Separating Controller Design from Closed-Loop Design: A New Perspective on System-Level Controller Synthesis

Abstract

We show that given a desired closed-loop response for a system, there exists an affine subspace of controllers that achieve this response. By leveraging the existence of this subspace, we are able to separate controller design from closed-loop design by first synthesizing the desired closed-loop response and then synthesizing a controller that achieves the desired response. This is a useful extension to the recently introduced System Level Synthesis framework, in which the controller and closed-loop response are jointly synthesized and we cannot enforce controller-specific constraints without subjecting the closed-loop map to the same constraints. We demonstrate the importance of separating controller design from closed-loop design with an example in which communication delay and locality constraints cause standard SLS to be infeasible. Using our new two-step procedure, we are able to synthesize a controller that obeys the constraints while only incurring a 3% increase in LQR cost compared to the optimal LQR controller.

Realization, Internal Stability, and Controller Synthesis

“…Email: shtseng@caltech.edu grams/implementations, 1 describe how a system can be built from some interconnection of basic blocks/transfer functions. It is well known, also shown by recent studies [13], [14], that the same controller can admit multiple different realizations, even under the same parameterization scheme. We would then wonder if we can only handle those realizations case by case, or if there is a unified framework to study them.…”

“…In Section III, we revisit controller synthesis theories, including Youla [2], input-output [10], system level [6], [7], and some mixed parameterizations [12], and verify the equivalence results in [11]. In Section IV, we derive the original SLS realization and alternative realizations from [13] and [14] using the realization-stability lemma. Finally, we conclude the paper in Section V.…”

“…We remark that Corollary 9 does not guarantee that S ∈ RH ∞ , and hence the authors in [14] propose to perform a posteriori stability check. According to the proof of Corollary 9, we can easily guarantee S ∈ RH ∞ by requiring S δx ∈ z −1 RH ∞ (to ensure S δδ = S δx (zI − A) ∈ RH ∞ ).…”

“…Instead of directly adopting the realization in Fig. 5, [14] found that it is possible to use much simpler transfer matrices to realize the same controller. The following corollary is from [14, Theorem 2] 3 .…”

Frontiers in Scalable Distributed Control: SLS, MPC, and Beyond

“…In a setting with minimal uncertainty, robust SLS can be used when overly strict controller-motivated constraints on {Φ x , Φ u } result in infeasibility during synthesis; this was the original goal of the robust SLS derivation. An alternative 'two-step' approach is presented in [16], whose results allow for separation of controller and closed-loop system constraints in the state-feedback setting.…”