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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 "consistency'', which measures the competitive ratio when predictions are accurate, and "robustness'', which bounds the competitive ratio when predictions are inaccurate. We propose a novel λ-confident controller and prove that it maintains a competitive ratio upper bound of 1 + min {O(λ2ε)+ O(1-λ)2,O(1)+O(λ2)} where λ∈ [0,1] is a trust parameter set based on the confidence in the predictions, and ε is the prediction error. Further, 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 ε and the variation of system perturbations and predictions. We show that its competitive ratio is bounded from above by 1+O(ε) /(Θ)(1)+Θ(ε))+O(μVar) where μVar measures the variation of perturbations and predictions. It implies that by automatically adjusting the trust parameter online, the self-tuning scheme ensures a competitive ratio that does not scale up with the prediction error ε.