pith. sign in

arxiv: 2605.20038 · v1 · pith:QQSJNL24new · submitted 2026-05-19 · 📡 eess.SY · cs.SY

A New Simple-to-Configure Self-Perturbing Multivariable Extremum-Seeking Controller

Timothy I. Salsbury , Min Gyung Yu This is my paper

Pith reviewed 2026-05-20 03:40 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords extremum-seeking controlstochastic relayMISO systemsgradient estimationparameter tuningstability proofdynamic systemsoptimization controller
0
0 comments X

The pith

A new stochastic relay-based controller optimizes multi-input systems with one tunable parameter per input channel.

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

The paper develops a simpler extremum-seeking controller for multi-input single-output systems. Stochastic relay gains identify gradients without complex tuning. For static maps only one parameter per input is needed and a stability proof is given. The dynamic version requires one extra parameter. Simulations show the approach works for both static and dynamic cases, potentially making such controllers easier to deploy in practice.

Core claim

The authors present a stochastic relay-based extremum-seeking controller for MISO systems that solves gradient identification via stochastic relay gains. It requires only one configurable parameter per input channel for the static case and one additional parameter for the dynamic version. A simple stability proof for the static case is presented, and simulation tests demonstrate performance for optimizing both static and dynamic systems.

What carries the argument

Stochastic relay gains that generate a gradient estimate whose average behavior drives convergence to the optimum.

Load-bearing premise

The average behavior of the stochastic relay gains supplies a usable gradient estimate sufficient for convergence under the stated conditions.

What would settle it

A static map satisfying all stability conditions but failing to converge when the controller is applied would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.20038 by Min Gyung Yu, Timothy I. Salsbury.

Figure 1
Figure 1. Figure 1: Multi-relay ESC for static map In developing the approach, we will assume p input chan￾nels and that the static map (cost function) is given by: y = Q(θ), θ = [θ1, θ2, . . . , θp] T , (1) with a local minimum at θ ∗ . The cost function is not known but can be measured via y at any point in time and θ can be manipulated. As a basis for extending the SISO relay approach to multiple inputs, we make use of the… view at source ↗
Figure 2
Figure 2. Figure 2: Multi-relay ESC for general MISO dynamic system [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Effect of minimum relay hold time The system dynamics also introduce auto-correlation into the signals meaning that the implicit assumption in Equa￾tion 3 for estimating the new relay directions of having independent samples populating the matrix rows is no longer valid. This problem can be addressed by extending the estimation window so that it includes enough samples over a long enough time to counteract… view at source ↗
Figure 6
Figure 6. Figure 6: Trajectory of objective function y for the static case with K = 0.01 towards optimum value [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Trajectory of objective function y for the dynamic case with K = 0.01 towards optimum value [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Trajectory of objective function y for the dynamic case with the adaptive gain approach towards opti￾mum value Test results for both static and dynamic cases are shown in [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
read the original abstract

This paper presents a new stochastic relay-based extremum-seeking controller (ESC) for multi-input-single-output (MISO) systems. The goal of this work was to create an algorithm that is much simpler to configure than alternative approaches making deployment to real-world problems easier. A solution is developed first for a static map and then adapted for a general class of dynamic systems. The number of configurable parameters is one per input channel for the static case and only one additional parameter is needed for the dynamic version. The problem of gradient identification is solved via the use of stochastic relay gains and a simple stability proof for the static case is presented. Simulation tests demonstrate the performance of the strategy for optimizing both static and dynamic systems.

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 manuscript presents a stochastic relay-based extremum-seeking controller for multi-input single-output (MISO) systems. For static maps it requires only one configurable parameter per input channel; for a general class of dynamic systems one additional parameter is introduced. Gradient identification is performed via stochastic relay gains, a stability proof is given for the static case, and simulation results are reported for both static and dynamic optimization problems.

