Beyond Bounded Noise: Stochastic Set-Membership Estimation for Nonlinear Systems

Abstract: In this paper, we derive a novel procedure for set-membership estimation of dynamical systems affected by stochastic noise with unbounded support. By employing a bound on the sample covariance matrix, we are able to provide a finite-sample uncertainty set containing the true system parameters with high probability. Our approach can be natively applied to a wide class of nonlinear systems affected by sub- Gaussian noise. Through our analysis, we provide conditions under which the proposed uncertainty set converges to the true system parameters and establish an upper bound on the convergence rate. The proposed uncertainty set can be used directly for the synthesis of robust controllers with probabilistic stability and performance guarantees. Concluding numerical examples demonstrate the advantages of the proposed formulation over established approaches.

• The paper develops a novel SME methodology that constructs finite-sample uncertainty sets for nonlinear systems exposed to unbounded, sub-Gaussian noise.
• It leverages non-asymptotic concentration bounds on noise covariance to parameterize a Quadratic Matrix Inequality, ensuring high-probability inclusion of true system parameters.
• The approach guarantees an O(N^(-1/2)) contraction rate for the uncertainty set, making it applicable for robust controller synthesis in both linear and nonlinear regimes.

Stochastic Set-Membership Estimation for Nonlinear Systems with Unbounded Noise

Problem Formulation and Motivation

Set-Membership Estimation (SME) is a central paradigm for robust system identification under bounded-noise assumptions, yielding parameter uncertainty sets that are directly applicable to robust control synthesis. However, most SME variants critically depend on the availability of a tight bound on the noise support, which precludes their application to systems with realistic stochastic disturbances—particularly those with unbounded support such as Gaussian or sub-Gaussian noise. Addressing this gap, the paper develops a new SME methodology for general (potentially nonlinear) systems subject to i.i.d. sub-Gaussian, isotropic noise with unbounded support.

For a discrete-time system governed by $x_{t+1} = \theta_* z_t + w_t$ (with $z_t$ a possibly nonlinear lifting of past state and input), the challenge is constructing a finite-sample uncertainty set of parameter matrices $\Theta_N$ that contains the true system parameters $\theta_*$ with high probability, without reliance on unrealistic support constraints on $w_t$.

Methodological Innovation

The key conceptual advance is leveraging high-probability bounds on the empirical sample covariance of the noise, derived via non-asymptotic results from high-dimensional probability theory. Specifically, the approach utilizes a probabilistic matrix inequality for the noise sample covariance, such as

$$\left\|\frac{1}{N}\sum_{t} w_t w_t^\top - I\right\| \leq \eta$$

with $\eta=O\left(\sqrt{\frac{n_x}{N}}\right)$, holding with probability at least $1-\delta$ for sub-Gaussian, isotropic noise. This result, adapted from Vershynin's concentration bounds [vershynin2012], underpins the construction of a high-probability set for the entire sequence of noise realizations $W_N$.

By exploiting the affine relationship $X_N = \theta Z_N + W_N$, the authors parameterize the set of system matrices compatible with both the observed data and the noise covariance bound as the solution set to a specific Quadratic Matrix Inequality (QMI):

$z_t$0

where $z_t$1 encapsulates the probabilistic upper bound on the empirical covariance, with explicit non-asymptotic dependence on the sample size $z_t$2, the state dimension $z_t$3, and user-defined failure probability $z_t$4.

This approach is directly compatible with nonlinear system representations via arbitrary (known) nonlinear liftings $z_t$5, extending robust identification methods to a significantly broader class of systems.

Statistical Guarantees: Consistency and Convergence Rate

Under standard Persistency of Excitation (PE) assumptions on the regressors (ensuring well-conditioned data), the analysis establishes two core statistical guarantees:

• Consistency: The parameter set $z_t$6 converges to the true parameter $z_t$7 asymptotically as $z_t$8, provided the sample covariance of the noise converges to its population value and the OLS estimator is consistent (i.e., for stable, persistently-excited systems).
• Convergence Rate: The uncertainty set contracts at an $z_t$9 rate, in the sense that:

$\Theta_N$0

This rate is provably slower than the $\Theta_N$1 statistical rate of the OLS estimator due to the additional relaxation required to accommodate the lack of noise support bounds. Nevertheless, it guarantees that the set-centric description for robust control becomes asymptotically tight.

The procedure also admits in-built hypothesis testing: If the assumed noise covariance is conservative (overestimated), the uncertainty set is conservative but not vacuous; if underestimated, the set becomes empty, indicating a breakdown in the underlying model or assumptions.

Numerical Validation

Empirical results are provided for both linear and nonlinear systems, illustrating key behaviors:

• For LTI systems, the approach yields parameter uncertainty sets whose volume and convergence properties match the theoretical rate and reliably contain the true system with high probability, provided noise model assumptions are valid.
• For nonlinear systems (e.g., a nonlinear pendulum), the method naturally extends via a nonlinear regressor lifting, preserving the same qualitative features.
• Compared to state-of-the-art SME-based techniques using support bounds, the proposed method is robust to model misspecification regarding noise variance, offering a distinct qualitative diagnostic (empty set or non-convergent set) when model assumptions are invalid.

Implications for Robust Data-Driven Control

By formulating the parameter uncertainty set as a QMI, the method is amenable to direct deployment in robust control synthesis pipelines for both linear and nonlinear systems. Controllers designed for all systems within $\Theta_N$2 are certifiably robust with high probability, inheriting the probabilistic coverage of the estimation stage. This closes an open gap in the data-driven control literature, enabling robust synthesis under realistic stochastic excitation without requiring bounded noise.

Additionally, because the construction is independent of the specific nonlinear regressor $\Theta_N$3, the framework admits straightforward integration with kernel methods and other rich function class representations.

Theoretical and Practical Outlook

The results extend SME theory and robust identification methodology to stochastic regimes previously out of reach. Key theoretical contributions include: rigorous high-probability finite-sample guarantees, direct QMI-based uncertainty representations for nonlinear systems, and a tight characterization of the qualitative impact of mis-specified noise models.

Several avenues remain open:

• Convergence Rate Enhancement: Investigation into leveraging both upper and lower covariance matrix bounds for sharper rates, or combining with alternative concentration inequalities, may yield improved finite-sample performance.
• Unstable System Identification: For unstable LTI systems (where OLS is known to be inconsistent), further work is needed to reliably certify SME-based sets.
• Extension to Non-i.i.d. and Heavy-Tailed Noise: Accommodating more general stochastic disturbances is of significant practical interest.

Conclusion

This work provides a technically rigorous, broadly applicable mechanism for set-membership identification under sub-Gaussian stochastic noise, covering both linear and nonlinear dynamical systems. The probabilistic QMI formulation is both statistically principled and directly useful for robust controller design, with explicit control over the impact of noise model mis-specification. This development substantially broadens the applicability of SME-based robust identification and control to stochastic settings that align with realistic engineering scenarios.