# Integral Quadratic Constraints: Exact Convergence Rates and Worst-Case Trajectories

## TL;DR

This paper introduces a new spectral radius concept for linear systems with integral quadratic constraints, precisely characterizing stability and instability, and linking it to existing IQC theory.

## Contribution

It defines an exact spectral radius for IQC systems, establishing stability criteria and constructing unstable trajectories when the spectral radius equals one.

## Key findings

- Spectral radius less than one implies asymptotic stability for IQC systems.
- Spectral radius equal to one allows construction of an unstable IQC-satisfying trajectory.
- The new spectral radius aligns with and extends existing IQC stability analysis.

## Abstract

We consider a linear time-invariant system in discrete time where the state and input signals satisfy a set of integral quadratic constraints (IQCs). Analogous to the autonomous linear systems case, we define a new notion of spectral radius that exactly characterizes stability of this system. In particular, (i) when the spectral radius is less than one, we show that the system is asymptotically stable for all trajectories that satisfy the IQCs, and (ii) when the spectral radius is equal to one, we construct an unstable trajectory that satisfies the IQCs. Furthermore, we connect our new definition of the spectral radius to the existing literature on IQCs.