Hybrid Event-Triggering Mechanism

• Hybrid event-triggering mechanism is a control strategy that schedules updates based on state-dependent events, combining continuous dynamics with discrete jumps.
• It employs Lyapunov-based analysis to define flow and jump sets, ensuring stability by triggering control updates only when errors exceed precise thresholds.
• The framework integrates co-design of controllers and triggers, using auxiliary dynamics and dwell-time conditions to prevent Zeno behavior and enhance resource efficiency.

A hybrid event-triggering mechanism is a control strategy wherein the update or actuation instants of a dynamical system are determined by a combination of continuous evolution (“flows”) and discrete transitions (“jumps”) regulated by state-dependent or auxiliary variables, rather than periodic schedules. Such mechanisms are engineered to minimize resource consumption (e.g., communication, computation) by prompting control or measurement updates only when required to ensure performance or stability, with the hybrid model capturing both the continuous physical evolution and the logic-based event schedule.

1. Foundational Hybrid System Frameworks

The foundational premise in hybrid event-triggering is to represent the closed-loop system as a hybrid dynamical system, typically using the formalism:

• Flow dynamics: $\dot{x} = f(x, e)$ for $(x, e) \in C$
• Jump dynamics: $x^+ = g(x, e)$ for $(x, e) \in D$

Here, $x$ denotes the system state and $e$ the sampling-induced error due to aperiodic updates. The flow set $C$ is defined by a local “safe” region where the error is small relative to a Lyapunov functional, while the jump set $D$ collects states where the error is no longer negligible (e.g., $D = \{ (x, e): \tilde{y}(|e|) \geq V(x)\}$ with $V(x)$ a Lyapunov function and $\tilde{y}$ a threshold function). At each jump, the error variable is reset according to a predefined rule (often $e^+ = 0$) to ensure the Lyapunov functional can decrease during subsequent flows (Postoyan et al., 2011).

Hybrid modeling also encompasses auxiliary variables (e.g., $n$ evolving as $\dot{n} = -\delta(n)$), enabling temporal shaping of inter-event intervals and refining the trade-off between performance and update frequency.

2. Synthesis of Triggering Rules and Lyapunov-Based Analysis

Triggering conditions are synthesized via Lyapunov (or Input-to-State Stability) functions to guarantee stability and performance. A standard requirement is that the Lyapunov function $V(x)$ (or its hybrid extension $R(q)$) strictly decreases on flows and is non-increasing on jumps:

• During flows: $\dot{V}(x) \leq -a(V(x)) + y(|e|)$, where $a(\cdot)$ and $y(\cdot)$ are class-$K$ functions.
• Following a jump: $R(G(q)) \leq R(q)$.

Canonical triggering rules are derived by bounding the adverse effect of sampling-induced errors through inequalities of the form:

$V(x) \leq a^{-1}((1-o) y(|e|)), \quad o \in (0, 1)$

which is equivalent to $\tilde{y}(|e|) \leq V(x)$ for a suitable function $\tilde{y}$ (Postoyan et al., 2011). This establishes the explicit connection between the flow and jump sets:

$$C = \{ (x, e): \tilde{y}(|e|) \leq V(x) \}, \;\; D = \{ (x, e): \tilde{y}(|e|) \geq V(x) \}$$

To enforce a prescribed decay over intervals, auxiliary variables can be used with dynamic reset rules, leading to more general triggers such as $W(e) \geq \max\{n, V(x)\}$, ultimately affording direct control over the minimum inter-event time.

The composite Lyapunov function $R(q)$, with class-$K$ bounding properties and specific decrease estimates on flows and non-increase on jumps, underpins proofs of semiglobal asymptotic stability for the equilibrium $(x,e) = 0$ or appropriate target sets.

3. Resource Optimization and Generality

A critical feature of hybrid event-triggered mechanisms is their ability to minimize actuation or communication without sacrificing stability. This is achieved by ensuring transmissions only occur when the error surpasses a threshold that is itself “matched” to the instantaneous system energy (as quantified by the Lyapunov function). When the error remains small relative to the Lyapunov functional, no updates are scheduled, leading to potentially long intervals between transmissions in steady-state or near-equilibrium conditions (Postoyan et al., 2011).

Such mechanisms are not limited to static feedback control but also subsume dynamic controllers and a broad class of sampled-data implementations. The Lyapunov framework also unifies earlier strategies—such as static event-triggering and dynamic event-triggering (with time-varying thresholds or filters)—and enables further innovation through the introduction of new auxiliary variables or dynamic thresholds.

4. Extension to Co-Design and Output Feedback

Hybrid event-triggered schemes have been extended to the co-design of both controllers and triggering rules, particularly for linear systems. The plant, controller, and network-induced errors are modeled as a hybrid inclusion, where the control law and event-triggering parameters (e.g., gain matrices, threshold coefficients) are synthesized simultaneously using linear matrix inequalities.

For output feedback designs, network errors (differences between current and most recently transmitted outputs/controls) and a clock variable (tracking time since the last event) augment the system state. Triggering is based on a composite condition that blends the measurement error bound with a strict dwell-time, leading to partitions:

$$C = \{ (x, e, \tau): \gamma^2|e|^2 \leq \varepsilon_1|y|^2 \text{ or } \tau \in [0, T] \}$$

with transmissions forced when the error or time constraint is violated (Abdelrahim et al., 2014). The LMIs guarantee a strictly positive minimum inter-event time and can be optimized to maximize this interval, subject to performance constraints.

5. Auxiliary Dynamics and Avoidance of Zeno Phenomena

One technical challenge in aperiodic event-based schedules is the potential for Zeno behavior (an infinite number of triggering events in finite time). Hybrid event-triggering frameworks address this by enforcing dwell-time conditions: the existence of a uniform lower bound on inter-event intervals (excluding the equilibrium), which is enforced either via explicit timer variables, dynamic thresholds, or Lyapunov contraction rates. For example, ensuring that the decay in the Lyapunov functional over an inter-event period overcomes any destabilizing influence from the sampling-induced error precludes Zeno executions.

Use of auxiliary “filter” variables (e.g., $n(t)$ or decaying timers) allows fine-grained adjustment of inter-event times and further enables shaping of transmission patterns for network or computational resource optimization.

6. Implementation Strategies, Trade-Offs, and Applications

Implementation of hybrid event-triggered mechanisms requires selection of:

• The Lyapunov function(s) and corresponding classes $a$, $y$ to encode the stability/performance trade-off.
• Threshold functions and auxiliary dynamics governing triggering.
• Tuning parameters such as decay rates, auxiliary variable dynamics, and dwell-time lower bounds.
• LMI-based synthesis (especially for output feedback and co-design strategies).

Different approaches offer trade-offs: static triggers yield conservative (more frequent) updates, while dynamic triggers with auxiliary states or timers allow improved resource performance at the expense of increased design complexity. Proper selection and tuning enable balancing between control performance, stability margins, and resource minimization.

The mechanisms developed in these frameworks have been applied to:

• Nonlinear and linear systems in embedded and networked control contexts (Postoyan et al., 2011, Abdelrahim et al., 2014).
• Reducing bandwidth, computation, and energy expenditure.
• Enabling extensions to safety-critical systems and distributed multi-agent scenarios.

The hybrid event-triggering paradigm enables systematic synthesis of triggering rules that adapt to runtime system behavior, offer rigorous stability guarantees via Lyapunov analysis, and substantially improve efficiency in resource-constrained settings. The framework’s high degree of generality encompasses prior event-based strategies and continues to inform modern networked and cyber-physical system designs.