We consider the problem of row-column sparse linear quadratic controller (LQC) design. An optimization problem is formulated in which the quadratic performance loss is minimized subject to satisfaction of m+n sparsity constraints to obtain the row-column (r, c)-sparse LQC design where m and n refer to the number of inputs and states, respectively and r/c represent the maximum allowed density level for each row/column of controller. It is expressed that the obtained nonconvex optimization problem can equivalently be reformulated as a rank-constrained problem with m+n+1 rank constraints. After applying the non-fragility notion provided by [1] to such a rank-constrained problem, bi-linear rank penalty technique is deployed to find a sub-optimal row-column (r, c)-sparse LQC design which fulfills the rank constraint with desired tolerance. At last, to verify our proposed algorithm, given a randomly generated system, a sub-optimal row-column (r, c)-sparse LQC design is proposed and subsequently, the fundamental trade-off between r/c and quadratic performance loss is visualized.
