sin(x11(t))-0.7x12(t) +sat(u1),D0.96x22(t)=-0.25sin(x21(t))-0.1x22(t) +0.5+sat(u2),D0.96x32(t)=-0.25sin(x31(t))-0.6x32(t) +0.3cos(t)+sat(u3),D0.96x42(t)=-0.25sin(x41)-0.1x42(t) +0.2sin(t)+sat(u4),D0.96x52(t)=-0.25sin(x51(t))-0.2x52(t) -0.5sin(2t)+sat(u5),D0.96ω61(t)=ω62(t),D0.96ω71(t)=ω72(t),D0.96ω62(t)=-0.25sin(ω61)-0.1ω62(t)+0.5,D0.96ω72(t)=-3sin(ω71(t))-0.2ω72(t)+0.1,

Due to the unknown dynamics of the navigator and the presence of uncertainties and input saturation phenomena in the considered system, the methods proposed in literature [22–26] are not applicable. However, by using the method proposed in this article, we were able to achieve containment control. Design controller (20)-(23), where parameter boundary Z_M = 10, ɛ_z = 1.1, and control constraint -40 ≤ u_i (t) ≤30 (i = 1, 2, …, 5) . The fuzzy membership function is taken as the following form

μ_Mi11(xi1)=exp(-(xi1+0.3)22·102),μ¯Mj1(xj)=exp(-(xi1+0.3)22·152),μ_Mi12(xi1)=exp(-(xi1+0.1)22·102),μ¯Mj2(xj)=exp(-(xi1+0.1)22·152),μ_Mi13(xi1)=exp(-(xi1-0.1)22·102),μ¯Mj3(xj)=exp(-(xi1-0.1)22·152),μ_Mi14(xi1)=exp(-(xi1-0.3)22·102),μ¯Mj4(xj)=exp(-(xi1-0.3)22·152),μ_Mi21(xi2)=exp(-(xi2+0.3)22·102),μ¯Mi21(xj)=exp(-(xi2+0.3)22·152),μ_Mi22(xi2)=exp(-(xi2+0.1)22·102),μ¯Mi22(xj)=exp(-(xi2+0.1)22·152),μ_Mi23(xi2)=exp(-xi222·102),μ¯Mi23(xj)=exp(-xi222·152),μ_Mi24(xi2)=exp(-(xi2-0.1)22·102),μ¯Mi24(xj)=exp(-(xi2-0.1)22·152),μ_Mi25(xi2)=exp(-(xi2-0.4)22·102),μ¯Mi25(xj)=exp(-(xi2-0.4)22·152),μ_Mi26(xi2)=exp(-(xi2-0.6)22·102),μ¯Mi26(xj)=exp(-(xi2-0.6)22·152),

where i = 1, 2, …, 5.

There are constants k = 1, λ = 1, τ = 4 and matrix

P=[0.6439 0.1439 0.1733 0.1073 0.11130.1439 0.3243 0.2027 0.1486 0.11530.1733 0.2027 0.7027 0.2838 0.19950.1073 0.1486 0.2838 0.3919 0.19710.1113 0.1153 0.1995 0.1971 0.6971],

which makes conditions (18) and (19) hold.

The simulation results are shown in Figs. 2–4. As shown in Fig. 2, it can be seen that the position tracking error of the follower converges to zero, indicating that the position of the follower will eventually converge to the polyhedron formed by the position of leaders. As shown in Fig. 3, it can be seen that the velocity tracking error of the follower converges to zero, indicating that the velocity of the follower will eventually converge to the polyhedron formed by the velocity of the leader. Figure 4 shows the condition where the input of the system satisfies bounded constraints.

Fig. 2

Consensus tracking error e₁.

Fig. 3

Consensus tracking error e₂.

Fig. 4

Saturation control input sat(u).

6Conclusion

In this paper, a distributed IT2 adaptive fuzzy controller is designed to solve the containment control problem of nonlinear fractional-order multi-agent systems. The problems studied take into account unknown functions, uncertainties and input constraint. The results obtained have a wider range of applications. The simulation results also show that the proposed methods are effective. The next step will consider the situation that the communication topology is time-varying.

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