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Minimal Experiments for Robust Stabilization: Information, Spectral Geometry, and Duration

Alexey Peregudin, Ngoc Tuan Dinh

arXiv:2610.02187v1eess.SYcs.SYmath.OC

TL;DR

The paper asks how robustly experiments can stabilize controllable linear systems, beyond simply collecting enough information to compute a gain. It compares shortest predetermined experiments with a plant-informed causal benchmark and finds that they are competitive under spectral conditions, but slow actuation can make longer experiments necessary.

  • Problem

    Whether an experiment that contains enough information for stabilization also tolerates substantial errors is a quantitative question.

    Source: §I · p. 1

  • Method

    The paper compares shortest universal predetermined experiments with a causal oracle that knows the plant, observes exact states, and may choose any finite duration.

    Source: A. Principal results · p. 1

  • Results

    Shortest experiments need mn+1 transitions with exact states and m(n+1) with positive measurement uncertainty, and under bounded spectral radius and suitable root separation achieve a fixed fraction of oracle tolerance.

    Source: A. Principal results · p. 1

  • Takeaways & Limitations

    For A = I+hG with controllability depth ν ≥2, shortest-experiment tolerance scales as h^(ν−1), while duration of order 1/h is necessary and sufficient for constant-factor competitiveness.

    Source: A. Principal results · p. 1

    The spectral condition is sufficient but not necessary for competitive experiment families.

    Source: B. A uniform lower bound and… · p. 5

Abstract

from arXiv · show

On broad classes of linear systems, the shortest experiments are almost as good as the best possible ones. For $n$ states and $m$ inputs, the shortest input sequences that support robust data-driven stabilization of every controllable plant have $mn+1$ steps with exact states and $m(n+1)$ with noisy states. We show that, when the spectral radius is bounded and the spectrum is well separated near the unit circle, these sequences tolerate a fixed fraction of the error level achievable by any experiment, even one designed with full plant knowledge and allowed to use any finite duration. This constant-factor comparison can fail for slowly actuated systems. For $A=I+hG$ with controllability depth $\nu\ge2$, short experiments lose a factor of order $h^{\nu-1}$, and duration of order $1/h$ is both necessary and sufficient to recover a fixed fraction of the optimal tolerance.

I. INTRODUCTION

The paper asks how much bounded error experiments for data-driven stabilization can tolerate, not merely whether they contain enough information to produce a gain. It distinguishes minimum information from robustness: shortest experiments are competitive under spectral conditions, while slowly actuated systems can require longer durations.

II. PROBLEM AND ROBUSTNESS CRITERION

  • Robust stabilization requires every record to support one gain and quadratic Lyapunov function valid for all models consistent with that record.The setup assumes a known zero initial state, bounded inputs, and bounded process and measurement errors without a probability model.Source: §I · p. 1, §II · pp. 2–3

A. Principal results

  • With exact states, the shortest universal predetermined experiment needs mn+1 transitions; with any positive measurement uncertainty, it needs m(n+1).Sensor error increases the rank requirement.Source: A. Principal results · p. 1
  • Under bounded spectral radius and suitable root separation near the unit circle, shortest universal experiments attain a fixed fraction of the tolerance of plant-informed causal experiments of any finite duration.The comparison requires no matrix-norm or eigenvector-conditioning bound.Source: A. Principal results · p. 1
  • For A = I+hG with controllability depth ν ≥2, shortest-experiment tolerance scales as h^(ν−1) while the benchmark remains order one; competitiveness requires and is achievable at duration order 1/h.The loss reflects slow controllability: the shortest experiment can end before relevant dynamics develop.Source: A. Principal results · p. 1

B. Related work and scope

  • The paper situates its guarantees among data-driven stabilization and experiment-design work, while noting that adaptive multi-input minimum lengths remain open and are not ruled out.Its minimum-length claims concern inputs fixed before collection, a known zero initial state, and positive error tolerance on every controllable plant.Source: B. Related work and scope · pp. 1–2

A. Experiment tolerance and the causal oracle

The section defines robust experiment tolerance through certification of every physical record below a threshold and compares experiments with a causal oracle. It also relates controllability-matrix singular values to experiment tolerances.

