Observability of Sparse Initial State

In this chapter, we establish theoretical guarantees for the recovery of the sparse initial state of a linear dynamical system using the algorithms presented in Chapter 2 . We rely on the concept of the restricted isometry property and prove that if the initial state vector has a sparse representation, the number of necessary observations can be significantly reduced through random projections. Our analysis provides sufficient conditions for the observability matrix to possess the restricted isometry property, thereby ensuring the system’s observability. These results depend solely on the characteristics of the system’s transfer and observation matrices, employing tools from probability theory and compressed sensing.