An oracle for the discrete-time integral quadratic constraint problem

Abstract

In the theory of integral quadratic constraints (IQCs), the condition for robust performance is given in terms of a frequency-domain inequality. This inequality can be transformed into an equivalent finite-dimensional linear matrix inequality by the application of the Kalman–Yakubovich–Popov (KYP) lemma and the introduction of an auxiliary matrix variable. Thus, the IQC problem can be expressed as a semi-definite program (SDP) and solved using convex optimization tools such as general-purpose SDP solvers based on interior point methods. However, by introducing the auxiliary variable, the size of the IQC problem significantly increases, and the aforementioned solvers may fail due to the high computational cost of the SDP. To address this issue, specialized SDP solvers have been developed to efficiently solve the SDP by exploiting its structure. An alternative approach consists of using cutting plane methods to directly solve the frequency-domain inequality without resorting to the KYP lemma. Cutting plane methods require an oracle that confirms if a given candidate solution satisfies the frequency-domain inequality. If the candidate solution is not feasible, the oracle generates a hyperplane that separates the candidate solution from the set of feasible solutions. In this paper, the analytic center cutting plane method is applied to the discrete-time IQC problem. The oracle for the discrete-time IQC problem is based on the spectrum of an extended symplectic matrix pencil and does not require an invertibility assumption that holds in continuous-time but not necessarily in discrete-time. An illustrative example is provided that shows the efficacy of the proposed method and points out directions for future research.