Robustness and Consistency in Linear Quadratic Control with Untrusted Predictions

We study the problem of learning-augmented predictive linear quadratic control. Our goal is to design a controller that balances textit"consistency", which measures the competitive ratio when predictions are accurate, and textit"robustness", which bounds the competitive ratio when predictions are inaccurate. We propose a novel λ-confident policy and provide a competitive ratio upper bound that depends on a trust parameter λ∈ [0,1] set based on the confidence in the predictions and some prediction error varepsilon. Motivated by online learning methods, we design a self-tuning policy that adaptively learns the trust parameter λ with a competitive ratio that depends on varepsilon and the variation of system perturbations and predictions. We show that its competitive ratio is bounded from above by 1+O(varepsilon)/(Theta(1)+Theta(varepsilon))+O(μ_mathsfVar) where μ_mathsfVar measures the variation of perturbations and predictions. It implies that when the variations of perturbations and predictions are small, by automatically adjusting the trust parameter online, the self-tuning scheme ensures a competitive ratio that does not scale up with the prediction error varepsilon.