Computational Design of Interval Type-2 Fuzzy Control for Formation and Containment of Multi-Agent Systems with Collision Avoidance Capability

1  Introduction

Prompted by the rapid development of intelligent equipment, control problems under the Multi-Agent System (MAS) framework have proliferated in recent decades [1]. Since the agents can achieve common goals through their own communication, these control issues have become an important milestone for the cooperative control of multiple autonomous vehicles [2]. In various control scenarios, the cooperation of multiple autonomous devices offers a faster and more efficient solution than using only one device. More importantly, multiple autonomous devices equipped with low-cost onboard systems can serve as a substitute for a single device that relies on high-end equipment [3]. This advantage significantly reduces the cost when the devices require maintenance and are even destroyed. In the broader context of control issues in MASs, cooperative behaviors are primarily discussed in terms of consensus, formation, and containment [4–6]. Consensus control serves as a foundational application for multiple agents to coordinate and reach a common state. Under the leader-follower control framework, consensus means that all follower agents reach the same state as the leader [4]. When there is more than one leader, the consensus problem escalates to the containment problem. Containment control is commonly found in control scenarios where agents with lower capabilities need to be protected by those with higher capabilities and guided into a specific region [5]. Usually, this specific region is defined through the formation control of leaders. Formation control refers to making the agents adhere to a predetermined order to achieve the formation [6]. This feature makes formation control a dominant topic in the research of MASs.

The Formation and Containment (F-and-C) issues have contributed to more efficient achievement of common objectives and broadened the application scope of multi-device control. An increasing number of researchers have also developed composite applications of the Formation Controller (FC) and Containment Controller (CC) using the leader-follower framework [7–9]. However, the safety of agents, especially the leaders, in MASs remains a critical concern due to the dynamic and ever-changing working environments in which autonomous devices operate. Given this consideration, an increasing number of researchers have placed emphasis on the issue of collision avoidance in the control of MASs. Several researchers have also incorporated the collision avoidance mechanism into the design of FC and CC [10,11]. It should be noted that the path planning algorithm plays a crucial role in collision-avoidance methods. Over the past decades, various motion or path planning algorithms have been developed and remained active research topics in collision avoidance scenarios, such as the A* algorithm [12], the dynamic window approach [13], the velocity obstacle method [14], and the Artificial Potential Field (APF) [15]. The collision avoidance mechanisms based on the APF offers advantages such as an intuitive design process, low computational complexity, and reliance on local environmental information. These features make APF particularly suitable for scenarios involving multiple unit encounters and situations requiring rapid emergency responses. For the design of FC and CC, the APF method has also been employed to achieve control objectives while maintaining the capability to avoid collisions between agents [10,11]. Up to now, only a few studies have investigated the collision avoidance issue using the APF in the F-and-C control of MASs.

However, the inherent nonlinearities in system dynamics must be considered due to the complex working environments and the control precision requirements of autonomous devices. With consideration of the interaction protocol, it can be observed that nonlinear dynamic behaviors in Nonlinear MASs (NMASs) significantly complicate the controller design process [4–9]. It is further worth noting that the complexities and challenges are exacerbated when multiple control objectives are required, such as incorporating the collision avoidance issue into F-and-C [10,11]. Solving the control and analysis challenges of systems with nonlinearities, the Takagi-Sugeno Fuzzy Model (T-SFM) has been proposed as an efficient method to convert nonlinear control problems into linear [16–19]. In this manner, nonlinear systems can be linearized into multiple linear subsystems based on a set of fuzzy rules by appropriately selecting premise variables and operating ranges. This characterization enables the design of fuzzy controllers within the linear framework and efficiently reduces the design complexity associated with nonlinear control methods. According to [20], the kinematic components of the mathematical model for the mobile robot exhibit clear nonlinear characteristics. The researchers used the T-SFM to achieve trajectory tracking along with obstacle avoidance capability. Since fuzzy logic and fuzzy inference systems serve as a powerful approach in decision making, fuzzy theory has gradually been integrated into various obstacle avoidance methods [21,22]. Researchers have also extended the fuzzy-based obstacle avoidance mechanism into the FC design using the APF method [23,24]. However, the collision avoidance using fuzzy logic for the F-and-C control of MASs remains an open issue, especially when considering the integration of APF.

Nevertheless, the so-called type-1 T-SFM used in previous studies [16–20] is insufficient for comprehensively characterizing uncertainties in nonlinear systems. This limitation also results in the corresponding fuzzy controller design methods being unable to handle uncertainties. In widely adopted applications of NMASs, such as unmanned vehicles, uncertainty problem becomes even more critical. These uncertainties may be attributed to discrepancies in the design of membership grades from a linguistic perspective, and structural changes from an application perspective [25,26]. As a result, Type-2 (T-2) fuzzy sets and fuzzy systems have been developed to more comprehensively incorporate uncertain factors into system modeling [27]. A typical T-2 MF consists of primary and secondary membership grades, which results in a three-dimensional structure. However, this structure significantly increases the computational complexity and burden compared to the two-dimensional type-1 MFs [28]. To improve practicality, Interval Type-2 (IT-2) MFs were introduced by setting all secondary membership grades to the maximum value of 1 [29], which reduces computational burden and still maintains a reasonable level of descriptive capability. The IT-2 T-SFM offers better capability for modeling nonlinear systems with uncertainties than type-1 T-SFM, while requiring lower computational resources than the general T-2 T-SFM. Since computational efficiency remains a key concern in real-world control systems, especially in autonomous devices with dynamic uncertainty, this advantage makes IT-2 fuzzy systems a widely adopted solution [30]. Furthermore, due to the close relationship between collision avoidance and decision-making, IT-2 fuzzy logic controllers have been successfully applied to address uncertainties in practical avoidance scenarios [31]. In addition, IT-2 fuzzy control theory provides a powerful framework for NMASs, which involve high computational demands, to mitigate the effects of dynamic uncertainties [32,33].

