Probabilistic Reachability Analysis of Stochastic Control Systems

TL;DR

The objective is to present a unified framework that characterizes the reachable set of a dynamic system in the presence of both stochastic disturbances and deterministic disturbances, and introduces a novel technique that probabilistically bounds the difference between a stochastic trajectory and its deterministic counterpart.

Abstract

We address the reachability problem for continuous-time stochastic dynamic systems. Our objective is to present a unified framework that characterizes the reachable set of a dynamic system in the presence of both stochastic disturbances and deterministic inputs. To achieve this, we devise a strategy that effectively decouples the effects of deterministic inputs and stochastic disturbances on the reachable sets of the system. For the deterministic part, many existing methods can capture the deterministic reachability. As for the stochastic disturbances, we introduce a novel technique that probabilistically bounds the difference between a stochastic trajectory and its deterministic counterpart. The key to our approach is introducing a novel energy function termed the Averaged Moment Generating Function that yields a high probability bound for this difference. This bound is tight and exact for linear stochastic dynamics and applicable to a large class of nonlinear stochastic dynamics. By combining our innovative technique with existing methods for deterministic reachability analysis, we can compute estimations of reachable sets that surpass those obtained with current approaches for stochastic reachability analysis. We validate the effectiveness of our framework through various numerical experiments. Beyond its immediate applications in reachability analysis, our methodology is poised to have profound implications in the broader analysis and control of stochastic systems. It opens avenues for enhanced understanding and manipulation of complex stochastic dynamics, presenting opportunities for advancements in related fields.

Figures (6)

• Figure 1: An illustration of $\delta$-PRS at time $t$. Here $\mathcal{R}_{\delta,t}$ is a $\delta$-PRS of the stochastic system \ref{['eq:stochastic']}, whose trajectories are in color, and $\mathcal{R}_t$ is the DRS of the associated deterministic system \ref{['eq:associate-deterministic']}, whose trajectories are in black.
• Figure 2: An illustration of separation strategy. Here $\mathcal{R}_{\delta,t}$ is a $\delta$-PRS of the stochastic system \ref{['eq:stochastic']}, whose trajectory is $X_t$ in red. $\overline{\mathcal{R}}_t$ is an over-approximation of the DRS of the associated deterministic system \ref{['eq:associate-deterministic']}, 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. In Figure \ref{['fig: Lin set']}(a), each curve represents an independent trajectory of $X_t$. In Figure \ref{['fig: Lin set']}(b), each solid curve is an independent trajectory of $\|X_t-x_t\|$. The blue envelope and the blue dashed curve correspond to our bound \ref{['eq: thm1']}.
• Figure 4: Illustration of the tightness of $r_{\delta,t}$ w.r.t. $\delta,n$. In Figure \ref{['fig: r-delta r-n']}(a), 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 $\hat{r}_{\delta,t}^2$. In Figure \ref{['fig: r-delta r-n']}(b), 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 $\hat{r}_{\delta,t}^2$.
• Figure 5: Left: The solid blue lines show the boundary of $\delta$-PRS with $\delta=10^{-3}$ at times $t=1,2,4$ for the stochastic inverted pendulum \ref{['sys: SP']} starting from $\overline{\mathcal{X}_0}\supset \mathcal{X}_0$ obtained using Proposition \ref{['prop:cbr']}. The dashed blue lines are the boundary of the ellipsoids that over-approximate reachable sets of the associated deterministic system \ref{['eq:ip-determine']}. The red dots are $2000$ random trajectories of the inverted pendulum \ref{['sys: SP']} starting from $\overline{\mathcal{X}}_0\supset \mathcal{X}_0$ at times $t=1,2,4$. Right: The solid blue lines show the boundary of $\delta$-PRS with $\delta=10^{-3}$ at times $t=1,2,4$ for the inverted pendulum \ref{['sys: SP']} starting from $T^{-1}\overline{\mathcal{Y}}_0\supset \mathcal{X}_0$ obtained using Theorem \ref{['thm: PRS']} and interval-based reachability of the transformed system. The dashed blue lines are the boundary of the parallelotopes obtained from the interval analysis that over-approximates the reachable sets of the associated deterministic system \ref{['eq:ip-determine']}. The red dots are $2000$ random trajectories of the inverted pendulum \ref{['sys: SP']} starting from $T^{-1}\overline{\mathcal{Y}}_0\supset \mathcal{X}_0$ at times $t=1,2,4$.

Theorems & Definitions (21)

• Definition 2.1: DRS
• Definition 2.2: Matrix Measure
• Lemma 2.1
• Definition 3.1: $\delta$-PRS
• Proposition 1: Separation strategy
• proof
• Definition 5.1
• Lemma 5.1
• Definition 5.2: AMGF
• Lemma 5.2: Properties of $\Phi_{n,\lambda}$