Distributed Event-Triggered Formation Control

• Distributed event-triggered formation control is a coordination strategy for multi-agent systems that updates actions on state-dependent events, reducing unnecessary communication.
• The approach leverages hybrid dynamics and decentralized architectures, using local measurements and dynamic thresholds to ensure robust formation stability.
• Practical implementations in multi-robot and autonomous vehicle systems demonstrate significant communication savings and resilient performance under resource constraints.

A distributed event-triggered formation controller is a control strategy in which a group of physically distributed agents—such as robots, autonomous vehicles, or networked sensors—coordinate their states to achieve and maintain a prescribed geometric formation, but communicate and update their control actions only when certain state-dependent “events” occur rather than at fixed time intervals. This approach aims to optimize resource utilization (notably communication bandwidth and actuation effort) while ensuring rigorous stability properties of the closed-loop network dynamic. Formal frameworks for such controllers have been developed for various plant dynamics, hybrid system models, and under both centralized and fully distributed scheduling protocols, with applications ranging from multi-robot systems to cyber-physical networks (Postoyan et al., 2011, Mulla et al., 2013, Yi et al., 2016, Nowzari et al., 2014, Viel et al., 2017).

1. Fundamental Principles and Hybrid System Formulation

Event-triggered formation control replaces periodic (time-triggered) information exchange and actuation with condition-driven updates. The agents’ plant and controller dynamics are typically modeled as

$\dot{x}_p = f_p(x_p, u), \quad u = g_c(x_c, x_p)$

where $x_p$ and $x_c$ denote the plant and controller states, respectively, and $u$ the control input (Postoyan et al., 2011). The aggregate system—comprising all agents—exhibits hybrid dynamics. Here, the system flows according to continuous dynamics and jumps—i.e., triggers communication and control updates—whenever a state-dependent event condition is satisfied. A typical event-triggering law compares a measure of the network-induced error, often denoted $e$, to a Lyapunov function $V(x)$ associated with the state $x$, for example,

$\tilde{\omega}(W(K,e)) \geq V(x)$

where $W(K,e)$ represents a network-induced error function parameterized by the scheduling protocol, and $\tilde{\omega}(·)$ is a monotonically increasing threshold function (Postoyan et al., 2011).

In distributed settings, not all agents reset their local errors at each event; only the triggering node updates. To address this, auxiliary variables (thresholds, clocks) are introduced into the event rules. For example, introducing a decaying auxiliary variable $n$ yields systems of the form: $$\dot{n} = -\delta(n), \quad n^+ = \tilde{\omega}(W(K,e))$$ The hybrid state $(x, e, K, n)$ evolves over "flow" sets (no event) and "jump" sets (event triggered), defined to guarantee a minimum dwell-time between transmissions and thus exclude Zeno behavior (Postoyan et al., 2011).

2. Distributed and Decentralized Architectures

A distributed event-triggered controller typically operates as follows:

• Each agent observes its local state (and/or relative state with neighbors) and computes a triggering function based on locally available information.
• An agent transmits its state to neighbors (or updates its own control) only when the triggering function exceeds a design threshold.
• The controller and event-generator modules utilize only local state information, directly measured inter-agent distances, or locally estimated errors (Yi et al., 2016, Sun et al., 2019, Psomiadis et al., 15 Sep 2025).

For instance, in distance-based rigid formation control, an agent $i$ triggers a control update when

$\| e_i(t) \|^2 = \sigma_i(...)$

where $e_i(t)$ is the measurement error since the last control update and $\sigma_i$ is a tunable threshold. Alternative implementations use relative errors with neighbors: $\| e_j - e_i \| \geq \beta_i \delta_{ij}$ with $\delta_{ij}$ the measured inter-agent distance (Psomiadis et al., 15 Sep 2025).

Decentralized computation is reinforced in protocols where agents adjust coupling (feedback gain) parameters using only local or neighboring information, as in the distributed pole placement and formation encoding via eigenstructure methods (Mulla et al., 2013). Event-driven recomputation of local controller gains or weights upon significant state changes (e.g., leader updates) further reduces unnecessary communication.

3. Triggering Mechanisms, Guarantees, and Stability Analysis

Distributed event-triggered formation control necessitates careful design of triggering rules to ensure closed-loop properties such as convergence, robustness, and the exclusion of Zeno execution (infinitely many events in finite time).

Typical techniques include:

• Auxiliary threshold variables with strictly decreasing dynamics, ensuring a lower bound on inter-event intervals (Postoyan et al., 2011, Yi et al., 2017).
• Dynamic triggering laws employing internal variables or clocks, which dynamically accumulate or “budget” the monitored error and guarantee positive dwell times via Lyapunov-type decay conditions (Yi et al., 2017).
• Exponential decaying thresholds in error triggering, e.g., $\|e_i(t)\| \leq \alpha e^{-\beta t}$, where $\alpha,\beta > 0$ enforce increasingly tighter error caps as time progresses (Yi et al., 2016).
• Trigger functions combining state and error norms: e.g.,

$f_i(t) = \|e_i(t)\|^2 - \theta_i (\ldots)$

where $\theta_i$ is a dynamically tuned parameter based on local state evolution (Yi et al., 2017).

Lyapunov stability analysis is leveraged to show that the candidate Lyapunov function $V(x)$ (often incorporating both formation error and velocity/error energy) decays along trajectories, is non-increasing at jump instants, and that the triggering conditions enforce a strictly positive dwell time, thereby excluding Zeno phenomena (Postoyan et al., 2011, Yi et al., 2016, Yi et al., 2017, Psomiadis et al., 15 Sep 2025).