A. Experiment tolerance and the causal oracle

  • The error level ε is normalized by the square root of duration, so a fixed per-step process-error level remains admissible at every duration.For per-step errors bounded by ε, the whole-record error norm is bounded by ε√T.Source: A. Experiment tolerance and the causal… · p. 3
  • A tolerance threshold requires every physical record at every smaller error level to be certifiable, with one shared gain and Lyapunov matrix per record.Success on one error-free record or at the supremum itself is not required by the definition.Source: A. Experiment tolerance and the causal… · p. 3
  • The causal oracle knows the plant when choosing its policy and duration but must certify every compatible alternative; predetermined plant-informed experiments are included.The oracle observes exact states, making it a benchmark for measured-state experiments.Source: A. Experiment tolerance and the causal… · p. 3

B. Controllability scales

  • C_j stacks the first j input-response blocks, and its minimum singular value measures how strongly they reach the weakest state direction.For a controllable pair with minimal-polynomial degree d, σ_min(C_d) > 0.Source: B. Controllability scales · p. 3

III. INFORMATION AND EXACT MINIMUM DURATION

Positive robustness requires state information of full rank, with a stronger rank condition when states are measured noisily; the resulting universal predetermined pulse sequences have exact minimum durations. The supplied certificate tests connect record data and error bounds to stabilization guarantees and to bounds on what records can certify.

III. INFORMATION AND EXACT MINIMUM DURATION

  • For an error-free controllable record, positive-robustness certification requires rank X = n with exact states and the stronger condition (9) with measured states.If the measured-state condition fails, no single gain stabilizes all compatible models for any positive sensor-error bound, even with zero process error.Source: §III · p. 3
  • The measured-state condition requires the state rows to add n independent directions beyond the input row space, so sensor uncertainty changes the information requirement.Exact states can determine a stabilizing gain without identifying the plant; with sensor uncertainty, the record must determine state dynamics and used input directions.Source: §III · p. 3
  • The shortest universal predetermined sequences require mn+1 transitions with exact states and m(n+1) with any positive measurement uncertainty.The pulse sequences Uexact and Umeas attain these minima; any positive sensor uncertainty costs exactly m−1 additional transitions.Source: §III · p. 4

IV. ROBUSTNESS OF THE SHORTEST EXPERIMENTS

  • The shortest universal sequences are compared with Γ, a plant-informed causal benchmark without a duration limit, using the controllability scale σmin(C).The supplied text introduces record tests as the link between data errors and the subsequent robustness comparison.Source: §IV · p. 4, A. Certificates from a single record · p. 4

A. Certificates from a single record

  • The record margin rrec certifies noisy records below its threshold and bounds what an error-free record can certify.A Lipschitz bound transfers positive margin to nearby noisy records, while a pencil bound supplies upper bounds.Source: A. Certificates from a single record · p. 4
  • For measured states, σ⊥ quantifies whether the state data add independent directions beyond the inputs: σ⊥ > 0 exactly when rank Z = n + rank U.When δv < σ⊥, the conditional measured-state enclosure admits a certificate under condition (18), which is exact for the enclosure but only sufficient for the measured compatibility set.Source: A. Certificates from a single record · pp. 4–5

B. A uniform lower bound and comparison with the oracle

With bounded spectral radius and separated roots near the unit circle, the shortest pulse experiments retain a fixed fraction of the causal oracle's tolerance. The comparison does not require bounds on the plant norm or eigenvector conditioning, and the spectral condition is sufficient but not necessary.

