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The Source of Disturbance Matters: External, Internal, and Control-Generated Noise in Adaptive Regulation

Veronique Ziegler

arXiv:2609.26428v1cs.AIeess.SYnlin.AO

TL;DR

Adaptive regulation can be perturbed by external, persistent internal, or control-generated disturbance, so the study compares these sources under different regulatory timing. In replicated simulations, persistent internal disturbance has the largest exposure and regulatory burden within the tested grid, while increasing induced cost is associated with amplification followed by suppression.

  • Problem

    Environmental-noise tests alone do not assess sensitivity to persistent internal fluctuations or disturbance generated by corrective action.

    Source: §1 · p. 2

  • Method

    The study uses an IRAM-Ω-Q simulation testbed to compare disturbance sources and regulation-first versus disturbance-first timing while distinguishing exposure from regulatory burden.

    Source: §1 · p. 2

  • Results

    Persistent internal disturbance produces the largest exposure and burden within the tested parameter grid, while induced-cost increases show amplification followed by suppression.

    Source: §9 · p. 13

  • Takeaways & Limitations

    The results motivate testing distinct disturbance sources and assessing regulatory activity alongside exposure; increased between-run variability is a candidate diagnostic whose predictive value needs testing.

    Source: §7 · p. 13

    The findings apply to the tested parameter grid and outcomes; they do not directly measure task performance or establish a stability criterion, and the induced-cost function is only one possible form.

    Source: §8 · p. 13

Abstract

from arXiv · show

Adaptive regulation can itself perturb the state it is intended to stabilize. In replicated simulations of an adaptive agent, we compare external disturbance, persistent internally generated disturbance, and control-generated disturbance under regulation-first and disturbance-first ordering. Persistent internal disturbance produces the largest exposure and regulatory burden within the tested parameter grid. When positive controller updates generate an immediate disturbance cost, increasing that cost produces a nonmonotonic response: effective disturbance initially rises, variability across stochastic runs increases over an intermediate range, and corrective activity becomes strongly suppressed at higher costs. The results show how disturbance source and timing shape exposure and controller burden in this model. They motivate testing adaptive agents with distinct disturbance sources and assessing regulatory activity alongside exposure.

1 Introduction: the effect of the disturbance source structure

Adaptive regulation can be disturbed by the environment, persistent internal fluctuations, or regulatory action itself, and these sources need not affect a regulated system equivalently. This study compares their effects and examines how disturbance source and timing relate to exposure and controller effort.

1 Introduction: the effect of the disturbance source structure

  • The study asks whether an adaptive agent responds differently to external, internal, and control-generated disturbance, with consequences for exposure, controller effort, and ordering sensitivity.An aggressive corrective response can itself add disturbance, as illustrated by the mobile-robot example.Source: §1 · pp. 1–2
  • The IRAM-Ω-Q simulation varies disturbance source and timing while holding the remaining dynamics fixed, using a scalar control level µ(t) to regulate structured uncertainty.The authors present it as a transparent testbed, not a claim about a particular deployed system.Source: §1 · p. 2
  • Prior work models control-dependent motor noise and broader control-dependent, state-dependent, and internal noise; this study compares a same-cycle tightening cost with external and persistent internal disturbance.The comparison uses regulation-first and disturbance-first timing and examines exposure and regulatory burden.Source: §1 · p. 2
  • The study distinguishes raw from effective disturbance and evaluates both exposure and regulatory burden, since similar exposure can require different controller activity.Burden includes stronger or more variable controller activity.Source: §1 · p. 2

2 Model and disturbance timing

The model represents an adaptive agent with a cognitive state, regulation level, and disturbance, and separates disturbance sources from the timing of their application. Regulation can reduce non-induced disturbance before exposure, whereas the induced component is not suppressed by the update that creates it.

2 Model and disturbance timing

  • The published IRAM-Ω-Q framework supplies the background state and regulation definitions; this study focuses on disturbance composition and placement relative to the regulatory update.The cited framework defines the entropy-driven gain update and clipping, and the coherence gap [4].Source: §2 · p. 2
  • The agent has a cognitive state, regulation level µ(t), and disturbance η(t), which is decomposed into non-induced and induced components.Non-induced disturbance includes external and autonomous internal sources; induced disturbance comes from positive controller tightening.Source: §2 · pp. 2–3
  • Regulation can reduce non-induced disturbance if it acts before exposure, but the induced component is not suppressed by the same update that generated it.Regulation-first applies the non-induced disturbance after the controller update; disturbance-first applies it before the reactive update.Source: §2 · p. 3

3 Disturbance-source models

The disturbance-source models differ in whether disturbance is external, persistent within the agent, generated by controller tightening, or mixed. Their specifications also distinguish memory carryover, induced cost, and the timing of regulation.

