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arxiv: 1907.01861 · v1 · pith:2YE6FU5Fnew · submitted 2019-07-03 · 📡 eess.SY · cs.SY

Self-triggered stabilizing controllers for linear continuous-time systems

Fairouz Zobiri , Nacim Meslem , Brigitte Bidegaray-Fesquet This is my paper

Pith reviewed 2026-05-25 10:12 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords self-triggered controlevent-triggered controllinear continuous-time systemspseudo-Lyapunov functionroot-finding algorithmstability
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The pith

Self-triggered controllers predict update times for linear systems via a pseudo-Lyapunov function to reduce communications while preserving stability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a self-triggered control method for linear continuous-time systems that avoids constant monitoring of the system state. It generates control updates when a pseudo-Lyapunov function reaches a preset upper limit, predicting those instants from the system model and prior data. A hybrid root-finding algorithm that merges bisection for reliability with Newton-Raphson for speed, started from a minimized initial guess, locates the next crossing. The design achieves considerably fewer controller-to-plant communications without sacrificing closed-loop stability.

Core claim

A self-triggered version of an event-triggered controller is presented in which events occur when a pseudo-Lyapunov function increases to a given limit; the future intersection times are computed by a root-finding procedure that combines the global convergence of bisection with the rapid convergence of Newton-Raphson, the initial iterate being obtained through a minimization step, thereby reducing intercommunications while guaranteeing stability.

What carries the argument

The pseudo-Lyapunov function (PLF) whose crossings with an upper bound trigger control updates, located by a hybrid bisection-Newton-Raphson root-finding algorithm whose initial point is supplied by minimization.

If this is right

  • Control updates are issued only at the predicted instants rather than after continuous monitoring.
  • The closed-loop system remains stable under the self-triggered law.
  • The volume of communications between controller and plant is considerably smaller than in the underlying event-triggered scheme.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same prediction technique could be tested on systems with modest model mismatch to quantify robustness.
  • In bandwidth-limited networks the reduced update rate may translate directly into lower energy or channel usage.

Load-bearing premise

The system model is accurate enough that the future trajectory of the pseudo-Lyapunov function can be predicted without continuous state feedback.

What would settle it

Applying the predicted update instants to a physical or simulated linear plant whose state evolution deviates from the model and checking whether the closed-loop trajectory becomes unstable.

Figures

Figures reproduced from arXiv: 1907.01861 by Brigitte Bidegaray-Fesquet, Fairouz Zobiri, Nacim Meslem.

Figure 1
Figure 1. Figure 1: The pseudo-Lyapunov function and the upper limit. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Shape of the PLF for different choices of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Locating the root inside an interval. Algorithm 2 Interval Finding 1: function Pre-processing(tk) 2: t1 ← ρk 3: if Z(t1) < 0 then 4: θ ← −κ2(ρk − tk), 0 < κ2 ≤ 0.5 5: else 6: θ ← κ2(ρk − tk) 7: t2 ← ρk + θ 8: while Z(t1)Z(t2) ≥ 0 do 9: t2 ← t2 + θ 10: if t2 ≤ tk then 11: t2 ← t2 − θ, θ ← θ/2 12: t2 ← t2 + θ 13: tmax ← max(t1, t2) 14: tmin ← tmax − |θ| 15: return tmin, tmax The root-finding algorithm can on… view at source ↗
Figure 4
Figure 4. Figure 4: Simulation results of self-triggered control. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The running times of the self-triggered algorithm versus the inter [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

Self-triggered control is an improvement on event-triggered control methods. Unlike the latter, self-triggered control does not require monitoring the behavior of the system constantly. Instead, self-triggered algorithms predict the events at which the control law has to be updated before they happen, relying on system model and past information.\\ In this work, we present a self-triggered version of an event-triggered control method in which events are generated when a pseudo-Lyapunov function (PLF) associated with the system increases up to a certain limit. This approach has been shown to considerably decrease the communications between the controller and the plant, while maintaining system stability. To predict the intersections between the PLF and the upper limit, we use a simple and fast root-finding algorithm. The algorithm mixes the global convergence properties of the bisection and the fast convergence properties of the Newton-Raphson method. \\ Moreover, to ensure the convergence of the method, the initial iterate of the algorithm is found through a minimization algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper develops a self-triggered stabilizing controller for linear continuous-time systems. It extends a prior event-triggered scheme that generates updates when a pseudo-Lyapunov function (PLF) reaches a design threshold; the self-triggered version replaces continuous monitoring by predicting the next crossing time via a hybrid bisection–Newton-Raphson root finder whose initial guess is obtained by a separate minimization. The claimed benefit is a substantial reduction in controller-to-plant communications while preserving closed-loop stability.

