Input Design for System Identification via Convex Relaxation

TL;DR

This paper introduces a convex relaxation framework for optimizing excitation inputs in system identification, effectively handling amplitude constraints and providing near-optimal solutions with theoretical guarantees.

Contribution

It presents a novel convex relaxation approach for input design in system identification that accounts for amplitude constraints and offers provable bounds on solution quality.

Findings

Convex relaxation yields an upper bound within 2/π of the true maximum.

Randomized algorithm produces feasible solutions expected to be at least 2/π as informative.

Exact global optimum recovered under power constraint scenarios.

Abstract

This paper proposes a new framework for the optimization of excitation inputs for system identification. The optimization problem considered is to maximize a reduced Fisher information matrix in any of the classical D-, E-, or A-optimal senses. In contrast to the majority of published work on this topic, we consider the problem in the time domain and subject to constraints on the amplitude of the input signal. This optimization problem is nonconvex. The main result of the paper is a convex relaxation that gives an upper bound accurate to within $2/\pi$ of the true maximum. A randomized algorithm is presented for finding a feasible solution which, in a certain sense is expected to be at least $2/\pi$ as informative as the globally optimal input signal. In the case of a single constraint on input power, the proposed approach recovers the true global optimum exactly. Extensions to situations with both power and amplitude constraints on both inputs and outputs are given. A simple simulation example illustrates the technique.