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Event-triggered Control and Online Learning for Networked Systems under Computational Delays

Xiaobing Dai, Armin Lederer, Zewen Yang, Sihua Zhang, Lu Wan, Yang Tang, Sandra Hirche

arXiv:2608.29576v1cs.LGcs.ROeess.SY

TL;DR

Online learning can improve control of uncertain systems but introduces computational delays that challenge resource-limited implementations. The paper uses remote in-network learning-based control, derives delay-aware tracking guarantees, and designs asynchronous event-triggered control and learning. The proposed strategy preserves the desired control performance, excludes Zeno behavior, and improves communication and computation efficiency in simulations.

  • Problem

    Online inference and model updates create non-negligible computational delays, especially when learning-based control runs on systems with limited local computational resources.

  • Method

    The paper deploys the learning-based controller remotely, derives tracking-error bounds under computational delay, and designs asynchronous event triggers for control and online learning.

  • Results

    The proposed event-triggered strategy achieves the same control performance as the time-triggered scenario while excluding Zeno behavior and improving communication and computation efficiency.

  • Takeaways & Limitations

    Delay-aware in-network control can support diverse communication and computation strategies while meeting a desired control performance.

Abstract

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Online learning-based control is a promising approach to control uncertain systems, where unknown components are identified during operation to improve control performance. However, resource-intensive online learning algorithms introduce non-negligible computational delays, especially when executed on systems with limited local computational resources. To mitigate this, an in-network online learning-based control structure is employed by deploying the learning-based controller on a remote computation node and connecting it via a communication channel. In this paper, control performance guarantee is first established by deriving tracking error bound for the in-network control architecture, while accounting for computational delays. The derived tracking error bound allows for diverse communication and computation strategies under a specific condition, including time-/event-triggered mechanisms. Additionally, the trade-off between communication and computation performances is shown for a given desired control performance. Furthermore, to enhance the efficiency in both communication and computation, an efficient control framework with an asynchronous event-triggered mechanism in both control and online learning is devised under the existence of computational delay. The proposed event-triggered strategy is proven to achieve the same control performance as time-triggered scenario while excluding Zeno behavior. Finally, we derive an explicit expression of the proposed event-trigger condition for exponentially stabilizable systems, and demonstrate its effectiveness through simulations.

I. INTRODUCTION

The paper addresses uncertain-system control when online learning improves prediction but imposes computational demands and delays. It proposes an in-network architecture and asynchronous event-triggered control and learning to analyze and improve communication and computation efficiency.

  • Motivation: Online learning updates data-driven models during operation, but inference and model updates impose substantial computational and storage burdens.These burdens can prevent deployment on systems with limited computational resources.
  • Motivation: The in-network architecture places the learning-based controller on a remote computation node connected to the local control system through communication.The architecture is intended to mitigate delays from computationally intensive machine learning.
  • Contributions: The paper analyzes tracking-error bounds for networked learning-based control under computational delays and permits diverse control and learning strategies satisfying a specific condition.The analysis includes time-triggered and event-triggered mechanisms.
  • Contributions: A trade-off between communication and computation performance is established for a prescribed control performance.A time-triggered strategy is provided as an example satisfying the condition used in the performance analysis.
  • Contributions: An asynchronous event-triggered framework coordinates control and online learning under computational delay while excluding Zeno behavior.The framework is designed for higher efficiency in both communication and computation.

B. Networked Learning and Control Setting with Delay

The networked control setting separates computationally intensive learning from the plant, while prediction and model-update delays affect the information available to control. The paper models these delays, analyzes their impact, and discusses assumptions and practical boundaries of the setting.

  • Delay Setting: The delayed prediction used for control compensates the unknown system component because the current data-driven prediction is unavailable during model evaluation.The controller therefore uses a delayed version of µ(x(·)).
  • Scope and Assumptions: The paper notes that computational delays can exceed high-frequency control time budgets, motivating their explicit inclusion in analysis and design.Reported examples include approximately 40 ms for SkyGP and 200 ms for LoG-GP versus a 1 ms sampling period.
  • Delay Setting: Online learning adds processing time because new data pairs can be incorporated into the data-driven model during operation.The data set affects inference accuracy, which in turn affects control performance.
  • Delay Setting: Prediction and model-update computation times are represented separately and assumed to be bounded by known positive constants.The bounded-time assumption requires termination of the inference and update algorithms after finitely many mathematical operations.
  • Scope and Assumptions: The analysis focuses on computational delays from online Gaussian process inference and model updating rather than explicitly modeling communication-induced delays.Communication effects may matter in wireless or mobile applications and are identified for future joint treatment.
  • Networked Implementation: The network transmits control information only at discrete times, with a strictly positive lower bound on the transmission interval.The lower bound depends on the power of the communication channel.

