On the observability of linear systems from random, compressive measurements

Recovering or estimating the initial state of a high-dimensional system can require a potentially large number of measurements. In this paper, we explain how this burden can be significantly reduced for certain linear systems when randomized measurement operators are employed. Our work builds upon recent results from the field of Compressive Sensing (CS), in which a high-dimensional signal containing few nonzero entries can be efficiently recovered from a small number of random measurements. In particular, we develop concentration of measure bounds for the observability matrix and explain circumstances under which this matrix can satisfy the Restricted Isometry Property (RIP), which is central to much analysis in CS. We also illustrate our results with a simple case study of a diffusion system. Aside from permitting recovery of sparse initial states, our analysis has potential applications in solving inference problems such as detection and classification of more general initial states.