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Event-Triggered $H_\infty$ Control: a Switching Approach
Anton Selivanov, Emilia Fridman
TL;DR
The paper addresses the tension between Zeno behavior in continuous event-triggering and redundant transmissions in periodic event-triggering. It introduces a switching-based waiting-time method, extends it to perturbed systems with delays, and reports substantially fewer transmitted measurements in examples.
Problem
Continuous event-triggering can generate infinitely many sampling instants in finite time, whereas periodic event-triggering avoids this behavior but does not use all continuously available measurements.
Method
The paper switches between periodic sampling and continuous event-triggering, selecting a stability-preserving maximum sampling period as the waiting time.
Results
The switching trigger reduces the average amount of sent measurements by almost 40% in the reported example.
Takeaways & Limitations
The approach guarantees a positive lower bound on inter-event times and extends to L2-gain and ISS analysis of perturbed networked systems with delays.
Takeaways & Limitations
The approach assumes bounded network-induced delays and can account for bounded consecutive packet losses when acknowledgement of successful reception is available.
Abstract
from arXiv · showhide
Event-triggered approach to networked control systems is used to reduce the workload of the communication network. For the static output-feedback continuous event-trigger may generate an infinite number of sampling instants in finite time (Zeno phenomenon) what makes it inapplicable to the real-world systems. Periodic event-trigger avoids this behavior but does not use all the available information. In the present paper we aim to exploit the advantage of the continuous-time measurements and guarantee a positive lower bound on the inter-event times by introducing a switching approach for finding a waiting time in the event-triggered mechanism. Namely, our idea is to present the closed-loop system as a switching between the system under periodic sampling and the one under continuous event-trigger and take the maximum sampling preserving the stability as the waiting time. We extend this idea to the $L_2$-gain and ISS analysis of perturbed networked control systems with network-induced delays. By examples we demonstrate that the switching approach to event-triggered control can essentially reduce the amount of measurements to be sent through a communication network compared to the existing methods.
1 Introduction
Networked control systems must transmit sampled measurements, creating a trade-off between communication workload and use of continuous measurements. Continuous triggering can exhibit Zeno behavior, while periodic triggering avoids it but may send redundant packets.
- Motivation: Networked control systems transmit measurements from sensors to controllers through communication networks only at discrete time instants.The systems include sensors, actuators, and controllers connected through a communication network.
- Motivation: Periodic sampling sends measurements at every kh, including when output fluctuations do not significantly change the control signal.Its sampling period h must be chosen to preserve stability.
- Motivation: Continuous event-triggering avoids redundant packets by transmitting when an output-based condition is violated.The condition uses the weighted output difference and a parameter ε.
- Problem: Continuous static output-feedback triggering can generate infinitely many events in finite time, producing the Zeno phenomenon.This makes the mechanism inapplicable to networked control systems.
- Existing approaches: Periodic event-triggering guarantees inter-event times of at least h and applies when sensors measure only sampled outputs y(ih).It avoids Zeno behavior but does not exploit continuously available measurements.
2 A switching approach to event-trigger
The paper models event-triggered control as switching between periodic sampling and continuous event-triggering. Stability conditions based on Lyapunov analysis yield an appropriate waiting time and support larger sampling periods under the proposed trigger.
- Switching mechanism: Separating sampling and triggering errors makes stability analysis more tractable than treating both errors together.Periodic sampling contributes a sampling error, whereas continuous triggering contributes a triggering error.
- Switching mechanism: The proposed trigger waits at least h seconds after each transmission, then continuously checks the event condition until it is violated.The waiting phase follows periodic sampling dynamics, while the subsequent phase follows continuous event-trigger dynamics.
- Stability analysis: Theorem 1 establishes exponential stability with decay rate δ when the stated matrix conditions are feasible.The proof uses different Lyapunov functions for the two switching modes and matches their values at switching instants.
- Stability analysis: Periodic event-triggering is also exponentially stable with decay rate δ under its corresponding feasibility conditions.The proposed conditions are developed by extending the stability proof to the periodic event-trigger formulation.
- Comparison: Feasibility of the proposed conditions implies feasibility for periodic event-triggering, allowing not smaller h and ε values.For identical h, ε, and Ω, periodic event-triggering can deliberately send fewer measurements because it checks only at periodic instants.
3 Event-trigger under network-induced delays and disturbances
The paper extends its switching event-trigger design to delayed, disturbed networked systems, using LMIs to certify stability and L2-gain while accounting for measurement noise and network delays.