Although the F-and-C controllers for NMASs have been proposed by some researchers [34,35], the nonlinear frameworks reduce practical applicability and cost-effectiveness. Leveraging the IT-2 T-SFM representation [36], nonlinear control design problems can be recast as the linear problems, while uncertainties are also better handled. As the pioneering work applying IT-2 T-SFM to UNMASs, the researchers in [37] have solved the containment problem of UNMASs under the leader-follower framework. Nevertheless, the leaders are considered as the open-loop systems, which lead to the divergence of the entire UNMAS since the leaders are unstable even though the followers achieve containment. In [38,39], the authors developed the IT-2 fuzzy F-and-C controller design method, which not only guarantees leaders’ stability but also achieves the formation objective. In contrast to the existing formation-containment literatures, the design concept of the IT-2 fuzzy FC for leaders is proposed by developing the individual tracking controller. This FC design strategy eliminates the need for communication between the leaders that are farthest apart from each other in the F-and-C problem and enables specifying the entire UNMAS dynamics and time-varying formation without the additional control term in the FC framework. The detailed advantages of the IT-2 F-and-C design method can be found in [39]. However, the safety of agents in UNMASs can only be ensured if the agents possess collision avoidance capabilities. It can be inferred that, since the IT-2 fuzzy F-and-C controller design method for UNMASs was first proposed in [38,39], the collision avoidance issue under this framework has not yet been investigated. Moreover, a novel collision avoidance algorithm is proposed in this research by combining the fuzzy logic and the APF method. Extending the application of fuzzy theory, very few studies have used fuzzy logic to allocate control efforts between modes [40], and research using fuzzy logic to allocate between different APF types remains limited. However, combining different types of APF is a common practice in collision avoidance. Moreover, these studies on the fuzzy APF method still focus on single control systems. Based on the above statements, the contributions of this research can be summarized as follows:

(1)   Building upon the advantages of the IT-2 fuzzy FC design concept in [38,39], an IT-2 fuzzy F-and-C controller design approach combined with a collision avoidance method is first proposed for UNMASs.

(2)   A new collision avoidance algorithm is developed by combining the APF method and fuzzy theory, and integrated into the IT-2 fuzzy FC design under the IT-2 T-SFM framework.

(3)   Fuzzy logic is used to manage the switching between different APF types, including the vortex APF and the source APF, as well as between the trajectory tracking mode and the collision avoidance mode. In addition, the Distance at Closest Point of Approach (DCPA), commonly applied in avoidance scenarios, is selected to serve as the criterion within the fuzzy logic for determining the switching between modes.

Following the discussion of contributions, the design procedure for this research is presented as follows. First, the IT-2 T-SFM is constructed to more comprehensively represent UNMASs by extending the results in [36]. Leveraging the IT-2 T-SFM and the Imperfect Premise Matching (IPM) concept, the IT-2 fuzzy FC is designed with the tracking control approach, and the IT-2 fuzzy CC is designed with the interaction protocol. Most importantly, the collision avoidance algorithm is proposed by combining fuzzy theory and the APF method into the FC design for the leaders. Since the followers all comply with the dynamics of the leaders, the entire UNMAS can avoid potential collision and obstacle risks if the leaders have the avoidance capability. In this research, a vortex flow serves to steer the agents toward the right-hand side, whereas a source flow serves to repel the agents away from collision points. For the closed-loop IT-2 T-SFM, the IT-2 MF-dependent stability analysis methods are proposed to achieve F-and-C for leaders and followers according to Lyapunov theory. It is worth noting that the desired trajectories calculated by the APF-based method must also be tracked in the avoidance mode, which can efficiently be solved by the IT-2 tracking controller designed for the formation purpose. Finally, the stability conditions are converted into a Linear Matrix Inequality (LMI) problem for the IT-2 fuzzy F-and-C controller design. To verify the feasibility and applicability, a simulation considering multiple ships under the UNMAS structure is conducted.

The organization of this research is as follows. In Section 2, the IT-2 T-SFM and IPM-based fuzzy F-and-C controller are presented for UNMASs. The vortex and source flows are also introduced for the APF method. In Section 3, the stability criteria and IT-2 fuzzy F-and-C controller design with the collision avoidance algorithm are proposed. In Section 4, the simulation results of a multi-ship UNMAS are presented to verify the formation, containment, and collision avoidance performances. In Section 5, some conclusions are presented for the proposed IT-2 fuzzy F-and-C control method.