Significance. If the stability argument is rigorous and the simulations are representative, the approach could meaningfully reduce the configuration burden of multivariable extremum-seeking control, which is a practical barrier to deployment. The low parameter count and the explicit stability claim for the static map are potentially valuable contributions to the adaptive-control literature, provided the averaging step is shown to close without hidden assumptions on the relay statistics or time-scale separation.

major comments (2)
  1. [Stability proof for the static case] Stability proof for the static case (abstract and the section presenting the proof): the central claim rests on the stochastic relay gains producing an averaged vector field whose only equilibrium is the extremum. The manuscript must explicitly compute the expectation (or correlation) of the relay output with respect to the measured cost and demonstrate that it is proportional to the true gradient for a general static map; without this derivation the averaging loop does not close and the stability result is not yet established.
  2. [Method description of the stochastic relay] Method description of the stochastic relay (the paragraph introducing the relay gain): the paper states that one parameter per input suffices, yet the distribution, correlation time, or memory properties of the stochastic process are not specified. If the switching statistics do not separate from the map evaluation, bias terms can appear in the averaged dynamics; this assumption is load-bearing for the gradient-estimate claim.
minor comments (3)
  1. [Abstract] The abstract should briefly indicate the class of dynamic systems for which the extension is claimed (e.g., strict-feedback, minimum-phase, etc.).
  2. [Method section] Notation for the stochastic relay gain and its expectation should be introduced consistently in the method section to avoid ambiguity when the averaging argument is later invoked.
  3. [Simulation results] Simulation figures would benefit from multiple Monte-Carlo runs or shaded variability bands so that the reader can assess repeatability under the single-parameter tuning.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the stability argument and the specification of the stochastic process.

read point-by-point responses

  1. Referee: Stability proof for the static case (abstract and the section presenting the proof): the central claim rests on the stochastic relay gains producing an averaged vector field whose only equilibrium is the extremum. The manuscript must explicitly compute the expectation (or correlation) of the relay output with respect to the measured cost and demonstrate that it is proportional to the true gradient for a general static map; without this derivation the averaging loop does not close and the stability result is not yet established.

    Authors: We agree that an explicit derivation of the expectation is necessary to rigorously close the averaging argument. The original manuscript stated the stability result for the averaged system but did not include the intermediate step computing the correlation of the relay output with the cost function. In the revision we will insert a dedicated derivation (new subsection or appendix) that shows, for a general static map and under the stated assumptions on the relay process, that the expected relay output is proportional to the gradient vector. This will make the passage from the stochastic system to the averaged vector field fully explicit. revision: yes

  2. Referee: Method description of the stochastic relay (the paragraph introducing the relay gain): the paper states that one parameter per input suffices, yet the distribution, correlation time, or memory properties of the stochastic process are not specified. If the switching statistics do not separate from the map evaluation, bias terms can appear in the averaged dynamics; this assumption is load-bearing for the gradient-estimate claim.

    Authors: The referee correctly notes that the statistical properties of the stochastic relay must be stated to justify time-scale separation and the absence of bias. We will revise the paragraph introducing the relay gain to specify the probability distribution, correlation time, and memory properties of the process. With these details added, we will also briefly verify that the chosen statistics permit the required separation from the map dynamics, thereby confirming that the averaged gradient estimate remains unbiased. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with independent stability argument.

full rationale

The paper introduces a stochastic relay-based ESC design for MISO systems, presenting the relay gains as a mechanism to solve gradient identification and then providing a separate stability proof for the static map case. Configurable parameters are explicitly design choices (one per input for static, one extra for dynamic). No equations reduce a prediction to a fitted input by construction, no self-citation chain is load-bearing for the central claim, and the stability argument is asserted as independently derived rather than imported or renamed from prior results. The derivation chain remains non-circular on inspection of the abstract and method description.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the approach relies on standard assumptions from extremum-seeking control literature plus the new stochastic relay mechanism; no explicit free parameters beyond the advertised tunable gains are mentioned.

free parameters (1)
  • stochastic relay gain per input
    One configurable value per input channel stated as the main tuning knob for the static case.
axioms (1)
  • domain assumption The average behavior of the stochastic relay produces a usable gradient estimate for convergence.
    Invoked to support the stability claim for the static map.

pith-pipeline@v0.9.0 · 5653 in / 1238 out tokens · 30612 ms · 2026-05-20T03:40:40.043770+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    2013 European Control Conference (ECC) , pages=

    Building temperature distributed control via explicit MPC and Trim and Respond methods , author=. 2013 European Control Conference (ECC) , pages=. 2013 , organization=

    work page 2013

  2. [2]

