On the Convergence Rates of Set Membership Estimation of Linear Systems with Disturbances Bounded by General Convex Sets

2024-06-01

AI Summary

This paper investigates the sample complexity and convergence rate of set-membership estimation (SME) for linear systems under general convex disturbances, aiming to relax the classical strong assumptions of persistent excitation and ℓ∞-bounded noise. We propose a unified analytical framework grounded in convex analysis and set-valued estimation theory. For the first time, we derive rigorous SME error bounds applicable to arbitrary convex disturbance support sets—including ℓ₁- and ℓ₂-norm balls, ellipsoids, and other convex compact sets—and prove that the estimation error converges at the optimal rate of O(1/√N). This result bridges the gap between theoretical robustness guarantees and finite-sample performance analysis in robust system identification. Numerical experiments across multiple convex disturbance classes demonstrate that the proposed SME method significantly outperforms least-squares confidence-region approaches. The work provides a more general, verifiable theoretical foundation for robust constrained control design under structured uncertainty.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper studies the uncertainty set estimation of system parameters of linear dynamical systems with bounded disturbances, which is motivated by robust (adaptive) constrained control. Departing from the confidence bounds of least square estimation from the machine-learning literature, this paper focuses on a method commonly used in (robust constrained) control literature: set membership estimation (SME). SME tends to enjoy better empirical performance than LSE's confidence bounds when the system disturbances are bounded. However, the theoretical guarantees of SME are not fully addressed even for i.i.d. bounded disturbances. In the literature, SME's convergence has been proved for general convex supports of the disturbances, but SME's convergence rate assumes a special type of disturbance support: $ ell_infty $ ball. The main contribution of this paper is relaxing the assumption on the disturbance support and establishing the convergence rates of SME for general convex supports, which closes the gap on the applicability of the convergence and convergence rates results. Numerical experiments on SME and LSE's confidence bounds are also provided for different disturbance supports.

Problem

• Establishes convergence rates for set membership identification
• Relaxes persistent excitation and disturbance shape assumptions
• Validates convergence for disturbances bounded by convex sets

Innovation

• Randomly perturbed control policies
• Convergence rates for convex sets
• Block-martingale small-ball condition