  We consider the problem of optimal sparse output feedback controller
synthesis for continuous linear time invariant systems when the feedback gain
is static and subject to specified structural constraints. Introducing an
additional term penalizing the number of non-zero entries of the feedback gain
into the optimization cost function, we show that this inherently non-convex
problem can be equivalently cast as a rank constrained optimization, hence, it
is an NP-hard problem. We further exploit our rank constrained approach to
define a structured output feedback control feasibility test with global
convergence property, then, obtain upper/lower bounds for the optimal cost of
the sparse output feedback control problem. Moreover, we show that our problem
reformulation allows us to incorporate additional implementation constraints,
such as norm bounds on the control inputs or system output, by assimilating
them into the rank constraint. We propose to utilize a version of the
Alternating Direction Method of Multipliers (ADMM) as an efficient method to
sub-optimally solve the equivalent rank constrained problem. As a special case,
we study the problem of designing the sparsest stabilizing output feedback
controller, and show that it is, in fact, a structured matrix recovery problem
where the matrix of interest is simultaneously sparse and low rank.
Furthermore, we show that this matrix recovery problem can be equivalently cast
in the form of a canonical and well-studied rank minimization problem. We
finally illustrate performance of our proposed methodology using numerical
examples.