4. Control Law Synthesis and Performance

The control law structure in distributed event-triggered formation controllers varies with agent dynamics and formation objectives:

• Distance-Based Control: Agents apply control inputs to minimize the squared difference between measured and desired inter-agent distances, typically:

$$u_i(t_k^i) = -\alpha \sum_{j \in \mathcal{N}_i} ( \|x_j - x_i\|^2 - \tilde{\delta}_{ij}^2 )(x_i - x_j)$$

where $t_k^i$ denotes agent $i$'s most recent trigger time and $\tilde{\delta}_{ij}$ is the desired distance (Psomiadis et al., 15 Sep 2025, Sun et al., 2019).

• Potential Function–Based and Gradient Laws: Control inputs are syntheses of local gradients of a global formation potential, possibly composed with damping or state feedback, as in manipulator formation (Wu et al., 2021).
• Pole Placement and Linear Feedback: For agents modeled as single integrators, feedback matrices can be designed to assign desired closed-loop poles and formation shapes via eigenstructure assignment $A F=0$ (where $F$ denotes the formation vector) (Mulla et al., 2013).

These laws, when operated in event-driven fashion, continue to guarantee exponential convergence (or practical stabilization) of formation errors, and, in many frameworks, network connectivity preservation (Yi et al., 2016).

Performance comparisons with periodic (fixed-interval) control demonstrate substantial reductions in updating and transmission rates—often by factors of 2–3—without loss of convergence speed or accuracy (Psomiadis et al., 15 Sep 2025, Yi et al., 2016, Nowzari et al., 2014). This is a central motivation for event-triggered mechanisms in resource-constrained or communication-limited networked systems.

5. Extensions: Self-Triggered, Team-Triggered, and Fault-Tolerant Approaches

To address the overhead of continuous system monitoring, self-triggered and team-triggered approaches have been developed:

• Self-triggered control uses model-based predictions to schedule the next transmission based on the current state, without continuous evaluation of the event condition. Transmission times are calculated using Lie derivative conditions:

$$\sum_{i=0}^{n-1} S_i t^*{}^i \mathcal{L}_f^i T(\phi(t, j)) = 0$$

where $T(\cdot)$ is a triggering function and $\phi(t, j)$ is the flow trajectory from the last event (Postoyan et al., 2011).

• Team-triggered coordination generalizes event- and self-triggered control by having agents exchange “promises” about future behavior. Agents schedule requests and issue warnings if promises cannot be maintained, enabling a hybrid trigger based on promised and observed state evolution. This yields significant communication savings while maintaining theoretical performance guarantees even under delays, packet loss, and uncertainty (Nowzari et al., 2014).
• Fault-tolerant and robust event-triggered controllers, such as neuroadaptive schemes with dynamic filtering, accommodate non-affine nonlinearities and sensor failures. Event-triggered sampling and neural network weights are designed to compensate for polluted state measurements and unknown dynamics, ensuring that internal signals remain semi-globally uniformly ultimately bounded (Sun et al., 2023).

6. Practical Architectures and Implementation Considerations

Distributed event-triggered formation control has been validated through both large-scale simulations and physical robotic testbeds (e.g., GRITSBot X robots) (Psomiadis et al., 15 Sep 2025). Practical implementation considerations emphasized in the literature include:

• Measurement Modality: Controllers requiring only relative distance measurements are advantageous for systems lacking a global reference frame or where only inter-agent range sensors are available (Sun et al., 2019, Psomiadis et al., 15 Sep 2025).
• Prediction-based Trigger Checking: Continuous monitoring can be avoided by prediction between events; each agent broadcasts its state/control at triggering times and locally simulates neighbor evolution until the next trigger (Yi et al., 2016).
• Robustness to Communication Delays and Packet Loss: Techniques such as set-valued promises, Lie derivative–based time estimation, and adaptive threshold adjustment offer robustness to real-world network effects (Nowzari et al., 2014, Viel et al., 2017).
• Scalability: Analytical results for scalable asymptotic stability have been derived for distributed port-Hamiltonian representations, confirming that formation performance is robust with respect to network size (Javanmardi et al., 2022).

These features yield resource-efficient, flexible formation controllers suitable for large-scale, heterogeneous multi-agent systems in both static and time-varying environments.

7. Applications and Representative Results

Distributed event-triggered formation controllers have been successfully applied to:

• Multi-robot formation (planar and 3D) with rigid distance constraints (Sun et al., 2019, Psomiadis et al., 15 Sep 2025).
• Swarms with only distance or local-frame measurements, enabling practical formation stabilization in the absence of global localization (Wang et al., 2019).
• Autonomous vehicle platoons with event-driven estimation and adaptive control to handle uncertain dynamics (Wang et al., 7 Jun 2025).
• Networked control systems where energy, bandwidth, or computation must be strictly managed and communication takes place over shared, scheduled channels (Postoyan et al., 2011, Alavi et al., 2019).

Reported empirical results uniformly state that event-triggered schemes reduce communication and actuation rates compared to periodic control by 50% or more, with negligible degradation in steady-state formation accuracy. Exponential convergence, network connectivity preservation, and practical robustness to noise and parameter uncertainties are documented in both theoretical proofs and real-world deployments (Psomiadis et al., 15 Sep 2025, Yi et al., 2016, Viel et al., 2017).

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Distributed event-triggered formation controllers represent a class of formally analyzed, practically validated, and increasingly mature strategies for orchestrating multi-agent formations under strict resource constraints. Hybrid system modeling, rigorous Lyapunov analysis, decentralized computation, and robust event detection are core elements enabling scalable, reliable, and efficient cooperative coordination in modern networked dynamical systems.