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Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis

Yingying Li, Jing Yu, Lauren Conger, Taylan Kargin, Adam Wierman

TL;DR

The paper introduces non-asymptotic convergence rates for set membership estimation (SME) in unknown linear dynamical systems with bounded disturbances, offering a direct method to bound uncertainty sets rather than relying on LSE confidence regions. It provides two analytic paths: SME with a known disturbance bound yields instance-dependent rates and, for many common distributions, a diametric decay of \\tilde{O}( n_x^{1.5}(n_x+n_u)^2 / T)$; when the bound is unknown, a UCB-SME framework achieves comparable decay in T at the cost of increased dimensional dependence, with \\theta^* \\in \\hat{\\Theta}^{\\text{ucb}}_T$ with high probability and \\text{diam}(\\hat{\\Theta}^{\\text{ucb}}_T) \\leq \\tilde{O}(\,\sqrt{n_x}\,\\delta_T)$. The analysis hinges on a novel stopping-time construction and a block-martingale small-ball (BMSB) property to manage correlations between data and disturbances. The results enable non-asymptotic performance guarantees for robust adaptive control schemes (e.g., robust MPC and SLS) and are corroborated by numerical experiments showing SME and UCB-SME outperform LSE confidence regions in terms of uncertainty diameter and data-efficiency, with practical implications for safety-critical control under bounded disturbances.

Abstract

This paper studies uncertainty set estimation for unknown linear systems. Uncertainty sets are crucial for the quality of robust control since they directly influence the conservativeness of the control design. Departing from the confidence region analysis of least squares estimation, this paper focuses on set membership estimation (SME). Though good numerical performances have attracted applications of SME in the control literature, the non-asymptotic convergence rate of SME for linear systems remains an open question. This paper provides the first convergence rate bounds for SME and discusses variations of SME under relaxed assumptions. We also provide numerical results demonstrating SME's practical promise.

Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis

TL;DR

The paper introduces non-asymptotic convergence rates for set membership estimation (SME) in unknown linear dynamical systems with bounded disturbances, offering a direct method to bound uncertainty sets rather than relying on LSE confidence regions. It provides two analytic paths: SME with a known disturbance bound yields instance-dependent rates and, for many common distributions, a diametric decay of \\tilde{O}( n_x^{1.5}(n_x+n_u)^2 / T) with high probability and \\text{diam}(\\hat{\\Theta}^{\\text{ucb}}_T) \\leq \\tilde{O}(\,\sqrt{n_x}\,\\delta_T)$. The analysis hinges on a novel stopping-time construction and a block-martingale small-ball (BMSB) property to manage correlations between data and disturbances. The results enable non-asymptotic performance guarantees for robust adaptive control schemes (e.g., robust MPC and SLS) and are corroborated by numerical experiments showing SME and UCB-SME outperform LSE confidence regions in terms of uncertainty diameter and data-efficiency, with practical implications for safety-critical control under bounded disturbances.

Abstract

This paper studies uncertainty set estimation for unknown linear systems. Uncertainty sets are crucial for the quality of robust control since they directly influence the conservativeness of the control design. Departing from the confidence region analysis of least squares estimation, this paper focuses on set membership estimation (SME). Though good numerical performances have attracted applications of SME in the control literature, the non-asymptotic convergence rate of SME for linear systems remains an open question. This paper provides the first convergence rate bounds for SME and discusses variations of SME under relaxed assumptions. We also provide numerical results demonstrating SME's practical promise.
Paper Structure (44 sections, 31 theorems, 102 equations, 4 figures)

This paper contains 44 sections, 31 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Problem Formulation
  4. Set Membership Estimation (SME)
  5. Set Membership Convergence Analysis
  6. Convergence Rate of SME with Known $w_{\max}$
  7. SME with Unknown $w_{\max}$
  8. Proof Sketch of Theorem \ref{['thm: estimation err bdd']}
  9. Applications to Robust Adaptive Control
  10. Numerical Experiments
  11. Comparison of SME, SME with loose bound, UCB-SME, and LSE.
  12. Scaling with dimension.
  13. Concluding Remarks
  14. Additional notations
  15. More discussions on least square and set membership
  16. ...and 29 more sections

Key Result

Theorem 3.1

For any $m>0$ any $\delta>0$, when $T>m$, we have where $a_1 = \frac{\sigma_{z} p_{z}}{4}$, $a_2=\frac{64 b_z^2}{\sigma_{z}^2 p_{z}^2}$, $a_3= \frac{p_{z}^2}{8}$, $a_4=\frac{4b_z \sqrt{n_x}}{a_1}$, $p_z, \sigma_z, b_z$ are defined in Assumption ass: BMSB, bounded xt, $\lceil\cdot\rceil$ denotes the ceiling function, and $\textup{diam}(\cdot)$ is de

Figures (4)

  • Figure 1: A visualized toy example of uncertainty set comparison between SME in \ref{['eq:membership_set']} and LSE confidence regions in simchowitz2020naiveabbasi2011regret for a one-dimensional system $x_{t+1}=A^*x_t +B^* u_t +w_t$, with $w_t,\, u_t \in [-1, 1]$ generated i.i.d. from a truncated Gaussian distribution. Detailed experiment settings are in \ref{['append: simulation']}. Figure (a) compares the diameters of the uncertainty sets from SME and LSE 90% confidence bounds. Figure (b) and (c) visualize the the uncertainty sets after $T=5$ and $T=250$ data points.
  • Figure 2: Figures (a)-(b) compares the diameters of SME, UCB-SME, and SME with loose disturbance upper bounds that are 2, 3, 5, and 10 times larger than the true disturbance bound $w_{\max}$, as well as the baseline uncertainty set from the 90% confidence region of LSE. Figure (c) shows the convergence to the true bound $w_{\max}$ of the lower estimation $\bar{w}_{\max}$ in \ref{['equ: define wbarmax']} and the UCB $\hat{w}_{\max}$ generated by the UCB-SME algorithm in Figures (a)-(b).
  • Figure 3: Diameters of the uncertainty sets constructed by SME, UCB-SME, and LSE for systems with different dimensions.
  • Figure 4: Linear quadratic tracking of robust adaptive MPC based on SME, LSE's confidence regions, and the accurate model (OPT).

Theorems & Definitions (54)

  • Definition 2.1: Diameter of a set of matrices
  • Definition 2.3: Persistent excitation
  • Definition 2.4: BMSB simchowitz2018learning
  • Example 1: Robust (adaptive) MPC
  • Theorem 3.1: Convergence rate of SME
  • Corollary 3.2: Estimation error bound when $q_w(\epsilon)=\Omega(\epsilon)$
  • Corollary 3.3: Convergence rate with $B^*=0$ (informal)
  • Theorem 3.4: Convservative bound on $w_{\max}$
  • Theorem 3.5
  • Corollary 3.6
  • ...and 44 more