Separable and Localized System-Level Synthesis for Large-Scale Systems

Abstract

A major challenge faced in the design of large-scale cyber-physical systems, such as power systems, the Internet of Things or intelligent transportation systems, is that traditional distributed optimal control methods do not scale gracefully, neither in controller synthesis nor in controller implementation, to systems composed of millions, billions or even trillions of interacting subsystems. This paper shows that this challenge can now be addressed by leveraging the recently introduced System Level Approach (SLA) to controller synthesis. In particular, in the context of the SLA, we define suitable notions of separability for control objective functions and system constraints such that the global optimization problem (or iterate update problems of a distributed optimization algorithm) can be decomposed into parallel subproblems. We then further show that if additional locality (i.e., sparsity) constraints are imposed, then these subproblems can be solved using local models and local decision variables. The SLA is essential to maintaining the convexity of the aforementioned problems under locality constraints. As a consequence, the resulting synthesis methods have O(1) complexity relative to the size of the global system. We further show that many optimal control problems of interest, such as (localized) LQR and LQG, H2 optimal control with joint actuator and sensor regularization, and (localized) mixed H2/L1 optimal control problems, satisfy these notions of separability, and use these problems to explore tradeoffs in performance, actuator and sensing density, and average versus worst-case performance for a large-scale power inspired system.

A System-Level Approach to Controller Synthesis

“…This example also serves to illustrate the importance of locality in achieving scalability of controller implementation. Our companion paper [47] shows how locality further leads to scalability of controller synthesis.…”

“…As such, we provide here a description of the largest known class of constrained stabilizing controllers that admit a convex parameterization. Further, as we show in our companion paper [47], it is this ability to impose locality constraints on the controller structure via convex constraints that allows us to scale the methods proposed in [10], [21] to large-scale systems.…”

“…In particular, in order to implement the state feedback policy for the ith actuator u i , only those states x j for which A ij = 0 need to be collected -thus only those states corresponding to immediate neighbors of node i in the graph G, i.e., only local states, need to be collected to compute the corresponding control action, leading to a localized implementation. As we discuss in our companion paper [47], the idea of locality is essential to allowing controller synthesis and implementation to scale to arbitrarily large systems, and hence such a structured controller is desirable. Now suppose that we naively attempt to solve optimal control problem (30) by converting it to its equivalent H 2 model matching problem (5) and constraining the controller K to have the same support as A, i.e., K =…”

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

“…Important exceptions to this include the subclass of funnelcausal and quadratically-invariant problems (see e.g. [9], [10], and a more recent approach [11]).…”

Distributed Design for Decentralized Control Using Chordal Decomposition and ADMM

“…Recent work has started to use distributed optimization techniques to realize distributed synthesis in the dissipative framework [13]. Furthermore, the recently proposed system level approach has been promoted to address distributed design of dynamic distributed controllers [14].…”