Worst-Case Finite-Horizon Analysis of Perturbed LQR via Performance Estimation Problem

Submitted to 2027 European Control Conference (ECC)

Authors

Anthony Couthures

Gianluca Bianchin

Julien M. Hendrickx

Published

2026

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 citation:
@misc{couthures2026,
  author = {Couthures, Anthony and Bianchin, Gianluca and Hendrickx,
    Julien M.},
  title = {Worst-Case {Finite-Horizon} {Analysis} of {Perturbed} {LQR}
    via {Performance} {Estimation} {Problem}},
  date = {2026},
  url = {https://anthonycouthures.github.io/publications/2026_preprint_pep-perturbed-lqr.html},
  doi = {10.48550/arXiv.2610.29910},
  langid = {en},
  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.}
}
For attribution, please cite this work as:
Anthony Couthures, Gianluca Bianchin, and Julien M. Hendrickx. 2026. “Worst-Case Finite-Horizon Analysis of Perturbed LQR via Performance Estimation Problem.” arXiv preprint. https://doi.org/10.48550/arXiv.2610.29910.