Sparsity measures for spatially decaying systems

Abstract

We consider the omnipresent class of spatially decaying systems, where the sensing and controls is spatially distributed. This class of systems arises in various applications where there is a notion of spatial distance with respect to which couplings between the subsystems can be quantified using a class of coupling weight functions. We exploit spatial decay property of the dynamics of the underlying system in order to introduce system-oriented sparsity measures for spatially distributed systems. We develop a new mathematical framework, based on notions of quasi-Banach algebras of spatially decaying matrices, to relate spatial decay properties of spatially decaying systems to sparsity features of their underlying information structures. Moreover, we show that the inverse-closedness property of matrix algebras plays a central role in exploiting various structural properties of spatially decaying systems. We show that the quadratically optimal state feedback controllers for spatially decaying systems are sparse and spatially localized in the sense that they have near-optimal sparse information structures. Finally, our results are applied to quantify sparsity and spatial localization features of a class of randomly generated power networks.

Localized LQR optimal control

“…a chain), the QI condition requires that each controller share its measurements with the whole network -this becomes a limiting factor as systems scale to larger and larger size. Although several attempts have been made in the literature to find sparse controllers (which lead to tractable implementations), such as sparsity-promoting control [14], [15], and spatial truncation [16], [17], the synthesis procedure is still centralized. In this paper and our previous work [18], we advocate the importance of locality (i.e.…”

Wiener's Lemma for Infinite Matrices of Gohberg-Baskakov-Sjöstrand Class

“…So · Cq,w is a quasi-norm [4,14]. Therefore (C q,w , · Cq,w ) forms a quasiBanach algebra by Proposition 2.1.…”

Controller architectures: Tradeoffs between performance and structure

“…For spatially invariant systems, the optimal controllers with respect to quadratic performance indices (e.g., LQR, H 2 , H ∞ ) are also spatially invariant and they exponentially discount information with spatial distance [2]. Moreover, it has been suggested that optimal controllers for spatially-decaying systems over general graphs also possess spatially-decaying property [84,85]. This motivates the search for inherently localized controllers and suggests that localized information exchange in the distributed controller may provide a viable strategy for controlling large-scale systems.…”

“…For spatially invariant systems, the quadratic optimal controllers are also spatially invariant and the information from other subsystems is exponentially discounted with the distance between the controller and the subsystems [2]. For systems on graphs, this spatially decaying property was studied in [84,85] and it motivates the search for inherently localized controllers.…”