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. The indirect approach has a higher rate of convergence, but its suboptimality gap does not vanish as the size of the dataset increases. This reveals the existence of two distinct regimes of performance as the size of the dataset of input-output behaviors is increased. We show that the indirect approach, by relying on the identification of a lower dimensional model, has lower variance and outperforms the direct approach for smaller datasets, while it incurs an asymptotic bias as a result of the process noise and a (possibly) incorrect assumption on the order of the identified model. The direct approach, however, does not incur an asymptotic bias, and outperforms the indirect approach for larger 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.