1. Introduction
In recent decades, the control towards multi-agent systems (MASs) has become a leading research subject, stemming from the superior efficiency of multiple agents collaborating to execute tasks compared to an individual agent [
1]. MASs hold significant applications spanning various domains, including service robotics [
2,
3], hazardous environment detection [
4], and unmanned aerial vehicle formation flying [
5]. Consensus control of MASs is a fundamental and core issue based on tracking control [
6,
7]. A significant amount of research has emerged on the consensus of MASs [
8,
9,
10,
11,
12]. Ren [
8] constructed MASs with second-order integrator dynamics by analyzing the swarming model and designed a consensus protocol. Tian and Liu [
9] attained two decentralized consensus conditions of MASs with diverse input and communication delays. Wen et al. [
10] introduced an innovative protocol designed by using synchronous intermittent local feedback for second-order consensus of MASs. Zhang et al. [
11] proposed event-trigger output feedback control approaches, enabling that all connected communication graphs reach consensus. Tan et al. [
12] derived the consensus criteria for cyber-physical systems under sampled data control, employing suitable Lyapunov function. The above studies predominantly concentrate on achieving consensus of MASs with integer order, which is difficult to describe the actual systems in nature and industry.
Fractional order systems (FOSs) are capable of more accurately modeling and computing genetic and memory effects in various complex processes than integer order systems [
13]. Similarly, the consensus of fractional order MASs (FOMASs) has attracted widespread interest [
14,
15,
16,
17,
18,
19]. Su and Ye proposed a control strategy with input delays to achieve the consensus of general linear and nonlinear FOMASs under event-triggered in [
14,
15], respectively. Yang et al. [
16] considered the consensus of nonlinear distributed and input delayed FOMASs, and further explored the performance of FOMASs in terms of leader-following and leaderless global consensus in [
17]. Hu et al. [
18] developed an adaptive controller that employed an event-triggered scheme without Zeno behavior, aiming to realize the consensus of FOMASs. Bahrampour et al. [
19] proposed new Lyapunov-based LMIs conditions to determine the state feedback controller gains on the distributed consensus control of heterogeneous FOMASs with interval uncertainties. However, many practical MASs exhibit multiple time-scale characteristics, which refer to the coupled coexistence of fast dynamics and slow dynamics. The design of controllers for these systems frequently encounters difficulties due to the presence of high dimensionality and pathological values [
20,
21].
Singular perturbation systems (SPSs) have multiple time-scale and inherently pathological dynamical properties [
22,
23,
24]. SPSs with certain parasitic parameter
are modeled to describe real systems. In power system modeling,
is used to represent transient phenomena in machine reactors or voltage regulators [
25]. In industrial control systems, it signifies small time constants between control and response [
26]. Numerous scholars have intensively studied SPSs [
27,
28,
29,
30,
31,
32]. On one hand, two commonly employed strategies for solving control problems of SPSs are the quasi-steady-state method [
27] and the block diagonalization method [
28], which decompose the system into slow and fast subsystems. But these methods rely on the assumption that the fast subsystem matrix is nonsingular and are not applicable to non-standard SPSs that cannot be easily decomposed. On the other hand, Yang et al. [
29], Gao et al. [
30] and Liu et al. [
31] proposed the integral sliding mode control method for full-order SPSs with mismatched disturbances, uncertainty and nonlinear input, respectively. Their methods are based on a full-order model, which significantly eliminates the need to decompose the system. Furthermore, techniques such as Lyapunov functions and LMIs are also applied to system analysis. Fridman [
32] derived the LMIs criteria for the stability of SPSs for delay proportional to
and delay independent of
, respectively. Additionally, for singular perturbation MASs (SPMASs), both Ben Rejeb et al. [
33] and Tognetti et al. [
34] designed the decentralized controllers, enabling systems to synchronize and ensuring global performance. Xu et al. [
35] presented the sliding-mode controller with memory output for addressing consensus of SPMASs in finite time. Zhang et al. [
36] achieved global Mittag-Leffler consensus tracking for fractional SPMASs modeled by discontinuous function with nondecreasing property. However, in practical application, the exact value of the parameter
is often difficult to obtain directly. By analyzing the background information of specific problems in depth, the reasonable change range of
is effectively estimated. Given
in known interval , the design of controllers for achieving consensus of nonlinear FOSPMASs remains an open problem in the field of control theory.