B. A uniform lower bound and comparison with the oracle

  • A uniform lower bound on b∂(A), together with bounded root moduli, keeps distinct root factors apart near the unit circle; a single root may reach it.The condition allows repeated and defective eigenvalues away from the unit circle; it is sufficient but not necessary for competitiveness.Source: B. A uniform lower bound and… · p. 5
  • The measured pulse sequence retains at least α/(max{1, κ}√m β) of Γ when the stated spectral condition holds; exact states drop max{1, κ}.The constants require neither a bound on ∥A∥ nor a bound on eigenvector conditioning.Source: B. A uniform lower bound and… · p. 5
  • The oracle upper bound applies to every finite duration and causal policy by splitting polynomial roots into stable, unstable, and near-unit-circle groups.Filtered responses, actuator-perturbation robustness, and pencil bounds control the groups; the argument avoids estimating eigenvector matrices or spectral projectors.Source: B. A uniform lower bound and… · p. 5

C. Stable plants and the stability gap

The stable-plant comparison depends on the stability gap and minimal-polynomial degree, while slow actuation can make short experiments uncompetitive. For controllability depth ν ≥2, the theorem identifies duration of order 1/h as necessary and sufficient to recover a fixed fraction of the benchmark tolerance.

C. Stable plants and the stability gap

  • For stable plants with ρ(A) ≤ r < 1, the upper bound depends on the stability gap 1−r and minimal-polynomial degree d.The fraction bound uses a gap power with exponent d−1; its constant is not claimed optimal, and exact states omit the factor max{1, κ}.Source: C. Stable plants and the stability gap · pp. 5–6

V. THE DURATION NEEDED FOR COMPETITIVENESS

  • When ν ≥2, directions reached only through successive powers of G have weak responses over fixed durations and become visible on the time scale 1/h.The controllability depth ν counts the input-response blocks needed to span the state space, and ν ≤ d, with equality for single-input pairs.Source: §V · p. 6
  • A disturbance construction produces the upper bound for every deterministic causal policy, so the duration obstruction is not specific to a certificate choice.The construction fits the observed record with a second plant having an uncontrollable eigenvalue at one.Source: §V · p. 6
  • Every experiment attaining a fixed positive fraction of Γ needs duration Ω(1/h), while a sequence of duration O(1/h) attains some fixed positive fraction.A predetermined sequence holds e1, . . . , em in turn for N = ⌈1/h⌉ steps, with duration mN; it requires knowing the time scale 1/h. The result covers stable, unstable, semisimple, and defective generators.Source: §V · p. 6
  • The stable-gap exponent d−1 is sharp even for semisimple plants, as the diagonal single-input example has σmin(Cd) ≍ h^(d−1) while Γ ≍ 1.Here d = ν = n and 1−ρ(I+hG) ≍ h.Source: §V · p. 6

A. Weak links and slow actuation

Weak links and slow actuation can both make the shortest experiments weakly robust, but only slow actuation lets longer experiments recover a fixed fraction of the oracle tolerance. A nonnormal oscillator provides a contrasting case in which the shortest sequence remains competitive despite growing matrix norm.

A. Weak links and slow actuation

  • For the weak-link family, longer experiments cannot improve the tolerance order because the weak link limits even a plant-informed experiment.Here σmin(C) = μ, the minimal polynomial is z², and b∂(Aμ) = 1.Source: A. Weak links and slow actuation · p. 6
  • For the slow-actuation family, Eκ_meas/Γ ≍ h → 0, while transitions of order 1/h are necessary and sufficient to recover a fraction of Γ bounded away from zero.The family has G = diag(−1, −2) and controllability depth ν = 2.Source: A. Weak links and slow actuation · p. 6
  • The weak-link and slow-actuation families share weak finite controllability and the same shortest universal experiment, but only slow actuation gains an unbounded factor from longer duration.This separates absolute robustness from the cost of using the shortest possible experiment.Source: A. Weak links and slow actuation · p. 7

B. A nonnormal oscillator

  • The shortest sequence remains uniformly competitive for the nonnormal oscillator even as ∥Ak∥ → ∞, and increasing k reduces its tolerance and the oracle tolerance at the same rate.The lower-bound analysis uses a disturbance that keeps x2,t = 0 while x1,t+1 = ut.Source: B. A nonnormal oscillator · p. 7

VII. CONCLUSIONS

  • The conclusions distinguish weak controllability, which can limit every experiment, from slow controllability, where short experiments end before relevant dynamics develop.The paper leaves open adaptive multi-input minimum lengths, sharper competitive constants, and duration adaptation to an unknown time scale.Source: §VII · p. 7

APPENDIX

The appendices establish the information and certification results through rank-based arguments, explicit record constructions, and an SDP characterization. They also organize the oracle-bound proof around a stable-disk transfer identity.

