An Iterative Algorithm to Estimate Invariant Sets for Uncertain Systems

Peter J. Seiler

Abstract: This paper develops an iterative algorithm to estimate an invariant set for uncertain systems. The uncertain system is given as a connection of a nominal linear time-invariant system and a perturbation. The input/output behavior of the perturbation is described by integral quadratic constraints (IQCs). The proposed approach incorporates IQCs into a dissipation inequality formulation. One issue is that it is often useful to specify the IQC in the frequency domain or, equivalently, in the time-domain as a “soft” infinite-horizon constraint. However, the dissipation inequality formulation requires constraints that are valid over all finite time horizons. The main technical result is a finite-horizon bound on soft IQCs constructed using a state-feedback transformation. This forms the basis for the proposed iterative algorithm to estimate invariant sets. A simple example is provided to demonstrate the proposed approach.