公式章 1 节 1 Prescribed-Time Observer-Based Fuzzy Adaptive Practical Prescribed-Time
Formation Control of Nonlinear Multi-Agent Systems with Unmodeled Dynamics
Yuqi Gao1, Tao Li1,*, Yuanmei Wang2, Daixv Luo1, Zhiwei Wang1, Yuxiang She1
1School of Electrical Engineering and
Automation, Hubei Normal University, Huangshi 435002, China
2 School of Electronic Information and Electrical Engineering, Yangtze
University, Jingzhou 434023, China
*Corresponding author: taohust2008@163.com
公式章 1 节 1 Prescribed-Time Observer-Based Fuzzy Adaptive Practical Prescribed-Time
Formation Control of Nonlinear Multi-Agent Systems with Unmodeled Dynamics
Yuqi Gao1, Tao Li1,*, Yuanmei Wang2, Daixv Luo1, Zhiwei Wang1, Yuxiang She1
1School of Electrical Engineering and
Automation, Hubei Normal University, Huangshi 435002, China
2 School of Electronic Information and Electrical Engineering, Yangtze
University, Jingzhou 434023, China
*Corresponding author: taohust2008@163.com
|
Abstract
This article investigates the fuzzy adaptive practical
prescribed-time formation control of nonlinear multi-agent systems with
unmodeled dynamics based on a prescribed-time observer. At first, a prescribed-time observer is
developed for followers to estimate the leader’s states. Then, fuzzy logic
systems combined with adaptive methods are employed to approximate unknown
continuous nonlinear functions. To compensate for the effects of unmodeled
dynamics, a sliding-mode control strategy is incorporated. On these bases, a fuzzy
adaptive practical prescribed-time formation control protocol is designed. This protocol
guarantees that the formation error converges to a small neighborhood around
zero within an arbitrary prespecified time. Finally, simulation results are provided
to demonstrate the effectiveness of the proposed approach.
Keywords
Nonlinear multi-agent
systems; Practical prescribed-time formation control; Unmodeled dynamics;
Prescribed-time observer; Fuzzy adaptive method
1. Introduction
Distributed cooperative control of multi-agent systems (MASs) has
attracted significant attention due to its broad applications [1-4]. Among
various distributed cooperative control problems, formation control has been
extensively investigated [5-8].
In recent years, increasing attention has been devoted to the study of
practical formation control for multi-agent systems [9-12]. Reference [9] investigated the
predefined-time practical formation control of MUSVs with lumped uncertainties.
Reference [10] studied the fault-tolerant practical formation-containment
control for multi-agent systems based on directed graphs. Practical
prescribed-time bipartite time-varying formation control for multi-agent
systems was investigated [11]. Reference [12] investigated the practical robust
formation control for nonlinear multi-agent systems.
In practical applications, certain dynamical behaviors of the system
cannot be accurately modeled, or even modeled at all, which inevitably gives
rise to unmodeled dynamics [13]. However, the presence of unmodeled dynamics is
not addressed in references [9-12].
Therefore, the practical formation control of multi-agent systems with
unmodeled dynamics deserves further in-depth investigation. Besides, nonlinear
multi-agent systems (MASs) offer greater flexibility in capturing such
complexities and have attracted significant attention in practical applications
[14-16]. Consequently, the
investigation of practical formation control for nonlinear multi-agent systems
with unmodeled dynamics is challenging.
Fuzzy logic systems (FLSs) are
effective tools for handling nonlinearities and unmodeled dynamics [17-19]. The combination of adaptive control and
unmodeled dynamics has also become a focus of wide spread attention among
scholars [20-22]. The
integration of adaptive control and fuzzy logic systems has been extensively
investigated to address nonlinearities and unmodeled dynamics of MASs [23-26]. Therefore, this paper addresses the
practical formation control problem of nonlinear multi-agent systems with unmodeled
dynamics by employing a combination of adaptive control and fuzzy logic
systems.