2  System Description and Problem Statements

In this section, the T-SFM is established for UNMASs based on interval type-2 fuzzy sets. To achieve the tracking objective, an IT-2 T-SFM is also constructed for the target trajectory of UNMASs. By dividing the IT-2 T-SFM into leader and follower models, the IT-2 fuzzy F-and-C controller are respectively developed according to the IPM concept. Subsequently, the definitions of vortex and source flows are introduced to aid the APF-based collision avoidance design. First, the IT-2 T-SFM is presented as follows.

x˙α(t)=∑ℓ=1rΩ~ℓ(υα(t)){Aℓxα(t)+Bℓuα(t)} for α=1,2,…,m+n(1)

where xα(t)∈ℜs and uα(t)∈ℜh are the state and input vectors, υα(t) is the premise variable, Aℓ and Bℓ are the constant matrices, ℓ indicates the index of fuzzy rules from 1 to r, α indicates the agent numbers with the leaders α=1,2,…,m and the followers α=m+1,m+2,…,m+n. For the IT-2 T-SFM (1), the IT-2 MF Ω~ℓ(υα(t)) is constructed as follows.

Ω~ℓ(υα(t))=ε¯ℓ(υα(t))Ω¯ℓ(υα(t))+ε_ℓ(υα(t))Ω_ℓ(υα(t))(2)

where ∑ℓ=1rΩ~ℓ(υα(t))=1, Ω¯ℓ(υα(t))=∏q=1zω¯ℓq(υαq(t))≥0 and Ω_ℓ(υα(t))=∏q=1zω_ℓq(υαq(t))≥0 are the upper bound and lower bound grades of membership which satisfy the relationship 1≥Ω¯ℓ(υα(t))≥Ω_ℓ(υα(t))≥0, ω¯ℓq(υαq(t)) and ω_ℓq(υαq(t)) are the upper and lower bound MFs inferred from If-Then fuzzy rules, which satisfy the relationship 1≥ω¯ℓq(υαq(t))≥ω_ℓq(υαq(t))≥0, q indicates the index of premise variables from 1 to z. Note that ε¯ℓ(υα(t)) and ε_ℓ(υα(t)) in (2) are nonlinear functions associated with the uncertain factors and are not required to be known. These functions also satisfy the relationship 1≥ε¯ℓ(υα(t))≥ε_ℓ(υα(t))≥0 and ε¯ℓ(υα(t))+ε_ℓ(υα(t))=1. The detailed modeling information of the IT-2 T-SFM can be found in [36].

For the formation of leaders under the tracking control framework, the IT-2 T-SFM of the target trajectory is constructed in a similar manner to (1).

x˙αd(t)=∑ℓ=1rΩ~ℓ(υα(t)){Aℓxαd(t)} for α=1,2,…,m(3)

For the followers in UNMASs, the interaction protocol is considered through communication between agents to achieve the containment purpose. Because of this reason, the definition of the interaction relationship for the followers in IT-2 T-SFM (1) is provided based on graph theory as follows.

Definition 1: For an undirected graph G=(M,C,𝒜), the node set is defined for all the nodes as M={μ1,μ2,…,μm+n}, and the edge set is defined as C⊆M×M. Thus, (μα,μγ)∈C indicates the existence of an edge between nodes μα and μγ. For the neighbor nodes of node μα, the corresponding set is defined as N={μγ∈M:(μα,μγ)∈C}. Based on the definitions of M, C and N, the adjacency matrix is formulated as 𝒜=[aαγ]∈ℜ(m+n)×(m+n) to represent the relationships among all nodes, in which the elements aαγ with the values 1 and 0 respectively indicate (μα,μγ)∈C and (μα,μγ)∉C. Then, the degree matrix 𝒟=[dαα]∈ℜ(m+n)×(m+n) is also formulated by these elements with the calculation dαα=∑γ=1m+naαγwhere α≠γ. By subtracting the adjacent matrix from degree matrix, the so-called Laplacian matrix is obtained as L=𝒟−𝒜.

Based on Definition 1, the agents in UNMASs are denoted as nodes while the interaction relationships are denoted as edges. In most leader-follower structures, the leaders serve as units to detect environmental changes and potential dangers, subsequently commanding the followers for control purposes. That is to say, the leaders do not acquire information from other agents. As a result, the Laplacian matrix L is further divided into the following form.

L=[0m×m0m×nL2L1](4)

where L1 and L2 represent the interactions between followers and from leaders to followers, respectively. Note that the Laplacian matrix in (4) is widely used to incorporate interaction relationship into the control design of leader-follower MASs.

Referring to [37], the properties of the Laplacian matrix (4) are presented in Lemma 1.

Lemma 1 [37]: All eigenvalues of matrix L1 have positive real parts. The nonnegative matrix −L1−1L2 has the property that the sum of each row equals one.

