A State Alignment-Centric Approach to Federated System Identification: The FedAlign Framework

This paper presents FedAlign, a Federated Learning (FL) framework particularly designed for System Identification (SYSID) tasks by aligning state representations. Local workers can learn State-Space Models (SSMs) with equivalent representations but different dynamics. We demonstrate that directly aggregating these local SSMs via FedAvg results in a global model with altered system dynamics. FedAlign overcomes this problem by employing similarity transformation matrices to align state representations of local SSMs, thereby establishing a common parameter basin that retains the dynamics of local SSMs. FedAlign computes similarity transformation matrices via two distinct approaches: FedAlign-A and FedAlign-O. In FedAlign-A, we represent the global SSM in controllable canonical form (CCF). We apply control theory to analytically derive similarity transformation matrices that convert each local SSM into this form. Yet, establishing global SSM in CCF brings additional alignment challenges in multi input - multi output SYSID as CCF representation is not unique, unlike in single input - single output SYSID. In FedAlign-O, we address these alignment challenges by reformulating the local parameter basin alignment problem as an optimization task. We determine the parameter basin of a local worker as the common parameter basin and solve least square problems to obtain similarity transformation matrices needed to align the remaining local SSMs. Through the experiments conducted on synthetic and real-world datasets, we show that FedAlign outperforms FedAvg, converges faster, and provides improved stability of the global SSM thanks to the efficient alignment of local parameter basins.