State reconstruction for linear time-invariant systems with binary-valued output observations

This paper focuses on state reconstruction problems for systems whose outputs are measured only by binary-valued sensors. A binary-valued sensor is characterized by its switching threshold or post-switching location and by its output that indicates only whether the system output is above or below the threshold. The traditional passive-type state reconstruction that does not require input design will fail in general, even if the system has a full-rank observability matrix. It is shown that under controllability conditions and bounded uncertainty on the initial state, it is always possible to construct a causal input that will cause the output to cross the sensor threshold in a designated time interval. Information on the time instants of threshold crossing is then used to reconstruct the initial state. It is proved that if the system is observable, such a reconstruction is always possible if the eigenvalues of the system are all real valued. If the eigenvalues of the system contain only purely imaginary and non-repeating values, it is sufficient that threshold crossing occurs within a relatively small time interval. In general without constraints on system eigenvalues, an input can always be randomized to ensure that the state can be reconstructed with probability one. These results lead to an active state reconstruction algorithm.