Overview
Research addresses the challenge of safety verification for linear time-invariant systems with bounded inputs operating in continuous time. The established methodology for this involves a two-stage process: an initial discretization of time, followed by a forward analysis conducted within the resulting discretized system. Conventional algorithms in this domain typically employ either a fixed time step or an adaptive time step that adjusts based on the approximation error relative to the continuous system. This work introduces an efficient reachability algorithm designed to adapt its time step specifically in response to a given safety property. The core principle is to utilize the largest permissible time step while still maintaining the ability to rigorously prove safety.
Research Context
Safety verification for linear time-invariant systems with bounded inputs in continuous time relies on reachability analysis. Current techniques for this analysis commonly involve discretizing the continuous system before performing a forward analysis. The effectiveness and accuracy of these methods are influenced by the choice of time step. Existing approaches vary in their time-step management, utilizing either a uniform, predetermined time step or one that dynamically adjusts in proportion to the discrepancy between the discrete approximation and the continuous system's behavior.
Approach
The proposed algorithm for reachability analysis implements a property-driven time-step adaptation. Instead of adjusting the time step based solely on approximation error relative to the continuous system, this method directly links time-step size to the specific safety property under consideration. The algorithm aims to identify and employ the maximum possible time step that still permits a definitive proof of safety.
To enhance scalability and practical utility, several optimizations are incorporated:
- Avoidance of repeated, computationally intensive calculations of the matrix exponential during the discretization phase.
- A strategic balance in managing the approximation error, considering contributions from both system states and inputs. This careful management allows the algorithm to maintain a moderate approximation error even when employing larger time steps.
This design choice enables the algorithm to necessitate fewer total steps compared to prior methodologies, contributing to its efficiency.
Findings
The developed algorithm demonstrates effectiveness and scalability when evaluated against the large-scale SLICOT benchmark suite. Across these benchmarks, the algorithm consistently exhibited superior performance compared to other state-of-the-art approaches. The property-driven time-step adaptation, coupled with the discussed optimizations, facilitates the use of larger time steps while maintaining moderate approximation errors, thereby reducing the total number of steps required for analysis.
Why This Matters
The ability to efficiently perform safety verification for linear time-invariant systems is critical. By introducing an algorithm that adapts its time step based on a safety property and incorporates optimizations to manage computational expense, this research offers a method that can potentially improve the practical applicability of reachability analysis for system safety guarantees.