The object of this thesis is the study of constrained measurement systems of signals having low-dimensional structure using analytic tools from Compressed Sensing (CS). Realistic measurement systems usually have architectural constraints that make them differ from their idealized, well-studied counterparts. Nonetheless, these measurement systems can exploit structure in the signals that they measure. Signals considered in this research have low-dimensional structure and can be broken down into two types: static or dynamic. Static signals are either sparse in a specified basis or lying on a low-dimensional manifold (called manifold-modeled signals). Dynamic signals, exemplified as states of a dynamical system, either lie on a low-dimensional manifold or have converged onto a low-dimensional attractor. In CS, the Restricted Isometry Property (RIP) of a measurement system ensures that distances between all signals of a certain sparsity are preserved. This stable embedding ensures that sparse signals can be distinguished one from another by their measurements and therefore be robustly recovered. Moreover, signal-processing and data-inference algorithms can be performed directly on the measurements instead of requiring a prior signal recovery step. Taking inspiration from the RIP, this research analyzes conditions on realistic, constrained measurement systems (of the signals described above) such that they are stable embeddings of the signals that they measure. Specifically, this thesis focuses on four different types of measurement systems. First, we study the concentration of measure and the RIP of random block diagonal matrices that represent measurement systems constrained to make local measurements. Second, we study the stable embedding of manifold-modeled signals by existing CS matrices. The third part of this thesis deals with measurement systems of dynamical systems that produce time series observations. While Takens' embedding result ensures that this time series output can be an embedding of the dynamical systems' states, our research establishes that a stronger stable embedding result is possible under certain conditions. The final part of this thesis is the application of CS ideas to the study of the short-term memory of neural networks. In particular, we show that the nodes of a recurrent neural network can be a stable embedding of sparse input sequences.