I. Introduction
Classical state-space solutions to optimal centralized control problems do not scale well as the dimension increases [1]. Moreover, structural constraints such as locality and delay are ubiquitous in real-world controllers. The optimal decentralized control problem (ODC) has been proposed in the literature to bridge this gap. On the one hand, ODC can have nonlinear optimal solutions even for linear systems and is NP-hard in the worst case [2], [3]. On the other hand, the existence of dynamic structured feedback laws is completely captured by the notion of fixed modes [4], and several works have discovered structural conditions on the system and/or the controller under which the ODC problem admits tractable solutions. The conditions include spatially invariance [5], partially nestedness [6], positiveness [7], and quadratic invariance [8]. More recently, the System Level Approach [9] has convexified structural constraints at the expense of working with a series of impulse response matrices. Promising approximation [10]–[12] and convex relaxation techniques [13]–[16] also exist in the literature.