On Fundamental Limitations of Dynamic Feedback Control in Regular Large-Scale Networks

Abstract

We study fundamental performance limitations of distributed feedback control in large-scale networked dynamical systems. Specifically, we address the question of whether dynamic feedback controllers perform better than static (memoryless) ones when subject to locality constraints. We consider distributed linear consensus and vehicular formation control problems modeled over toric lattice networks. For the resulting spatially invariant systems we study the large-scale asymptotics (in network size) of global performance metrics that quantify the level of network coherence. With static feedback from relative state measurements, such metrics are known to scale unfavorably in lattices of low spatial dimensions, preventing, for example, a 1-dimensional string of vehicles to move like a rigid object. We show that the same limitations in general apply also to dynamic feedback control that is locally of first order. This means that the addition of one local state to the controller gives a similar asymptotic performance to the memoryless case. This holds unless the controller can access noiseless measurements of its local state with respect to an absolute reference frame, in which case the addition of controller memory may fundamentally improve performance. In simulations of platoons with 20-200 vehicles we show that the performance limitations we derive manifest as unwanted accordion-like motions. Similar behaviors are to be expected in any network that is embeddable in a low-dimensional toric lattice, and the same fundamental limitations would apply. To derive our results, we present a general technical framework for the analysis of stability and performance of spatially invariant systems in the limit of large networks.

Localized high-order consensus destabilizes large-scale networks

“…Proof: Follows the derivations in [14]. The bound (14) also holds for any subgraph of the r-fuzz lattice, that is, any graph that is embeddable in the lattice. This follows from the following important lemma:…”

Noise-induced limitations to the scalability of distributed integral control

“…In this case, the situation differs depending on whether there is absolute feedback from additional states (such as damping terms) or only relative feedback. Relative localized feedback, as we considered here, does not scale well to large networks with high-order dynamics -stability tends to be lost [19,Chapter 5]. It is, however, an open question whether more intricate nodal dynamics (with further absolute states) could be exploited to alleviate performance limitations.…”

“…To simplify the reading of this short letter, we will avoid the multiindex notation. A more detailed treatment of technicalities related to the states' dimensionality is found in [9]. 2 Alternatively, any control on the form u = F u x + G u v, where F u and G u satisfy Assumptions A1-A4 is possible.…”

“…We remark that the analysis here is not limited to nearest-neighbor feedback, but we assume that measurements are available from a neighborhood of width 2q. As in [1,9], we make the following additional assumptions:…”

“…In such more general network topologies, it is often possible to derive performance bounds on similar forms as the results herein using embedding arguments. See [9] for a more elaborate discussion. This means that performance limitations that apply to lattices also apply to general networks that can be embedded in them.…”