# Complexity reduction through a Schur-based decomposition for reachability analysis of linear time-invariant systems

@article{Kaynama2011ComplexityRT, title={Complexity reduction through a Schur-based decomposition for reachability analysis of linear time-invariant systems}, author={Shahab Kaynama and Meeko Oishi}, journal={International Journal of Control}, year={2011}, volume={84}, pages={165 - 179} }

This article presents a method for complexity reduction in reachability analysis and safety-preserving controller synthesis via Schur-based decomposition. The decomposition results in either decoupled or weakly-coupled (lower dimensional) subsystems. Reachable sets, computed independently for each subsystem, are back-projected and intersected to yield an overapproximation of the actual reachable set. Moreover, applying this technique to a class of unstable LTI systems, we show that when certainâ€¦Â

## 17 Citations

Reachability Analysis of Linear Hybrid Systems via Block Decomposition

- Mathematics, Computer ScienceIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
- 2020

This article improves the continuous-post operator by performing computations in high-dimensional state space only for time intervals relevant for the subsequent application of the discrete-posts operator and introduces a new discrete- post operator that performs low-dimensional computations.

Scalable techniques for the computation of viable and reachable sets : safety guarantees for high-dimensional linear time-invariant systems

- Computer Science, Mathematics
- 2012

Two separate approaches that improve the scalability of the computation of the minimal reachable tube and the viability kernel for high-dimensional systems are proposed and a scalable algorithm using ellipsoidal techniques for reachability is proposed.

Hamilton-Jacobi reachability: A brief overview and recent advances

- Computer Science2017 IEEE 56th Annual Conference on Decision and Control (CDC)
- 2017

This tutorial presents an overview of basic HJ reachability theory and provides instructions for using the most recent numerical tools, including an efficient GPU-parallelized implementation of a Level Set Toolbox for computing reachable sets.

Set Propagation Techniques for Reachability Analysis

- Mathematics, Computer ScienceAnnual Review of Control, Robotics, and Autonomous Systems
- 2021

A class of methods for computing a guaranteed overapproximation of the reachable set of continuous and hybrid systems, relying predominantly on set propagation; starting from the set of initial states, these techniques iteratively propagate a sequence of sets according to the system dynamics.

A Novel Unified Framework for Solving Reachability, Viability and Invariance Problems

- MathematicsArXiv
- 2021

In the proposed method, the reachable, viable and invariant sets are characterized by some non-zero level sets of different value functions, and the value functions are approximated by recursion and interpolation.

Reachability of weakly nonlinear systems using Carleman linearization

- MathematicsRP
- 2021

A solution method for a special class of nonlinear initial-value problems using set-based propagation techniques using Carleman linearization to leverage recent advances of highdimensional reachability solvers for linear ordinary differential equations based on the support function.

Methods for Reachability-based Hybrid Controller Design

- Computer Science, Mathematics
- 2012

This dissertation describes several efforts aimed towards the development of theoretical analysis tools and computational synthesis techniques to facilitate the systematic design of feedback control policies satisfying safety and target attainability specifications with respect to subclasses of hybrid system models.

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

- MathematicsIEEE Transactions on Automatic Control
- 2013

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 are studied.

Hamiltonâ€“Jacobi Reachability: Some Recent Theoretical Advances and Applications in Unmanned Airspace Management

- Computer ScienceAnnual Review of Control, Robotics, and Autonomous Systems
- 2018

This review presents recent work showing how high-dimensional reachability verification can be made more tractable by focusing on two areas of development: system decomposition for general nonlinear systems, and traffic protocols for unmanned airspace management.

Scalable synthesis of safety certificates from data with application to learning-based control

- Computer Science, Mathematics2018 European Control Conference (ECC)
- 2018

Efficient techniques for the synthesis of a safe set and control law are introduced, which offer improved scalability properties by relying on approximations based on convex optimization problems.

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