Integral Quadratic Constraints: Exact Convergence Rates and Worst-Case Trajectories
Provides a precise stability criterion for a broad class of systems with IQCs, addressing a fundamental problem in control theory.
The paper defines a new spectral radius for linear systems with integral quadratic constraints (IQCs) that exactly characterizes stability: when <1, asymptotic stability holds; when =1, an unstable IQC-satisfying trajectory exists.
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.