On Direct vs Indirect Data-Driven Predictive Control

In this work, we compare the direct and indirect approaches to data-driven predictive control of stochastic linear time-invariant systems. The distinction between the two approaches lies in the fact that the indirect approach involves identifying a lower dimensional model from data which is then used in a certainty-equivalent control design, while the direct approach avoids this intermediate step altogether. Working within an optimization-based framework, we find that the suboptimality gap measuring the control performance w.r.t. the optimal model-based control design vanishes with the size of the dataset only with the direct approach, while the indirect approach incurs an asymptotic bias. On the other hand, the indirect approach, by relying on the identification of a lower dimensional model, has lower variance and outperforms the direct approach for smaller datasets. Ultimately, by revealing the existence of two non-asymptotic regimes for the performance of direct and indirect data-driven predictive control designs, our study suggests that neither approach is invariably superior and that the choice of design must, in practice, be informed by the available dataset.