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 $\lambda$-confident policy and provide a competitive ratio upper bound that depends on a trust parameter $\lambda\in [0,1]$ 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 $\lambda$ 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 $1+{O(\varepsilon)}/({{\Theta(1)+\Theta(\varepsilon)}})+O(\mu_{\mathsf{Var}})$ where $\mu_\mathsf{Var}$ 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 $\varepsilon$.