C. Problem Formulation

The paper formulates tracking-error guarantees for in-network online learning control under computational delay, then designs a Zeno-free event-triggered framework using certified prediction bounds. Gaussian-process assumptions and bounded training sets support the required prediction-error guarantees.

  • C. Problem Formulation: The first problem is to derive a high-probability tracking-error bound for uncertain systems controlled through an in-network learning architecture.
  • C. Problem Formulation: The second problem is to design a Zeno-free event-triggered control and online-learning framework that preserves a desired tracking-error bound under computational delay.
  • A. Gaussian Process Regression: Gaussian process regression models each unknown function component as a Gaussian distribution whose posterior mean predicts the function and posterior variance estimates prediction performance.
  • A. Gaussian Process Regression: Conservative RKHS-norm bounds may enlarge certified learning and tracking bounds and increase triggering events, although the control law itself does not depend on that norm bound.
  • A. Gaussian Process Regression: Lemma 1 provides a probabilistic upper prediction-error bound, with an offline-computable constant matrix making online evaluation depend on posterior variance.
  • A. Gaussian Process Regression: After an online GP update, Corollary 1 gives a prediction-error bound independent of the complete data set and usable as the desired inference performance.
  • A. Gaussian Process Regression: A bounded number of training samples can be maintained through data forgetting, including deleting the oldest, farthest, or least informative data points.
  • A. Gaussian Process Regression: The GP guarantees assume unknown functions belong to RKHSs with bounded norms and Lipschitz stationary kernels, while bounded data sets support bounded prediction and update computation times.

B. Performance of Networked Learning-based Control

The performance analysis bounds tracking error when control and online-learning discrepancies remain below a sufficient threshold. An exemplar bound combines prediction, delay, and triggering effects, yielding guaranteed transient and ultimate error behavior.

  • The closed-loop analysis studies in-network learning-based control under computational delay through the system’s closed-loop dynamics.
  • Lemma 2 bounds the difference between the actual control input and the ideal input using prediction error, delay-related terms, and system and controller Lipschitz constants.
  • The input-discrepancy bound applies to any control-update and online-learning strategy satisfying the stated prediction and delay conditions.
  • The performance guarantee is achieved by the proposed event-triggered strategy, whose positive minimum trigger interval ensures strictly positive inter-event times.
  • Theorem 1 guarantees, with probability at least 1 −δ, that tracking error remains below a bound whenever the discrepancy ψ(t) does not exceed ϕ.
  • The theorem characterizes guaranteed transient behavior through the comparison solution and gives an ultimate tracking-error bound determined by ϕ.
  • The derived bound is a general comparison form, whereas earlier works obtain analytical ultimate bounds for specific system structures and control laws.
  • The tracking guarantee requires ψ(t) ≤ϕ for all t, and the following strategy is introduced to satisfy this premise.

C. Exemplar Control and Online Learning via Time-trigger

The time-triggered strategy updates control and online learning at scheduled times while satisfying conditions that preserve the tracking-error bound under computational delay. Its performance is practically implementable, but it does not optimize communication or computation efficiency.

  • C. Exemplar Control and Online Learning via Time-trigger: Time-triggered online learning updates the dataset with newly collected data at scheduled instances, supporting the required learning-performance bound under bounded computation time.The strategy uses the online-learning performance guarantee and the bounded computation-time assumption to ensure the relevant prediction-error condition.
  • C. Exemplar Control and Online Learning via Time-trigger: Corollary 2 provides a probabilistic bound e(t) ≤ ē(t) for all t ∈ R0,+ with probability at least 1 − δ.The result applies when the theorem assumptions hold and the time-triggered control and online-learning strategy is used.
  • C. Exemplar Control and Online Learning via Time-trigger: The required initial condition can be realized by adding {x(0), y(0)} to D(0) and generating f̂−1 from the resulting dataset.This makes the time-triggered strategy practically implementable.
  • C. Exemplar Control and Online Learning via Time-trigger: Although the time-triggered strategy achieves the desired tracking performance, it does not address learning and control efficiency.The subsequent design targets improved communication and computation efficiency under computational delay.