In this research, it is assumed that there is at least one interaction from each leader to the followers. Then, dividing the IT2 T-SFM (1) into the leader and follower parts, the following tracking error model for the leader is obtained by subtracting (3) from the leader’s T-SFM.

e˙α(t)=∑ℓ=1rΩ~ℓ(υα(t)){Aℓeα(t)+Bℓuα(t)} for α=1,2,…,m(5)

where eα(t)=xα(t)−xαd(t) represents the tracking error of system states.

Therefore, the tracking objective can be achieved by designing the control input uα(t) to ensure the stability of the error model (5) for each leader. This implies that the leaders’ dynamics will converge to the desired target dynamics. By appropriately assigning different target dynamics to each leader, the formation objective can thus be accomplished.

For the error model (5) of leaders and the followers in the IT-2 T-SFM (1), the IT-2 fuzzy controller is designed as follows to achieve the F-and-C objectives.

uα(t)=∑τ=1vΨ~τ(υα(t)){Fτeα(t)} for α=1,2,…,m(6)

uα(t)=∑τ=1vΨ~τ(υα(t)){Kτ∑γ∈Naαγ(xα(t)−xγ(t))} for α=m+1,m+2,…,m+n(7)

where Fτ and Kτ are the control gains for the F-and-C objectives, Ψ~τ(υα(t)) is the IT-2 MF of fuzzy controller derived as follows.

Ψ~τ(υα(t))=ϑ¯τ(υα(t))Ψ¯τ(υα(t))+ϑ_τ(υα(t))Ψ_τ(υα(t))∑o=1v(ϑ¯o(υα(t))Ψ¯o(υα(t))+ϑ_o(υα(t))Ψ_o(υα(t)))(8)

where ∑τ=1vΨ~τ(υα(t))=1, Ψ¯τ(υα(t))=∏q=1zψ¯τq(υαq(t))≥0 and Ψ_τ(υα(t))=∏q=1zψ_τq(υαq(t))≥0 are the upper bound and lower bound grades of membership, ψ¯τq(υαq(t)) and ψ_τq(υαq(t)) are the upper and lower bound MFs. Note that these membership grades and MFs satisfy the same condition as those in IT-2 T-SFM (1). Different from the IT-2 T-SFM, ϑ¯τ(υα(t)) and ϑ_τ(υα(t)) are the predefined functions since the fuzzy controller is developed by the designers.

It is worth noting that the MFs as well as the fuzzy rules of FC and CC in (6) and (7) can be designed differently due to the IPM concept. This feature makes the fuzzy controller design process more flexible and helps reduce application complexity by using fewer rules in the controller. Accordingly, the closed-loop error model for leaders can be obtained by substituting (6) into (5) as follows.

e˙α(t)=∑ℓ=1r∑τ=1vΩ~ℓ(υα(t))Ψ~τ(υα(t)){(Aℓ+BℓFτ)eα(t)} for α=1,2,…,m(9)

From (9), it can be observed that if the tracking error converges through the design of gain Fρ, the formation objective can be achieved by assigning different target trajectories to different leaders. This formation control strategy eliminates the need for communication between leaders and effectively mitigates faults arising from signal transmission. Obviously, leaders are the most critical components in UNMASs, and the enhancement of safety has therefore become a crucial issue.

To achieve the containment objective for followers, the closed-loop IT-2 T-SFM is obtained as follows by substituting (7) into (1).

x˙F(t)=∑ℓ=1r∑τ=1vΩ~ℓ(υF(t))Ψ~τ(υF(t)){(In⊗Aℓ+L1⊗BℓKτ)xF(t)+(L2⊗BℓKτ)xL(t)}(10)

where I is the identity matrix with a dimension determined by the number of followers, ⊗ is the Kronecker product, the state vectors are xF(t)=[xm+1(t)xm+2(t)⋯xm+n(t)]T∈ℜ(s×n) and xL(t)=[x1(t)x2(t)⋯xm(t)]T∈ℜ(s×m).

Nevertheless, the analysis process for fuzzy CC design may become increasingly conservative as both the number of agents and the fuzzy rules escalate due to the growing number of stability conditions. To solve the problem, an analysis approach for the containment analysis of the IT-2 T-SFM (10) as follows.

Lemma 2 [41]: Let λα denote the eigenvalues and JL1=W−1L1W denote the Jordan canonical form of the matrix L1, where W is a nonsingular matrix. The eigenvalues are reordered following the relationship Re{λm+1}<Re{λm+2}<…<Re{λm+n} in which Re {•} denotes the real part of ⋅. Thus, the following result is obtained.

IftheconditionΘ1+Re{λ~β}Θ2+Im{λ~β}Θ3<0forβ=1,2,3,4issatisfied,thentheΘ1+Re{λα}Θ2+Im{λα}Θ3<0forα=m+1,m+2,…,m+nisalsosatisfied.(11)

where λ~1,2=Re{λm+1}±jmax{Im{λα}} and λ~3,4=Re{λm+n}±jmax{Im{λα}}, Im {•} denotes the imaginary part of ⋅, Θ1, Θ2 and Θ3 are the real symmetric matrices.

Owing to the representation of IT-2 T-SFM, Lemma 2 can be utilized for the followers in UNMASs. From Lemma 2, it can be seen that the conditions in (11) are presented in LMI form. Moreover, the condition (11) indicates that only four LMI conditions need to be solved during the containment analysis process, regardless of how many followers are present in the UNMASs. This reduction in the number of stability conditions significantly benefits the controller design method based on the T-SFM, since the number of conditions tends to increase with the number of rules.

