  We address the problem of designing optimal linear time-invariant (LTI)
sparse controllers for LTI systems, which corresponds to minimizing a norm of
the closed-loop system subject to sparsity constraints on the controller
structure. This problem is NP-hard in general and motivates the development of
tractable approximations. We characterize a class of convex restrictions based
on a new notion of Sparsity Invariance (SI). The underlying idea of SI is to
design sparsity patterns for transfer matrices Y(s) and X(s) such that any
corresponding controller K(s)=Y(s)X(s)^-1 exhibits the desired sparsity
pattern. For sparsity constraints, the approach of SI goes beyond the notion of
Quadratic Invariance (QI): 1) the SI approach always yields a convex
restriction; 2) the solution via the SI approach is guaranteed to be globally
optimal when QI holds and performs at least as well as considering a nearest QI
subset. Moreover, the notion of SI naturally applies to designing structured
static controllers, while QI is not utilizable. Numerical examples show that
even for non-QI cases, SI can recover solutions that are 1) globally optimal
and 2) strictly more performing than previous methods.