APPENDIX

  • Theorem 5’s uniform lower bound and oracle upper bound are proved in separate appendices, with the stable-disk proof developing the transfer identity used for the oracle argument.Source: APPENDIX · p. 7

A. Proof of Theorem 1

  • For exact states, a rank-deficient state record admits compatible plants that defeat a common stabilizing gain, while full row rank yields a controllable realization and a gain robust to sufficiently small perturbations.Source: A. Proof of Theorem 1 · p. 7
  • With sensor uncertainty, scaling the underlying trajectory creates alternative explanations that rule out a common gain when the state rows fail to add independent directions; the full-rank condition supports a common robust gain.Source: A. Proof of Theorem 1 · p. 7

B. Proof of Theorem 2

  • The pulse sequences establish attainability; for one input, every causal run with T≤n has rank X≤T−1<n, yielding the exact-state duration lower bound, while a nonzero-state-row relation makes the measured-state record non-certifiable for any δv>0.Source: B. Proof of Theorem 2 · pp. 7–8

C. Proof of Lemma 3

  • The SDP margin characterizes record certification: strict margin above the process-error bound yields a certificate, and failure holds at equality.The proof establishes both sufficiency and necessity of the strict threshold.Source: C. Proof of Lemma 3 · p. 8

D. Proof of Lemma 4

The proof bounds compatible model changes by projecting the data equation onto the regressor’s row space, then uses an enclosing set to transfer a certificate to the measured record.

D. Proof of Lemma 4

  • Projecting the data equation onto the row space of Z yields a bound on ∥A′∥ in terms of ∥bA∥ and ∥∆Z∥/σ⊥.Source: D. Proof of Lemma 4 · pp. 8–9
  • The enclosing set has a zero-residual explanation, so its certificate transfers to the original measured record by containment.Source: D. Proof of Lemma 4 · p. 9
  • The certificate argument is algebraic: the fitted blocks need not themselves form a shifted state trajectory.Source: D. Proof of Lemma 4 · p. 9

E. Proof of Theorem 5: the short-experiment lower bound

At the fixed experiment durations, sufficiently small normalized errors preserve positive robust-certification margins uniformly. The proof obtains these guarantees by controlling perturbations in compact, data-normalized coordinates.

E. Proof of Theorem 5: the short-experiment lower bound

  • Uniform rank margins for the error-free pulse regressors persist under sufficiently small normalized perturbations.Compactness of the controllable-pair family and the error-free pulse ranks provide the positive least-singular-value margins.Source: E. Proof of Theorem 5: the… · p. 9
  • Data normalization keeps the records uniformly close to a compact family of error-free normalized records.The argument allows every admissible data factor and does not require choosing the factor continuously.Source: E. Proof of Theorem 5: the… · p. 9
  • Sufficiently small ζ makes the robust-certification test hold for every physical record, for both exact and measured states.The exact-state argument does not require full row rank of [Xc; U].Source: E. Proof of Theorem 5: the… · p. 9
  • Under (5), the normalized error scale is ε√Texact/σmin(C) for exact states and max{1, κ}ε√Tmeas/σmin(C) for measured states.Absorbing the fixed durations into a positive constant gives the lower bounds.Source: E. Proof of Theorem 5: the… · p. 9

F. Proof of Corollary 6

The proof bounds certified levels using an input-cancellation obstruction and then compares those bounds with the shortest-experiment lower bounds. Taking suprema over certified levels, durations, and causal policies yields the competitive-fraction conclusion.