In practical formation control problems, the states of the leaders may
be unavailable, or only a subset of followers can directly access the leaders’
states. In response to this, reference [27] proposed a distributed asymptotic
observer to estimate the leader's state. An asymptotic extended state observer
was proposed to estimate the leader's state and reject external disturbances [28].
Although various finite-time observers and fixed-time observers have been
proposed [29-32], guaranteeing state estimation within a prescribed time remains a
challenging problem.
The convergence time of the MASs is an important index to measure the
control performance. Traditional asymptotic practical formation control
guarantees convergence only as time approaches infinity, which may not satisfy
the requirements of real-time applications. To accelerate the convergence time,
finite-time practical formation control is proposed [33-35]. But the settling time of finite-time
practical formation control depends on the initial conditions. To address the
limitation, fixed-time practical formation control is proposed [36-38]. However, the settling time of fixed-time
practical formation control depends on the controller design parameters.
Therefore, prescribed-time practical formation control has recently received
increasing attention because the convergence time can be arbitrarily assigned
and the convergence time is independent of both initial conditions and
controller parameters.
Motivated by the above discussion, this article focuses on the
prescribed-time observer-based fuzzy adaptive practical prescribed-time
formation control design of nonlinear multi-agent systems with unmodeled
dynamics. The rest parts of this paper are outlined as follows. Section 2
presents the problem formulation and necessary preliminaries. Section 3
constructs a prescribed-time observer, designs a fuzzy adaptive practical
prescribed-time formation control protocol, develops an adaptive law, and
conducts a rigorous stability analysis of the overall system. Section 4
provides simulation studies to validate the effectiveness of the proposed
method. Finally, Section 5 concludes this article and outlines future research
directions.
2. Problem formulation and some preliminaries
2.1.
Graph theory
Consider the directed graph 
, where 
is the set of nodes and 
is the set of edges. An edge 
of the graph 
indicates that agent 
can receive
information from agent 
. The set of all
neighbors is denoted by
. The weighted adjacency matrix
, where 
if can send information
to , otherwise 
. Self-loops are not allowed, i.e.,
.
The in-degree matrix is defined
as 
, where 
. The Laplacian matrix is defined as 
. A directed graph is said to contain a spanning tree if
there exists a root node. Let 
denote the leader
adjacency matrix, where 
if agent can directly access
the leader, and 
otherwise.
2.2. A time-varying function
A time-varying function is
proposed as follows [39]
where 
is a real number
satisfying 
, 
and 
are the initial time
and user-given settling time.
Consider the following system
where 
is the state, 
is a nonlinear vector
bounded in time, 
is the initial state.
2.3. Fuzzy logic systems
IF-THEN rules
: If 
is 
is
, then 
is 
,
Combining singleton fuzzifier,
product inference, and center average defuzzifier, one has
where 
, 
and 
are the FLS input and
output, respectively. 
and 
are membership
functions of fuzzy sets 
and. 
is the number of
inference rules.
Define 
as
where 
and 
.
The FLS can be rewritten as
2.4. Problem formulation
First, consider the following
nonlinear MASs with unmodeled dynamics of the ith follower
where 
are the position and
velocity of the ith follower, respectively; 
is the input of the
ith follower; 
is unknown smooth
nonlinear functions with
; 
is an unknown
nonlinear function; and 
represents unmodeled
dynamics.
Moreover, the dynamics of the
leader is given as
where 
are the position and velocity, respectively, 
is the control input
or the acceleration of the leader.
Assumption
1. For the unmodeled dynamics in (6), there exists a Lyapunov function 
satisfying that
where 
, 
, and 
are functions of class
, 
, and 
.
Definition 1. For an
arbitrary prespecified time 
, the practical prescribed-time
formation control of MASs (6) and (7) can be realized if the following
condition holds:
where 
is a positive bounded
constant, and 
denotes the relative position between the
leader and the agent 
.
2.5.
