Observability With Random Observations

Recovery of the initial state of a high-dimensional system can require a large number of measurements. In this paper, we explain how this burden can be significantly reduced when randomized measurement operators are employed. Our work builds upon recent results from Compressive Sensing (CS). In particular, we make the connection to CS analysis for random block diagonal matrices. By deriving Concentration of Measure (CoM) inequalities, we show that the observability matrix satisfies the Restricted Isometry Property (RIP) (a sufficient condition for stable recovery of sparse vectors) under certain conditions on the state transition matrix. For example, we show that if the state transition matrix is unitary, and if independent, randomly-populated measurement matrices are employed, then it is possible to uniquely recover a sparse high-dimensional initial state when the total number of measurements scales linearly in the sparsity level (the number of non-zero entries) of the initial state and logarithmically in the state dimension. We further extend our RIP analysis for scaled unitary and symmetric state transition matrices. We support our analysis with a case study of a diffusion process.