After the F-and-C objectives are ensured for leaders and followers, the collision avoidance requirement must also be considered for the safety improvement of UNMASs. By combining the fuzzy theory with the APF method, the collision avoidance algorithm is incorporated into the IT-2 fuzzy F-and-C controller design process. However, one of the most critical issues in the APF method is the deadlock problem. Based on the results in [42], a velocity potential field design approach is presented to efficiently avoid deadlock in collision avoidance situations.

Remark 1: Referring to [42], no deadlock exists within the region if the velocity potential function Φ is a harmonic function and continuous throughout the entire region.

Since the prerequisite information required by the APF method is difficult to be directly provided based on the state vector of the IT-2 T-SFM (9) and (10), the states of the leader agents are preliminarily assumed as follows for the application. Specifically, the X and Y positions, yaw angle on the earth-fixed frame, as well as the longitudinal and lateral velocities and yaw angular velocity on the body-fixed frame, are typically considered in trajectory tracking problems. Therefore, the states are set as (xα1(t),xα2(t))=(Xα(t),Yα(t)) and (xα4(t),xα5(t))=(Vxα(t),Vyα(t)) for α=1,2,…,m in the theoretical part, which can also be seen in the simulation section. Considering Remark 1, the velocity potential functions of vortex and source flows are respectively introduced as follows for the application of APF method.

Definition 2: The vortex potential field and the corresponding gradient are given as follows.

Φvσ=kvθσ and ∇Φvσ=(vvxσ,vvyσ)=(−kv(xα2(t)−yσc(t)Rσ),kv(xα1(t)−xσc(t)Rσ))(12)

Definition 3: The source potential field and the corresponding gradient are given as follows.

Φsσ=ksln⁡(Rσ) and ∇Φsσ=(vsxσ,vsyσ)=(ksRσcos(θσ),ksRσsin(θσ))(13)

where Rσ=(xα1(t)−xσc(t))2+(xα2(t)−yσc(t))2, θσ=atan2(xα2(t)−yσc(t)xα1(t)−xσc(t)), xσc(t) and yσc(t) are the X and Y positions of collisions to be avoided, the index σ denotes the σ-th collision, vr is the radial velocity and vθ is the tangential velocity, kv and ks are respectively the intensities of vortex and source flows, the symbol ∇ denotes the gradient operator.

Note that the vortex potential field is defined with a counterclockwise direction to adjust the agents toward the right-hand side in this research. Moreover, the source potential field serves as an APF to repulse the agents away from the collision events. According to the theory of APF method, it is known that if only the source APF is applied in the collision avoidance mechanism, the agent may become stuck at a point in front of the obstacle when the target is located directly behind the obstacle. In addition, if only the vortex APF is applied, the agent may keep circling around the obstacle without reaching the target. Based on the definitions in (12) and (13), these undesired behaviors become more significant as the agent gets closer to the obstacle, which corresponds to the center of the APFs. To overcome these problems, a combined application of the source APF and the vortex APF is proposed in this research through the use of fuzzy logic. The key function of fuzzy logic is that the abrupt changes caused by the switching between different APFs can be reduced, and smoother avoidance behavior can be realized.

In addition to the APF method, the Time to Closest Point of Approach (TCPA) and Distance at Closest Point of Approach (DCPA) are commonly used for the assessment of collision situations and risks. The concepts of TCPA and DCPA have been widely applied in collision avoidance for autonomous vehicles. Incorporating these factors into the development of collision avoidance algorithms enables agents to achieve more precise avoidance performance. Based on the above setting for the system states, the TCPA and DCPA are presented in Definition 4.

Definition 4: The TCPA and DCPA for all potential collisions are defined using the positions xα1(t), xα2(t) and velocities xα4(t), xα5(t) as follows.

TCPAσ:

−R~σ⋅V~σ|V~σ|2=−(xσc(t)−xα1(t))(Vxσc(t)−xα4(t))+(yσc(t)−xα2(t))(Vyσc(t)−xα5(t))(Vxσc(t)−xα4(t))2+(Vyσc(t)−xα5(t))2(14)

DCPAσ:

|R~σ+TCPAσ⋅V~σ|=(xσc(t)−xα1(t)+TCPAσ(Vxσc(t)−xα4(t)))2+(yσc(t)−xα2(t)+TCPAσ(Vyσc(t)−xα5(t)))2(15)

where the vectors R~σ=(xσc(t)−xα1(t),yσc(t)−xα2(t)) and V~σ=(Vxσc(t)−xα4(t),Vyσc(t)−xα5(t)) are the relative positions and velocities.

Within the collision avoidance framework, the TCPA and DCPA parameters are utilized to assess whether a potential collision event is imminent within the detection range. Specifically, when the conditions based on TCPA and DCPA are satisfied, the leaders transition into collision avoidance mode; otherwise, the leaders maintain tracking mode. Upon entering collision avoidance mode, the target trajectories used in the IT-2 fuzzy formation controller (6) are generated by combining the vortex APF in Definition 2 and the source APF in Definition 3. Moreover, fuzzy logic is applied in this research to achieve smoother adjustment between the two APFs.