    Ashrae Journal , volume=

    Resetting setpoints using trim & respond logic , author=. Ashrae Journal , volume=

    work page

  3. [3]

    ASHRAE journal , volume=

    Increasing efficiency with VAV system static pressure setpoint reset , author=. ASHRAE journal , volume=

    work page

  4. [4]

    Energy and Buildings , volume=

    Fault-tolerant supervisory control of VAV air-conditioning systems , author=. Energy and Buildings , volume=

    work page

  5. [5]

    Renewable and Sustainable Energy Reviews , volume=

    A review of recent developments and technological advancements of variable-air-volume (VAV) air-conditioning systems , author=. Renewable and Sustainable Energy Reviews , volume=

    work page

  6. [6]

    Energy Conversion and Management , volume=

    Minimization of energy consumption in HVAC systems with data-driven models and an interior-point method , author=. Energy Conversion and Management , volume=

    work page

  7. [7]

    reality , author=

    Advanced HVAC control: Theory vs. reality , author=. IFAC Proceedings Volumes , volume=

    work page

  8. [8]

    2006 , publisher=

    Sequences of Operation for Common HVAC Systems , author=. 2006 , publisher=

    work page 2006

  9. [9]

    Modelica Buildings Library , url=

    work page

  10. [10]

    Control Engineering Practice , volume=

    Self-perturbing extremum-seeking controller with adaptive gain , author=. Control Engineering Practice , volume=

    work page

  11. [11]

    2007 IEEE International Symposium on Industrial Electronics , pages=

    Analysis and comparison of extremum seeking control techniques , author=. 2007 IEEE International Symposium on Industrial Electronics , pages=. 2007 , organization=

    work page 2007

  12. [12]

    An exposition of adaptive control , year=

    Extremum-seeking regulators , author=. An exposition of adaptive control , year=

    work page

  13. [13]

    IFAC-PapersOnLine , year=

    Fixed-Time Newton-Like Extremum Seeking , author=. IFAC-PapersOnLine , year=

    work page

  14. [14]

    2018 IEEE Conference on Decision and Control (CDC) , year=

    Cooperative Extremum Seeking Control via Sliding Mode for Distributed Optimization , author=. 2018 IEEE Conference on Decision and Control (CDC) , year=

    work page 2018

  15. [15]

    ArXiv , year=

    On the Design and Analysis of Multivariable Extremum Seeking Control using Fast Fourier Transform , author=. ArXiv , year=

    work page

  16. [16]

    2015 American Control Conference (ACC) , year=

    Multiparameter extremum seeking with output delays , author=. 2015 American Control Conference (ACC) , year=

    work page 2015

  17. [17]

    2022 16th International Workshop on Variable Structure Systems (VSS) , year=

    Multivariable Sliding Mode-based Extremum Seeking Control via Periodic Switching Function and Cyclic Search* , author=. 2022 16th International Workshop on Variable Structure Systems (VSS) , year=

    work page 2022

  18. [18]

    Proceedings of the 29th Chinese control conference , pages=

    Extremum seeking from 1922 to 2010 , author=. Proceedings of the 29th Chinese control conference , pages=. 2010 , organization=

    work page 1922

  19. [19]

    2003 , publisher=

    Real-time optimization by extremum-seeking control , author=. 2003 , publisher=

    work page 2003

  20. [20]

    2022 , address=

    Monthly Energy Review August 2022 , author=. 2022 , address=

    work page 2022

  21. [21]

    IEEE transactions on aerospace and electronic systems , volume=

    MPPT of photovoltaic systems using extremum-seeking control , author=. IEEE transactions on aerospace and electronic systems , volume=

    work page

  22. [22]

    Proceedings of the 2000 American Control Conference , volume=

    Analysis of multivariable extremum seeking algorithms , author=. Proceedings of the 2000 American Control Conference , volume=. 2000 , organization=

    work page 2000

  23. [23]

    Proceedings of the 2002 American Control Conference , volume=

    Analysis and design of multivariable extremum seeking , author=. Proceedings of the 2002 American Control Conference , volume=. 2002 , organization=

    work page 2002

  24. [24]

