Event-Based Impulsive Control System

• Event-based impulsive control systems are hybrid frameworks that integrate continuous-time dynamics with state-triggered, discrete impulses.
• They employ Lyapunov and control barrier functions to ensure stability, safety, and optimal performance under uncertainties and delays.
• Applications range from robotics and aerospace to networked resource allocation, with design methods addressing convergence, robustness, and prevention of Zeno phenomena.

An event-based impulsive control system is a hybrid dynamical framework in which continuous-time system trajectories evolve under ordinary differential equations except at discrete event times—triggered by explicit state-dependent or output-dependent conditions—when instantaneous, typically discontinuous, state jumps (impulses) are applied. These mechanisms combine continuous feedback laws with an impulse-triggering policy, yielding enhanced performance, resource efficiency, and robustness, especially in the presence of uncertainties, delays, constraints, or safety requirements.

1. Mathematical Formulation and Hybrid System Models

Event-based impulsive control systems are characterized by a partition of the state space into a flow set $\mathcal{C}$ (where the state evolves continuously) and a jump set $\mathcal{D}$ (where impulses are triggered), as in the canonical framework: $$\begin{cases} \dot{\mathbf{x}} = f(\mathbf{x}) & \mathbf{x} \in \mathcal{C}, \ \mathbf{x}^{+} \in \mathbf{J}(\mathbf{x}) & \mathbf{x} \in \mathcal{D}, \end{cases}$$ where $f$ is the continuous vector field, and $\mathbf{J}(\mathbf{x})$ defines the post-jump state. The jump set $\mathcal{D}$ is determined by guards based on system state, often representing zero-crossings or more general event surfaces.

A typical example for second-order systems with damping uncertainty is: $$m\ddot{x} + (d(t) + D)\dot{x} + Kx = 0,\quad d(t)\in[\underline{d},\overline{d}],$$ where state-dependent impulses act whenever $x = 0$ or $\dot{x} = 0$ via: $$u(\mathbf{x}) = -\alpha\,\delta_{x=0}\,\mathrm{sign}(\dot{x}) - \beta\,\delta_{\dot{x}=0}\,\mathrm{sign}(x),$$ with parameters $\alpha$, $\beta$ tuned to guarantee robust convergence under prescribed uncertainty bounds (Ruderman, 2017).

Hybrid and impulsive control extends to general nonlinear, delayed, switched, or stochastic systems by specifying tailored flow/jump maps, impulsive resets, and event triggers (Clark et al., 2021, Zhang et al., 2019, Zhang et al., 2022, Yang et al., 27 Oct 2025).

2. Event-Triggered Impulse Mechanisms

Impulse actions are initiated not periodically but by online monitoring of explicit event functions—typically based on Lyapunov certificates, state zero-crossings, output thresholds, or violation of safety constraints. Key classes of event functions include:

• Zero-crossing events: In hybrid motion control, impulses are fired at $x=0$ or $\dot{x}=0$, providing robustness against parameter variations (Ruderman, 2017).
• CBF/CLF violations: Impulses are triggered as soon as the state approaches the boundary of a safety set (defined via a control barrier function $h(x)\ge0$) or whenever a Lyapunov certificate $V(x)$ is about to increase, thus enforcing forward invariance or stability (Liu et al., 13 Mar 2025, Breeden et al., 2023).
• Threshold-based rules: In resource allocation or AIMD-style algorithms, impulses are initiated each time the state variable crosses a prescribed threshold $\theta^*(\lambda)$ (Avrachenkov et al., 2018).
• Minimum inter-event dwell: To avoid accumulation of infinitely many events in finite time (the Zeno phenomenon), hybrid controllers enforce lower bounds $h > 0$ between events, sometimes by forcibly injecting impulses if the event condition would hold too soon (Zhang et al., 2019, Zhang et al., 2022, Zhang et al., 2023).

