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arxiv: 2605.04419 · v1 · submitted 2026-05-06 · 🧮 math.OC

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A Numerical Investigation of Extremum-Seeking-Based Command Generation for Adaptively Controlled Systems

Jhon Manuel Portella Delgado , Aidan Rice , Jacob C. Vander Schaaf , Dennis S. Bernstein
Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:59 UTC · model grok-4.3

classification 🧮 math.OC
keywords extremum seekingadaptive controlcommand generationpredictive controlstabilizationdisturbance rejectionnumerical simulation
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The pith

Extremum-seeking command generation combined with predictive cost adaptive control optimizes unknown measurable costs during stabilization and command following.

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

The paper develops and tests through simulation an integrated method that pairs an extremum-seeking command generator with indirect adaptive control to optimize a cost that can be measured but whose form is unknown. The adaptive component identifies the plant online and applies constrained predictive control to enforce stability and tracking. This setup is examined for its ability to stabilize the loop, follow commands, and reject disturbances without explicit models of either the cost or the dynamics. Readers would care because many practical control problems involve observable performance metrics whose underlying equations are unavailable or change.

Core claim

The paper shows through numerical examples that the ECG/PCAC combination generates commands that drive an unknown cost toward its minimum while the closed-loop system remains stable, tracks references, and attenuates disturbances, with system identification performed by recursive least squares with variable-rate forgetting and constraint handling by quadratic programming.

What carries the argument

The ECG/PCAC framework, where extremum-seeking generates commands that asymptotically optimize the measured cost and predictive cost adaptive control performs online identification plus constrained optimization.

If this is right

  • Commands are produced that asymptotically minimize the unknown cost.
  • Closed-loop stability and command following are maintained through continuous model updates.
  • Output constraints are enforced by the quadratic program inside the adaptive controller.
  • Disturbance effects are attenuated as the identified model improves.

Where Pith is reading between the lines

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

  • The same structure could be tested on plants whose dominant dynamics change faster than the forgetting schedule can track.
  • If cost measurements contain noise, the extremum-seeking loop may require additional filtering whose effect on convergence rate remains unexamined.
  • The approach suggests a route to hybrid controllers that switch between different cost functions without retuning the identification layer.

Load-bearing premise

The cost function can be measured in real time even though its mathematical expression is unknown, and the plant dynamics admit sufficiently accurate online identification by recursive least squares.

What would settle it

A simulation run in which the measured cost fails to decrease toward a minimum or the closed-loop outputs violate stability or tracking specifications despite the identification algorithm converging and the extremum seeker operating.

Figures

Figures reproduced from arXiv: 2605.04419 by Aidan Rice, Dennis S. Bernstein, Jacob C. Vander Schaaf, Jhon Manuel Portella Delgado.

Figure 2
Figure 2. Figure 2: shows that ECG is based on the extremum￾seeking controller described in [48], [49]. The dynamics of ECG are given by yh,k = (1 − ωhTs)yh,k−1 + Jk − Jk−1, (1) yd,k = 2bes aes ωlTs sin(kωesTs)yh,k, (2) yl,k = (1 − ωlTs)yl,k−1 + ωlTsyd,k, (3) yes,k = yes,k−1 + Kes,k yl,k, (4) rk = yes,k + aes sin(kωesTs), (5) where yh,k is the output of the highpass filter at step k, yl,k is the output of the lowpass filter a… view at source ↗
Figure 1
Figure 1. Figure 1: ECG/PCAC sampled-data control architecture for adap￾tive stabilization, command following, and disturbance rejection. III. Extremum-Seeking-Based Command Generator view at source ↗
Figure 3
Figure 3. Figure 3: shows the closed-loop response of the un￾damped oscillator (43), (44) under PCAC, together with the command rk generated by the ECG scheme (5) and the command-following error |ek|. The yellow region shows that the modulation (46) asymptotically reduces the oscillations view at source ↗
Figure 4
Figure 4. Figure 4: shows uk obtained from the constrained optimization (38)–(42), the estimated model coefficients θk given by (18), and the forgetting factor λk given by (21) view at source ↗
Figure 5
Figure 5. Figure 5: Undamped oscillator: This plot shows the values of J(rk) computed using the command rk obtained from ECG. The step￾heat map shows that rk converges to the optimal command r ⋆ = 2 denoted by the vertical dotted line. The cost function J(rk) is shown by the dashed black line. B. Double integrator Consider the double integrator x˙ 1 = x2, (47) x˙ 2 = u + w, (48) where x1 is the position of the particle, x2 is… view at source ↗
Figure 7
Figure 7. Figure 7: shows uk obtained from solving the con￾strained optimization problem (38)–(42), the estimated model coefficients θk given by (18), and the forgetting factor λk given by (21) view at source ↗
Figure 8
Figure 8. Figure 8: Double integrator with constant disturbance. This plot shows the values of J(rk) computed using the command rk obtained from (5). The step-heat map shows that rk converges to r ⋆ denoted by the vertical dotted line. The cost function J(rk) is shown by the dashed black line. In this example, we use the normalized gradient gain Kes,k = Kes,0 ϵ|yl,k| , (53) where Kes,k is the gradient gain at step k; Kes,0 is… view at source ↗
Figure 9
Figure 9. Figure 9: shows the closed-loop response of the ex￾ponentially unstable system system (43), (44) under PCAC, together with the command rk generated by ECG (5) and the command-following error |ek|. The yellow region shows that (54), (55) asymptotically reduces the oscillations view at source ↗
Figure 10
Figure 10. Figure 10: shows uk obtained from solving the con￾strained optimization problem (38)–(42), the estimated model coefficients θk given by (18), and the forgetting factor λk given by (21) view at source ↗
Figure 11
Figure 11. Figure 11: Exponentially unstable system. This plot shows the values of J(rk) computed using the command rk obtained from (5). The step-heat map shows that rk converges to r ⋆ denoted by the vertical dotted line. The cost function J(rk) is shown by the dashed black line. VI. Conclusions This work integrated extremum-seeking-based com￾mand generation (ECG) with predictive cost adaptive control (PCAC) within a closed-… view at source ↗
read the original abstract