Necessary lemmas
Lemma 1 [40]. There
exists a positive definite matrix 
such that 
is positive definite, where 
, 
, and 
.
Lemma 2 [41]. There exists a Lyapunov
function 
for (2) such that
where
,
, 
is defined as
then the
origin of system (2) is globally prescribed-time stable with the
prescribed-time 
. Especially, the
following solution can be obtained.
Lemma 3 [42]. For
a continuous function 
and 
on a compact set 
, there exists an FLS such that
Lemma 4. For the
Lyapunov function 
, if
holds, the
nonlinear MASs (6) and
(7) can achieve practical prescribed-time formation control and satisfy 
, where 
, 
, and 
are constants. The
settling time is 
.
Proof. Let
, and the proof process can be divided into two parts.
(1)
If 
, according to the Young's inequality, one has
It follows
from (15) that
Then, (16)
and (17) imply that
From (18),
one knows that 
is strictly monotonically decreasing when 
. Let 
be the time that first goes into the region 
(note that 
). Integrating (18) from 
to , one has
It obtains
Let 
with 
,one can obtain
Let 
, one knows that 
and
. Thus, there exists a constant 
such that 
, and if 
, according to (17), one knows that 
. That is, 
for all 
.
(2) If 
, similar to the above proof, one knows that for all 
.
To sum up, 
is a prescribed time
for going into the region 
. This ends the proof.
Lemma 5 [43]. For
the Lyapunov function 
satisfying (9), there
exists a finite time 
. For 
, the measurement of a dynamic signal can be represented as
and for 
where 
is a negative
function, 
, and 
.
Lemma 6 [44]. For 
, 
, 
, and 
, it holds that
Lemma 7 [45]. For 
and 
, one has
where 
is a constant that
satisfies 
i.e., 
.
3.
Main result
3.1. Prescribed-time
observer
In this section, a prescribed-time
observer is developed to estimate the leader’s states. The prescribed-time
observer is designed as
where
are the observed
states of 
. 
are the observer
gains, are the control
parameters. 
(
) is defined as
where 
.
Theorem 1.
Using the prescribed-time observer (24) for
system (6) and (7), if
then the
prescribed-time observer errors converge to zero within 
, where 
,
.
Proof.
The convergence analysis will be implemented by following two steps. First of
all, define 
, 
. Then, the first
equation of (24) can be rewritten as 
.
Consider the following Lyapunov
function
Taking the
derivative of 
yields
From (25), it obtains that 
. It concludes that 
. Then,
substituting these inequalities into (26), we obtain
where 
, i.e., 
. It thus follows that 
converges to zero
within 
.
Similar to the steps (25) and (26),
it can be obtained that 
converges to zero
within . It can be derived that when 
, all the followers have access to the stat es of the leader.
Define the tracking error as
where
is denoted the tracking error; 
is the observed states of 
.
The
formation error is defined as
where
.
3.2.
Fuzzy adaptive practical prescribed-time formation control
protocol
In this section, a fuzzy adaptive
practical prescribed-time formation control protocol is developed for the nonlinear
MASs in (6) and (7). The design details
are as follows.
Define the sliding mode functions
as
where 
, 
, 
, 
is the signum
function, and 
is designed as
where 
is a positive constant.
Then, the derivative of 
is
To establish stability and
synthesize the control protocol, the design is carried out in two steps. The step 1 is that we construct the Lyapunov function 
based on the sliding
surface to analyze the formation error dynamics, and the control protocol is
synthesized by integrating fuzzy approximation, adaptive methods, and sliding
mode compensation. The specific details are as follows.
Consider the Lyapunov function as follows
where 
.
The
derivative of 
is
where
,
,
,
,
and
.
From (33)
and (34), one gets
where 
,
,
and 
.
Applying
Lemma 5 and Assumption 1, one gets
where
.
Substituting
(36) into (35), it obtains
where 
, 
, and
.
Using Lemma 3 to approximate 
, we obtain
where 
and 
represents the fuzzy
basis function. 
is an ideal parameter vector of 
. For simplicity, we use 
and 
instead of and in subsequence writing.