Therefore, an IT-2 fuzzy F-and-C controller design approach combined with the collision avoidance algorithm is proposed for UNMASs based on the IT-2 T-SFMs (9) and (10) and the APF method from Definitions 2–4. Moreover, Lemma 2 is also applied to obtain a more relaxed containment analysis process for followers.

3  Interval Type-2 Fuzzy Controller Design for Multiple Objectives

According to the IT-2 T-SFM (9) of tracking error, a more convenient stability criterion is proposed to achieve FC design objectives in this section. By virtue of this design scheme, a collision avoidance algorithm combining the APF method with fuzzy theory is also proposed for the leaders in UNMASs. After the collision-free formation controller has been designed, the IT-2 fuzzy CC design approach can be proposed for followers based on the interaction relationship among all agents. Therefore, the entire UNMAS can avoid potential collisions through the IT-2 fuzzy FC design and the control commands of leaders. Since the followers are ensured to converge into the leaders’ formation, their safety is certainly guaranteed by IT-2 fuzzy CC. First, a stability criterion for the IT-2 fuzzy FC design is proposed as follows based on the Lyapunov theory and the IPM concept.

Theorem 1: If there exist the positive definite matrices QL, HℓτL, the symmetric matrix SL and the matrices Gτ such that the following conditions are all satisfied based on the given scalars η¯ℓτi1i2⋯izκL and η_ℓτi1i2⋯izκL, the error dynamic in the IT-2 T-SFM (9) is asymptotically stable.

QL>0 and HℓτL>0(16)

AℓQL+QLAℓT+BℓGτ+GτTBℓT−HℓτL+SL<0 for ℓ=1,..,r,τ=1,..,v(17)

∑ℓ=1r∑τ=1v(η_ℓτi1i2⋯izκL(AℓQL+QLAℓT+BℓGτ+GτTBℓT)−(η_ℓτi1i2⋯izκL−η¯ℓτi1i2⋯izκL)HℓτL+η_ℓτi1i2⋯izκLSL)−SL<0for i1,i2⋯iz=1,2,κ=1,2,…,δL(18)

where Gτ=FτQL and QL=PL−1, η¯ℓτi1i2⋯izκL and η_ℓτi1i2⋯izκL are the IT-2 MF-dependent parameters, in which i1 to iz are the parameters associated with the number of premise variables, and κ denotes the division of the sub-state space in the MF-dependent analysis method. The detailed definition and information of these parameters will be provided later in the derivation.

Proof of Theorem 1: In this research, the formation control objective is achieved through individual trajectory tracking controllers for each leader. That is, due to the homogeneity among UNMASs, the stability analysis is only required to be developed for one leader. Accordingly, the Lyapunov function is defined as follows for the stability analysis.

VL(e1(t))=e1T(t)PLe1(t)(19)

where PL is the positive definite matrix. Then, the following derivative is obtained from (19).

V˙L(e1(t))=∑ℓ=1r∑τ=1vΩ~ℓ(υ1(t))Ψ~τ(υ1(t)){e1T(t)(AℓTPL+PLAℓ+FτTBℓTPL+PLBℓFτ)e1(t)}(20)

It is known that the Lyapunov function (19) is defined for the tracking errors from an energy perspective. That is, if the derivative (20) of the Lyapunov function, which denotes the rate of energy change, is ensured to be negative, then the energy of the tracking error dynamics will converge to zero. This also implies that the leader dynamics can approach the desired target dynamics.

To simplify the IT-2 MF-dependent analysis method, Γ~ℓτ(υ1(t))=Ω~ℓ(υ1(t))Ψ~τ(υ1(t)) is adopted to define the MF throughout the rest of the research. Usually, the analysis process of the type-1 T-SFM involves making the matrix within the error vector in (20) negative definite. However, this approach is too conservative for the IT-2 T-SFM that a more detailed discussion can be found in [36]. Because of this reason, the IT-2 MF-dependent analysis method is presented as follows. First, the MF is represented in the following form.

Γ~ℓτ(υ1(t))=ϖ¯ℓτ(υ1(t))Γ¯ℓτ(υ1(t))+ϖ_ℓτ(υ1(t))Γ_ℓτ(υ1(t))(21)

where ϖ¯ℓτ(υ1(t)) and ϖ_ℓτ(υ1(t)) are functions that also do not need to be known because of ε¯ℓ(υ1(t)) and ε_ℓ(υ1(t)), and the relationship 1≥ϖ¯ℓτ(υ1(t))≥ϖ_ℓτ(υ1(t))≥0 is also satisfied.

Subsequently, the upper and lower bound MFs in (21) can be defined as follows.

