Efficient Computation of Minimal and Maximal Backward Reachable Sets for Perturbed Continuous-Time Linear Systems

Core Concepts

The core message of this article is to present efficient algorithms for computing inner and outer approximations of the minimal and maximal backward reachable sets for perturbed continuous-time linear systems. The proposed approaches scale polynomially with the state dimension, enabling the analysis of high-dimensional systems.

Abstract

The article focuses on the computation of backward reachable sets for continuous-time linear time-invariant (LTI) systems with control inputs and disturbances. It presents algorithms to efficiently approximate the minimal and maximal backward reachable sets, which are important for safety verification and controller synthesis, respectively.
Key highlights:

For the minimal backward reachable set:

An inner and outer approximation for the time-point solution (Section V-A)
An outer approximation for the time-interval solution (Section V-B)

For the maximal backward reachable set:

An inner and outer approximation for the time-point solution (Section VI-A)
An inner approximation for the time-interval solution (Section VI-B)

All proposed algorithms scale polynomially with the state dimension, in contrast to the exponential scaling of existing Hamilton-Jacobi-based methods.
The approximation errors of the computed sets are analyzed, showing that they can be made arbitrarily small in the limit of the discretization step size.
Simplifications for the unperturbed case are discussed, where the runtime complexity and approximation error are further improved.

Stats

The article does not provide explicit numerical data or statistics. However, it mentions that the proposed algorithms enable the analysis of systems with well over a hundred states, which is a significant improvement over the state-of-the-art methods.

Quotes

"Crucially, the proposed algorithms scale only polynomially with the state dimension."
"Our numerical examples demonstrate the tightness of the obtained backward reachable sets and show an overwhelming improvement of our proposed algorithms over state-of-the-art methods regarding scalability, as systems with well over a hundred states can now be analyzed."

Key Insights Distilled From

Backward Reachability Analysis of Perturbed Continuous-Time Linear Systems Using Set Propagation

by Mark Wetzlin... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2310.19083.pdf

Deeper Inquiries

How can the proposed algorithms be extended to handle nonlinear dynamics or time-varying systems

The proposed algorithms can be extended to handle nonlinear dynamics by incorporating techniques from reachability analysis for nonlinear systems. One approach is to use techniques like Taylor models or interval arithmetic to approximate the dynamics of the nonlinear system within each time step. By linearizing the system around different points or using higher-order approximations, the nonlinear dynamics can be approximated by a sequence of linear systems, allowing the application of the backward reachability algorithms designed for linear systems.
For time-varying systems, the algorithms can be adapted to consider the time-varying nature of the dynamics. This can be achieved by discretizing the time-varying system into a sequence of time-invariant systems over small time intervals. The particular solutions for each time interval can then be computed separately and combined to obtain the overall backward reachable set for the time-varying system.

What are the potential applications of the computed minimal and maximal backward reachable sets beyond safety verification and controller synthesis

The computed minimal and maximal backward reachable sets have various potential applications beyond safety verification and controller synthesis.

Reachability Analysis in Autonomous Systems: The sets can be used to analyze the reachability of states in autonomous systems, helping in decision-making processes and trajectory planning.

Anomaly Detection: By comparing the actual system behavior with the computed reachable sets, anomalies or deviations from expected behavior can be detected, leading to early warnings and preventive actions.

Resource Allocation: The sets can aid in optimizing resource allocation in systems where safety and efficiency are critical, such as in transportation systems, energy grids, and manufacturing processes.

Risk Assessment: By analyzing the regions of the state space that are reachable under different conditions, the sets can be used for risk assessment and mitigation strategies in complex systems.

System Design: The information from the backward reachable sets can guide system design by identifying critical states and potential failure modes, leading to more robust and reliable systems.

Can the set representations and operations used in the algorithms be further optimized to improve the computational efficiency

The set representations and operations used in the algorithms can be further optimized to improve computational efficiency in the following ways:

Adaptive Grid Refinement: Implementing adaptive grid refinement techniques to focus computational resources on regions of the state space that are most critical for the analysis, reducing the overall computational burden.

Parallel Processing: Utilizing parallel processing capabilities to distribute the computational workload across multiple processors or cores, enabling faster computation of the reachable sets.

Sparse Representation: Using sparse representations for the sets to reduce memory usage and computational complexity, especially in high-dimensional state spaces.

Algorithmic Improvements: Developing more efficient algorithms for set operations, such as Minkowski sums and differences, to reduce the overall computational cost of the reachability analysis.

By incorporating these optimizations, the algorithms can achieve faster computation times and handle larger and more complex systems with improved efficiency.