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Finite Sample Identification of Analytic Nonlinear Systems

Negin Musavi, Ziyao Guo, Geir E. Dullerud, Yingying Li

arXiv:2608.29908v1eess.SYmath.OC

TL;DR

The paper addresses whether non-active exploration can identify general LPN systems after prior counterexamples showed failure in some nonlinear settings. It develops BMSB-based convergence analyses for LSE and SME under real-analytic features, constructs non-real-analytic counterexamples, and validates the results numerically.

  • Problem

    Prior work showed that non-active exploration may be insufficient for general LPN systems, leaving its scope between bilinear and piecewise-affine systems unresolved.

  • Method

    The paper establishes BMSB conditions and analyzes non-asymptotic convergence rates for LSE and SME under non-active exploration, including single-trajectory and perturbed-policy settings.

  • Results

    Non-active exploration is sufficient for LPN systems with real-analytic features under the paper's conditions, while counterexamples show possible failure for non-real-analytic systems, including infinitely differentiable ones.

  • Takeaways & Limitations

    Real analyticity is important for the paper's non-active identification guarantee, and numerical experiments support the theoretical convergence results.

Abstract

from arXiv · show

This paper studies the identification of linearly parameterized nonlinear (LPN) systems. Although LPN systems share the same linear parameterization structure as linear systems, they are more challenging to identify. In particular, previous work has shown, through a counterexample based on a piecewise-affine system, that non-active exploration is generally insufficient for LPN system identification. In this paper, we consider LPN systems with real-analytic feature functions. We show that non-active exploration is sufficient for the identification of this class of systems by establishing non-asymptotic convergence rates of least-squares estimation and set-membership estimation. In addition, we provide counterexamples to show that non-active exploration may not be sufficient for system identification for non-real-analytic systems, even if those systems are infinitely differentiable. We present numerical experiments to further support and validate our theoretical results.

I. INTRODUCTION

The paper addresses whether non-active exploration can identify LPN systems, despite prior counterexamples for general LPN systems. It establishes sufficiency under real-analytic features and analyzes least-squares and set-membership estimation.

  • LPN systems use linear parameterization with nonlinear feature functions and apply to robotics, power systems, and transportation.
  • Prior work showed that non-active exploration can fail for general LPN systems, including a piecewise-affine counterexample under bounded process noise.
  • The paper asks when non-active exploration remains effective, motivated by the gap between infinitely differentiable bilinear systems and non-differentiable piecewise-affine systems.
  • Non-active exploration has practical value for single-trajectory online learning because control inputs must balance control objectives with information acquisition.
  • Under real-analytic feature functions and semi-continuous noises, the paper establishes non-asymptotic convergence rates for least-squares estimation and set-membership estimation.
  • Counterexamples show that non-active exploration may remain insufficient for non-real-analytic systems, even when those systems are infinitely differentiable.
  • Numerical experiments compare theoretical and empirical convergence rates for LSE and SME, along with active exploration and a fast SME algorithm.
  • The paper also connects its analysis to related work on bilinear systems, nonlinear regression, uncertainty-set estimation, and randomly perturbed closed-loop policies.

Conditions and Assumptions

The analysis assumes real-analytic features, semi-continuous bounded noise, and local input-to-state stability. These conditions support bounded trajectories and can be applied to examples including pendulums, drones, and GELU-based models.

  • Feature functions: Real analyticity means each feature function has a convergent local power-series representation; the paper assumes this property for every feature.
  • Feature functions: The real-analytic class includes polynomial, trigonometric, and exponential functions, and the property is preserved under addition and multiplication.
  • Feature functions: The global analyticity assumption can be relaxed to local analyticity on the reachability set.
  • Examples: A pendulum model fits the LPN form because its features combine linear terms with sin(αt), which is real analytic.
  • Examples: A discretized drone model is an LPN system with third-order polynomial features and unknown mass and inertia parameters.
  • Examples: GELU-based models are LPN systems when only the final linear-layer weights adapt, because GELU-generated features are real analytic.
  • Noise distributions: Semi-continuity excludes purely discrete noise distributions while allowing all continuous distributions and mixtures containing a continuous component.
  • Noise distributions: The noise and control perturbation sequences are mutually independent, i.i.d., zero-mean sub-Gaussian, bounded, and semi-continuous.

