[Paper Review] A System Level Approach to Controller Synthesis

Yuh-Shyang Wang, Nikolai Matni+1 more|arXiv (Cornell University)|Oct 16, 2016

Fault Detection and Control Systems47 references19 citations

TL;DR

This paper introduces a system-level approach to controller synthesis that redefines controller design by parameterizing closed-loop system responses rather than controllers directly. By leveraging System Level Parameterizations (SLPs), System Level Constraints (SLCs), and System Level Synthesis (SLS), the framework enables convex optimization over the broadest known class of structured, constrained controllers—generalizing quadratic invariance and enabling tractable solutions for large-scale, distributed cyber-physical systems with sparsity, delay, and locality constraints.

ABSTRACT

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.

Motivation & Objective

• Address the challenge of designing constrained, decentralized controllers for large-scale cyber-physical systems with limited, sparse, and distributed communication and computation.
• Overcome the limitations of traditional Youla parameterization in distributed settings, where structural constraints often lead to NP-hard problems.
• Develop a unified framework that preserves convexity in controller synthesis even under complex structural constraints such as sparsity, delay, and locality.
• Generalize quadratic invariance (QI) by introducing a broader class of stabilizing controllers that admit convex characterization through system-level responses.
• Enable co-design of controllers, system responses, and communication/sensing architectures via a unified optimization formulation.

Proposed method

• Propose System Level Parameterizations (SLPs) that parameterize all achievable and stable closed-loop system responses (from sensors to actuators) via transfer matrices R and M, forming an isomorphism between responses and stabilizing controllers.
• Introduce System Level Constraints (SLCs) to embed structural properties such as sparsity, delay, and locality directly into the response space, enabling natural enforcement of communication and computational constraints.
• Formulate System Level Synthesis (SLS) problems as convex optimization problems over the intersection of affine constraints (on responses) and a constraint set S, enabling efficient solution via convex programming.
• Establish a generalized notion of stabilizability and detectability based on the existence of solutions to affine equations involving transfer matrices, extending classical definitions to the system-level framework.
• Derive a novel controller implementation (Equation 18) that realizes the desired system response using only local information and communication, ensuring structural compliance.
• Leverage the structure of the SLS problem to transparently analyze computational complexity and enable convex relaxations when non-convexity arises.

Experimental results

Research questions

• RQ1 Can a parameterization of closed-loop system responses be developed that enables convex synthesis of constrained controllers in large-scale, distributed systems?
• RQ2 How can structural constraints such as sparsity, delay, and locality in communication and computation be naturally embedded into the controller design process?
• RQ3 To what extent does the proposed system-level framework generalize quadratic invariance (QI) and preserve convexity in structured controller synthesis?
• RQ4 Can the framework support co-design of controllers, system responses, and network architectures in large-scale networked systems?
• RQ5 What is the computational complexity of the resulting optimization problem, and how does the formulation enable tractable solution via convex programming?

Key findings

• The system-level approach parameterizes all internally stabilizing controllers through the closed-loop system response (R, M), generalizing the Youla parameterization to distributed settings.
• The framework identifies the broadest known class of constrained stabilizing controllers that admit a convex characterization, extending beyond quadratic invariance.
• System Level Synthesis (SLS) problems are formulated as convex optimization problems over affine constraints and a constraint set S, enabling efficient solution via convex programming.
• The approach naturally incorporates SLCs such as sparsity, delay, and locality in both communication and computation, enabling structured controller design with provable stability.
• The framework enables a rich separation structure between stabilizability/detectability and performance, generalizing classical notions to the system-level context.
• The proposed controller implementation (Equation 18) realizes the desired response using only local information, ensuring structural compliance and scalability in large networks.