AI Surrogate Speeds Up Stability Optimization for Delayed Vibration Control, With Exact Spectral Checks Still in the Loop

Engineers fighting vibration in mechanical systems often rely on feedback that is deliberately delayed, a strategy that can tame oscillations but turns the mathematics of stability into a formidable challenge. A new study in Results in Engineering proposes a hybrid optimization framework that pairs a neural-network surrogate with exact spectral computation to make this design problem faster to solve without abandoning the rigorous model behind it. The work, led by Feleke Tsegaye Yareshe and colleagues, targets delayed vibration suppression in systems whose stability is governed by infinitely many characteristic roots, and it reports that artificial intelligence can serve as a computational accelerator inside a physics-consistent loop rather than as a replacement for the underlying spectral model.

The core quantity at stake is the spectral abscissa, defined as the maximum real part of the characteristic roots of the closed-loop system. It directly determines the exponential decay rate of vibrations: the more negative the spectral abscissa, the faster the oscillations die out. Minimizing it is therefore a natural design goal, but the authors note that this objective depends nonsmoothly and nonconvexly on the controller parameters, making direct optimization computationally difficult. To sidestep that obstacle, the framework adopts the H2-smoothed spectral abscissa, a differentiable approximation introduced in earlier literature that converges to the true spectral abscissa as the smoothing parameter epsilon approaches zero, while larger values of epsilon progressively emphasize H2-oriented performance.

The benchmark system is a one-degree-of-freedom oscillator with delayed velocity feedback, in which a mass-spring-damper structure receives a control force proportional to its velocity measured a moment earlier. The delay injects infinite-dimensional dynamics into the closed loop, so the characteristic quasi-polynomial admits infinitely many roots. The team evaluates the spectral abscissa using the Quasi-Polynomial Mapping-based Rootfinder, or QPmR, which computes all characteristic roots within a prescribed region of the complex plane. For each candidate gain and delay pair, the smoothed spectral abscissa is then obtained by solving an implicit integral equation through a safeguarded bracketing and bisection procedure, with unstable or numerically invalid candidates assigned a large penalty.

The optimization itself unfolds as a fixed-epsilon continuation. The researchers solve the smoothed problem for a sequence of smoothing parameters, using the solution from one level as the warm start for the next, and adaptively expand the sequence until the optimized smoothed spectral abscissa turns nonnegative. This sweep also yields an automatic estimate of the critical smoothing threshold epsilon-star, the largest smoothing level for which the optimized design remains stabilizing. For the considered delayed velocity-feedback system, the threshold was bracketed between 6.3995857244 and 6.3997210974, with the refined lower estimate retained. At that refined threshold the optimized gain and delay reached approximately 3.0 and 0.01, the true spectral abscissa remained strictly negative at about minus 0.3495, and the smoothed value sat numerically close to zero at roughly minus 4.45 times ten to the minus six, a difference the authors highlight as evidence of the criterion’s conservative character near the transition point.

Where the artificial intelligence enters is in the local surrogate layer. Rather than training a network over the entire admissible design domain, the team constructs a local neural-network model of the smoothed objective inside a small box centered on a coarse QPmR-based search result or, for subsequent smoothing levels, on the validated solution from the preceding continuation step. This non-circular strategy means the surrogate never requires prior knowledge of the true optimum, which is computed independently and retained only as a numerical benchmark. A feedforward network with two hidden layers is trained on exact-model samples within the local region, and the surrogate optimum is then minimized over that box to propose a candidate controller.

Critically, the surrogate-generated candidate is never accepted directly. Each one is validated against the true delayed-system model, and if the candidate is unstable, nonfinite, or numerically inconsistent, a fallback mechanism substitutes the nearest valid point from the local training dataset. The validated candidate then serves as the initialization for an AI-refined stage, a second true-model optimization that minimizes the exact smoothed spectral abscissa rather than the network prediction. The authors argue that this structure keeps stability assessment tied to the characteristic roots of the original delayed system while the neural network functions as a computationally inexpensive local approximation during the candidate search.

