Self-Triggered Communication in Networks

• Self-triggered communication is a protocol where each agent autonomously determines its next update based on its local state and neighbor sampling, reducing overhead.
• The mechanism uses a hybrid system framework with piecewise constant controls and countdown timers, applying state-based rules for stability and Zeno exclusion.
• It enhances resilience in multi-agent consensus by filtering out extreme values, ensuring robust performance even under adversarial or noisy conditions.

A self-triggered communication mechanism is a control and coordination protocol in networked systems where each agent independently determines its next communication or update time based on its own state (and possibly sampled neighbor states) at the current triggering instant, rather than relying on periodic or event-driven strategies with continuous monitoring. This paradigm enables agents to only sample, broadcast, or actuate when necessary, thereby reducing communication overhead and improving resource efficiency in distributed control, estimation, and coordination tasks. The mechanism is typically formalized within a hybrid system framework: agents maintain piecewise constant controls, periodically poll neighbor states, and compute the next inter-event interval according to a local rule that ensures stability, performance guarantees, and avoidance of Zeno behavior.

1. Formal Structure and Typical Workflow

Self-triggered mechanisms are generically implemented as follows: each agent $i$ evolves its local state $x_i(t)$ under a control $u_i(t)$, which is piecewise constant between communication triggers. A local timer or countdown variable $\theta_i(t)$ decrements at a fixed rate. When $\theta_i(t)$ reaches zero, agent $i$ polls (its neighbors or local sensors), computes a condition-specific average or discrepancy, and updates both its control and $\theta_i$ according to a self-triggering rule parameterized by a consensus or performance metric (often with a tunable sensitivity threshold $\epsilon$). The next trigger instant is set by the computed $\theta_i$, which is lower-bounded to prevent Zeno.

One canonical realization in resilient consensus is:

• State: $x_i(t)\in\mathbb{R}$, control $u_i(t)\in\{-1,0,+1\}$, local clock $\theta_i(t)\ge0$
• Between triggers: $\dot x_i=u_i$, $u_i$ constant, $\dot\theta_i=-1$
• On trigger at $t=t_i^k$: poll neighbors, run filtering (e.g., F-MSR), compute robust average $\mathrm{ave}_i^M$, update

$$u_i(t_i^{k+}) = \mathrm{sign}(f_\epsilon(\mathrm{ave}_i^M)), \quad \theta_i(t_i^{k+}) = \max\{|\mathrm{ave}_i^M|, \epsilon\}$$

and set $t_{i}^{k+1}=t_{i}^k+\theta_i(t_i^{k+})$ (Matsume et al., 2020).

2. Trigger Rule and Parameterization

The self-triggering interval is determined through local evaluation of a performance-related function, often linked to the magnitude of local disagreement or state error. Common parameterizations include

• Linear scaling: $\Delta t = |\mathrm{ave}_i|/C$ for $|\mathrm{ave}_i|\ge\epsilon$, else $\epsilon/C$, ensuring $\Delta t\ge\epsilon/C>0$ (Persis et al., 2012).
• Adaptive thresholds based on local state magnitude, quantized or noisy measurements (Shi et al., 2018).
• Lyapunov-based rules, where the next trigger is set to the first time instant where a predicted metric (e.g., error/cost/consensus spread) would violate a contractive or monotonicity condition (often derived from Lyapunov decay) (Bahavarnia et al., 2018, Gleizer et al., 2020, Trimpe, 2016).

Controllers utilize ternary or quantized updates (sign functions with dead-zones), sample-and-hold implementations, or optimized feedback gains. In more advanced settings, triggering instants are optimized via convex programs subject to reachability, safety, or performance constraints, e.g., via LMI or SDP relaxations (Bahavarnia et al., 2018, Hashimoto et al., 2018).

3. Robustness, Resilience, and Filtering

Robustness is achieved by combining self-triggered scheduling with filtering algorithms such as mean subsequence reduced (MSR) filtering, which remove extreme neighbor values to guard against adversarial or faulty nodes. In adversarial consensus:

• Agents operate under an $F$-total malicious model; up to $F$ unknown adversaries may inject synchronized corrupt data.
• At each trigger, regular agents filter up to $F$ largest and $F$ smallest neighbor values before computing local averages.
• The protocol guarantees $c$-approximate consensus if the communication graph is $(2F+1)$-robust and $\epsilon$ is selected per derived bounds (Matsume et al., 2020).
• Filtering, encoded in a local rule, ensures the agents only react to benign neighbor data while ignoring worst-case inputs.

