An indirect data-driven method certified stability with about 800 samples in a numerical benchmark, compared with about 2,200 samples for a direct method. Both figures were reported at 95% confidence, using the same test: the upper bound on the system’s joint spectral radius, or JSR, had to fall below 1. The thresholds are approximate, but the gap points to a sample-efficiency advantage for the indirect route in this test.
The result comes from a methods preprint comparing two ways of turning noisy trajectory data into a probabilistic stability certificate for a switched linear system. The benchmark represented a hidden-network consensus problem as a five-dimensional system with three modes. It used trajectory data and compared three noise settings: Gaussian noise, Gaussian-mixture noise with outliers, and bounded noise.
In the study’s setup, the mode and noise realizations remained latent, so the observed information consisted of input-output pairs. The theoretical data used independently drawn experiment triplets, while sample counts varied across the numerical comparisons. That made the certification step central: the methods had to turn noisy observations into an upper bound on the system’s JSR.
Two routes to one certificate
Under the indirect route, the researchers first identify a switched linear model. They then build a robust quadratic Lyapunov function, a mathematical certificate used to track whether the system contracts, and carry the model’s identification error through a sensitivity analysis. The result is a finite-sample JSR upper bound that is stated to cover the true JSR with confidence 1 − β over the sampled data.
The indirect experiment used trimmed k-LinReg. It alternated between assigning observations to modes and fitting each mode by least-squares regression, then discarded observations with the largest residuals, or mismatches between the observed and fitted behavior. A separate training and testing split was used when bounding the identification error. These choices are part of the reported implementation, not a guarantee that every indirect identification method would behave the same way.
The direct route took a different path. Rather than first estimating the hidden switched model, it solved a quasi-convex scenario-optimization problem for a quadratic norm and a contraction rate, then converted that rate into a probabilistic upper bound on the JSR. Its output was likewise stated to upper-bound the true JSR with confidence 1 − β. In the benchmark, a bound below 1 was the point at which stability was certified.
The two approaches therefore differ in where they place the main calculation. The indirect framework estimates the system before constructing its certificate, while the direct framework works through a scenario-optimization problem on the observed data. The comparison was designed to examine not only the resulting bound, but also the sample count and computational burden needed to reach the stability threshold.
Where the methods separated
Under Gaussian noise, the indirect bound fell below 1 at around 800 samples, while the direct bound did so at around 2,200. Those are not exact sample requirements: the thresholds were read approximately from the reported comparison, and no interval or repeated-run uncertainty was supplied. Even so, the difference suggests that the indirect method was better suited to a limited-data regime in this benchmark.
The comparison also showed a computational gap. The direct method was substantially more expensive because it solves a linear matrix inequality, or LMI. The indirect route primarily incurs the cost of k-LinReg and alternating least squares. Exact timing values were not reported in the supplied analysis, so the evidence supports a relative cost difference rather than a precise speed multiplier.
Outliers produced the sharpest contrast. With Gaussian-mixture noise containing outliers, the indirect approach produced a meaningful stability guarantee below 1, whereas the direct approach failed to yield an informative bound. The authors partly attribute this to trimming: the indirect routine discards observations with the largest residuals before constructing the model-based certificate. The direct formulation, by contrast, enforces Lyapunov decrease along every sampled trajectory, a requirement the authors say makes it sensitive to corrupted observations.
The bounded-noise comparison gave another point of separation. The paper’s bound fell below 1 at around 7,500 samples, while the bound from the cited comparator did not certify stability. Because that crossing point is approximate and figure-based, it should be read as a benchmark result rather than an exact requirement for data-driven control.
A result with boundaries
These guarantees are probabilistic. Both routes state that their output is an upper bound on the true JSR with confidence 1 − β; the study reports a certificate rather than an exact JSR value. In the Gaussian setup, the stated parameters were δ = 5, β = 5%, W = 3σ = 4 × 10−3, p(W) approximately 0.006, and ε = 5 × 10−3. Here, β = 5% is the complement of the 95% confidence level used in the sample comparison.
The study’s scope limits how far the comparison can be carried. Its numerical evidence comes from one hidden-network consensus benchmark represented as a five-dimensional switched system with three modes, and the indirect test uses trimmed k-LinReg rather than a range of identification algorithms. The reported analysis therefore leaves open how the ranking would change with other systems, dimensions, mode distributions, or identification procedures.
For researchers choosing a stability-certification pipeline, the practical message is conditional. In the reported benchmark, indirect identification followed by quadratic certification looked more sample- and compute-efficient and more tolerant of outliers than the direct formulation. But the study does not show that indirect methods are universally superior, always produce tighter bounds, or that trimmed k-LinReg is the best available identification procedure. Robust direct scenario optimization and adaptive sampling remain open directions rather than tested results in this comparison.
The document is a preprint submitted to Automatica. The project reports funding from the European Research Council under the European Union’s Horizon 2020 programme, through grant agreement No 864017 – L2C.
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
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