- System model: The delayed closed-loop model uses zero-order-hold updates at actuator times t_k=s_k+η_k, with bounded network delays satisfying t_k≤t_{k+1}.The overall delay from sensor to actuator is η_k≤η_M.
- Switching mechanism: The controller switches from periodic sampling during an h-second waiting interval to continuous event-triggering afterward.If communication delays make t_k+h≥t_{k+1}, no switching occurs.
- Delay representation: A fictitious delay ¯η(t) represents the event-trigger phase, with τ(t) bounded by τ_M=h+η_M.The linear interpolation of ¯η(t) connects the delays at the switching and next update times.
- L2-gain analysis: Feasible LMIs provide sufficient conditions for internal exponential stability with decay rate δ and L2-gain less than γ.The result applies to the disturbed system with measurement noise and network-induced delays.
- ISS analysis: When C1=D1=0, the same conditions imply input-to-state stability with respect to process and delayed measurement disturbances.The proof derives a dissipation inequality of the form V˙≤−2δV+γ^2Δ^2 for bounded disturbances.
- Scope: The method handles polytopic-type uncertainties because the theorem LMIs are affine in A, B, B1, and B2.The approach can also account for bounded consecutive packet losses when successful reception is acknowledged.
4 Numerical examples
The numerical examples compare periodic sampling, periodic event-triggering, and the proposed switching event-trigger across systems with and without network delays. The switching approach reduces sent measurements in several examples while preserving the specified convergence rate.
- Example 1: Periodic sampling was best in Example 1, while both event-triggering methods failed to reduce the network workload.The reported periods were h = 0.356 for δ = 0.24 and h = 0.424 for δ = 0.001.
- Example 2: The switching event-trigger reduced the average sent measurements by almost 40% in Example 2, whereas the periodic event-trigger offered no significant improvement.The comparison used δ = 0.24 and Tf = 20.
- Example 2: The switching trigger produced larger inter-sampling times and preserved the decay rate, unlike the periodic trigger, which skipped only two measurements in the simulation.Figures 3 and 4 used ε = 4.6 × 10−3, h = 1.115 and ε = 0.555, h = 0.899, respectively.
- Delayed systems: With network delays, the periodic event-trigger again gave no improvement, while switching reduced sent measurements by almost 20% for ηM = 0.1 and preserved δ.The delays were randomly selected subject to the stated delay condition.
- Example 3: In the disturbed, delayed system, both event-triggers reduced network workload, with switching providing more than 11% reduction versus periodic event-triggering.The comparison was reported for γ = 100.
5 Conclusion
The paper proposes switching between periodic sampling and continuous event-triggering to exploit continuous measurements while guaranteeing a positive inter-event-time lower bound. It extends the analysis to perturbed delayed systems and reports reduced network workload for uncertain linear systems.
- 5 Conclusion: The proposed switching approach guarantees a positive lower bound on inter-event times while reducing communication-network workload.The approach is based on switching between periodic sampling and continuous event-triggering.
- 5 Conclusion: The method is extended to L2-gain and ISS analyses of perturbed networked control systems with network-induced delays.
- 5 Conclusion: The results apply to linear systems with polytopic-type uncertainties, while extension to nonlinear networked control systems is identified as future research.
Appendix Proof of Theorem 2
The proof constructs a composite Lyapunov functional for the switched delayed system and derives matrix inequalities for its operating modes. These conditions establish exponential stability and the stated performance bound.
- Lyapunov construction: The Lyapunov functional is composed of VP, VS0, VS1, VR0, and VR1.
- χ(t) = 0 mode: For χ(t) = 0, the proof differentiates the component functionals and handles delayed states using Jensen’s inequality and Park’s theorem.
- χ = 1 mode: For χ = 1 and τ(t) ∈ (ηM, τM], the proof applies Jensen’s inequality and Park’s theorem to compensate the delayed state.
- Mode relation: For χ = 1 and τ(t) ∈ [0, ηM], the system is represented with χ = 0 and e(t) = 0, so the same condition guarantees the inequality.
- χ = 1 mode: The resulting dissipation inequality is expressed using the vector ψ(t) and matrix Ψ, with Ψ ≤ 0 guaranteeing the required inequality.
- Conclusion of proof: With zero disturbances and measurement noise, the dissipation inequality implies internal exponential stability with decay rate δ; integration yields the stated performance result.