A System Level Approach to Controller Synthesis

Biological and advanced cyberphysical control systems often have limited, sparse, uncertain, and distributed communication and computing in addition to sensing and actuation. Fortunately, the corresponding plants and performance requirements are also sparse and structured, and this must be exploited to make constrained controller design feasible and tractable. We introduce a new “system level” (SL) approach involving three complementary SL elements. System Level Parameterizations (SLPs) generalize state space and Youla parameterizations of all stabilizing controllers and the responses they achieve, and combine with System Level Constraints (SLCs) to parameterize the largest known class of constrained stabilizing controllers that admit a convex characterization, generalizing quadratic invariance (QI). SLPs also lead to a generalization of detectability and stabilizability, suggesting the existence of a rich separation structure, that when combined with SLCs, is naturally applicable to structurally constrained controllers and systems. We further provide a catalog of useful SLCs, most importantly including sparsity, delay, and locality constraints on both communication and computing internal to the controller, and external system performance. The resulting System Level Synthesis (SLS) problems that arise define the broadest known class of constrained optimal control problems that can be solved using convex programming. An example illustrates how this system level approach can systematically explore tradeoffs in controller performance, robustness, and synthesis/implementation complexity.

💡 Research Summary

The paper addresses the challenge of designing optimal controllers for large‑scale biological and cyber‑physical systems that operate under severe communication, computation, sensing, and actuation constraints. While classical optimal control relies on the Youla parameterization—an isomorphism between stabilizing controllers and closed‑loop input‑output maps—this framework only permits structural constraints on the controller’s I/O map and fails to capture many practical distributed architectures, especially when the underlying plant is densely connected. To overcome these limitations, the authors introduce a “system‑level” approach (SLA) built around three complementary concepts: System‑Level Parameterizations (SLPs), System‑Level Constraints (SLCs), and System‑Level Synthesis (SLS).

SLPs provide a complete parameterization of all internally stabilizing controllers by directly describing the closed‑loop response from disturbances to states, control actions, and regulated outputs. Mathematically, the SLP characterizes the set of transfer matrices Φₓ₍w₎, Φᵤ₍w₎, Φ_z₍w₎ that satisfy linear equations derived from the plant dynamics (e.g., AΦₓ₍w₎+B₂Φᵤ₍w₎=I, C₂Φₓ₍w₎+D₂₂Φᵤ₍w₎=0) and belong to RH∞, guaranteeing stability.

SLCs are convex sets imposed on these response matrices. By formulating sparsity, delay, locality, finite‑impulse‑response (FIR), and architectural constraints as linear or convex matrix inequalities on Φ, the designer can enforce exactly the same structural properties on the internal realization of the controller. Because the constraints act on the response rather than on the controller directly, the approach sidesteps the quadratic invariance (QI) condition that previously limited convex synthesis to subspaces where K P₂₂ K∈C for all admissible K. Consequently, even for strongly connected (dense) plants, one can obtain localized or sparse controllers via convex programming.

The SLS problem combines SLPs and SLCs into a single convex optimization: minimize a performance norm (typically H₂ or H∞) of the closed‑loop map T₁₁+T₁₂QT₂₁ subject to Q∈RH∞ and the chosen SLCs (or equivalently, Φ∈SLC). When the SLCs are expressed as FIR or sparsity constraints, the problem reduces to a finite‑dimensional semidefinite program that can be solved efficiently, and distributed algorithms such as ADMM can be employed for very large systems. The authors show that the classical QI‑based distributed optimal control formulation and recent localized optimal control frameworks are special cases of SLS, thereby establishing SLS as the most general convexly tractable formulation known to date.

A numerical example on a large power‑grid model demonstrates how SLS can systematically explore trade‑offs among performance, robustness, communication delay, and implementation complexity. The results illustrate that SLS yields controllers with comparable or better performance than QI‑based designs while respecting strict locality and sparsity requirements that QI cannot accommodate.

In summary, the system‑level approach redefines the controller synthesis paradigm: by parameterizing the entire closed‑loop response, imposing convex structural constraints directly on that response, and solving the resulting convex synthesis problem, it extends the scope of tractable distributed optimal control far beyond the quadratic‑invariance frontier, offering both theoretical insight and practical tools for the design of scalable, implementable controllers in modern networked systems.