IV. ASYNCHRONOUS EVENT-TRIGGERED CONTROL AND ONLINE LEARNING UNDER COMPUTATIONAL DELAY

The paper develops an asynchronous event-triggered framework that decides whether to update the controller or learning model based on predicted Lyapunov-function evolution under computational delay. The design uses separate update decisions while preserving the required performance conditions.

  • IV. ASYNCHRONOUS EVENT-TRIGGERED CONTROL AND ONLINE LEARNING UNDER COMPUTATIONAL DELAY: The proposed framework uses event triggers for both online learning and control to improve communication and computation efficiency under computational delay.The section analyzes delay effects before designing and analyzing the asynchronous mechanism.
  • IV. ASYNCHRONOUS EVENT-TRIGGERED CONTROL AND ONLINE LEARNING UNDER COMPUTATIONAL DELAY: The trigger decision asks whether to update the controller or machine-learning model when control performance is insufficient.The decision is based on the instantaneous evolution trend of the Lyapunov function.
  • IV. ASYNCHRONOUS EVENT-TRIGGERED CONTROL AND ONLINE LEARNING UNDER COMPUTATIONAL DELAY: The prediction uses a conservative upper bound for an unavailable future delay, enabling trigger design without access to that future time instance.The future delay is unknown at the current time, but its upper bound follows from the bounded-delay assumption.
  • IV. ASYNCHRONOUS EVENT-TRIGGERED CONTROL AND ONLINE LEARNING UNDER COMPUTATIONAL DELAY: When V(t) exceeds its desired bound, the control input is updated to reset the elapsed time since the latest control update.The control update is one of the two perspectives used to formulate the asynchronous trigger.
  • IV. ASYNCHRONOUS EVENT-TRIGGERED CONTROL AND ONLINE LEARNING UNDER COMPUTATIONAL DELAY: A Gaussian-process model update is required when it ensures that a later control update can achieve sufficiently good performance.This requirement is formulated using the predicted Lyapunov-function behavior over intervals affected by computational delay.

1) Predictive Model Update:

The predictive model-update mechanism evaluates whether the current Gaussian-process model will provide sufficiently good future performance. It uses Lyapunov-function estimates and delay-aware quantities to formulate the learning trigger.

  • 1) Predictive Model Update:: The online-learning decision at a prior update time determines the model’s predicted performance over the subsequent interval.The prediction accounts for the influence of the learning decision on the future Lyapunov-function evolution.
  • 1) Predictive Model Update:: The estimated Lyapunov function is constructed using the time-triggered controller as a baseline over the interval where the model prediction is evaluated.This provides predefined control behavior for estimating future Lyapunov-function values.
  • 1) Predictive Model Update:: The predictive mechanism is evaluated using delay-aware expressions whose components include prior learning decisions, elapsed times, and prediction quantities.The delay terms reflect the future interval over which the model must remain adequate.
  • 1) Predictive Model Update:: For safety-critical control, the Gaussian-process event trigger must ensure upper bounds on the Lyapunov function at the relevant update times.This requirement is tied to maintaining the tracking-error bound from Theorem 1.
  • 1) Predictive Model Update:: Property 1 formalizes sufficiently good Gaussian-process performance as a condition that guarantees the desired Lyapunov-function bound.The property is stated for a prescribed bound V̄(·) from Theorem 1.

2) Instant Control Update:

The instant control-update mechanism uses the current Lyapunov-function estimate to determine when a control command must be refreshed. Its guarantee complements the predictive model-update condition in the asynchronous design.

  • 2) Instant Control Update:: An event-triggered control update occurs immediately when the current control action is insufficient for maintaining the estimated performance bound.The update sets the relevant control-update time to the current time.
  • 2) Instant Control Update:: The control strategy guarantees V(t) ≤ V̂p,1 for the relevant interval following the model-update time.This is the first requirement retained for the estimated Lyapunov function because the alternative bound is more conservative.
  • 2) Instant Control Update:: The asynchronous control and online-learning mechanism is designed from the predictive model-update and instant control-update properties with performance analysis.The two properties provide the respective conditions used by the combined event-triggered strategy.

B. Event-trigger Design and Performance Analysis

The framework designs asynchronous event triggers for control and Gaussian-process updates under computational delay. Its analysis guarantees bounded tracking performance while ensuring strictly positive inter-event times.