T-S fuzzy models possess the capability of approximating nonlinear dynamics, thereby the well-established control methods for linear systems is extended to the analysis and design of nonlinear systems. Therefore, numerous scholars have done extensive research endeavors focusing on T-S fuzzy SPSs [
37,
38,
39,
40]. Yang and Zhang [
37] proposed a design method of state feedback controller depending on
for T-S fuzzy SPSs. Chen et al. [
38] focused on nonlinear SPSs and presented two novel methods to design static output feedback
controller based on LMIs. Visavakitcharoen et al. [
39] designed an event-triggered controller based on integral feedback for nonlinear SPSs with a fuzzy model. Zhang and Han [
40] proposed two diverse feedback controllers aiming to attain the stabilization criteria of fuzzy FOSPSs with order
. Nevertheless, the research on the consensus control of fuzzy fractional order singular perturbation MASs (FOSPMASs) is still relatively limited.
Inspired by previous discussions, this paper focuses on filling this research gap. The following is an overview of the main contributions of this research:
To provide a more accurate portrayal of complex systems in practice, a T-S fuzzy FOSPMAS with is formulated to reduce the difficulty of directly studying nonlinear systems. Compared to integer order systems, the constructed model exhibits more enhanced accuracy and complexity. A fuzzy FOSPS with error as a variable is derived by designing a fuzzy observer-based controller.
The fuzzy FOSPS is analyzed by transforming it into a fuzzy SFOS using the system augmentation method. In comparison to the existing work [
41], the proposed approach not only relaxes the assumption that the fast subsystem matrix must be nonsingular, but also avoids the ill-conditioned issue arising from the parameter
.
The consensus conditions for fuzzy FOSPMASs with and are formulated in this study for any , where and are given lower and upper boundaries, respectively. The results are presented based on LMIs without equality constraints, reducing solution difficulties. It is demonstrated through an RLC circuit model that the proposed methods are effective in practice.
The remaining parts are structured in the following manner:
Section 2 provides foundational definitions in graph theory and correlative lemmas. The establishment of system model and the primary findings on the consensus of FOSPMASs are detailed in
Section 3.
Section 4 presents two practical examples. Lastly,
Section 5 summarizes the study.
Notations: and signify that the matrix X is positive definite and positive semi-definite, respectively. stands for transpose of the matrix X, and . spec is the spectrum of . Symbol * is the symmetric element of a matrix. ⊗ denotes the Kronecker product. For , , denotes . represents a diagonal matrix. stands for rounding up to the nearest integer.
4. Numerical Examples
This section presents two demonstrative instances that highlight the effectiveness of control protocol in achieving the consensus of fuzzy FOSPMASs with order in and , respectively.
Example 1. Consider a T-S fuzzy FOSPMAS composed of one leader and four followers, and the behavior of each agent is described by the fractional order RLC circuit model as shown in
Figure 1.
The capacitor and inductor have fractional characteristics with order
.
L represents a very small parasitic inductance.
denote the resistance of corresponding resistors.
is a diode and its characteristic function is
. It is known that the relationships between voltages are
and
. The dynamic of each agent is subsequently described by
Let
,
,
, then the circuit model (
56) is reformulated as follows:
where
.
The parameters are chosen as , , , , . It is assumed that belongs to . Subsequently, the fuzzy rules are set as follows:
Rule 1: If the value of
is approximately 0, then
Rule 2: If the value of
is approximately
, then
where
The fuzzy weighting function is selected as
,
. The communication network between these agents is depicted in
Figure 2.
With
, by solving LMIs (
50)-(
55) in Theorem 4, the feasible solutions are obtained as
Let
, then the tracking errors and estimation errors are depicted in
Figure 3 and
Figure 4. The consensus errors to zero means that each follower converges toward the leader, which demonstrates that system achieves the consensus by using the observer-based protocol (
9). It indicates the practical applicability and efficacy of the proposed method in
.
Example 2. Considering the T-S fuzzy FOSPMAS (
6)-(
7) within the topology in
Figure 2, the fuzzy rules of the system are established in the manner outlined below:
Rule 1: If
is
, then
Rule 2: If
is
, then
The remaining parameters are proposed as follows:
Assuming that the state belongs to , the fuzzy weighting function is selected as , .
Considering
, the feasible solutions are presented based on the LMIs (
40)-(
43) in Theorem 2:
Figure 5 and
Figure 6 present the simulation results of the error system (
14) with
. As depicted in
Figure 5, the state of follower agents exhibits a successful tracking of the state of leader agent, indicating that the consensus issue of fuzzy FOSPMAS (
6)-(
7) with
is solved by the criteria in Theorem 2.