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Concentration of Stochastic System Trajectories with Time-varying Contraction Conditions

Zishun Liu, Liqian Ma, Hongzhe Yu, Yongxin Chen

Abstract

We establish two concentration inequalities for nonlinear stochastic system under time-varying contraction conditions. The key to our approach is an energy function termed Averaged Moment Generating Function (AMGF). By combining it with incremental stability analysis, we develop a concentration inequality that bounds the deviation between the stochastic system state and its deterministic counterpart. As this inequality is restricted to single time instance, we further combine AMGF with martingale-based methods to derive a concentration inequality that bounds the fluctuation of the entire stochastic trajectory. Additionally, by synthesizing the two results, we significantly improve the trajectory-level concentration inequality for strongly contractive systems. Given the probability level $1-δ$, the derived inequalities ensure an $\mO(\sqrt{\log(1/δ))}$ bound on the deviation of stochastic trajectories, which is tight under our assumptions. Our results are exemplified through a case study on stochastic safe control.

Concentration of Stochastic System Trajectories with Time-varying Contraction Conditions

Abstract

We establish two concentration inequalities for nonlinear stochastic system under time-varying contraction conditions. The key to our approach is an energy function termed Averaged Moment Generating Function (AMGF). By combining it with incremental stability analysis, we develop a concentration inequality that bounds the deviation between the stochastic system state and its deterministic counterpart. As this inequality is restricted to single time instance, we further combine AMGF with martingale-based methods to derive a concentration inequality that bounds the fluctuation of the entire stochastic trajectory. Additionally, by synthesizing the two results, we significantly improve the trajectory-level concentration inequality for strongly contractive systems. Given the probability level , the derived inequalities ensure an bound on the deviation of stochastic trajectories, which is tight under our assumptions. Our results are exemplified through a case study on stochastic safe control.

Paper Structure

This paper contains 8 sections, 5 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Concentration of System State at Single Time
  4. Concentration of Stochastic Trajectories
  5. Martingale-Based Concentration Inequality
  6. Improvement for Strongly Contracting Systems
  7. Case Study
  8. Conclusion

Key Result

Lemma 1

Consider the function $\Phi_M(X)$ in Definition def: AMGF, where $X\in\mathbb{R}^n$ and $M\in\mathbb{R}^{n\times n}_{SPD}$, then it holds that: $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Comparison of the single-time bound $r_{\delta}$ (Left) and the trajectory-level bound $\overline{r}_{\delta,t}$ (Right).
  • Figure 2: Comparison of the $\overline{r}_{\delta,t}$ derived by Theorem \ref{['thm: traj_bound']} (Left) and Thorem \ref{['thm: traj c<0']}. The experiment is done on a linear system $\mathrm{d}X_t=A_tX_t+\Sigma\mathrm{d}W_t$ with strongly contracting $A_t$. Each figure contains 5000 independent trajectories of $\|X_t-x_t\|_{M_t}$, and use $\delta=0.001$, $\varepsilon=15/16$.
  • Figure 3: Safe planning for the PVTOL system. The robot must reach the goal region while avoiding obstacles. Left: The obstacles are enlarged according to the computed probabilistic bound \ref{['eq: r traj c<0']}. The nominal trajectory of the PVTOL system remains collision-free and reaches the shrunken goal region. Right: $10^4$ stochastic rollouts of the TVLQR-controlled system.

Theorems & Definitions (11)

  • Definition 1: Contracting System, tsukamoto2021contraction
  • Definition 2
  • Lemma 1
  • Theorem 1
  • proof
  • Definition 3: Affine Martingale, liu2025safety
  • Lemma 2
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more