Fixed-Time Observation and Control for Network Systems: A Distributed Event-Based Saturation Adaptive Method
Academic Background
Complex Networks (CNs) play a crucial role in fields such as sociology, engineering, and natural sciences, and are widely applied in scenarios like power distribution, traffic dispatching, and multi-agent collaboration. However, due to factors such as communication packet loss, sensor noise, and environmental uncertainties, obtaining accurate state information of both the leader and individuals in the network has become a challenging issue. Particularly in distributed systems, how nodes can effectively synchronize their states to achieve consensus is a critical problem. Traditional observation and control methods often rely on continuous sampling and computation, which not only increases communication overhead but also places pressure on network infrastructure. To address these issues, Liang Feng, Cheng Hu, Juan Yu, and Quanxin Zhu proposed a fixed-time observation and control method based on an event-triggered mechanism and a saturation adaptive strategy.
The primary goal of this research is to design a distributed observer and controller that can achieve synchronization of complex networks within a fixed time while reducing the consumption of communication and computational resources. By introducing an event-triggered mechanism and a saturation adaptive algorithm, the research team not only eliminated the dependency on network topology information but also significantly improved the convergence speed of the synchronization error system.
Source of the Paper
The paper was co-authored by Liang Feng, Cheng Hu, Juan Yu, and Quanxin Zhu, affiliated with the College of Mathematics and System Sciences, Xinjiang University and the School of Mathematics and Statistics, Hunan Normal University. The paper was accepted by the journal Nonlinear Dynamics on February 23, 2025, and published in the same year. The DOI of the paper is 10.1007/s11071-025-11041-2.
Research Process and Results
1. Design of the Distributed Event-Triggered Saturation Adaptive Observer
The research team first designed a Distributed Event-triggered Fixed-time Distributed Adaptive Observer (EFDA Observer) to identify the output state of the leader system within a fixed time. The core innovation of this observer lies in the introduction of a power-law-based saturation adaptive technique, which reduces the frequency of signal updates through the event-triggered mechanism while avoiding the need for continuous sampling.
Step 1: Observer Design
The dynamic equation of the observer is as follows:
[
\dot{s}{oi}(t) = f(s{oi}(t)) + b \left( q{i1}(t^k) + p{i1}(t^k) + q{i2}(t^k) + p{i2}(t^k) \right)
]
where (s{oi}(t)) and (y{oi}(t)) represent the estimated state and output estimate of the leader system for the (i)-th node, respectively, (b) is the coefficient matrix, and (q{i1}), (q{i2}), (p{i1}), and (p{i2}) are sampling values based on local communication.
Step 2: Event-Triggered Condition
The trigger condition is determined by the following rule:
[
t_{k+1}^i = \inf \left{ t > tk^i : \frac{1}{2} \left( | e{i1}(t) |^2 + | e_{i2}(t) |^2 \right) > \mu_i | \Gamma_i(t) | \right}
]
where (\Gamma_i(t)) is the dynamic threshold function, and (\mu_i) is the trigger parameter.
Step 3: Adaptive Algorithm
To eliminate the dependency on network topology information, the research team designed a saturation adaptive algorithm:
[
\dot{c}{i1}(t) = \beta{i1} + \beta{i2} \tilde{c}{i1}^\alpha(t) + \delta \sum{j=1}^n a{ij} | y{oj}(t) - y{oi}(t) |
]
where (\tilde{c}{i1}(t) = c{i1}^* - c{i1}(t)), and (c{i1}^*) is the saturation parameter.
Results and Analysis
Through theoretical analysis and numerical simulation, the research team proved that the observer can accurately identify the output state of the leader system within a fixed time, and the observation time is not affected by any network parameters. The simulation results show that the observation error converges to zero within 1.50 seconds, verifying the effectiveness of the method.
2. Design of the Event-Triggered Adaptive Synchronization Controller
Based on the observer, the research team further designed an Event-triggered Adaptive (ETA) controller to achieve fixed-time synchronization of complex networks.
Step 1: Controller Design
The dynamic equation of the controller is as follows:
[
ui(t) = b \left( q{i3}(ts^i) + q{i4}(ts^i) \right)
]
where (q{i3}(t) = -\kappa c_{i3}(t) \epsiloni(t)), and (q{i4}(t) = -\beta_{i3} \text{sign}(\epsilon_i(t)) \exp { | \epsilon_i(t) |^{\alpha_1} }).
Step 2: Event-Triggered Condition
The trigger condition is determined by the following rule:
[
t_{s+1}^i = \inf \left{ t > ts^i : \frac{1}{2} \left( | e{i3}(t) |^2 + | e_{i4}(t) |^2 \right) > \nu_i | \Phi_i(t) | \right}
]
where (\Phi_i(t)) is the dynamic threshold function, and (\nu_i) is the trigger parameter.
Step 3: Adaptive Algorithm
The adaptive algorithm of the controller is as follows:
[
\dot{c}{i3}(t) = \beta{i3} + \beta{i4} \tilde{c}{i3}^{\alpha_1}(t) + \tilde{\delta} \kappa \epsilon_i^T(t) \epsiloni(t)
]
where (\tilde{c}{i3}(t) = c{i3}^* - c{i3}(t)), and (c_{i3}^*) is the saturation parameter.
Results and Analysis
Through theoretical analysis and numerical simulation, the research team proved that the controller can achieve synchronization of complex networks within a fixed time, and the synchronization time is determined solely by the adaptive parameters, independent of the network structure. The simulation results show that the synchronization error converges to zero within 2.90 seconds, verifying the effectiveness of the method.
Conclusions and Value
This research proposes a fixed-time observation and control method based on an event-triggered mechanism and a saturation adaptive strategy, addressing the synchronization issues in complex networks caused by communication limitations and environmental uncertainties. By introducing the saturation adaptive algorithm, the research team successfully eliminated the dependency on network topology information, significantly improving the system’s robustness and anti-interference capabilities. Additionally, the method reduces communication and computational resource consumption by decreasing the frequency of signal updates, demonstrating significant practical application value.
Research Highlights
- Innovative Algorithm: For the first time, the saturation adaptive algorithm was combined with the event-triggered mechanism, proposing a fixed-time observer and controller, significantly enhancing the synchronization performance of complex networks.
- Strong Robustness: The observation and synchronization times are unaffected by network parameters, exhibiting strong robustness and anti-interference capabilities.
- Resource Efficiency: The event-triggered mechanism reduces the frequency of signal updates, lowering the consumption of communication and computational resources.
- Wide Applicability: This method can be widely applied in fields such as power systems, traffic dispatching, and multi-agent collaboration, demonstrating significant practical application value.
Other Valuable Information
The research team also improved the traditional adaptive algorithm by introducing saturation parameters, avoiding the issue of excessively large adaptive coefficients, and further reducing the operational burden of the controller. Simulation results show that, compared to traditional methods, the adaptive coefficients of this method remain within the preset range, verifying its superiority.