The numerical results illustrate the distinct roles of the three strategies. For smoothing parameters from ten to the minus five up to one, the true optimization produced an almost unchanged design with a gain near 2.2517, a delay near 0.0170, and a spectral abscissa near minus 2.0411, indicating a robust interior optimum governed mainly by spectral-abscissa minimization. As epsilon increased beyond unity, the optimal delay grew at epsilon values of 2 and 4, and the gain eventually reached the upper admissible bound of 3, a boundary-seeking behavior the authors caution should not be read as evidence that arbitrarily large gains and tiny delays are physically optimal; it reflects the interaction between the smoothed objective and the prescribed design domain. At epsilon equal to 8, the optimized smoothed spectral abscissa turned positive at roughly 0.0627, signaling that the optimization had crossed the stability boundary in the smoothed formulation and enabling the threshold estimate.

Computational accounting is handled with deliberate care. The AI-assisted stage, once a surrogate has been constructed, required only about one second of online optimization time for most smoothing levels in the one-degree-of-freedom benchmark, with a single exact validation evaluation afterward, compared with runtimes ranging from roughly 0.8 to 35 seconds for the direct true method. However, the authors stress that the small AI-assisted counts exclude the offline cost of generating the surrogate dataset, which for the benchmark involved a 15 by 15 sampling grid of up to 225 exact-model samples per smoothing level. The AI-refined stage, meanwhile, is explicitly not a cheaper replacement for direct optimization; it reintroduces exact evaluations and for some smoothing levels took longer than the direct method, its purpose being to recover true-model accuracy from a surrogate-generated start rather than to guarantee savings.

The framework also proved extensible to a more demanding coupled two-degree-of-freedom system with three design variables: two delayed velocity-feedback gains and the shared delay. Fixing epsilon at ten to the minus two, an eight-cubed coarse QPmR search over the three-dimensional domain identified a best stable point, and a subsequent multi-start exact optimization of 102 runs established a true optimum with a spectral abscissa of minus 3.711490, a process that consumed roughly 4730.72 seconds in total. A local surrogate built from 689 valid stable samples out of 729 candidate points, requiring about 217.27 seconds of dataset generation, initially showed a substantial prediction discrepancy, triggering validation-driven adaptive retraining over two rounds. The final AI-assisted candidate predicted a smoothed value of minus 3.706979, but exact validation at the same design yielded only minus 3.176051, a gap the authors cite as demonstrating why an exact-model validation stage remains essential in higher dimensions. The subsequent exact refinement reduced the objective difference relative to the true optimum from about 14.42 percent to about 3.59 percent, at an additional cost of 149.41 seconds, while the online surrogate optimization itself took just 3.19 seconds.

Time-domain simulations and robustness checks rounded out the evaluation. For the one-degree-of-freedom benchmark, all three optimized controllers produced rapidly decaying responses without visible oscillatory behavior, with the AI-assisted and AI-refined transients nearly overlapping and slightly faster initial decay than the true-optimization design. Against a simplified baseline that fixes the smoothing evaluation at the imaginary axis, the proposed formulation achieved an estimated decay-rate improvement of approximately 45 percent for representative small and moderate smoothing values. Under mechanical-parameter perturbations of plus or minus 20 percent in mass, plus 50 percent in damping, and plus 30 percent in stiffness, the three strategies showed broadly similar robustness trends, with the stiffness increase producing the largest variation, roughly 46 to 49 percent faster estimated decay alongside about 28 percent shorter settling time and about 23 percent lower integral absolute error for smoothing levels at or below one. The authors conclude that the neural network acts as a computational surrogate rather than a replacement for the spectral model, that exact validation remains essential as dimensionality grows, and that the framework offers a transparent accounting of the tradeoff between surrogate-assisted efficiency and high-fidelity accuracy in delayed vibration-control design.

Subject of Research: Hybrid AI-assisted optimization of delayed vibration suppression using the H2-smoothed spectral abscissa with exact spectral validation

Article Title: AI-refined neural-network surrogate optimization for delayed vibration suppression using the smoothed spectral abscissa

Article References: Yareshe, F. T., Pekař, L., Roba, M. T., Wolde, M. K., & Wendimu, A. A. (2026). AI-refined neural-network surrogate optimization for delayed vibration suppression using the smoothed spectral abscissa. Results in Engineering, 32, Article 113722. https://doi.org/10.1016/j.rineng.2026.113722

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113722

Keywords: delayed vibration suppression, spectral abscissa, neural-network surrogate, time-delay systems, QPmR rootfinder, H2 smoothing, controller optimization, stability analysis, vibration control, hybrid AI framework, computational efficiency, AI-refined