  To control structural responses under various actions, the growing use of
supplementary damping systems in modern civil engineering structures
necessitates inspecting and evaluating their operational performance
postinstallation. However, due to the dispersed placement and complex
nonlinearities of these devices, difficulties arise in determining minimal
sensor configuration. This is inherently connected to a pivotal challenge:
establishing a reliable input-output mapping, which comprises both the
mathematical model and sensor arrangements. Prior work indicates this can be
achieved through theoretical observability analysis or Lie symmetries analysis,
both of which provide different perspectives on the existence of a way to
access the solutions of a system identification problem uniquely (at least
locally). The present study introduces a unified framework, enhanced by
algorithm realization as an application guide, for analyzing the observability
and Lie symmetries of a given input-output mapping. We demonstrate its
implementation via examples of a building structure with various damping
systems under different conditions such as seismic loads, wind loads, and
operational vibrations. Finally, we present a case study for an isolation
building with an inerter damper and minimal sensor arrangement under seismic
action. The results demonstrate that the unscented Kalman filter, a system
identification method, can precisely estimate structural responses and assess
damping device performance once a reliable input-output mapping is established.
