Researchers studied how long an experiment must be to collect data that guarantee a single feedback law will stabilize a whole family of linear systems despite bounded errors. They show that for an n‑state system with m inputs, the shortest predetermined input sequences that work for every controllable plant use mn+1 time steps when states are measured exactly, and m(n+1) steps when measurements are noisy. Under further spectral assumptions, these very short experiments tolerate a fixed fraction of the error level that the best possible, plant‑informed experiment can tolerate.

The experiments they consider start from a known zero state and use bounded inputs. Process and measurement errors are only assumed to be bounded; no probability model is used. The authors compare experiments by the largest error level they can certify. “Certify” means: from the observed data you can produce one feedback gain K and one positive definite matrix P (a quadratic Lyapunov function) that prove stability for every model consistent with the data and the error bounds. The error bounds are normalized by the square root of the experiment duration so that short and long experiments are compared on the same scale.

To judge how good a short, predetermined experiment is, the paper uses a strong benchmark called the causal oracle. The oracle knows the true plant when it plans inputs and may choose inputs adaptively while still needing to certify robustness across all models compatible with the observed record. The main positive result is that, when the system’s spectral radius is bounded and its eigenvalues are reasonably separated near the unit circle, the minimal fixed‑input experiments above achieve a constant fraction of the oracle’s error tolerance. In plain terms: if the system is not on the verge of instability and its modes are not tightly clustered near marginal stability, very short experiments are almost as good as lengthy or adaptive ones for robust stabilization.