FedAlign: Federated Learning with State Alignment for System Identification

This paper introduces FedAlign, a novel Federated Learning (FL) framework with state alignment tailored for System Identification (SYSID) tasks. We show that when local workers learn equivalent representations of State-Space Models (SSMs) but differing parameter basins, direct aggregation via FedAvg leads to a global SSM with distorted system dynamics. To overcome this issue, we introduce FedAlign, which preserves the dynamics of local SSMs by leveraging similarity transformation matrices to align their state representations. In FedAlign, we define two distinct methods to obtain similarity transformation matrices: (1) data-free analytical method which defines the common parameter basin of the global SSM in controllable canonical form and uses linear control theory to compute the transformation matrices that convert local SSMs into this canonical form; and (2) data-driven optimization-based method which defines the common parameter basin as the parameter basin of one local worker and estimates the similarity transformation matrices by solving an optimization problem to align the local SSMs of the remaining workers with the common parameter basin. We present comparative SYSID results that demonstrate that FedAlign achieves higher performance compared to FedAvg with faster convergence and enhanced stability of the global SSM, demonstrating the efficacy of aligning local SSMs in FL-SYSID.