III. THEORETICAL RESULTS

The paper establishes BMSB for real-analytic LPN feature vectors and uses it to derive non-asymptotic LSE rates under non-active exploration, including randomly perturbed closed-loop policies.

  • Block-Martingale Small-Ball Condition: The section develops BMSB for LPN systems as the key condition underlying finite-sample identification analysis.BMSB provides directional anti-concentration and is related to persistent excitation.
  • Block-Martingale Small-Ball Condition: Under the stated assumptions and i.i.d. random inputs, LPN feature vectors satisfy a BMSB condition with block length 1 and horizon-independent positive parameters.The parameters are sϕ > 0 and 0 < pϕ < 1.
  • Theoretical Results: The BMSB result supports convergence-rate analyses for both least square estimation and set-membership estimation.The paper explicitly uses the condition as a foundation for analyzing these methods.
  • Least Square Estimation: The LSE guarantee holds with probability at least 1 −3δ and requires a sufficiently large T whose lower bound is constant in T.The estimator also assumes uniformly bounded feature vectors.
  • Least Square Estimation: LSE converges to the true parameter matrix at rate 1/√T under real-analytic features and semi-continuous noise distributions.This rate matches the reported dependence on the sampling horizon for linear and bilinear systems under i.i.d. random inputs.
  • Least Square Estimation: The convergence analysis has explicit dimension dependence √nx + nϕ, while the implicit dependence of BMSB parameters on dimension remains unresolved for general nonlinear systems.The paper identifies this dependence as a future research direction.

C. Randomly Perturbed Control Policies

The paper extends its exploration and convergence results from open-loop stability to systems stabilized by a known real-analytic controller, using randomly perturbed closed-loop policies.

  • Motivation and Assumptions: The closed-loop extension targets nonlinear systems that require a known stabilizing controller because they are not open-loop stable.The policy is assumed to make the closed-loop system locally input-to-state stable.
  • Motivation and Assumptions: Assumption 4 permits any real-analytic policy that stabilizes the closed-loop system, with the zero policy recovering the open-loop setting.This broadens the applicability beyond open-loop stability.
  • Results: Under a randomly perturbed stabilizing policy, feature vectors satisfy a BMSB condition with positive parameters and remain uniformly bounded.The resulting constants are denoted ˜sϕ, ˜pϕ, and ˜ϕmax.
  • Results: LSE retains the O(1/√T) convergence rate under the randomly perturbed closed-loop policy.The proof is deferred to the appendix, while the rate follows from the closed-loop BMSB and boundedness results.
  • Proof Strategy: The BMSB proof uses real-analyticity, semi-continuous noise distributions, and anti-concentration of feature projections.Real-analyticity makes nontrivial zero sets measure zero, supporting positive small-ball probabilities.
  • Proof Strategy: The analysis establishes positive, horizon-independent terms that yield valid BMSB parameters.The construction concludes with Term 1 > 0 and pϕ > 0 independent of T.

B. Proof of Theorem 2

The LSE proof combines the BMSB condition with a general correlated-data linear-regression bound under sub-Gaussian noise and adapted regressors.

  • Proof Strategy: The estimation-error proof is based on the BMSB condition and a general error bound for linear regression with correlated data.This connects the system-specific exploration result to a general statistical estimation proposition.
  • Assumptions: The regression proposition assumes zero-mean conditionally sub-Gaussian noise with proxy κw.The noise is conditioned on the natural filtration.
  • Assumptions: It also requires the regressors to satisfy a (k, Γsb, p)-BMSB condition and an additional finite-probability condition.The supplied proposition states these as Conditions II and III.
  • Conclusion: Applying the proposition yields the stated high-probability LSE error bound for the estimator ˆΘT.The result holds with probability at least 1 −3δ.

V. CONVERGENCE RATES OF SET MEMBERSHIP ESTIMATION

The paper analyzes SME for LPN systems by bounding the uncertainty-set diameter and failure probability, obtaining a ˜O(1/T) rate under a boundary-visiting noise condition.