  • Event-trigger design: The general event-trigger design coordinates control and online-learning updates using Lyapunov-function performance requirements.The trigger construction is motivated by estimating future Lyapunov-function evolution and deciding whether control or model updates are needed.
  • Control triggering: The control trigger preserves boundedness of the actual Lyapunov function under the stated assumptions.Lemma 3 establishes boundedness using the Lyapunov-function estimate for any online-learning strategy.
  • Online-learning triggering: The event-triggered online-learning mechanism maintains the desired Lyapunov-function upper bound during model-update intervals.Lemma 4 gives the required bound for the estimated Lyapunov function after event-triggered learning updates.
  • Performance guarantee: The joint strategy guarantees e(t) ≤ ¯e(t) with probability at least 1 −δ for all t and gives both trigger intervals strictly positive lower bounds.Thus, the event-triggered strategy attains the tracking-error guarantee under computational delay.
  • Performance guarantee: Strictly positive inter-event times exclude Zeno behavior, preventing infinitely many transmissions within any finite time interval.The positivity condition applies to control transmissions and supports the event-triggered implementation.
  • Communication strategy: Applying the control trigger on the sensor-to-controller channel reduces state-transmission frequency, but evaluating it continuously can increase local computational burden.The controller-to-actuator strategy can instead transmit after state reception or GP updates.

C. Event-Trigger for Exponentially Stabilizable System

For exponentially stabilizable nominal control laws, the paper derives closed-form Lyapunov, tracking-error, and event-trigger expressions. These forms support explicit trigger implementation and input-to-state stability of the tracking error.

  • Exponential stability assumption: Assumption 7 models the stability functions with quadratic α1 and α2, linear α, and quadratic γ forms.The assumption corresponds to exponential stability when the unknown function is fully known.
  • Closed-form analysis: The specific stability functions yield an analytical expression for the solution S(t|s0, ϕ).This expression is the basis for closed-form performance and trigger formulas.
  • Closed-form analysis: Closed-form expressions are obtained for the Lyapunov-function upper bound and tracking-error bound in the main theorems.The resulting bounds support input-to-state stability of the tracking error.
  • Trigger implementation: The control and online-learning trigger functions also obtain analytical expressions under exponential stability.The event-triggered approach for exponentially stable nominal control laws is summarized in Algorithm 1.

V. SIMULATIONS ON FRANKA EMIKA PANDA

The simulation evaluates the proposed asynchronous event-triggered controller on a 7-DoF Franka Emika Panda manipulator tracking a joint reference trajectory. The setup models unknown dynamics with Gaussian processes and includes computation-time delays.

  • Simulation setup: The experiment uses a commercial Franka Emika Panda manipulator with 7 degrees of freedom.The joint-space dynamics follow an Euler–Lagrange model.
  • System model: The state comprises seven joint positions and seven joint velocities, while joint torque is the control input.The mass matrix is known, whereas Coriolis force, gravity, and joint friction are treated as unknown.
  • System model: Gaussian processes model the seven unknown dynamics components represented by −M^-1(q)fq(x).Each dimension uses a squared exponential kernel with automatic relevance determination.
  • Delay model: The computation time is modeled as ∆p(t) = ∆u(t) = c∆|D(t)|^2 with c∆ = 2.5×10^-7, giving maximum prediction and update times of 0.01.The dataset size is bounded by |D(t)| ≤ 200.
  • Tracking task: The task tracks a joint reference trajectory generated from an end-effector reference through inverse kinematics.Figure 3 compares the end-effector reference trajectory with the trajectory generated by the proposed asynchronous event-triggered controller.
  • Controller and implementation: The controller uses K = [−100I7, −50I7], a Lyapunov function based on P, and control-trigger interval ε = 5 × 10^-3.The test combines MATLAB and CoppeliaSim for joint simulation.

B. Control and Learning Performance

The four learning-based control methods are compared through control, prediction, communication, and computation performance under the proposed event-triggered framework. Simulations show comparable control and prediction performance while highlighting efficiency gains and conservatism in the trigger design.

  • Evaluation methods: The evaluation compares time-triggered and event-triggered control and online-learning strategies, including the proposed asynchronous Double Trigger.The baselines are Time Trigger, Control Trigger, GP Trigger, and Double Trigger.
  • Control performance: Both the Lyapunov function V(·) and tracking error e(·) are similar across all four methods and remain bounded by their respective bounds.These control measures are reported in Fig. 4.
  • Prediction performance: The Double Trigger prediction closely follows the true unknown function, with prediction error bounded as established in Lemma 1.The prediction result is shown in Fig. 5.
  • Limitations and future directions: The trigger design is conservative because worst-case prediction, delay-induced error, and Lyapunov-function evolution enlarge the triggering thresholds.The paper identifies conservatism as a limitation and proposes more accurate prediction bounds and local-Lipschitz estimation as future directions.
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