F. Proof of Corollary 6

  • A unit vector aligned with the least singular direction of Cd yields an input-cancellation bound on every certified level.The argument extends the input by zeros and uses stability and Parseval’s identity to bound the first T samples.Source: F. Proof of Corollary 6 · p. 9
  • The obstruction applies for complex |τ| ≤ 1, and zero sensor error makes it applicable for every κ.The proof chooses τ from the modal relation according to whether |λ| is at most or at least one.Source: F. Proof of Corollary 6 · pp. 9–10
  • The competitive fraction follows by taking suprema over certified levels, durations, and causal policies and using σmin(Cd) ≤ σmin(C).The proof establishes both bounds in a chain before this comparison.Source: F. Proof of Corollary 6 · pp. 9–10

G. Proof of Theorem 5: the oracle ceiling

The oracle ceiling proof controls responses by decomposing the characteristic polynomial into spectral groups. Uniformly bounded coefficient systems let the groupwise estimates combine without requiring bounded spectral projectors in the original coordinates.

G. Proof of Theorem 5: the oracle ceiling

  • For dimension one, every row is modal, and a least-singular-vector choice for B gives the stated upper bound.The comparison uses σmin(B) ≤ σmin(C).Source: G. Proof of Theorem 5: the oracle… · p. 10
  • Roots near the unit circle are assumed simple and separated from the other roots, while stable and unstable groups are separated from that circle by θ.The factorization groups roots below 1 − θ, above 1 + θ, and remaining simple roots.Source: G. Proof of Theorem 5: the oracle… · p. 10
  • Stable and unstable polynomial factors have boundary magnitudes bounded using θ, yielding a constant β0 depending on n, M, and θ.The stable-group estimate uses the transfer identity and a record obstruction; the one-root estimate follows from the corresponding bound.Source: G. Proof of Theorem 5: the oracle… · p. 10
  • Uniform invertibility of the Chinese-remainder coefficient systems bounds the recombined coefficient without bounding original-coordinate spectral projectors.The argument uses compactness of fixed-degree families with separated groups; collisions within a group are allowed.Source: G. Proof of Theorem 5: the oracle… · p. 11

H. Proof of Theorem 7

For A = I + hG with controllability depth ν, the shortest experiments have a tolerance gap of order h^(ν−1). Holding each input coordinate for order 1/h steps attains a constant tolerance, while a causal obstruction gives a matching duration-scale necessity.

H. Proof of Theorem 7

  • The controllability matrix scale is bounded below by α0h^(ν−1), with a matching upper estimate from a vector orthogonal to lower-order G^jB directions.The proof relates hB, h^2GB, …, h^νG^(ν−1)B to the full-row-rank controllability matrix by a fixed triangular transformation.Source: H. Proof of Theorem 7 · p. 11
  • Holding each coordinate input for N = ⌈1/h⌉ steps gives an experiment of duration mN, while a causal obstruction applies when hT ≤ 1.For hT > 1, input-cancellation disturbance supplies a uniform constant ceiling.Source: H. Proof of Theorem 7 · p. 11
  • The finite unit-interval excitation has a positive controllability Gramian, giving σmin(Z0 h) ≥ α1h^(−1/2).The proof uses controllability to show that a vanishing quadratic integral forces all state and input coefficients to vanish.Source: H. Proof of Theorem 7 · p. 11
  • For sufficiently small process and sensor errors, one continuous-time Lyapunov certificate yields a gain hKct certifying every alternative for all sufficiently small h.The argument first keeps the scaled compatible pairs close to (G, B).Source: H. Proof of Theorem 7 · pp. 11–12
  • The shortest-experiment and optimal tolerances satisfy Eexact, Eκmeas ≍ h^(ν−1), while duration of order 1/h recovers a fixed fraction of the optimum.The attaining sequence has Γ ≍ 1; the stated stable-gap sharpness follows as well.Source: H. Proof of Theorem 7 · p. 12
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