  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 $\varepsilon$.
