Optimal input design via Frank-Wolfe
Abstract
We study optimal input design over a finite horizon for linear dynamical systems. The goal is to minimize a weighted inverse-covariance (information) criterion subject to an energy budget. The set of covariances achievable by causal policies is convex but lacks a tractable explicit description, ruling out projection-based methods. We show that Frank--Wolfe applies naturally: each linear minimization subproblem is a budget-constrained finite-horizon linear quadratic (LQ) problem, solvable by a Riccati recursion and one-dimensional bisection over a Lagrange multiplier. Using smoothness of the objective over the feasible set, we establish an 𝒪(1/M) convergence rate for the objective value, while strong convexity yields an 𝒪(1/√M) rate for the iterates. We further extend the framework to input design for system identification with unknown dynamics and adaptive online LQR, and illustrate the approach numerically.
Related
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BibTeX
@misc{bencherki2026optimal,
title = {{Optimal input design via Frank-Wolfe}},
author = {Fethi Bencherki and Bruce Lee and Nikolai Matni and Anders Rantzer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08751},
eprint = {2609.08751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08751},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Optimal input design via Frank-Wolfe
AU - Fethi Bencherki
AU - Bruce Lee
AU - Nikolai Matni
AU - Anders Rantzer
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08751
UR - https://arxiv.org/abs/2609.08751
AB - We study optimal input design over a finite horizon for linear dynamical systems. The goal is to minimize a weighted inverse-covariance (information) criterion subject to an energy budget. The set of covariances achievable by causal policies is convex but lacks a tractable explicit description, ruling out projection-based methods. We show that Frank--Wolfe applies naturally: each linear minimization subproblem is a budget-constrained finite-horizon linear quadratic (LQ) problem, solvable by a Riccati recursion and one-dimensional bisection over a Lagrange multiplier. Using smoothness of the objective over the feasible set, we establish an 𝒪(1/M) convergence rate for the objective value, while strong convexity yields an 𝒪(1/√M) rate for the iterates. We further extend the framework to input design for system identification with unknown dynamics and adaptive online LQR, and illustrate the approach numerically.
ER -