Then, using
Young's inequality, one has
where 
and 
.
Construct the fuzzy adaptive practical
prescribed-time formation control
protocol 
as
where 
is the minimum
eigenvalue of the matrix 
, 
, 
. 
and 
are positive constants
with 
and 
. 
is the estimate of 
. 
and 
are the sum of the 
th column of matrices 
and 
.
Substituting
(36)-(39) and Lemma 7 into (35), we obtain
where
,
,
. 
represents a 
matrix obtained by
taking the 
of each element.
The step 2 is that we introduce
the Lyapunov function 
for the adaptive
parameters and develop an adaptive law to ensure boundedness and convergence of
the estimation errors. The specific contents are as follows.
Consider the Lyapunov function as
The
derivative of is
Construct the adaptive law 
as
Substituting
(44) into (43), we obtain
3.3. Stability analysis
In this
section, we discuss the stability of the system, from which Theorem 2 can be
derived.
Theorem 2.
Considering nonlinear MASs (6) and (7) with Assumptions 1 under directed graph, the fuzzy adaptive practical prescribed-time
formation control protocol (40) and adaptive laws in (44) can guarantee
that the practical prescribed-time formation control of MASs (6) and (7) can be
realized within any prespecified time 
.
Proof. Construct
the Lyapunov function 
as
The
derivative of is
Similar to
(39), one gets
Then
Substituting
(48)-(50) into (47), we obtain
Then, one
has
Substituting (52) into (51), one obtains
where
and 
Substituting
Lemma 6 into (53), one gets
where 
.
From the
definitions of 
and 
, one has
Then, one
gets
As 
approaches zero, both 
and 
approach zero. From
the error dynamics and the designed control protocol, it can be derived that
the time derivative of the tracking error is bounded.
Specifically, by combining MASs (6) and (7), the observer (24), and the control
protocol (40), together with Assumption 1 and Lemma 5, there exists a positive
constant 
such that 
Considering that the
two regions where the sliding-mode term dominates are different, the analysis
is conducted in two cases.
(1)
when 
, the sliding mode function is dominated by the
term 
. So we have:
This gives
the bound on
(2)
when 
, the sliding mode function is dominated by the
term 
. So have:
This gives
the bound on
Combine (58)
and (60), one gets
Therefore,
the error can be simplified as
Substituting (56) into (62), one obtains
where 
is defined as: 
.
From the definition of formation
error (29) and the prescribed-time observer, we have
Hence, the practical prescribed-time formation control of MASs (6) and (7) is achieved
within the prescribed-time , where 
is an arbitrary prespecified time.
4.
Numerical simulation 
Consider the following nonlinear
MASs
where 
.
The initial values of followers
are selected as
, 
The leader’s trajectory is
expressed as
The initial values of leader is selected
as
The communication topology is
shown in Fig. 1.
Fig 1. Communication topology.
The Laplacian matrix 
is given as follows
, 
Corresponding eigenvalues are
obtained as
, 
and 
. Let
, 
,
, 
, 
,
. It can be concluded that 
,
. Fig. 2 and Fig. 3 shows that the observer errors 
and 
converge to zero
within the prescribed-time 
, 
.
Fig. 2.
Evolutions of the observer errors about velocity 
.
Fig. 3. Evolutions of the observer errors
about position 
.
The fuzzy membership functions
are defined as
,
The 
are
The fuzzy basis function vector
is 
The parameters are selected as 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
with
, 
. The unmodeled dynamics 
with 
and 
.
It is shown from Fig. 4-6 that the practical prescribed-time
formation control can be realized within the prescribed time 
through the scheme designed.
Fig. 4 illustrates the evolutionary
trajectories of formation tracking. Fig. 5 shows that the formation
error 
can converges to a
small neighborhood around the zero within the prescribed time . In addition, the trajectory of the adaptive parameter 
is illustrated in Fig.
