Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Probabilistic Reachability of Discrete-Time Nonlinear Stochastic Systems

Zishun Liu, Saber Jafarpour, Yongxin Chen

Abstract

In this paper we study the reachability problem for discrete-time nonlinear stochastic systems. Our goal is to present a unified framework for calculating the probabilistic reachable set of discrete-time systems in the presence of both deterministic input and stochastic noise. By adopting a suitable separation strategy, the probabilistic reachable set is decoupled into a deterministic reachable set and the effect of the stochastic noise. To capture the effect of the stochastic noise, in particular sub-Gaussian noise, we provide a probabilistic bound on the distance between a stochastic trajectory and its deterministic counterpart. The key to our approach is a novel energy function called the Averaged Moment Generating Function, which we leverage to provide a high probability bound on this distance. We show that this probabilistic bound is tight for a large class of discrete-time nonlinear stochastic systems and is exact for linear stochastic dynamics. By combining this tight probabilistic bound with the existing methods for deterministic reachability analysis, we propose a flexible framework that can efficiently compute probabilistic reachable sets of stochastic systems. We also provide two case studies for applying our framework to Lipschitz bound reachability and interval-based reachability. Three numerical experiments are conducted to validate the theoretical results.

Probabilistic Reachability of Discrete-Time Nonlinear Stochastic Systems

Abstract

In this paper we study the reachability problem for discrete-time nonlinear stochastic systems. Our goal is to present a unified framework for calculating the probabilistic reachable set of discrete-time systems in the presence of both deterministic input and stochastic noise. By adopting a suitable separation strategy, the probabilistic reachable set is decoupled into a deterministic reachable set and the effect of the stochastic noise. To capture the effect of the stochastic noise, in particular sub-Gaussian noise, we provide a probabilistic bound on the distance between a stochastic trajectory and its deterministic counterpart. The key to our approach is a novel energy function called the Averaged Moment Generating Function, which we leverage to provide a high probability bound on this distance. We show that this probabilistic bound is tight for a large class of discrete-time nonlinear stochastic systems and is exact for linear stochastic dynamics. By combining this tight probabilistic bound with the existing methods for deterministic reachability analysis, we propose a flexible framework that can efficiently compute probabilistic reachable sets of stochastic systems. We also provide two case studies for applying our framework to Lipschitz bound reachability and interval-based reachability. Three numerical experiments are conducted to validate the theoretical results.
Paper Structure (25 sections, 7 theorems, 63 equations, 6 figures)

This paper contains 25 sections, 7 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Statement and Preliminaries
  3. Deterministic Reachable Set
  4. Probabilistic Reachable Set
  5. Separation Strategy and Main Challenge
  6. Expectation Bound and Limitations
  7. Expectation Bound on Stochastic Deviation
  8. Limitation of The Expectation Bound
  9. Probabilistic Bound on Stochastic Deviation
  10. Motivation: Moment Generating Function
  11. Averaged Moment Generating Function
  12. Theoretical Analysis
  13. Tightness of The Bound
  14. Comparison with Worst-Case Analysis
  15. Probabilistic Reachable Set
  16. ...and 10 more sections

Key Result

Proposition 1

Consider the stochastic system sys: d-t ss with its associated deterministic system sys: d-t ds. Given $\mathcal{X}_0\subseteq \mathbb{R}^n$, $\mathcal{U}\subseteq \mathbb{R}^p$, let $X_t$ and $x_t$ be the associated trajectories with inputs $u_t\in\mathcal{U}$ and the initial state $x_0\in\mathcal{ then for any over-approximation $\overline{\mathcal{R}}_t$ of the DRS $\mathcal{R}_t$ of sys: d-t d

Figures (6)

  • Figure 1: An illustration of $\delta$-PRS at time $t$. Here $\mathcal{R}_{\delta,t}$ is the $\delta$-PRS of the stochastic system \ref{['sys: d-t ss']}, whose trajectories are in color. $\mathcal{R}_t$ is the DRS of the associated deterministic system \ref{['sys: d-t ds']}, whose trajectories are in black.
  • Figure 2: An illustration of separation strategy. Here $\mathcal{R}_{\delta,t}$ is the $\delta$-PRS of the stochastic system \ref{['sys: d-t ss']}, whose trajectory is $X_t$ in red. $\overline{\mathcal{R}}_t$ is the over-approximation of the DRS of the associated deterministic system \ref{['sys: d-t ds']}, whose trajectory is $x_t$ in black. The Minkowski sum corresponds to Proposition \ref{['prop: separation']}.
  • Figure 3: Probabilistic bound of stochastic deviation for a linear system. Left: each curve represents an independent trajectory of $X_t$ The radius of the blue envelope at time $t$ is the bound \ref{['eq: thm1']}. Right: each solid curve is an independent trajectory of $\|X_t-x_t\|$. The blue dashed curve is the bound \ref{['eq: thm1']}. The red dashed line is the bound \ref{['eq: sd by worst']}.
  • Figure 4: Illustration of the tightness of $r_{\delta,t}$ w.r.t. $\delta,n$. Left: the solid line shows the dependence of $r_{\delta,t}^2$ over $1/\delta$ and the dotted line in the same color is the corresponding simulated bound $\hat{r}_{\delta,t}^2$. The time is fixed as $t=25$. Right: the solid line shows the dependence of $r_{\delta,t}^2$ over $n$, and the dotted line in the same color is the corresponding simulated $\hat{r}_{\delta,t}^2$. $\delta=10^{-4}$ is fixed.
  • Figure 5: Evolution of the 2000 independent trajectories of the system \ref{['eq: eco sys']} and the $\delta$-PRS at $t=1$(upper-left), $t=2$(upper-right), $t=3$(lower-left) and $t=5$(lower-right). The purple circles are $\delta$-PRS calculated by Proposition \ref{['prop: case 1']}. The blue dashed circles are Lipschitz-bound DRS. The red points are the state of the trajectories at different times.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 2.1: DRS
  • Definition 2.2: sub-Gaussian
  • Definition 2.3: $\delta$-PRS
  • Proposition 1
  • Proposition 2
  • Lemma 3.1: Norm Concentration
  • Definition 4.1: AMGF
  • Lemma 4.1: Properties of AMGF
  • Lemma 4.2
  • Remark 4.1
  • ...and 3 more