Similar constructs can be adapted for the presence of clock skews, communication delays, or quantization errors, by tuning the self-trigger function and dead-zone parameters to maintain contractiveness and ensure convergence to neighborhoods of consensus (Persis et al., 2012, Shi et al., 2018).

4. Theoretical Guarantees and Zeno Exclusion

Proven properties include:

• Safety: regular agent states remain in the convex hull of initialization.
• Approximate agreement: state spread shrinks below $c$ in finite or asymptotic time.
• Absence of Zeno: the self-triggered interval is always lower bounded ($\tau_i^k\ge\epsilon$) due to the threshold in the scheduling rule, precluding infinitely many triggers in finite time.
• Monotonicity: Lyapunov functions decrease (or fail to increase) across triggers, ensuring convergence.

For example, under $(2F+1)$-robustness, the $c$-consensus condition holds for suitable $\epsilon$, with constructed chain gaps and matrix contraction yielding finite-time entry into the consensus tube (Matsume et al., 2020). Explicit bounds on convergence time, communication costs, and consensus errors are derived as functions of network parameters, delays, quantization levels, and filter conservativeness (Persis et al., 2012, Yi et al., 2016, Bahavarnia et al., 2018).

5. Comparison with Event-Triggered and Time-Triggered Protocols

Self-triggered communication fundamentally differs from event-triggered or periodic approaches:

• Self-triggered: the agent computes, at trigger time, when the next trigger will occur, based on its own state, yielding deterministic, aperiodic, a priori knowledge of future communication needs.
• Event-triggered: the agent monitors a continuous or sampled error condition and transmits whenever it violates a threshold; adversaries may induce frequent triggers and state changes.
• Time-triggered: agent updates are scheduled at uniform intervals regardless of system state.

Advantages of self-triggered strategies include:

• Strict decoupling from exogenous messages—agents are immune to adversary-induced flooding since triggers are local and agent-specific (Matsume et al., 2020).
• Communication savings—update counts do not increase with the number of adversaries, while event-triggered counts may blow up under malicious conditions.
• Asynchronous, fully distributed operation—each agent maintains its own internal schedule without global clocks.

Empirical studies confirm that self-triggered variants achieve comparable consensus accuracy but significantly lower update rates and greater resilience against growing $F$ (Matsume et al., 2020).

6. Extensions, Variations, and Generalizations

Recent developments extend self-triggered communication to:

• Multilevel gossip or one-neighbor polling protocols (Persis et al., 2012).
• Time-varying controller and threshold schedules for asymptotic convergence (Persis et al., 2012).
• Robust, sparse sensing and control in high-dimensional, networked multi-agent systems via $\ell_1$-regularized design (Bahavarnia et al., 2018).
• Data-driven scenario: trajectory prediction using Hankel data, enabling self-triggered rules without explicit plant model (Liu et al., 2022).
• Output-feedback and safety-critical applications, where ellipsoidal reachability sets encode all uncertainties and predict worst-case future events without continuous sensing (Gleizer et al., 2020, Kim et al., 2018).
• Practical cyber-physical testbeds and multi-hop wireless protocols with advance slot scheduling (Baumann et al., 2019).
• Safety and invariance by means of set iteration and partitioning, yielding decentralized hybrid countdown-to-communication rules (Kim et al., 2018, Hashimoto et al., 2018).

The common algorithmic blueprint is the computation of the next trigger interval as a function of the current state (and, where applicable, filtered or predicted neighbor data), local thresholds, and robustness parameters, often encapsulated in contractive or Lyapunov-inspired rules.

7. Practical Impact and Application Domains

Self-triggered communication has been widely demonstrated to:

• Attain high communication efficiency in multi-agent consensus, formation control, and resource-aware IoT feedback loops.
• Enhance robustness against time-varying delays, communication faults, quantization, noise, and adversarial attacks.
• Lower duty cycles and transmission rates in networked control systems without sacrificing closed-loop performance (Matsume et al., 2020, Yi et al., 2016, Bahavarnia et al., 2018, Baumann et al., 2019).
• Provide scalable mechanisms for large-scale formations, safety-critical collision avoidance, and distributed estimation and tracking.

In the context of adversarial multi-agent consensus, the self-triggered framework ensures resilience, bounded error, and minimized update events, with parameterized guarantees that scale with network connectivity, adversary count, and protocol filtering strength (Matsume et al., 2020).