    Automatica , volume=

    Multivariable Newton-based extremum seeking , author=. Automatica , volume=

    work page

  25. [25]

    Wind Engineering , volume=

    Maximizing wind turbine energy capture using multivariable extremum seeking control , author=. Wind Engineering , volume=

    work page

  26. [26]

    IEEE Transactions on Control Systems Technology , volume=

    Power optimization for photovoltaic microconverters using multivariable newton-based extremum seeking , author=. IEEE Transactions on Control Systems Technology , volume=

    work page

  27. [27]

    2020 IEEE RSJ International Conference on Intelligent Robots and Systems (IROS) , pages=

    In-flight range optimization of multicopters using multivariable extremum seeking with adaptive step size , author=. 2020 IEEE RSJ International Conference on Intelligent Robots and Systems (IROS) , pages=. 2020 , organization=

    work page 2020

  28. [28]

    Applied Thermal Engineering , volume=

    Self-optimizing control of air-source heat pump with multivariable extremum seeking , author=. Applied Thermal Engineering , volume=

    work page

  29. [29]

    Journal of Dynamic Systems, Measurement, and Control , volume=

    Multivariable extremum seeking for joint-space trajectory optimization of a high-degrees-of-freedom robot , author=. Journal of Dynamic Systems, Measurement, and Control , volume=

    work page

  30. [30]

    Automatica , volume=

    Stability of extremum seeking feedback for general nonlinear dynamic systems , author=. Automatica , volume=

    work page

  31. [31]

    International Journal of Adaptive Control and Signal Processing , volume=

    Multivariable extremum seeking control via cyclic search and monitoring function , author=. International Journal of Adaptive Control and Signal Processing , volume=

    work page

  32. [32]

    Automatica , volume=

    Using sliding modes in static optimization and nonlinear programming , author=. Automatica , volume=

    work page

  33. [33]

    Automatica , volume=

    Output-feedback global tracking for unknown control direction plants with application to extremum-seeking control , author=. Automatica , volume=

    work page

  34. [34]

    Journal of the Franklin Institute , volume=

    Global real-time optimization by output-feedback extremum-seeking control with sliding modes , author=. Journal of the Franklin Institute , volume=

    work page

  35. [35]

    Control Engineering Practice , volume=

    Multivariable extremum-seeking by periodic switching functions with application to raman optical amplifiers , author=. Control Engineering Practice , volume=

    work page

  36. [36]

    International Journal of Adaptive Control and Signal Processing , volume=

    Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes , author=. International Journal of Adaptive Control and Signal Processing , volume=

    work page

  37. [37]

    Automatica , volume=

    100 years of extremum seeking: A survey , author=. Automatica , volume=

    work page

  38. [38]

    Ashrae Journal , volume=

    Resetting setpoints using trim & respond logic , author=. Ashrae Journal , volume=. 2015 , publisher=

    work page 2015

  39. [39]

    ASHRAE journal , volume=

    Increasing efficiency with VAV system static pressure setpoint reset , author=. ASHRAE journal , volume=. 2007 , publisher=

    work page 2007

  40. [40]

    Energy and Buildings , volume=

    Fault-tolerant supervisory control of VAV air-conditioning systems , author=. Energy and Buildings , volume=. 2001 , publisher=

    work page 2001

  41. [41]

    Renewable and Sustainable Energy Reviews , volume=

    A review of recent developments and technological advancements of variable-air-volume (VAV) air-conditioning systems , author=. Renewable and Sustainable Energy Reviews , volume=. 2016 , publisher=

    work page 2016

  42. [42]

    Energy Conversion and Management , volume=

    Minimization of energy consumption in HVAC systems with data-driven models and an interior-point method , author=. Energy Conversion and Management , volume=. 2014 , publisher=

    work page 2014

  43. [43]

    reality , author=

    Advanced HVAC control: Theory vs. reality , author=. IFAC Proceedings Volumes , volume=. 2011 , publisher=

    work page 2011

  44. [44]

    Sequences of Operation for Common HVAC Systems

    ASHRAE. Sequences of Operation for Common HVAC Systems

    work page

  45. [45]

    Modelica Buildings Library , url =

    work page

  46. [46]

