# Parabolic Set Simulation for Reachability Analysis of Linear Time Invariant Systems with Integral Quadratic Constraint

## TL;DR

This paper introduces a novel parabolic set simulation method for reachability analysis of LTI systems with integral quadratic constraints, providing a practical way to overapproximate reachable sets in energy-constrained systems.

## Contribution

It extends ellipsoidal techniques to parabolic sets for LTI systems with IQCs, including a method to generate supporting paraboloids for exact reachable set characterization.

## Key findings

- Overapproximates reachable sets with time-varying parabolic sets.
- Proves the relationship between paraboloids and reachable sets.
- Applicable to systems with delays, rate limiters, and energy constraints.

## Abstract

This work extends reachability analyses based on ellipsoidal techniques to Linear Time Invariant (LTI) systems subject to an integral quadratic constraint (IQC) between the past state and disturbance signals , interpreted as an input-output energetic constraint. To compute the reachable set, the LTI system is augmented with a state corresponding to the amount of energy still available before the constraint is violated. For a given parabolic set of initial states, the reachable set of the augmented system is overapproximated with a time-varying parabolic set. Parameters of this paraboloid are expressed as the solution of an Initial Value Problem (IVP) and the overapproximation relationship with the reachable set is proved. This paraboloid is actually supported by the reachable set on so-called touching trajectories. Finally , we describe a method to generate all the supporting paraboloids and prove that their intersection is an exact characterization of the reachable set. This work provides new practical means to compute overapproximation of reachable sets for a wide variety of systems such as delayed systems, rate limiters or energy-bounded linear systems.