Sparsity Invariance for Convex Design of Distributed Controllers

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 subjected 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 <inline-formula><tex-math notation="LaTeX">$\mathbf {Y}(s)$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathbf {X}(s)$</tex-math></inline-formula> such that any corresponding controller <inline-formula><tex-math notation="LaTeX">$\mathbf {K}(s)=\mathbf {Y}(s)\mathbf {X}(s)^{-1}$</tex-math></inline-formula> 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 and 2) the solution via the SI approach is guaranteed to be globally optimal when QI holds and performs at least, considering the 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.