6.
Fig. 4. The
evolutionary trajectories of formation tracking.
Fig. 5. Evolution
of formation error 
.
Fig. 6. Evolution
of adaptive parameters 
.
5. Conclusion
This article investigated the
practical prescribed-time formation control problem
of nonlinear MASs with unmodeled
dynamics. A prescribed-time observer
was designed to estimate the leader’s states. Fuzzy logic systems combined with adaptive methods were utilized to
approximate unknown nonlinear functions. The properties of the sliding mode surface were exploited to compensate for uncertainties caused by unmodeled
dynamics. The proposed fuzzy
adaptive practical prescribed-time formation control protocol, which is based
on observer information, guaranteed
that the formation error converges to a small neighborhood near zero within an
arbitrary prespecified time. Simulation results verified the effectiveness of
the proposed approach. Future work will focus on extending the proposed method
to multi-agent systems under DoS attacks.
Acknowledgements
This work is
supported in part by the National Natural Science Foundation of China under
Grant 62473135.
References
[1] Ge
M F, Liu Z W, Wen G, et al. Hierarchical controller-estimator for coordination
of networked Euler–Lagrange systems[J]. IEEE transactions on cybernetics, 2019,
50(6): 2450-2461.
[2] Liu
S, Jiang B, Mao Z, et al. Distributed adaptive finite-time fault-tolerant
cooperative control of heterogeneous multi-agent systems with saturation and
disturbances[J]. International Journal of Systems Science, 2025, 56(7):
1442-1456.
[3] Liu
S, He M, Han W, et al. Distributed Control Algorithm for Multi‐Agent
Cooperation: Leveraging Spatial Information Perception[J]. International
Journal of Robust and Nonlinear Control, 2026, 36(1): 312-328.
[4] Zhang
X, Li Y, Xiong S, et al. A Robust Cooperative Control Protocol Based on Global
Sliding Mode Manifold for Heterogeneous Nonlinear Multi-Agent Systems Under the
Switching Topology[C]. Actuators. MDPI, 2025, 14(2): 57.
[5] Li
L, Chen C, Gao Z, et al. Observer-based Hierarchical Formation Control for
Multi-agent System Subject to External Disturbance[J]. IEEE Access, 2026.
[6] Shi
Z, Feng Z, Wang Q, et al. Prescribed-time time-varying output formation
tracking for heterogeneous multiagent systems[J]. IEEE Internet of Things
Journal, 2024, 12(9): 11622-11632.
[7] Abbasi
M, Marquez H J. Dynamic event-triggered formation control of multi-agent
systems with non-uniform time-varying communication delays[J]. IEEE
Transactions on Automation Science and Engineering, 2024, 22: 8988-9000.
[8] Wang
J, Li Y, Wu Y, et al. Fixed-time formation control for uncertain nonlinear
multiagent systems with time-varying actuator failures[J]. IEEE Transactions on
Fuzzy Systems, 2024, 32(4): 1965-1977.
[9]
Wang S, Han T, Xiao B, et al. Practical
Predefined-Time Formation Tracking of MUSVs With Lumped Uncertainties Via
Estimator-Based Adaptive Fuzzy Control Algorithm[J]. IEEE Transactions on
Vehicular Technology, 2025.
[10] Gong
X, Li X. Fault-tolerant practical prescribed-time formation-containment control
of multi-agent systems on directed graphs[J]. IEEE Transactions on Network
Science and Engineering, 2023, 11(1): 352-365.
[11] Xuan
S, Liang H, Li T, et al. Practical prescribed-time bipartite time-varying
formation control for multiagent systems with adaptive self-triggered[J]. IEEE
Transactions on Automation Science and Engineering, 2025.
[12] Wen
N, Feroskhan M. Practical robust formation control for nonlinear multiagent
systems via generative adversarial learning framework: Theory and
experiment[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems,
2025.
