Finite Sample Identification of Analytic Nonlinear Systems

• LPN systems use linear parameterization with nonlinear feature functions and apply to robotics, power systems, and transportation.Source: §I · p. 1
• Prior work showed that non-active exploration can fail for general LPN systems, including a piecewise-affine counterexample under bounded process noise.Source: §I · p. 1
• The paper asks when non-active exploration remains effective, motivated by the gap between infinitely differentiable bilinear systems and non-differentiable piecewise-affine systems.Source: §I · p. 1
• Non-active exploration has practical value for single-trajectory online learning because control inputs must balance control objectives with information acquisition.Source: §I · p. 1
• Under real-analytic feature functions and semi-continuous noises, the paper establishes non-asymptotic convergence rates for least-squares estimation and set-membership estimation.Source: §I · p. 1
• Counterexamples show that non-active exploration may remain insufficient for non-real-analytic systems, even when those systems are infinitely differentiable.Source: §I · pp. 1–2
• Numerical experiments compare theoretical and empirical convergence rates for LSE and SME, along with active exploration and a fast SME algorithm.Source: §I · p. 2
• The paper also connects its analysis to related work on bilinear systems, nonlinear regression, uncertainty-set estimation, and randomly perturbed closed-loop policies.Source: §I · p. 2

Feature functions

• Real analyticity means each feature function has a convergent local power-series representation; the paper assumes this property for every feature.Source: Conditions and Assumptions · p. 3
• The real-analytic class includes polynomial, trigonometric, and exponential functions, and the property is preserved under addition and multiplication.Source: Conditions and Assumptions · p. 3
• The global analyticity assumption can be relaxed to local analyticity on the reachability set.Source: Conditions and Assumptions · p. 3

Examples

• A pendulum model fits the LPN form because its features combine linear terms with sin(αt), which is real analytic.Source: Conditions and Assumptions · p. 3
• A discretized drone model is an LPN system with third-order polynomial features and unknown mass and inertia parameters.Source: Conditions and Assumptions · p. 3
• GELU-based models are LPN systems when only the final linear-layer weights adapt, because GELU-generated features are real analytic.Source: Conditions and Assumptions · p. 3

Noise distributions

• Semi-continuity excludes purely discrete noise distributions while allowing all continuous distributions and mixtures containing a continuous component.Source: Conditions and Assumptions · p. 3
• The noise and control perturbation sequences are mutually independent, i.i.d., zero-mean sub-Gaussian, bounded, and semi-continuous.Source: Conditions and Assumptions · p. 4

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.Source: §III · p. 4, A. Block-Martingale Small-Ball Condition · pp. 4–5
• 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.Source: A. Block-Martingale Small-Ball Condition · p. 5

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.Source: A. Block-Martingale Small-Ball Condition · p. 5

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.Source: B. Convergence Rates of Least Square… · p. 5
• 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.Source: B. Convergence Rates of Least Square… · p. 5
• 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.Source: B. Convergence Rates of Least Square… · p. 5, A. Block-Martingale Small-Ball Condition · p. 5

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.Source: C. Randomly Perturbed Control Policies · p. 5
• 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.Source: C. Randomly Perturbed Control Policies · pp. 5–6

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.Source: C. Randomly Perturbed Control Policies · p. 6
• 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.Source: C. Randomly Perturbed Control Policies · p. 6

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.Source: A. Proof of Theorem 1 · pp. 6–7
• 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.Source: A. Proof of Theorem 1 · pp. 6–7

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.Source: B. Proof of Theorem 2 · p. 8

Assumptions

• The regression proposition assumes zero-mean conditionally sub-Gaussian noise with proxy κw.The noise is conditioned on the natural filtration.Source: B. Proof of Theorem 2 · p. 8
• 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.Source: B. Proof of Theorem 2 · p. 8

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.Source: B. Proof of Theorem 2 · p. 8

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.Source: §V · p. 8

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.Source: §V · p. 8
• 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.Source: §V · p. 8

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.Source: §V · p. 8
• 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.Source: §V · p. 9

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.Source: §V · p. 9

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.Source: §VI · p. 9
• The paper constructs counterexamples to demonstrate the importance of these two conditions.Source: §VI · p. 9
• The discussion specifically focuses on assumptions that support the paper’s finite-sample identification analysis.Source: §VI · p. 9

A. More Discussions on Real-Analytic Condition

• i.i.d. random exploration is insufficient to identify the counterexample system under bounded noises.Source: A. More Discussions on Real-Analytic… · p. 9
• 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.Source: A. More Discussions on Real-Analytic… · p. 9
• 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.Source: A. More Discussions on Real-Analytic… · p. 9
• 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].Source: A. More Discussions on Real-Analytic… · p. 9
• 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.Source: A. More Discussions on Real-Analytic… · p. 9, Figure 1 · p. 10
• For a multidimensional variant, informative states occur with exponentially small probability, producing exponentially large sample complexity in the state dimension.Source: A. More Discussions on Real-Analytic… · pp. 9–10
• The corresponding sample complexity for real-analytic functions under Gaussian noises is left for future studies.Source: A. More Discussions on Real-Analytic… · p. 10

• The counterexamples satisfy the theorem’s other assumptions but violate semicontinuity, demonstrating that parameters may remain unidentified under non-active exploration.Source: B. More Discussions on Semi-Continuous… · p. 10
• With disturbances uniform on {−π, π}, α_t remains in {−π, π}, making a* and b* unidentifiable because sin(α_t)=0.Source: B. More Discussions on Semi-Continuous… · p. 10
• With control inputs uniform on {−π, π}, d* is unidentifiable because sin(u_t)=0 for every t.Source: B. More Discussions on Semi-Continuous… · p. 10
• Figure 2 shows that LSE errors for selected parameters do not vanish, while SME uncertainty sets fail to shrink along unidentifiable directions.Source: B. More Discussions on Semi-Continuous… · p. 10, Figure 2 · p. 10
• The numerical experiments are intended to demonstrate the paper’s theoretical results.Source: §VII · p. 11

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.Source: A. Experiment Settings · p. 11

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.Source: B. Convergence Rates of LSE · p. 11

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.Source: C. Convergence Rates of SME · p. 11

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.Source: C. Convergence Rates of SME · p. 11, Figure 5 · p. 12, Figure 6 · p. 12

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.Source: D. Numerical Comparison with Other… · p. 12, Figure 7 · p. 13
• 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.Source: D. Numerical Comparison with Other… · p. 12, Table I · p. 12

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.Source: A. Proof of Corollary 1 · p. 13
• 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.Source: A. Proof of Corollary 1 · p. 13

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.Source: A. Proof of Corollary 1 · p. 13

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.Source: B. Semi-Continuity Property in the… · p. 13
• 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.Source: B. Semi-Continuity Property in the… · p. 13

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.Source: C. Proofs for Section V on SME · p. 14

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).Source: C. Proofs for Section V on SME · p. 14

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.Source: C. Proofs for Section V on SME · p. 14

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.Source: C. Proofs for Section V on SME · p. 14

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.Source: D. More Details for Numerical… · p. 14
• 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.Source: D. More Details for Numerical… · p. 15

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.Source: D. More Details for Numerical… · p. 15