Significance. If the numerical prediction is guaranteed to occur no later than the true first PLF crossing, the method supplies a concrete, implementable alternative to both periodic and event-triggered control for linear plants, with the hybrid solver offering a pragmatic balance of reliability and speed. The approach is directly falsifiable through simulation on benchmark linear systems and could be extended to other PLF-based designs.

major comments (2)
  1. [Root-finding algorithm (description and convergence analysis)] The stability argument rests on the computed root being strictly less than or equal to the first positive time at which the true PLF trajectory reaches its upper bound. No a-priori residual bound, Lipschitz constant on the PLF derivative, or isolation argument is supplied to guarantee that the hybrid solver (bisection + Newton-Raphson) returns a time that precedes the actual crossing, even under exact arithmetic. A small overshoot would allow the PLF to exceed its design limit before the next control update is applied.
  2. [Stability theorem and prediction step] The forward prediction of the PLF trajectory is performed with the nominal linear model. No robustness margin or sensitivity analysis is given with respect to model mismatch, discretization error in the numerical integration of the PLF, or floating-point effects; any of these could shift the true crossing earlier than the predicted time.
minor comments (3)
  1. [§2] Notation for the PLF and its upper limit should be introduced with a short self-contained recap in §2, rather than assuming familiarity with the referenced event-triggered paper.
  2. [Algorithm 1] The minimization procedure used to obtain the initial iterate for the root finder is described at a high level; a pseudocode listing or explicit statement of the objective function would improve reproducibility.
  3. [Numerical examples] Figure captions should state the system matrices, initial condition, and design parameters used, so that the reported inter-update intervals can be reproduced.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: Root-finding algorithm (description and convergence analysis): The stability argument rests on the computed root being strictly less than or equal to the first positive time at which the true PLF trajectory reaches its upper bound. No a-priori residual bound, Lipschitz constant on the PLF derivative, or isolation argument is supplied to guarantee that the hybrid solver (bisection + Newton-Raphson) returns a time that precedes the actual crossing, even under exact arithmetic. A small overshoot would allow the PLF to exceed its design limit before the next control update is applied.

    Authors: We agree that the manuscript does not supply an a-priori residual bound or Lipschitz argument guaranteeing that the hybrid solver never returns a time after the true crossing. The bisection step is intended to maintain a bracket containing the root while Newton-Raphson accelerates within that bracket, and the minimization provides the initial iterate. In revision we will expand the algorithm description to clarify these safeguards and note the practical conditions (monotonicity of the PLF) under which overshoot is avoided. A full theoretical guarantee with explicit constants would require additional analysis. revision: partial

  2. Referee: Stability theorem and prediction step: The forward prediction of the PLF trajectory is performed with the nominal linear model. No robustness margin or sensitivity analysis is given with respect to model mismatch, discretization error in the numerical integration of the PLF, or floating-point effects; any of these could shift the true crossing earlier than the predicted time.

    Authors: The stability theorem is stated for the exact nominal linear model, which is the standard setting for this class of self-triggered results. We will revise the manuscript to state this modeling assumption explicitly in the theorem statement and add a short paragraph acknowledging that model mismatch, integration errors, and floating-point effects are not analyzed, suggesting them as directions for future work. revision: yes

standing simulated objections not resolved
  • A complete a-priori convergence analysis with residual bounds for the hybrid root finder to ensure it never exceeds the true PLF crossing time.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes a self-triggered controller that predicts PLF crossing times via a hybrid bisection/Newton-Raphson root finder whose initial guess comes from a separate minimization. No equations, fitted parameters, or self-citations appear in the provided text that would make any stability claim or prediction reduce to its own inputs by construction. The approach explicitly relies on an external system model and past data, rendering the derivation self-contained rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the pseudo-Lyapunov function is referenced but not defined here.

pith-pipeline@v0.9.0 · 5705 in / 956 out tokens · 23968 ms · 2026-05-25T10:12:41.547423+00:00 · methodology

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Reference graph

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