[13]
Zhang Y, Ioannou P A. Robustness of nonlinear
control systems with respect to unmodeled dynamics[J]. IEEE Transactions on
automatic control, 1999, 44(1): 119-124.
[14] Liu
G, Sun Q, Su H, et al. Adaptive cooperative fault-tolerant control for
output-constrained nonlinear multi-agent systems under stochastic FDI
attacks[J]. IEEE Transactions on Circuits and Systems I: Regular Papers, 2025.
[15]
Li Y,
Zheng X, Li K. Prescribed-time adaptive intelligent formation controller for
nonlinear multiagent systems based on time-domain mapping[J]. IEEE Transactions
on Artificial Intelligence, 2023, 5(4): 1778-1790.
[16]
Li M,
Zhang K, Liu Y, et al. Prescribed-time consensus of nonlinear multi-agent
systems by dynamic event-triggered and self-triggered protocol[J]. IEEE
Transactions on Automation Science and Engineering, 2025.
[17] Ma
H, Zhou Q, Ren H, et al. Distributed estimator-based fuzzy containment control
for nonlinear multi-agent systems with deferred constraints[J]. IEEE
Transactions on Fuzzy Systems, 2025.
[18] Wu
Y, Fu R, Zhang J, et al. Practical PID protocol of T–S fuzzy positive
multi-agent systems under attacks[J]. International Journal of Fuzzy Systems,
2025: 1-24.
[19] Yang
T, Dong J. Practically predefined-time leader-following funnel control for
nonlinear multi-agent systems with fuzzy dead-zone[J]. IEEE Transactions on
Automation Science and Engineering, 2023, 21(3): 3886-3895.
[20] Cheng
T T, Niu B, Zhang J M, et al. Time-event-triggered adaptive neural asymptotic
tracking control of nonlinear interconnected systems with unmodeled dynamics
and prescribed performance[J]. IEEE Transactions on Neural Networks and
Learning Systems, 2021, 34(9): 6557-6567.
[21]
Y. Yu, J. Guo, C.
K. Ahn and Z. Xiang. Neural adaptive distributed formation control of nonlinear
multi-UAVs with unmodeled dynamics[J]. IEEE Transactions on Neural
Networks and Learning Systems, 2023, 34(11):
9555–9561.
[22] X.
Xia, T. Zhang, Y. Fang and G. Kang. Adaptive quantized control of output
feedback nonlinear systems with input unmodeled dynamics based on backstepping
and small-gain method[J]. IEEE Transactions on Systems, Man, and Cybernetics:
Systems, 2019, 51(9): 5686-5697.
[23] Lyu
Z, Liu Z, Xie K, et al. Adaptive fuzzy output-feedback control for switched
nonlinear systems with stable and unstable unmodeled dynamics[J]. IEEE
Transactions on Fuzzy Systems, 2019, 28(8): 1825-1839.
[24] Tong
S, Li K, Li Y. Robust fuzzy adaptive finite-time control for high-order
nonlinear systems with unmodeled dynamics[J]. IEEE Transactions on Fuzzy
Systems, 2020, 29(6): 1576-1589.
[25] Hu
X, Xu B, Hu C. Robust adaptive fuzzy control for HFV with parameter uncertainty
and unmodeled dynamics[J]. IEEE Transactions on Industrial Electronics, 2018,
65(11): 8851-8860.
[26]
Hua C, Liu G, Li L, et al. Adaptive fuzzy
prescribed performance control for nonlinear switched time-delay systems with
unmodeled dynamics[J]. IEEE Transactions on Fuzzy Systems, 2017, 26(4):
1934-1945.
[27]
Yu J, Dong X, Liang Z, et al. Practical
time‐varying formation tracking for high‐order nonlinear multiagent systems
with multiple leaders based on the distributed disturbance observer[J].
International Journal of Robust and Nonlinear Control, 2018, 28(9): 3258-3272.
[28]
Yu J, Dong X, Li Q, et al. Practical
time-varying formation tracking for high-order nonlinear multi-agent systems
based on the distributed extended state observer[J]. International Journal of
Control, 2019, 92(10): 2451-2462.
