On the Computation of Sparse Solutions to the Controllability Problem for Discrete-Time Linear Systems

In this work, we address the fundamental problem of steering the state of a discrete-time linear system to the origin after a given (finite) number of stages by means of the sparsest possible control sequence, that is, the sequence of inputs comprised of the maximum possible number of null elements. In our approach, the latter controllability problem is associated with the problem of finding either the minimum 1-norm solution or the minimum p-norm, with p taking values greater than zero and less than one, solution of an under-determined system of linear equations, which are both known to exhibit good sparsity properties under certain technical assumptions. Motivated by practical considerations, we compute approximate solutions to the latter optimization problems by utilizing the class of iteratively weighted least squares algorithms from the literature of compressive (or compressed) sensing. This particular choice of algorithms is motivated by (1) their straightforward implementation, which makes them appealing to the non-expert and (2) the fact that some of the most costly operations involved in their implementation can be carried out recursively by leveraging well-known properties of the controllability Grammian of a discrete-time linear system. Finally, we apply the proposed approach to a spacecraft proximity operation problem and in particular, a linearized impulsive fixed-time minimum-fuel rendezvous problem in which the 1-norm serves as a proxy to the fuel consumption at a given time interval.