  We consider the co-design problem of sparse output feedback and
row/column-sparse output matrix. A row-sparse (resp. column-sparse) output
matrix implies a small number of outputs (resp. sensor measurements). We impose
row/column-cardinality constraint on the output matrix and the cardinality
constraint on the output feedback gain. The resulting nonconvex, nonsmooth
optimal control problem is solved by using the proximal alternating
linearization method (PALM). One advantage of PALM is that the proximal
operators for sparsity constraints admit closed-form expressions and are easy
to implement. Furthermore, the bilinear matrix function introduced by the
multiplication of the feedback gain and the output matrix lends itself well to
PALM. By establishing the Lipschitz conditions of the bilinear function, we
show that PALM is globally convergent and the objective value is monotonically
decreasing throughout the algorithm. Numerical experiments verify the
convergence results and demonstrate the effectiveness of our approach on an
unstable system with 60,000 design variables.