Such mechanisms are customized for continuous, delayed, discrete-time, or networked settings, with triggers that may be evaluated continuously or on periodic sampling grids (Zhang et al., 2023, Xu et al., 2018).

3. Stability, Safety, and Performance Guarantees

Rigorous stability analysis of event-based impulsive control systems is rooted in Lyapunov and input-to-state stability (ISS) theory, frequently employing hybrid Lyapunov functions that decrease along flows and are non-increasing (or strictly decreasing) at jumps. Central results include:

• Global asymptotic stability (GAS): For properly chosen PD and impulse gains, all system trajectories converge to the equilibrium under event-based jumps (Ruderman, 2017).
• ISS and exponential convergence: In nonlinear and linear settings (including those with delays and disturbances), suitable Razumikhin-type or composite Lyapunov conditions—coupled with event-trigger laws and minimal dwell constraints—yield GAS, ISS, or practical exponential stability (Zhang et al., 2019, Zhang et al., 2022, Xu et al., 2018, Eilers et al., 22 Nov 2025).
• Safety-critical invariance: Control barrier function (CBF) conditions are enforced both along flows and across impulsive jumps, often via quadratic programming (QP) at impulses and adaptive gain adjustment, ensuring safety constraints are never violated during or after jumps (Liu et al., 13 Mar 2025, Breeden et al., 2023). Impulsive Timed CBF (ITCBF) theory provides conditions under which, after any impulse, the system remains in the safe set for at least the minimum dwell (Breeden et al., 2023).
• Optimality: In hybrid optimal control (Bolza problems), the hybrid Pontryagin Maximum Principle extends classical adjoint/optimality systems with state and costate jumps aligned to the event surface and reset map, preserving symplectic structure and allowing energy-matching at each event (Clark et al., 2021).

Absence of Zeno behavior—i.e., infinite events in finite time—is guaranteed by dwell-time constraints, positive minimum event separations, and, in geometric approaches, symplectic-volume preservation arguments indicating measure-zero Zeno sets (Ruderman, 2017, Clark et al., 2021, Zhang et al., 2019, Zhang et al., 2022, Breeden et al., 2023).

4. Representative Applications and Case Studies

Event-based impulsive control systems offer both theoretical tractability and practical efficiency in diverse domains:

• Motion and robotics: Second-order mechanical systems with uncertain dissipation are stabilized via hybrid zero-crossing triggers; simulation shows rapid, oscillation-free settling compared to continuous PD control (Ruderman, 2017).
• Networked resource allocation: G-AIMD impulsive control (e.g., for internet congestion, EV charging) uses threshold policies and Whittle indices to optimize utility and ensure fairness under resource constraints; analytic optimality and asymptotic performance guarantees are established (Avrachenkov et al., 2018).
• Safety-critical control: Robotic manipulators and spacecraft docking under impulsive actuation employ event-triggered QP controllers with CBFs and adaptive gains to guarantee safety post-jump and after each flow interval (Liu et al., 13 Mar 2025, Breeden et al., 2023).
• Spacecraft hovering: Single-impulse, event-triggered laws maintain spacecraft within safety-critical regions under thruster dead-zone/saturation, using univariate polynomial root-finding and logical triggers for minimal computation (Sanchez et al., 13 Jan 2025).
• Multiscale, stochastic, or multi-agent systems: PETIC frameworks achieve mean-square exponential consensus for time-varying, energy-constrained multi-agent topologies, with virtual-state embedding handling communication between heterogeneous agents (Yang et al., 27 Oct 2025).
• Neuromorphic control: Leaky integrate-and-fire (LIF) event-based controllers for continuous plants guarantee practical stability and establish a direct theoretical link to analogue feedback systems (Eilers et al., 22 Nov 2025).

Empirical studies consistently demonstrate significant reductions in control updates, resource consumption, and computational overhead, while providing equivalent or enhanced performance in constraint satisfaction and convergence.

