Convergence of APSMC from identity initialization under sensor-deficient settings

Determine whether, in scenarios where the initial system matrix A0 is significantly inaccurate and the number of sensors is less than the model order, initializing A0 as the identity matrix guarantees convergence of the Adaptive Physics-Informed System Modeling with Control (APSMC) algorithm to the optimal state-space model estimate that minimizes the APSMC loss function with embedded physical constraints.

Background

The paper introduces the APSMC framework for online identification of time-varying state-space models under noise and external inputs, leveraging adaptive filtering and proximal gradient updates with physical constraints.

In practice, the success of APSMC depends on the initialization of the system matrix A0 and the sensing configuration. The authors report that when A0 is highly inaccurate and the number of sensors is less than the model order, Kalman filtering can fail to provide accurate state estimates, hindering model updates.

Empirically, using the identity matrix for A0 in this sensor-deficient regime tends to yield prediction errors similar to an ERA baseline, but a theoretical guarantee of convergence to the optimal solution is not established.