Sparsity is an emerging topic in signal/ image processing, control systems, machine learning, statistics etc. Compressed sensing theory addresses the problem of recovering a relatively sparse entity from a limited number of measurements. A new formulation for both sparse regression and compressed sensing, CLOT (Combined L-One and Two) norm introduced in (Ahsen et al., 2017) that exhibits both Grouping Effect and Robust Sparse Recovery. This dissertation explains three projects related to compressed sensing.

Firstly, it is focused on extending the “best possible” bound introduced by (Cai and Zhang, 2014) on RIP constant that is required to achieve robust sparse recovery. This “best possible” bound is extended to the new formulation, CLOT. Various numerical examples showing the dual role of CLOT and the theorems on the “best possible” bound extended to CLOT formulation are stated.

The second part is focused on application of the theory introduced in first part to control systems. Maximum hands-off control is introduced in (Nagahara et al., 2016) which is the L0-optimal control (the control that has the minimum support length) among all feasible controls that are bounded by a fixed value and transfer the state from a given initial state to the origin within a fixed time duration. As obtaining the L0-optimal control is NP hard, its convex envelope L1-optimal control is used in (Nagahara et al., 2016). This is called LASSO control which is discontinuous and not recommended for practical use. Thus, we introduce a new control called CLOT control as CLOT norm exhibits both Grouping Effect and robust sparse recovery. Simulation results with theoretical backing supporting the statement that CLOT control is sparse (w.r.t. time) and continuous control is stated.

The third and final part deals with the sparsity of inputs to the control system. The sparsity of inputs, meaning reducing the number of actuators required by the system with LQR control. This deals with Group Sparsity unlike the previous project that deals with conventional sparsity. We consider the optimal selection of sensors and/or actuators for large-scale dynamical systems with an LQR performance criterion. By considering the classical control system, x ̇ = Ax + Bu, various problems such as state feedback control, state feedback observer and output feedback control are studied for actuator selection as well as sensor selection. By using all the sensors/actuators available in the system, we will achieve the best performance; however, reducing the number of actuators and/or sensors results in sub-optimal performance. Group sparsity techniques are used to promote sparsity in input (output) for actuator (sensor) selection problem. We also propose to use Sparse Group Lasso (SGL), a group sparsity technique that promotes sparsity both between the groups as well as within the groups, to lay down a way to get the feedback controller with as few inputs as possible, and where each input is based on as few states as possible.