5. Design Methodologies and Parameterization

Implementation and tuning typically follow structured procedures:

1. Lyapunov/CBF candidate selection: Choose a Lyapunov function or safety barrier. Establish continuous-flow and impulsive-jump inequalities quantifying decrease or invariance.
2. Event-trigger rule specification: Define trigger conditions (involving state, output, or error variables) linked to the chosen certificates such that they predict performance or constraint loss.
3. Impulse law and gain tuning: Set feedback/impulse gains ($\alpha,\,\beta$, etc.) within analytical bounds to ensure finite-time convergence or prescribed contraction rates (Ruderman, 2017).
4. Dwell-time enforcement: Select or compute explicit lower bounds on inter-event intervals to avoid pathological accumulation and ensure feasibility of actuation (Zhang et al., 2019, Zhang et al., 2022).
5. Hybrid/optimization routines: In safety-critical or optimal scenarios, formulate QP or one-shot constrained optimization for each impulse, incorporating CBF/CLF constraint satisfaction and possibly slack variables for feasibility (Liu et al., 13 Mar 2025, Breeden et al., 2023).
6. Discrete/periodic sampling: In digital or resource-limited settings, periodic evaluation grids or threshold decay laws reduce sensing and actuation requirements, with explicit stability conditions in terms of the sampling period, delay, and decay rates (Zhang et al., 2023, Xu et al., 2018).

Detailed illustrative examples provide guidance in selecting practical parameters, interpreting analytic sufficient conditions, and quantifying the trade-off between event frequency and convergence rate.

6. Theoretical Extensions and Research Directions

Recent advances in the theory and applications of event-based impulsive control include:

• Geometric control and symplectic methods: Application of geometric mechanics and symplectic geometry to hybrid control yields structure-preserving flows, with Zeno sets of measure zero in the generic case (Clark et al., 2021).
• Index policies and distributed control: Analytical tractability of Whittle-index policies in large networks, their asymptotic optimality, and extensions to multi-dimensional or coupled constraints (Avrachenkov et al., 2018).
• Hybrid observer-based and output-feedback PETC: Observer designs with triggering based on state/output error and explicit LMI-based characterization of ISS for incrementally quadratic nonlinearities (Xu et al., 2018).
• Adaptive and learning-based impulsive gains: Adaptive gain selection mechanisms, exploiting local state or jump magnitude, optimize recovery to safety sets and minimize over-actuation (Liu et al., 13 Mar 2025).

Ongoing challenges include robustness to measurement noise (addressed via hysteresis or bounds on excursions), actuation timing and delay compensation, and efficient numerical realization in high-dimensional and real-time scenarios.

7. Limitations and Open Issues

Notable limitations include:

• Actuation constraints: Infinite-power Dirac impulses are nonphysical; actuator bandwidth and saturation must be accounted for in real applications (Ruderman, 2017, Sanchez et al., 13 Jan 2025).
• Measurement noise and chattering: Zero-crossing and threshold-based triggers may provoke excessive chattering near guard surfaces; practical designs use hysteresis or minimum dwell to mitigate this (Ruderman, 2017, Avrachenkov et al., 2018).
• Global invariance: Local-invariance and reachability are sometimes only guaranteed in a region; outside, fallback to multi-impulse or global strategies is needed (Sanchez et al., 13 Jan 2025).
• Computational cost: Some schemes (CBF-QP, hybrid-PMP) require solution of online optimization or adjoint equations at each event, though typically at low update rates (Clark et al., 2021, Liu et al., 13 Mar 2025, Breeden et al., 2023).
• Zeno phenomena: Although positive dwell and geometric conditions generally rule out Zeno, pathological counterexamples exist if regularity or timing constraints are violated (Zhang et al., 2019, Clark et al., 2021, Zhang et al., 2022).

Future research is directed at unifying adaptive, learning-based, and distributed event-triggered impulsive methods for complex, networked, uncertain, or highly-constrained physical systems.