    Control Engineering Practice , volume=

    Self-perturbing extremum-seeking controller with adaptive gain , author=. Control Engineering Practice , volume=. 2020 , publisher=

    work page 2020

  47. [47]

    2022 , address =

    Monthly Energy Review August 2022 , author =. 2022 , address =

    work page 2022

  48. [48]

    IEEE transactions on aerospace and electronic systems , volume=

    MPPT of photovoltaic systems using extremum-seeking control , author=. IEEE transactions on aerospace and electronic systems , volume=. 2006 , publisher=

    work page 2006

  49. [49]

    Proceedings of the 2000 American Control Conference

    Analysis of multivariable extremum seeking algorithms , author=. Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No. 00CH36334) , volume=. 2000 , organization=

    work page 2000

  50. [50]

    Proceedings of the 2002 American Control Conference (IEEE Cat

    Analysis and design of multivariable extremum seeking , author=. Proceedings of the 2002 American Control Conference (IEEE Cat. No. CH37301) , volume=. 2002 , organization=

    work page 2002

  51. [51]

    Automatica , volume=

    Multivariable Newton-based extremum seeking , author=. Automatica , volume=. 2012 , publisher=

    work page 2012

  52. [52]

    Wind Engineering , volume=

    Maximizing wind turbine energy capture using multivariable extremum seeking control , author=. Wind Engineering , volume=. 2009 , publisher=

    work page 2009

  53. [53]

    IEEE Transactions on Control Systems Technology , volume=

    Power optimization for photovoltaic microconverters using multivariable newton-based extremum seeking , author=. IEEE Transactions on Control Systems Technology , volume=. 2014 , publisher=

    work page 2014

  54. [54]

    Applied Thermal Engineering , volume=

    Self-optimizing control of air-source heat pump with multivariable extremum seeking , author=. Applied Thermal Engineering , volume=. 2015 , publisher=

    work page 2015

  55. [55]

    Journal of Dynamic Systems, Measurement, and Control , volume=

    Multivariable extremum seeking for joint-space trajectory optimization of a high-degrees-of-freedom robot , author=. Journal of Dynamic Systems, Measurement, and Control , volume=. 2018 , publisher=

    work page 2018

  56. [56]

    Automatica-Kidlington , volume=

    Stability of extremum seeking feedback for general nonlinear dynamic systems , author=. Automatica-Kidlington , volume=. 2000 , publisher=

    work page 2000

  57. [57]

    International Journal of Adaptive Control and Signal Processing , volume=

    Multivariable extremum seeking control via cyclic search and monitoring function , author=. International Journal of Adaptive Control and Signal Processing , volume=. 2021 , publisher=

    work page 2021

  58. [58]

    Automatic and Remote Control , number=

    The use of the slip mode in problems of static optimization , author=. Automatic and Remote Control , number=

    work page

  59. [59]

    Automatica , volume=

    Using sliding modes in static optimization and nonlinear programming , author=. Automatica , volume=. 1974 , publisher=

    work page 1974

  60. [60]

    Automatica , volume=

    Output-feedback global tracking for unknown control direction plants with application to extremum-seeking control , author=. Automatica , volume=. 2011 , publisher=

    work page 2011

  61. [61]

    Journal of the Franklin Institute , volume=

    Global real-time optimization by output-feedback extremum-seeking control with sliding modes , author=. Journal of the Franklin Institute , volume=. 2012 , publisher=

    work page 2012

  62. [62]

    Control Engineering Practice , volume=

    Multivariable extremum-seeking by periodic switching functions with application to raman optical amplifiers , author=. Control Engineering Practice , volume=. 2020 , publisher=

    work page 2020

  63. [63]

    International Journal of Adaptive Control and Signal Processing , year=

    Multivariable extremum seeking control via cyclic search and monitoring function , author=. International Journal of Adaptive Control and Signal Processing , year=

    work page

  64. [64]

    International Journal of Adaptive Control and Signal Processing , volume=

    Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes , author=. International Journal of Adaptive Control and Signal Processing , volume=. 2007 , publisher=

    work page 2007

  65. [65]

    Automatica , volume=

    100 years of extremum seeking: A survey , author=. Automatica , volume=. 2024 , publisher=

    work page 2024