Γ¯ℓτ(υ1(t))=∑κ=1δ∑i1=12⋯∑iz=12∏q=1zχqiqκ(υq1(t))η¯ℓτi1i2⋯izκL(22)

Γ_ℓτ(υ1(t))=∑κ=1δ∑i1=12⋯∑iz=12∏q=1zχqiqκ(υq1(t))η_ℓτi1i2⋯izκL(23)

where χqiqκ(υq1(t)) is the so-called cross term for both upper and lower bound MFs satisfying the conditions 1≥χqiqκ(υq1(t))≥0 and χq1κ(υq1(t))+χq2κ(υq1(t))=1 for all iq, and the parameters satisfy the condition 1≥η¯ℓτi1i2⋯izκL≥η_ℓτi1i2⋯izκL≥0.

Then, the parameter κ is given from the δL connected sub-state space denoted as Mκ in the state space of interest, M=∪κ=1δLMκ. The detailed process of constructing the parameters in (21)–(23) can be found in (25)–(28) and Remarks 5 and 6 of [36]. Based on the definition of (21), the derivative of Lyapunov function can be derived as follows by multiplying PL−1 on the right and left-hand sides.

V˙L(e1(t))=∑ℓ=1r∑τ=1v(ϖ¯ℓτ(υ1(t))Γ¯ℓτ(υ1(t))+ϖ_ℓτ(υ1(t))Γ_ℓτ(υ1(t)))×{e1T(t)(QLAℓT+AℓQL+GτTBℓT+BℓGτ)e1(t)}(24)

In relation to the IT-2 MFs, the slack matrices are introduced as follows.

(∑ℓ=1r∑τ=1v(ϖ¯ℓτ(υ1(t))Γ¯ℓτ(υ1(t))+ϖ_ℓτ(υ1(t))Γ_ℓτ(υ1(t)))−1)SL=0(25)

−∑ℓ=1r∑τ=1v(1−ϖ¯ℓτ(υ1(t)))(Γ_ℓτ(υ1(t))−Γ¯ℓτ(υ1(t)))HℓτL≥0(26)

Based on the properties of the IT-2 MFs, it can be readily verified that equality (25) and inequality (26) hold. These slack matrices are combined into the stability analysis and stability conditions as additional variables to be solved using LMI techniques. It is known that increasing the number of variables in the stability conditions can reduce conservatism when solving the control problem under stability criteria. With the combination of (25) and (26), the relationship for (24) is derived as follows.

V˙L(e1(t))≤e1T(t)(∑ℓ=1r∑τ=1v(Γ_ℓτ(υ1(t))(QLAℓT+AℓQL+GτTBℓT+BℓGτ)−(Γ_ℓτ(υ1(t))−Γ¯ℓτ(υ1(t)))HℓτL+Γ_ℓτ(υ1(t))SL)−SL)e1(t)−∑ℓ=1r∑τ=1v(ϖ¯ℓτ(υ1(t))(Γ_ℓτ(υ1(t))−Γ¯ℓτ(υ1(t)))×e1T(t)(QLAℓT+AℓQL+GτTBℓT+BℓGτ−HℓτL+SL)e1(t))(27)

Consequently, the following results can be obtained by extracting the cross term from the right-hand side of inequality (27).

e1T(t)(χqiqκ(υq1(t))∑ℓ=1r∑τ=1v(η_ℓτi1i2⋯izκL(QLAℓT+AℓQL+GτTBℓT+BℓGτ)−(η_ℓτi1i2⋯izκL−η¯ℓτi1i2⋯izκL)HℓτL+η_ℓτi1i2⋯izκLSL)−SL)e1(t)−χqiqκ(υq1(t))∑ℓ=1r∑τ=1v(ϖ¯ℓτ(υ1(t))(η_ℓτi1i2⋯izκL−η¯ℓτi1i2⋯izκL)×e1T(t)(QLAℓT+AℓQL+GτTBℓT+BℓGτ−HℓτL+SL)e1(t))(28)

It is obvious that the negative definiteness of (28) can be achieved if the stability conditions (16)–(18) are satisfied by Theorem 1. This implies that the asymptotic stability of the error dynamics for Leader 1 is ensured based on the relationship in (27). In conclusion, the states of Leader 1 in the UNMAS are capable of tracking the desired state trajectory. □

By Theorem 1, the IT-2 fuzzy controller (6) can be designed for Leader 1. It is worth noting that the designed fuzzy controller can be directly extended to all other leaders in view of the homogeneity. In other words, the design approach of Theorem 1 enables the leaders to complete the formation task merely by properly assigning different target trajectories to each leader. However, the leaders are still required to provide a collision avoidance capability to lead the entire UNMAS away from potential dangers. Therefore, a collision avoidance algorithm combining the APF method with fuzzy theory is presented as follows.

Theorem 2: Based on the IT-2 fuzzy FC designed in Theorem 1, the controller (6) is formulated as follows to simultaneously achieve the trajectory tracking and collision avoidance objectives.

uα(t)=∑τ=1vΨ~τ(υα(t)){Fτ(xα(t)−x~αd(t))} for α=1,2,…,m(29)

where the desired state trajectories x~αd(t) are designed by Algorithm 1.