We develop an adaptive feedback control technique that combines an extremum-seeking-based command generator (ECG) with indirect adaptive control. In particular, ECG is used to generate commands that asymptotically optimize a cost function that is measured but whose functional form is unknown. For feedback control with command following and stabilization, the present paper combines ECG with predictive cost adaptive control (PCAC), which is an indirect adaptive control extension of model predictive control (MPC). PCAC extends generalized predictive control (GPC) by using quadratic programming to enforce output constraints and recursive least squares (RLS) with variable-rate forgetting (VRF) for system identification. The resulting ECG/PCAC framework combines command generation with closed-loop system identification and online optimization. The contribution of this paper is a numerical investigation of ECG/PCAC for adaptive stabilization, command following, and disturbance rejection

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 / 2 minor

Summary. The manuscript develops a combined ECG/PCAC framework in which an extremum-seeking command generator optimizes a measurable but unknown cost function while predictive cost adaptive control (an indirect adaptive extension of MPC) performs online system identification via RLS with variable-rate forgetting and enforces output constraints via quadratic programming. The central contribution is a numerical investigation demonstrating the framework's performance for adaptive stabilization, command following, and disturbance rejection.

Significance. If the numerical evidence is robust, the work offers a practical method for real-time optimization of unknown costs in adaptively controlled systems, integrating command generation with closed-loop identification. The approach could be relevant for applications where cost functions are observable but analytically unavailable, provided the online identification remains reliable under ECG-induced command variations.

major comments (2)
  1. [Section 4] Numerical investigation (Section 4): The reported simulations do not include quantitative details on the number of Monte Carlo trials, measurement-noise variances, or specific performance metrics (e.g., settling times, steady-state cost values, or constraint-violation rates). Without these, it is not possible to evaluate whether the ECG/PCAC combination reliably achieves stabilization and disturbance rejection across repeated realizations.
  2. [Section 4] Section 4, simulation setups: The examples do not stress-test the RLS-with-VRF identifier under the time-varying commands produced by ECG (e.g., low-persistent-excitation regimes or abrupt disturbance changes). Consequently, the adequacy of the identified model for the subsequent quadratic-programming step of PCAC is not demonstrated, leaving open the possibility that identification errors could produce suboptimal or unstable closed-loop behavior not captured in the presented runs.
minor comments (2)
  1. [Abstract] Abstract: The abstract states that numerical results support the claims but provides no quantitative highlights; adding one or two representative metrics would improve readability.
  2. [Section 2] Notation: The distinction between the extremum-seeking cost and the PCAC quadratic cost should be clarified with explicit symbols in the first use of each.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and outline the revisions we will make to strengthen the numerical investigation.

read point-by-point responses

  1. Referee: [Section 4] Numerical investigation (Section 4): The reported simulations do not include quantitative details on the number of Monte Carlo trials, measurement-noise variances, or specific performance metrics (e.g., settling times, steady-state cost values, or constraint-violation rates). Without these, it is not possible to evaluate whether the ECG/PCAC combination reliably achieves stabilization and disturbance rejection across repeated realizations.

    Authors: We agree that additional quantitative details are needed to better evaluate reliability. In the revised manuscript, we will specify the number of Monte Carlo trials (e.g., 50 independent runs per example), the measurement-noise variances used, and report averaged performance metrics such as settling times, steady-state cost values, and constraint-violation rates with standard deviations where appropriate. revision: yes

  2. Referee: [Section 4] Section 4, simulation setups: The examples do not stress-test the RLS-with-VRF identifier under the time-varying commands produced by ECG (e.g., low-persistent-excitation regimes or abrupt disturbance changes). Consequently, the adequacy of the identified model for the subsequent quadratic-programming step of PCAC is not demonstrated, leaving open the possibility that identification errors could produce suboptimal or unstable closed-loop behavior not captured in the presented runs.

    Authors: We acknowledge that the current examples focus on nominal conditions. In the revision, we will add dedicated simulation cases that introduce low-persistent-excitation regimes via ECG-generated commands and abrupt disturbance changes. These will include time histories of identification errors, model prediction accuracy, and closed-loop stability indicators to demonstrate the robustness of RLS-with-VRF and the QP step under such conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical investigation is self-contained

full rationale

The paper explicitly frames its contribution as a numerical investigation of the ECG/PCAC combination for adaptive stabilization, command following, and disturbance rejection. No derivation chain, uniqueness theorem, or fitted-parameter prediction is claimed; the reported simulation outcomes are generated independently and do not reduce by construction to any self-citation, ansatz, or input data. Prior concepts such as RLS with variable-rate forgetting and extremum seeking are referenced as background, but the central results stand on the presented numerical evidence rather than tautological reuse of those references. This satisfies the default expectation for an investigation-style paper with no load-bearing self-referential reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard assumptions from adaptive control and extremum-seeking; no new entities are introduced.

axioms (2)
  • domain assumption The plant is linear or can be locally approximated as linear for identification purposes.
    Implicit in the use of RLS and PCAC extensions of GPC.
  • domain assumption The measured cost function has a unique extremum that can be tracked asymptotically by the ECG.
    Required for the command generator to converge to an optimal command.

pith-pipeline@v0.9.0 · 5452 in / 1306 out tokens · 43246 ms · 2026-05-08T15:59:00.515909+00:00 · methodology

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Reference graph

Works this paper leans on

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