3DIOC: Direct Data-Driven Inverse Optimal Control for LTI Systems

This paper addresses the Direct Data-Driven Inverse Optimal Control (3DIOC) problem for linear time-invariant (LTI) systems under the linear quadratic (LQ) control. Unlike traditional approaches that require system identification, the proposed method learns the underlying objective function directly from measured input-output trajectories. Leveraging the input-output representation of LTI systems via the Fundamental Lemma, we derive a model-free optimality necessary condition (ONC) for the forward LQ problem, which forms the basis for formulating and solving an inverse optimal control problem. We also provide an identifiability condition to ensure the uniqueness of the inverse solution. While the ONC-based IOC approach is effective in the noise-free case, its performance is not promising when the data is corrupted with noises. We then reformulate the 3DIOC as a bi-level optimization problem, which is solved using iterative gradient descent and offers solution guarantees. Furthermore, we analyze the relationship between the solution sets of the two proposed formulations, providing practical insights into their selection. The simulation results validate the effectiveness and performance of our proposed methods.