A Modified Riccati Transformation for Decentralized Computation of the Viability Kernel Under LTI Dynamics

TL;DR

This paper introduces a modified Riccati transformation enabling decentralized computation of the viability kernel for LTI systems, significantly reducing complexity and allowing higher-dimensional systems to be analyzed safely.

Contribution

It proposes a novel isomorphism and decomposition method for LTI systems, facilitating conservative viability kernel approximation in a decentralized manner.

Findings

Decentralized computation reduces complexity for high-dimensional systems.

The method is validated on a 6D system example.

Conservative viability kernel approximation is achieved in transformed coordinates.

Abstract

Computing the viability kernel is key in providing guarantees of safety and proving existence of safety-preserving controllers for constrained dynamical systems. Current numerical techniques that approximate this construct suffer from a complexity that is exponential in the dimension of the state. We study conditions under which a linear time-invariant (LTI) system can be suitably decomposed into lower-dimensional subsystems so as to admit a conservative computation of the viability kernel in a decentralized fashion in subspaces. We then present an isomorphism that imposes these desired conditions, particularly on two-time-scale systems. Decentralized computations are performed in the transformed coordinates, yielding a conservative approximation of the viability kernel in the original state space. Significant reduction of complexity can be achieved, allowing the previously inapplicable tools to be employed for treatment of higher-dimensional systems. We show the results on two examples including a 6D system.