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Dynamic Triggering Mechanisms for Event-Triggered Control
Antoine Girard
TL;DR
Event-triggered control can avoid unnecessary periodic updates, but its triggering rules must preserve stability and provide useful execution-time guarantees. The paper introduces dynamic event-triggering mechanisms with an internal variable, proves stability, and analyzes parameter effects on decay, timing, and quadratic performance. It also identifies a limitation: a longer next execution time from a given state does not guarantee longer later execution times, and the Lyapunov function itself may increase.
Problem
Periodic execution can waste shared computational and communication resources, motivating event-triggering mechanisms that update inputs only when events occur.
Method
The paper enriches event-triggering mechanisms with an additional internal dynamic variable and analyzes the resulting nonlinear and linear closed-loop systems.
Results
The paper proves closed-loop stability, derives parameter-dependent inter-execution-time bounds for linear systems, and studies parameter effects on a quadratic integral performance index.
Takeaways & Limitations
Dynamic event-triggering mechanisms provide stability guarantees while allowing design parameters to shape Lyapunov decay and execution timing.
Takeaways & Limitations
A longer next execution time for a given state does not imply longer later execution times, and the Lyapunov function may increase even when an upper bound decreases.
Abstract
from arXiv · showhide
In this paper, we present a new class of event triggering mechanisms for event-triggered control systems. This class is characterized by the introduction of an internal dynamic variable, which motivates the proposed name of dynamic event triggering mechanism. The stability of the resulting closed loop system is proved and the influence of design parameters on the decay rate of the Lyapunov function is discussed. For linear systems, we establish a lower bound on the inter-execution time as a function of the parameters. The influence of these parameters on a quadratic integral performance index is also studied. Some simulation results are provided for illustration of the theoretical claims.
1. Introduction
The paper motivates dynamic event-triggered control as a way to reduce unnecessary periodic workloads while preserving stability and providing execution-time guarantees. It introduces an internal dynamic variable and studies its effects on Lyapunov decay, inter-execution times, and performance.
- Periodic controller execution can impose unnecessary computational and communication workloads when resources could serve other tasks.
- Dynamic event-triggering mechanisms introduce an additional internal dynamic variable, which motivates the paper’s terminology.
- The paper extends a nonlinear-control framework to dynamic mechanisms and proves asymptotic stability of the resulting closed-loop system.
- For dynamic mechanisms, the guaranteed lower bound on inter-execution times cannot be smaller than that of the corresponding static mechanism.
- For linear systems, the paper derives parameter-dependent inter-execution-time bounds and studies the parameters’ influence on a quadratic integral performance index.
- The paper also reviews class K and class KL function definitions and local Lipschitz continuity used in its control-theoretic analysis.
2. Triggering Mechanisms for Event-Triggered Control Systems
The paper develops dynamic event-triggering mechanisms that augment static state-error rules with an internal variable, proving stability while tuning decay and execution timing through design parameters.
- Motivation: Event-triggered control replaces periodic execution with state-dependent updates, reducing potentially unnecessary computational and communication workloads.Traditional time-triggered control may execute independently of system state, creating high workloads when resources could serve other tasks.
- Choice of parameters: The parameter σ tunes the guaranteed Lyapunov decay rate, while suitable β and η0=0 can recover the static ETM’s guaranteed decay rate.As σ approaches 0 in the static mechanism, the decay rate approaches that of the ideal error-free system; the dynamic mechanism can match the static guarantee under the stated parameter choice.
- Dynamic event-triggering mechanisms: The proposed dynamic ETM adds an internal variable η that filters σα(∥x∥)−γ(∥e∥), allowing this quantity to be nonnegative on average rather than continuously.Events are triggered so that η remains nonnegative, with β, σ, η0, and θ serving as design parameters.
- Stability analysis: The augmented Lyapunov function W decreases and guarantees asymptotic convergence of both x(t) and η(t) when execution times do not accumulate.The analysis establishes a positive-definite, radially unbounded W whose derivative is nonpositive under the dynamic triggering rule.
- Inter-execution times: The dynamic ETM’s minimum inter-execution time cannot be smaller than the static ETM’s, and smaller θ yields a larger minimum inter-execution time.The comparison is guaranteed for the minimum inter-execution time, although a longer next execution time for one state does not determine all subsequent execution times.