3.1 External disturbance

  • The external condition uses memoryless environmental disturbance at baseline scale η0, serving as a reference without internal carryover.It tests timing when disturbance is imposed from outside the agent.Source: §3.1 · p. 3

3.2 Persistent internal disturbance

  • The internal condition applies the absolute value of a carried fluctuation, with ρ controlling persistence: values near zero are nearly memoryless and values near one persist.In the fixed-innovation scan, σin stays fixed while increasing ρ also increases long-run variance, making this an accumulation stress test.Source: §3.2 · p. 3
  • The matched-scale internal scan controls long-run fluctuation scale to test whether carryover and regulation-first versus disturbance-first ordering still matter without the automatic variance increase.Source: §3.2 · p. 3

3.3 Control-generated disturbance

  • Control-generated disturbance charges positive controller tightening; downward changes are not penalized, and both timing orders incur the cost after their respective update.The induced term is therefore structurally present in both regulation-first and disturbance-first ordering.Source: §3.3 · p. 4

3.4 Mixed disturbance

  • The exploratory mixed condition combines external, internal, and control-generated disturbance to examine whether induced cost changes the RF–DF response in a system with persistent internal fluctuation.The authors state that interpretation of this condition is deliberately conservative.Source: §3.4 · p. 4

4 Simulation design and summary measures

The simulation scans five disturbance-source families under regulation-first and disturbance-first ordering, using replicated conditions and paired contrasts. Its measures are separated into exposure, burden, timing, and variability diagnostics.

4 Simulation design and summary measures

  • The full scan covers five source families under both orderings; induced-cost conditions use 50 stochastic replicates per condition, compared with 10 for other families.RF and DF use the same seed index for paired contrasts; full-grid figures report replicate-aggregated condition means, and paired contrasts report DF − RF with two-sided 95% Student-t confidence intervals.Source: §4 · p. 4
  • The measures are organized into exposure, burden, timing, and variability diagnostics rather than collapsed into a single noise measure.Source: §4 · p. 4

5 Results

Persistent internal disturbance creates the largest exposure and regulatory burden in the tested grid, while increasing induced cost produces an intermediate variability rise followed by suppressed corrective activity.

5.1 Disturbance families are not interchangeable

  • Across sampled conditions, persistent internal disturbance spans a wider and higher effective-disturbance range than the memoryless external baseline.Figure 1 maps parameter conditions with RF and DF shown separately, not a histogram of one condition; induced disturbance is analyzed separately because changing α also changes controller behavior.Source: §5.1 · p. 4, Figure 1 · p. 5

5.2 Persistent internal disturbance creates the largest burden

  • Figure 2’s fixed-innovation internal scan produces the largest exposure and regulatory burden within the tested parameter grid; DF generally lies above RF.As carryover and innovation scale increase, internal fluctuations persist and accumulate.Source: Figure 2 · p. 6, §5.2 · p. 6

5.3 The RF advantage remains when long-run internal scale is controlled

  • Figure 3 reduces innovation scale as carryover rises, keeping the long-run fluctuation scale approximately comparable across ρ.This matched-scale scan separates persistence from variance growth.Source: Figure 3 · p. 7

5.4 Control-generated disturbance reveals a transition band

  • At 21M–27M, paired DF−RF responses show increased variability; by 30M, mean regulatory activity is strongly suppressed in both orderings.The summaries do not establish distinct high- and low-regulation groups.Source: Figure 5 · p. 8, §5.4 · p. 8

5.5 Mixed-source disturbance

  • Figure 6 tracks regulatory diagnostics, with positive post-update controller change entering the induced-cost term.Source: Figure 6 · p. 9

6 Focused internal-disturbance contrast views

Focused contrasts show that disturbance-first ordering generally produces greater exposure and regulatory burden than regulation-first ordering under fixed-innovation internal disturbance. The regulation-first advantage remains when long-run fluctuation scale is matched.