[29] Wang
F, He S, Zhou C, et al. Distributed practical finite-time formation control of
quadrotor UAVs based on finite-time event-triggered disturbance observer[J].
IEEE Systems Journal, 2023, 18(1): 355-366.
[30] Xu
H, Cui G, Ma Q, et al. Fixed-time disturbance observer-based distributed
formation control for multiple QUAVs[J]. IEEE Transactions on Circuits and
Systems II: Express Briefs, 2023, 70(6): 2181-2185.
[31] Wang
Q, Dong X, Wang B, et al. Finite-Time Observer-Based 
Fault-Tolerant Output Formation Tracking Control for
Heterogeneous Nonlinear Multi-Agent Systems[J]. IEEE Transactions on Network
Science and Engineering, 2023, 10(4): 1822-1834.
[32] Ye
Z, Yu Z, Cheng Y, et al. Finite-time sliding-mode observer based distributed
robust fault tolerant control for multi-agent systems against actuator and
sensor faults[J]. International Journal of Systems Science, 2024, 55(4):
618-630.
[33] Wang
F, He S, Zhou C, et al. Distributed practical finite-time formation control of
quadrotor UAVs based on finite-time event-triggered disturbance observer IEEE
Systems Journal, 2023, 18(1): 355-366.
[34] Wang
Z, Wang J. A practical distributed finite-time control scheme for power system
transient stability[J]. IEEE Transactions on Power Systems, 2019, 35(5):
3320-3331.
[35]
Shang L, Cai M.
Adaptive practical fast finite-time consensus
protocols for high-order nonlinear multi-agent systems with full state
constraints[J]. IEEE Access, 2021,
9: 81554-81563.
[36]
Li J, Yu J, Hua Y, et al. Fully Distributed
Practical Fixed-Time Time-Varying Formation Tracking Control for General Linear
Multiagent Systems[C]. 2024 IEEE 18th International Conference on Control &
Automation (ICCA). IEEE, 2024: 827-832.
[37]
Gao K, Liu Y, Zhou Y, et al. Practical
fixed-time affine formation for multi-agent systems with time-based
generators[J]. IEEE Transactions on Circuits and Systems II: Express Briefs,
2022, 69(11): 4433-4437.
[38]
Huang J, Zhao T, Lu W, et al. Practical
Fixed‐Time Formation Control for Wheeled Robots: An Approach Based on Time‐Base
Generators[J]. International Journal of Robust and Nonlinear Control, 2026,
36(2): 816-829.
[39]
Song Y, Wang Y, Holloway J, et al. Time-varying
feedback for regulation of normal-form nonlinear systems in prescribed finite
time[J]. Automatica, 2017, 83: 243-251.
[40]
Zhang H, Li Z, Qu Z, et al. On constructing
Lyapunov functions for multi-agent systems[J]. Automatica, 2015, 58: 39-42.
[41]
Ren Y, Zhou W, Li Z, et al. Prescribed-time
cluster lag consensus control for second-order non-linear leader-following
multiagent systems[J]. ISA transactions, 2021, 109: 49-60.
[42]
Wang Q, Cao J, Liu H. Adaptive fuzzy control of
nonlinear systems with predefined time and accuracy[J]. IEEE Transactions on
Fuzzy Systems, 2022, 30(12): 5152-5165.
[43]
Liu Y. Robust adaptive observer for nonlinear
systems with unmodeled dynamics[J]. Automatica, 2009, 45(8): 1891-1895.
[44]
Qian C, Lin W. Non-Lipschitz continuous
stabilizers for nonlinear systems with uncontrollable unstable
linearization[J]. Systems & Control Letters, 2001, 42(3): 185-200.
[45]
Polycarpou M M. Stable
adaptive neural control scheme for nonlinear systems[J]. IEEE Transactions on
Automatic control, 2002, 41(3): 447-451.