- Choice of parameters: Unlike the static ETM, V(x(t)) may increase temporarily, but W remains decreasing and larger θ limits the potential increase.The paper relates this behavior to the prescribed decay-rate bound and treats θ as a tuning parameter for the dynamics of V.
3. The Case of Linear Systems
For linear systems, the dynamic event-triggering mechanism preserves asymptotic stability while tuning decay, inter-execution times, and quadratic performance through its parameters.
- Linear-system framework: The linear closed-loop dynamics use a stabilizing feedback controller and Lyapunov function V(x)=x⊤Px, with measurement error entering the implementation dynamics.The resulting dynamics are ẋ=Ax+BK(x+e), and V̇ includes the error term 2x⊤PBKe.
- Parameter influence: Choosing σ close to 0 makes the Lyapunov decay rate approach that of the ideal system, while θ does not affect the decay rate under the stated parameter choice.The decay rate can be tuned through σ, whereas θ affects the quadratic integral performance index.
- Event-triggering mechanism: The proposed dynamic ETM filters the triggering signal σx⊤Qx−2x⊤PBKe through an internal variable η.The mechanism introduces η as an additional design-dependent dynamic state, while the static ETM is recovered as θ→+∞.
- Stability and execution times: The dynamic ETM produces a larger next execution time than the static ETM for the same state, and smaller θ yields a larger next execution time.The lower-bound analysis shows that in some parameter regimes the dynamic bound is strictly larger than the static bound, while in the first case the bounds coincide.
- Stability and execution times: For all λ>0, σ∈(0,1), and η0,θ≥0, the dynamic ETM guarantees a positive inter-execution-time lower bound and asymptotic convergence of x(t) and η(t) to zero.The proof uses the augmented Lyapunov function W(x,η)=V(x)+η, whose decrease establishes convergence.
- Parameter influence: The quadratic performance bound increases with θ, so performance approaches the static ETM as θ→+∞ and can approach the ideal closed-loop performance by tuning σ and θ.A suggested strategy chooses η0=0 and λ=(1−σ)κ, then selects θ=1/(2a−λ) to balance execution-time and performance considerations.
4. Example
The example compares static and dynamic event-triggering mechanisms using simulations, showing how dynamic triggering affects inter-execution time, Lyapunov-function variation, and performance.
- Example: The simulations implement static ETM (18), dynamic ETM (21), and ETM (27) for a stabilizing linear-system example.For the dynamic ETM, η0 = 0, λ = (1 −σ)κ with κ = 0.48, and several θ values are tested.
- Example: Dynamic ETM (21) and ETM (27) produce significantly larger inter-execution times than static ETM (18) in the illustrated trajectory.The comparison uses σ = 0.1 and x(0) = [10 0]^⊤.
- Example: Across simulations, dynamic ETM (21) yields significantly larger average inter-execution times than the static ETM, especially for small σ.The simulations use 60 initial values over a 10-second frame and report mean times, variability coefficients, and normalized performance indices.
- Example: Dynamic ETM (21) provides a compromise between inter-execution time and the performance index as θ varies.For increasing θ, one branch improves performance while another reduces inter-execution time; θ = 1/(2a −λ) is close to the apparent optimum.
5. Conclusion
The conclusion presents dynamic event-triggering mechanisms as a new class with proved stability for nonlinear systems and parameter-dependent results for linear systems.
- Conclusion: The paper introduces a new class of dynamic ETMs for event-triggered control systems.The mechanisms use an internal dynamic variable.
- Conclusion: For nonlinear systems, the paper proves stability of the resulting closed-loop system.
- Conclusion: For linear systems, the paper derives parameter-dependent insight for choosing ETM parameters.
- Conclusion: The paper identifies potential applications in decentralized, output-based, and periodic event-triggered control.
Appendix
The appendix develops lower bounds on inter-execution times through comparison arguments and establishes parameter-dependent cases for the dynamic triggering mechanism.
- Appendix: The appendix compares dynamic and static ETMs through lower bounds on the time required for a triggering function to reach one.
- Appendix: The derivation uses bounds on the error and state derivatives together with a lower bound on η̇.
- Appendix: For θ ∈ [0, 1/(2a −λ)], the lower-bound integral increases with θ and matches the bound in (24) at θ = 1/(2a −λ).
- Appendix: The appendix derives the quadratic integral-cost relation by integrating inequalities obtained from the Lyapunov analysis.