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arxiv: 2605.04296 · v2 · submitted 2026-05-05 · 📡 eess.SY · cs.SY· math.OC

Recognition: no theorem link

Dynamic Quantum-Assisted Co-Design of Control Tuning and Lyapunov Stability Synthesis for Nonlinear Systems

Milad Hasanzadeh , Amin Kargarian , Mehdi Farasat
Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:38 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords quantum optimizationLyapunov stabilitynonlinear controlco-designIsing Hamiltonianimaginary time evolutionBlack-Hole calibrationonline tuning
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The pith

A quantum-assisted method jointly redesigns controller gains and Lyapunov certificates online for nonlinear systems by first shrinking the search region and then solving an Ising model derived from a quadratic surrogate.

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

This paper establishes a framework that redesigns both controller gains and Lyapunov function parameters together in real time for nonlinear closed-loop systems. It does so by repeatedly contracting the parameter search space around the current condition with a Black-Hole calibration step, then encoding the remaining candidates as a binary string that approximates the performance and stability costs via a quadratic pseudo-Boolean function. The encoded costs become an Ising-type Hamiltonian that quantum imaginary time evolution can optimize, after which the best bitstrings are decoded back to continuous parameters and checked against the true nonlinear dynamics. A sympathetic reader would care because conventional designs fix gains offline and check stability separately, while this approach embeds both tasks inside one online loop that can adapt as operating conditions change. The method is shown to work on consensus problems and motor drive control.

Core claim

The central claim is that controller parameters and Lyapunov-certificate parameters can be co-optimized at each decision epoch by first shrinking the admissible continuous region with Black-Hole calibration, building a finite binary representation over that region, fitting a local quadratic pseudo-Boolean surrogate to sampled closed-loop costs and Lyapunov penalties, converting the surrogate into an Ising Hamiltonian, solving it with quantum imaginary time evolution, decoding the resulting bitstrings, and finally re-evaluating the top candidates on the original nonlinear system before applying the update.

What carries the argument

The two-step computational structure consisting of Black-Hole-based calibration to contract the search region followed by construction of a finite binary representation and local quadratic pseudo-Boolean surrogate that yields an Ising-type Hamiltonian for quantum imaginary time evolution.

If this is right

  • The framework can accommodate different Lyapunov decay specifications simply by changing the stability penalty term.
  • Decoded candidates are always re-evaluated on the original nonlinear closed-loop system to limit dependence on the surrogate.
  • The same structure applies without modification to first-order nonlinear consensus, second-order nonlinear consensus, and induction-motor drive control.
  • Joint redesign of control and Lyapunov parameters occurs repeatedly at successive decision epochs rather than in separate offline stages.

Where Pith is reading between the lines

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

  • If the Black-Hole calibration reliably contracts the region, the method could scale to higher-dimensional parameter spaces where brute-force search becomes intractable.
  • The approach might be combined with model predictive control by treating Lyapunov parameters as additional decision variables inside each receding horizon.
  • Testing on systems with faster time-varying disturbances would show whether periodic recalibration maintains stability when the operating point drifts between epochs.
  • Replacing the quantum imaginary time evolution step with a classical Ising solver would allow direct measurement of any quantum advantage on these particular control instances.

Load-bearing premise

The local quadratic pseudo-Boolean surrogate, after Black-Hole calibration, sufficiently approximates the sampled nonlinear closed-loop evaluations so that the quantum solver produces candidates that improve performance and stability when re-evaluated on the original system.

What would settle it

Apply the method to the induction-motor drive example and check whether the re-evaluated closed-loop costs and Lyapunov penalties after each update are lower than those from a standard offline-tuned controller whose stability was verified separately.

Figures

Figures reproduced from arXiv: 2605.04296 by Amin Kargarian, Mehdi Farasat, Milad Hasanzadeh.