  • Set-Membership Estimation: SME directly estimates the parameter uncertainty set, and its convergence is studied through the set’s diameter.The true parameter is included whenever the disturbance set is valid.
  • Assumptions: SME implementation requires bounded disturbances, while the paper’s convergence-rate analysis additionally studies i.i.d. disturbances.The paper distinguishes algorithmic requirements from analytical assumptions.
  • Assumptions: Convergence requires boundary visiting: disturbances must enter every small neighborhood of both wmax and −wmax with positive probability.This tightness condition is identified as necessary for SME convergence.
  • Main Result: Under the stated assumptions and i.i.d. random inputs, Theorem 3 provides a convergence rate for SME’s uncertainty-set diameter and bounds its failure probability.The theorem applies for m ≥1, ϱ ∈(0, 1), and T > m.
  • Main Result: When qw(ℓ) = cwℓ, SME achieves a ˜O(1/T) rate in T under the additional boundary-visiting assumption.The paper compares this with the O(1/√T) LSE rate in sampling-horizon dependence.
  • Closed-Loop Extension: The same SME convergence guarantees extend to randomly perturbed closed-loop policies, including the ˜O(1/T) special case.The closed-loop result uses the modified parameters introduced for the perturbed policy.

VI. MORE DISCUSSIONS ON ASSUMPTIONS

The paper examines the real-analytic and semi-continuity assumptions because they are nonstandard in recent finite-sample system-identification literature and supports their importance with counterexamples.

  • VI. MORE DISCUSSIONS ON ASSUMPTIONS: The section discusses the real-analytic feature-function condition and the semi-continuity noise condition adopted in the paper.Both assumptions receive focused discussion because they are not standard in recent finite-sample system-identification research.
  • VI. MORE DISCUSSIONS ON ASSUMPTIONS: The paper constructs counterexamples to demonstrate the importance of these two conditions.
  • VI. MORE DISCUSSIONS ON ASSUMPTIONS: The discussion specifically focuses on assumptions that support the paper’s finite-sample identification analysis.

A. More Discussions on Real-Analytic Condition

The paper shows that higher-order differentiability alone does not ensure successful non-active identification: an infinitely differentiable, non-real-analytic counterexample remains unidentifiable under bounded random inputs, while Gaussian-noise implications remain unresolved.

  • A. More Discussions on Real-Analytic Condition: i.i.d. random exploration is insufficient to identify the counterexample system under bounded noises.
  • A. More Discussions on Real-Analytic Condition: The paper bridges prior results for non-differentiable piecewise-affine and real-analytic bilinear systems by studying the role of real analyticity in non-active exploration.
  • A. More Discussions on Real-Analytic Condition: The authors construct a counterexample system with feature functions that are infinitely differentiable but not real-analytic.The construction uses a standard infinitely differentiable, non-real-analytic function.
  • A. More Discussions on Real-Analytic Condition: Under bounded uniform process noise and inputs, the counterexample satisfies the other stated assumptions but violates the real-analytic feature condition.The parameters are a∗ = b∗ = 1, with process noise uniform on [−1, 1] and inputs uniform on [−2, 2].
  • A. More Discussions on Real-Analytic Condition: The state remains at most 3, so the informative region x_t > 4 is never reached and a∗ cannot be identified, although b∗ estimation can converge.Figure 1 reports the same persistent a∗ estimation error for both LSE and SME.
  • A. More Discussions on Real-Analytic Condition: For a multidimensional variant, informative states occur with exponentially small probability, producing exponentially large sample complexity in the state dimension.
  • A. More Discussions on Real-Analytic Condition: The corresponding sample complexity for real-analytic functions under Gaussian noises is left for future studies.

B. More Discussions on Semi-Continuous Condition

The section constructs counterexamples showing that violating semicontinuity in disturbances or control inputs can prevent parameter identification under non-active exploration. In both cases, estimation fails along unidentifiable parameter directions.

  • The counterexamples satisfy the theorem’s other assumptions but violate semicontinuity, demonstrating that parameters may remain unidentified under non-active exploration.
  • With disturbances uniform on {−π, π}, α_t remains in {−π, π}, making a* and b* unidentifiable because sin(α_t)=0.
  • With control inputs uniform on {−π, π}, d* is unidentifiable because sin(u_t)=0 for every t.
  • Figure 2 shows that LSE errors for selected parameters do not vanish, while SME uncertainty sets fail to shrink along unidentifiable directions.
  • The numerical experiments are intended to demonstrate the paper’s theoretical results.

A. Experiment Settings

Experiments evaluate LSE and SME convergence on pendulum and drone systems, then compare non-active and active identification methods using estimation error, uncertainty width, sample count, and wall-clock time.