In Algorithm 1, the conditions DCPAσ≤Rsafe and TCPAσ>0 for determining whether the agents should switch to the full collision avoidance mode in Step 4 are adopted from [43]. Moreover, the TCPA factor is utilized to evaluate and integrate the target trajectories associated with multiple simultaneously triggered collision events. As shown in (40), (41), (46) and (47), the final target trajectory in collision mode is computed using a weighted average approach, where the inverse of each TCPA value is used as the weighting factor. This is because a smaller TCPA indicates a more imminent collision risk. Consequently, events with smaller TCPA values are assigned higher weights, allowing the agent to prioritize avoidance of more dangerous obstacles. It is worth noting that, in Step 4, fuzzy logic is employed to replace the direct switching between different orientations generated by distinct APFs. This is because the desired directions calculated by the vortex and source APFs are significantly different, and fuzzy logic can be applied to largely prevent abrupt switching.

The advantage of fuzzy logic lies in its ability to make the collision avoidance behavior of agents smoother and more reasonable. In Fig. 1a, the MF is designed to combine two different APFs using fuzzy logic. It is observed that when the distance to the potential collision event is less than Rv, the source APF gradually activates. As the distance decreases further, the influence of the source APF increases. Once the distance falls below Rs, the target trajectory is fully determined by the source APF to strongly prevent the agent from colliding with the obstacle.

Figure 1: (a) MF associated with different APF strategies; (b) MF associated with different control modes

Similar to this concept, the fuzzy logic is applied in Step 6 when at least one collision event is detected, but no condition triggers the avoidance conditions, which prevents abrupt switching between the tracking mode and collision avoidance mode. Based on Theorem 2 and Algorithm 1, an IT-2 fuzzy FC design approach with the collision avoidance capability is proposed for leaders to lead the entire UNMAS away from potential risks.

Then, based on the control command for the leaders given in (29), the IT-2 fuzzy CC design approach is presented in the following theorem.

Theorem 3. If there exist the positive definite matrices Q~F, H~ℓτF, the symmetric matrix S~F, the matrices N~τ, and a minimized scalar ρ such that the following conditions are all satisfied based on the given scalars η¯ℓτi1i2⋯izκ~F, η_ℓτi1i2⋯izκ~F and ξ, the followers in the IT-2 T-SFM (10) can achieve the containment.

Q~F>0 and H~ℓτF>0(50)

[A~ℓQ~F+Q~FA~ℓT+ΞβB~ℓN~τ+N~τTB~ℓTΞβT+ξQ~F∗−B~ℓT−ξIh]−H~ℓτF+S~F<0for ℓ=1,..,rτ=1,..,vβ=1,2,3,4(51)

∑ℓ=1r∑τ=1v(η_ℓτi1i2⋯izκ[A~ℓQ~F+Q~FA~ℓT+ΞβB~ℓN~τ+N~τTB~ℓTΞβT+ξQ~F∗−B~ℓT−ξIh]−(η_ℓτi1i2⋯izκ−η¯ℓτi1i2⋯izκ)H~ℓτF+η_ℓτi1i2⋯izκSF)−SF<0for i1,i2⋯iz=1,2κ~=1,2,…,δFβ=1,2,3,4(52)

[Q~F∗Q~Fρ2Is](53)

where N~τ=K~τQ~F, Q~F=P~F−1, the abbreviation T ~=diag(T,T) stands for the matrices T=Aℓ,Bℓ,Nτ,QF,HℓτF,SF, diag(⋅) denotes the diagonal matrix with the elements ⋅, Ξβ=[Re{λ~β}−Im{λ~β}Im{λ~β}Re{λ~β}], the IT-2 MF-dependent parameters η¯ℓτi1i2⋯izκ~F and η_ℓτi1i2⋯izκ~F are obtained with the same process in Theorem 1, and κ~ is the division of sub-state space designed for the containment analysis.

Proof of Theorem 3: To carry out the containment analysis, the containment error is first defined as follows for the IT-2 T-SFM (10).

eF(t)=xF(t)+(L1−1L2⊗Is)xL(t)(54)

Based on the definition of the containment error in (54), the followers’ dynamics can converge within the leaders’ dynamics based on the properties of the Laplacian matrix in Lemma 2 if the error dynamics are stabilized. Then, the following derivative is obtained from (53).

e˙F(t)=∑ℓ=1r∑τ=1vΓ~ℓτ(υF(t)){(In⊗Aℓ+L1⊗BℓKτ)eF(t)+(L1−1L2⊗Bℓ)uL(t)}(55)

where uL(t)=[u1(t)u2(t)⋯um(t)]∈ℜ(h×m), Γ~ℓτ(υF(t))=Ω~ℓ(υF(t))Ψ~τ(υF(t)). Therefore, the containment objective can be achieved by ensuring the convergence of the error dynamics in (54).

However, the control input of leaders uL(t) obtained from (29) also has an influence on the error dynamics system (55). Leveraging the IT-2 T-SFM representation, the analysis approach for linear MASs in [44] can be extended to solve the containment analysis problem as follows.

According to properties of Laplacian matrix in Lemma 1, the item associated with the vector uL(t) in (55) can be reconstructed by the following process.

uαc(t)=∑γ=1mφαγuγ(t)forα=m+1,m+2,…,m+n(56)

where φαγ are the (α,γ)−th elements in the matrix L1−1L2, which satisfy the following condition.