Figure 1
Figure 1. Figure 1: Overview of the proposed online co-design framework the decay inequality associated with the desired stability notion. In this way, the proposed framework is not limited to asymptotic stability, but can incorporate different stability specifications directly into the dynamic co-design problem. III. TWO-STEP DYNAMIC QUANTUM CO-DESIGN FRAMEWORK In this section, we present the proposed two-step dynamic co-des… view at source ↗
Figure 2
Figure 2. Figure 2: A qubit in superposition and measurement outcomes it is obtained as a data-driven local approximation of the true encoded objective over the calibrated search region. After this fitting step, the exact discrete problem is replaced by the local QUBO-like problem min bk∈{0,1} nq Qk(bk). (44) To convert this binary quadratic model into Ising form, each binary variable is mapped to a spin variable according to… view at source ↗
Figure 3
Figure 3. Figure 3: Overview of the QITE-based encoded optimization This procedure is repeated for a prescribed number of imaginary-time steps, yielding a final parameter vector ϑ ⋆ and the corresponding state |ψ(ϑ ⋆ )⟩, which approximates a low-energy solution of the encoded Hamiltonian. This QITE-based search corresponds to the final block of Step II in view at source ↗
Figure 4
Figure 4. Figure 4: Closed-loop responses and online design-variable evolution view at source ↗
Figure 4
Figure 4. Figure 4: Closed-loop responses and online design-variable evolution [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Closed-loop responses and online design-variable evolution view at source ↗
Figure 6
Figure 6. Figure 6: Induction-motor drive closed-loop ω˙ = γpψiq − 1 J TL, (65) where ψ = q λ2 rα + λ 2 rβ (66) is the rotor-flux magnitude. The motor is initialized from x(0) = 0 0 0.9 0 0⊤ , (67) so that the rotor-flux magnitude starts at its nominal reference value and the motor speed starts from rest. The controller is constructed in a rotor-flux-oriented frame. Let ed = 1 ψ  λrα λrβ , eq = 1 ψ  −λrβ λrα  , (68) where… view at source ↗
Figure 7
Figure 7. Figure 7: Fixed-gain controller under the 50% Lm,p-mismatch rotor-flux-oriented controller. In this baseline case, the controller gains are kept constant over the full simulation horizon, with kψ = 100, kω = 100, (80) and no online redesign is performed. The purpose of this baseline is to show that the underlying controller can be tuned to track well under nominal conditions, while also demonstrating that fixed nomi… view at source ↗
Figure 6
Figure 6. Figure 6: Before evaluating the proposed online co-design strategy, we first consider a fixed-gain implementation of the same rotor-flux-oriented controller. In this baseline case, the controller gains are kept constant over the full simulation horizon, with kψ = 100, kω = 100, (80) and no online redesign is performed. The purpose of this baseline is to show that the underlying controller can be tuned to track well … view at source ↗
Figure 8
Figure 8. Figure 8: Closed-loop responses and online design-variable evolution view at source ↗
Figure 8
Figure 8. Figure 8: Closed-loop responses and online design-variable evolution [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

This paper proposes a dynamic quantum-assisted co-design framework for nonlinear closed-loop systems in which controller parameters and Lyapunov-certificate parameters are redesigned jointly at successive decision epochs. Unlike conventional nonlinear control designs that typically tune controller gains offline and verify stability separately, the proposed method embeds performance improvement and Lyapunov-based stability synthesis within a unified online optimization loop. The main novelty is a two-step computational structure that first contracts the continuous admissible search region around the current operating condition using a Black-Hole-based calibration procedure and then constructs a finite binary representation only over this calibrated region. The encoded objective is obtained from sampled nonlinear closed-loop evaluations and approximated by a local quadratic pseudo-Boolean surrogate, enabling an Ising-type Hamiltonian representation suitable for quantum-assisted optimization. Quantum imaginary time evolution is then used to explore the encoded Hamiltonian, and the resulting candidate bitstrings are decoded into continuous controller and Lyapunov parameters. To reduce dependence on the surrogate model, the decoded candidates are re-evaluated using the original nonlinear closed-loop cost and Lyapunov penalties before the final update is applied. The framework can accommodate different Lyapunov decay specifications by modifying the stability penalty and is validated on first-order nonlinear consensus, second-order nonlinear consensus, and induction-motor drive control examples. The implementation code used to generate the reported results is available at \href{https://github.com/LSU-RAISE-LAB/DQCLS-NS}{GitHub}.

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

3 major / 2 minor

Summary. The paper proposes a dynamic quantum-assisted co-design framework for nonlinear closed-loop systems in which controller parameters and Lyapunov-certificate parameters are jointly redesigned online at successive decision epochs. The core novelty is a two-step structure that first contracts the admissible parameter region around the current operating point via a Black-Hole-based calibration procedure, then encodes sampled nonlinear closed-loop evaluations into a local quadratic pseudo-Boolean surrogate that is mapped to an Ising Hamiltonian. Quantum imaginary time evolution explores this Hamiltonian; decoded candidate bitstrings are re-evaluated on the original nonlinear dynamics and Lyapunov penalties before the update is applied. The approach is illustrated on first-order nonlinear consensus, second-order nonlinear consensus, and induction-motor drive examples, with code released on GitHub.