  • Experimental systems: Pendulum simulations use feedback-plus-noise control, while drone simulations use a regulating controller plus exploration noise; both use ΔT = 0.01 seconds.The pendulum controller gain varies by figure, and the drone controller regulates altitude and Euler angles.
  • LSE convergence: O(1/T) empirical LSE error decay matches the theoretical rate across pendulum and drone scenarios, although theoretical bounds are conservatively larger.Errors are normalized by the nominal parameter l2 norm and averaged across 20 trials.
  • SME convergence: O(1/T) empirical SME convergence matches Corollary 2 for pendulum and is similarly observed for the drone cases.Theoretical and empirical quantities are normalized by the nominal parameter l2 norm, with averages over 10 trials.
  • Uncertainty sets: SME uncertainty sets contract as trajectory length increases while containing the true parameters in the pendulum and drone experiments.The pendulum experiment also reports shrinking set diameters over increasing trajectory lengths.
  • Method comparison: After the initial transient, SME achieves the smallest uncertainty width, with vanilla SME tightest and Fast SME closely approximating it.Active exploration reduces estimation error faster initially and consistently outperforms non-active LSE during that stage.
  • Method comparison: At ε = 10^-4, SME requires substantially fewer samples but higher per-sample cost, whereas LSE has the lowest per-update cost and needs many more samples.Active exploration improves non-active LSE sample efficiency, with wall time between the two approaches.

APPENDIX

The appendix supplies supporting arguments for closed-loop applicability, semi-continuity of joint noise distributions, and the measure-theoretic characterization used in the analysis.

  • Closed-loop reduction: A stabilizing controller converts the closed-loop dynamics into an equivalent LPN system with modified feature functions.The modified features are defined by composing the original features with the controller.
  • Closed-loop reduction: Real-analyticity is preserved under this composition, allowing the closed-loop system to satisfy the theorem assumptions.The stabilizing property also gives open-loop stability for the transformed system.
  • Closed-loop guarantees: The closed-loop system inherits a BMSB condition, bounded features, and an O(1/T) LSE convergence rate.These conclusions follow by applying Theorems 1 and 2 after verifying the transformed assumptions.
  • Semi-continuity: For mutually independent individually semi-continuous noises, the joint distribution is semi-continuous because it has a nonzero absolutely continuous component.The proof uses Lebesgue decomposition and independence of the two noise distributions.
  • Semi-continuity: A probability measure is semi-continuous if and only if its absolutely continuous component is nonzero.If that component vanishes, the measure is concentrated on a Lebesgue-null set.

C. Proofs for Section V on SME

The SME analysis applies a general linear-regression diameter bound under BMSB and bounded-feature conditions, yielding explicit finite-sample convergence rates for open- and closed-loop systems.

  • Proof strategy: SME convergence proofs combine the LPN BMSB condition with a general linear-regression SME result.Theorem 3 verifies the regression assumptions using the BMSB condition and boundedness established for the LPN features.
  • Noise conditions: The proof uses noise small-ball behavior of the form q_w(ℓ) = c_wℓ, including explicit constants for uniform and truncated-Gaussian disturbances.For uniform noise, c_w = 1/(2w_max).
  • Finite-sample guarantee: The diameter guarantee is obtained with probability at least 1 − 2δ after selecting the proof parameters m and ϱ appropriately.The bound is expressed as an uncertainty-set diameter no larger than ϱ.
  • Convergence rate: The resulting SME uncertainty-set diameter has a ˜O(1/T) convergence rate under the stated conditions.The closed-loop analogue follows from Corollary 1 and the same diameter-bound argument.

D. More Details for Numerical Experiments

The appendix describes numerical estimation of BMSB parameters and a fixed-complexity polytope approximation used by Fast SME.

  • BMSB estimation: The numerical procedure fixes s_ϕ, estimates p_ϕ, and decreases s_ϕ until 0 < p_ϕ < 1 satisfies the BMSB conditions.This provides a numerical pair of parameters for the theoretical analysis.
  • BMSB estimation: The p_ϕ estimate uses independent trajectories and Monte Carlo sampling of disturbances and inputs, computed as the fraction satisfying the BMSB condition.Samples are generated conditionally on the current state information.
  • Fast SME: Fast SME maintains a fixed-complexity polytopic outer approximation of the exact unfalsified set.Its constraint rows come from a regular polytope, while the right-hand side is tightened at each step.
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