Worst-Case Finite-Horizon Analysis of Perturbed LQR via Performance Estimation Problem
Abstract
We propose an exact formulation to quantify the finite-horizon robustness of a Linear Quadratic Regulator (LQR) subject to norm-bounded Linear Time-Invariant (LTI) parametric uncertainties, using the Performance Estimation Problem (PEP) framework. Originally developed for optimization, this method allows us to reformulate the worst-case performance search as a rank-constrained Semidefinite Program (SDP). By applying operator interpolation, this formulation retains the sequence variables, allowing the worst-case to be explicitly reconstructed when a rank-one solution is found. This approach also opens new possibilities for the finite-horizon certification of broader system classes, such as data-driven models or nonlinear components. Using this framework, we observe that finite-horizon LTI perturbations do not simply follow asymptotic strategies. Instead, they align slowly decaying invariant manifolds with specific initial conditions to drive transient growth, causing a bifurcation in the worst-case behavior as the horizon length changes. Finally, we show numerically that our performance bounds sit between the upper bounds of Integral Quadratic Constraint (IQC) relaxations and a lower bound derived from a pseudospectral heuristic.
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BibTeX
@misc{couthures2026worstcase,
title = {{Worst-Case Finite-Horizon Analysis of Perturbed LQR via Performance Estimation Problem}},
author = {Anthony Couthures and Gianluca Bianchin and Julien M. Hendrickx},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29910},
eprint = {2610.29910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29910},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Worst-Case Finite-Horizon Analysis of Perturbed LQR via Performance Estimation Problem
AU - Anthony Couthures
AU - Gianluca Bianchin
AU - Julien M. Hendrickx
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29910
UR - https://arxiv.org/abs/2610.29910
AB - We propose an exact formulation to quantify the finite-horizon robustness of a Linear Quadratic Regulator (LQR) subject to norm-bounded Linear Time-Invariant (LTI) parametric uncertainties, using the Performance Estimation Problem (PEP) framework. Originally developed for optimization, this method allows us to reformulate the worst-case performance search as a rank-constrained Semidefinite Program (SDP). By applying operator interpolation, this formulation retains the sequence variables, allowing the worst-case to be explicitly reconstructed when a rank-one solution is found. This approach also opens new possibilities for the finite-horizon certification of broader system classes, such as data-driven models or nonlinear components. Using this framework, we observe that finite-horizon LTI perturbations do not simply follow asymptotic strategies. Instead, they align slowly decaying invariant manifolds with specific initial conditions to drive transient growth, causing a bifurcation in the worst-case behavior as the horizon length changes. Finally, we show numerically that our performance bounds sit between the upper bounds of Integral Quadratic Constraint (IQC) relaxations and a lower bound derived from a pseudospectral heuristic.
ER -