Significance. If the surrogate fidelity after calibration is adequate and the re-evaluation step reliably recovers high-performing stable parameters, the framework offers a concrete route to embed quantum-assisted search inside an adaptive nonlinear control loop while preserving Lyapunov guarantees. The explicit separation between surrogate-guided search and final nonlinear verification, together with the open-source implementation, strengthens reproducibility and allows direct assessment of practical gains over purely classical co-design methods.

major comments (3)
  1. [§3.2] §3.2 (Surrogate construction): the local quadratic pseudo-Boolean approximation is stated to be built from sampled closed-loop evaluations inside the calibrated region, yet no quantitative fidelity metrics (hold-out MSE, ranking preservation of bitstrings, or worst-case pointwise error) are reported; because the quantum imaginary-time evolution operates exclusively on this surrogate, any systematic distortion directly affects which candidates are proposed for re-evaluation.
  2. [§5] §5 (Numerical examples): the three validation cases present qualitative trajectories and final parameter values but supply neither tabulated performance metrics (e.g., integrated cost, settling time, minimum Lyapunov decay rate) nor comparisons against classical nonlinear optimizers or Lyapunov-based tuning baselines; without these data it is impossible to determine whether the quantum-assisted candidates produce statistically meaningful improvements.
  3. [§4.1] §4.1 (Black-Hole calibration): the contraction mapping is described procedurally, but the manuscript does not quantify the resulting reduction in search-space volume or the sensitivity of the subsequent binary encoding to the calibration tolerance; these quantities are load-bearing for the claimed computational advantage of restricting the Ising Hamiltonian to a small calibrated region.
minor comments (2)
  1. [§3.2] Notation for the quadratic coefficients in the pseudo-Boolean surrogate is introduced without an explicit mapping from the continuous parameter vector to the binary variables; a small table or equation clarifying the encoding would improve readability.
  2. [§5] Figure captions for the example trajectories do not state the number of Monte-Carlo runs or the random-seed policy used to generate the plotted curves.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment point by point below, indicating the revisions we will make to improve the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Surrogate construction): the local quadratic pseudo-Boolean approximation is stated to be built from sampled closed-loop evaluations inside the calibrated region, yet no quantitative fidelity metrics (hold-out MSE, ranking preservation of bitstrings, or worst-case pointwise error) are reported; because the quantum imaginary-time evolution operates exclusively on this surrogate, any systematic distortion directly affects which candidates are proposed for re-evaluation.

    Authors: We agree that quantitative fidelity metrics for the surrogate would strengthen the presentation. Although the re-evaluation of decoded candidates on the original nonlinear dynamics and Lyapunov penalties is intended to mitigate surrogate inaccuracies and preserve stability, we recognize that explicit metrics would allow readers to assess the approximation quality directly. In the revised manuscript we will add hold-out MSE, bitstring ranking preservation rates, and worst-case pointwise errors to Section 3.2, computed from the sampling procedure used in the numerical examples. revision: yes

  2. Referee: [§5] §5 (Numerical examples): the three validation cases present qualitative trajectories and final parameter values but supply neither tabulated performance metrics (e.g., integrated cost, settling time, minimum Lyapunov decay rate) nor comparisons against classical nonlinear optimizers or Lyapunov-based tuning baselines; without these data it is impossible to determine whether the quantum-assisted candidates produce statistically meaningful improvements.

    Authors: We acknowledge that tabulated quantitative metrics and baseline comparisons are necessary to demonstrate the practical benefits. The revised version will include tables reporting integrated cost, settling time, and minimum Lyapunov decay rate for each of the three examples. We will also add comparisons against classical co-design methods (e.g., particle-swarm or gradient-based optimization of the same joint objective) to quantify any improvements in performance and stability margins. revision: yes

  3. Referee: [§4.1] §4.1 (Black-Hole calibration): the contraction mapping is described procedurally, but the manuscript does not quantify the resulting reduction in search-space volume or the sensitivity of the subsequent binary encoding to the calibration tolerance; these quantities are load-bearing for the claimed computational advantage of restricting the Ising Hamiltonian to a small calibrated region.

    Authors: We agree that explicit quantification of the search-space contraction and its sensitivity would better support the claimed computational advantage. We will revise Section 4.1 to report the reduction in admissible volume (as a ratio or percentage) achieved by the Black-Hole calibration and will include a sensitivity study showing how calibration tolerance affects binary encoding length, Ising Hamiltonian size, and final closed-loop performance across the examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained via external re-evaluation

full rationale

The paper's core loop encodes sampled nonlinear evaluations into a local quadratic surrogate for quantum optimization, then decodes and explicitly re-evaluates candidates on the original closed-loop system before acceptance. This breaks any potential reduction of final outputs to the surrogate fit. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim are present in the described structure. The Black-Hole calibration and Ising encoding are presented as procedural steps supported by external quantum techniques rather than derived from the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from nonlinear control and quantum optimization. No explicit free parameters or new physical entities are introduced in the abstract; the Black-Hole calibration and surrogate are algorithmic choices whose tuning details are not specified.

axioms (2)
  • domain assumption Lyapunov stability theory applies to the nonlinear closed-loop systems under consideration
    The framework synthesizes and penalizes Lyapunov certificates to enforce stability.
  • domain assumption Quantum imaginary time evolution can effectively minimize Ising-type Hamiltonians derived from the surrogate
    Used to explore the encoded objective for candidate bitstrings.

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