This study introduces an innovative framework for reconstructing structural responses using an enhanced measurement vector augmented with time-lagged measurements, as if obtained from virtual sensors, to tackle issues related to sparse instrumentation and complex system dimensions. By including delayed embeddings in the measurement vectors, this method greatly improves the observability of structural states, ensuring precise state estimation even when systems are sparsely monitored. The embedding approach adeptly counters the limitations posed by physical instrumentation, facilitating reliable monitoring even in demanding situations.

To further enhance the computational efficiency, the system dynamics are characterized in the modal domain, where mode shapes and natural frequencies are modulated with location-specific health variables. These variables facilitate the construction of a simplified state-space model, which is subsequently integrated into a Bayesian filtering-based estimation framework. By conducting filtering operations in modal coordinates, this strategy achieves notable computational efficiency without sacrificing precision. The framework undergoes testing on simplified linear time-invariant (LTI) systems utilizing the Kalman filter in the modal domain, with observability analysis informing sensor allocation. The findings demonstrate that utilizing time-lagged embedding along with reduced-order modeling within the modal domain can enhance the accuracy of state estimation, all while reducing the requirement for extensive instrumentation. The method's flexibility to adapt to more complex, time-variant systems emphasizes its promise for advanced structural health monitoring and parameter estimation.