A new preprint comparing two ways to certify stability in noisy switched linear systems reports a clear advantage for the indirect approach in its main simulation: it reached a 95% confidence certificate at around 800 samples, while the direct approach needed around 2,200. The indirect method was also substantially faster in the reported Gaussian-noise comparison.
The result comes with an important qualification. The authors report that indirect methods can outperform direct methods in several settings, particularly when datasets are small or contain outliers, but the paper does not establish that they are universally superior. Its evidence is a comparison of mathematical guarantees and numerical results under stated assumptions.
Two routes to the same stability question
The study compares direct and indirect approaches for switched linear systems, models that can move among several linear modes. Its key output is an upper bound on the joint spectral radius, or JSR. In ordinary terms, that is a bound on the system’s possible growth as its modes switch. The reported certificates use a bound below 1 as the relevant stability threshold.
The indirect route first identifies a switched linear model from the data. It then uses a robust quadratic Lyapunov function, a mathematical measure for tracking whether system behavior shrinks rather than grows, to upper-bound the JSR while accounting for error in the identified model. This creates a stability certificate from two linked steps: estimating the system and analyzing the estimate.
The direct route uses scenario-based quadratic synthesis to seek a certificate from the observations themselves. The paper extends that direct data-driven framework beyond bounded-noise settings to unbounded measurement noise, where the observations are not treated as having a fixed maximum error. That extension forms part of the basis for the noise comparisons reported in the study.
A small simulated test bed
The numerical benchmark was a five-dimensional system with three modes. Only noisy input-output pairs were observed; the mode labels and the actual noise realizations remained hidden. Initial states were assumed to be sampled uniformly on the unit sphere. These choices make the comparison a test of how the two methods work when important parts of the underlying system are not directly available.
Both methods come with finite-sample probability statements. Under their respective assumptions, each theorem says that the value returned by the method is an upper bound on the system’s true JSR with confidence written as 1 minus β. The study therefore treats a certificate as a quantified claim about the unseen system, rather than simply as a number produced by fitting the observed data.
That distinction shapes the comparison. The question is not only which method can produce a stability certificate, but how many samples it needs and whether its bound remains informative when the noise becomes more difficult. The experiments examine those issues under Gaussian noise, Gaussian-mixture noise with outliers and bounded noise.
The gap widened when outliers appeared
Under Gaussian noise, the reported 95% certification threshold arrived at around 800 samples for the indirect approach and around 2,200 for the direct approach. The authors also report a substantially lower computational cost for the indirect method in that comparison. The exact advantage in running time depends on the particular identification and optimization implementations used.
A second test added outliers through Gaussian-mixture noise. In that setting, the indirect approach produced meaningful bounds below 1 at 95% confidence, while the direct approach failed to provide informative bounds. Within this experiment, the indirect method therefore supplied a useful certificate in a situation where the direct method did not.
The bounded-noise comparison also favored the new bound, although it required a larger dataset. The new bound certified stability at around 7,500 samples, while the bound from Banse et al. (2024) failed to certify stability in the reported comparison. The result is a comparison of the bounds under that benchmark, not a general ranking of every stability method.
The researchers also compared two ways of converting the indirect method’s finite-sample information into a bound. Binomial-tail inversion was tighter than a Rademacher-complexity bound in the reported comparison. That finding concerns the tested calculation and may depend on tuning and data splitting.
Why the result needs careful reading
The strongest numerical evidence is concentrated on one simulated five-dimensional, three-mode benchmark. The authors’ conclusion is consequently conditional: indirect methods can significantly outperform direct methods in several settings, especially with small datasets or outliers, but the study does not show that the same pattern holds across all switched systems.
The indirect method also includes a model-identification step that the direct method does not use in the same way. Its final result can therefore reflect both the quality of the estimated model and the stability analysis applied afterward. The direct method, meanwhile, depends on its scenario-based synthesis and on the assumptions behind its extension to unbounded measurement noise.
The timing comparison likewise covers two different computational pipelines: identification followed by robust Lyapunov analysis for the indirect route, and scenario-based quadratic synthesis for the direct route. That makes the reported speed difference useful for the implementations tested, while leaving open how the comparison would look with other estimators, optimization procedures or data choices.
The paper’s evidence also does not establish performance on real-world systems or show that the certificates apply to every noise distribution, system dimension or identification procedure. Its contribution is narrower but concrete: it shows that, under the stated assumptions and in the reported simulations, an indirect pipeline can reach tighter or more useful certificates with fewer samples in several difficult settings.
A methods result, not a universal verdict
The document is a preprint version submitted to Automatica. It reports European Research Council funding under the European Union’s Horizon 2020 programme, grant agreement No 864017 – L2C, and lists FNRS positions for two authors.
For researchers choosing how to analyze noisy switched systems, the message is a practical one: the indirect approach deserves attention when data are scarce or contain outliers. But the result remains a benchmark-based methods finding. Broader system classes, additional identification procedures and further direct-method comparisons would be needed to determine how widely the reported advantage extends.
Paper data and sources
Original title: Direct vs. Indirect Data-Driven Control: Case-study of Switching Systems Stability
Authors: Alexis Vuille, Guillaume O. Berger, Raphaël M. Jungers
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text