6 Focused internal-disturbance contrast views

  • Focused contrast figures make the RF–DF ordering effect easier to read than the full-grid heatmaps.The heatmaps show broad response surfaces; the contrast figures plot selected paired differences directly.Source: §6 · p. 9
  • DF generally produces greater exposure and regulatory burden than RF, especially with high internal carryover and innovation scale.The fixed-innovation view treats internal disturbance as an accumulation stress test.Source: §6 · p. 9
  • The RF advantage persists after matching long-run fluctuation scale, so variance growth alone does not explain the ordering difference.The result is consistent with regulation occurring at different points relative to disturbance exposure.Source: §6 · p. 9

7 Discussion

Disturbance source structure changes both state exposure and controller behavior in this model. The discussion distinguishes source-specific patterns and emphasizes evaluating regulatory burden alongside exposure.

7 Discussion

  • Disturbance source structure changes both what reaches the state and how the controller behaves, making one generic noise parameter insufficient in this model.The discussion distinguishes external, persistent internal, and control-generated disturbance.Source: §7 · p. 9
  • Persistent internal disturbance can accumulate inside the agent, while external disturbance primarily creates a timing contrast between RF and DF.For control-generated disturbance, increasing cost scale α changes whether the controller tightens or suppresses corrective action.Source: §7 · pp. 9, 13
  • Figure 7 summarizes paired RF–DF contrasts across mixed-source conditions, averaging condition contrasts at each induced-cost value.Its faint points represent individual parameter conditions; foreground squares summarize condition means, with bars showing 1.96 standard errors across condition means.Source: Figure 7 · p. 10
  • Exposure alone is insufficient for evaluating adaptive stability because similar exposure may require strong or highly variable control.The analysis treats effective disturbance, mean regulation, controller adjustment, and coherence-gap variance as complementary diagnostics.Source: §7 · p. 13
  • The findings motivate testing distinct disturbance sources and induced-cost effects in other architectures, while treating transition-range variability as a candidate diagnostic, not a validated warning signal.Testing predictive value would require trajectories and an explicit prediction criterion.Source: §7 · p. 13

8 Limitations and scope

The findings are bounded by the tested parameter grid, the chosen induced-cost function, and the exploratory mixed-source analysis. Coherence-gap variance is a variability diagnostic, not a measure of mean exposure.

8 Limitations and scope

  • The results cover the tested parameter grid and do not establish a theorem across architectures or directly measure task performance or a stability criterion.The control-generated scan uses 50 replicates per condition, compared with 10 for other source families; further sensitivity studies remain needed.Source: §8 · p. 13
  • The induced-cost model uses a one-sided quadratic cost for positive tightening; other cost functions could shift or change the strength of suppression.Bounded, linear, or saturating functions are possible alternatives.Source: §8 · p. 13
  • The mixed-source analysis shows attenuation at high induced cost but does not establish a nonlinear interaction law.That claim would require a formal additive baseline from matched external-only, internal-only, and induced-only components.Source: §8 · p. 13
  • Coherence-gap variance measures dynamical variability, not mean exposure; it can be high in transition regions without maximal mean exposure.It reflects strong temporal fluctuation in the coherence diagnostic.Source: §8 · p. 13

9 Conclusion

The conclusion treats disturbance source as a structural feature of adaptive regulation and identifies persistent internal disturbance as the largest exposure and burden within the tested grid. It also reports a nonmonotonic response to increasing induced cost and calls for evaluation that separates sources and tracks regulatory burden.

9 Conclusion

  • Persistent internal disturbance produces the largest exposure and regulatory burden within the tested parameter grid because its fluctuations can carry forward and accumulate.RF–DF differences remain when long-run internal scale is controlled, so variance growth alone does not explain the ordering effect.Source: §9 · pp. 13–14
  • Increasing the cost scale for positive tightening first raises disturbance, increases between-run variability over an intermediate range, and strongly suppresses corrective activity at higher costs.The immediate same-cycle cost changes both exposure and the regulatory response.Source: §9 · p. 14
  • Adaptive agents should be evaluated with disturbance models that distinguish source, timing, and scale, alongside measures of regulatory burden and exposure.Implications for stability and task performance in other architectures remain open questions.Source: §9 · p. 14
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