System Level Synthesis via Dynamic Programming

System Level Synthesis (SLS) parametrization facilitates controller synthesis for large, complex, and distributed systems by incorporating system level constraints (SLCs) into a convex SLS problem and mapping its solution to stable controller design. Solving the SLS problem at scale efficiently is challenging, and current attempts take advantage of special system or controller structures to speed up the computation in parallel. However, those methods do not generalize as they rely on the specific system/controller properties. We argue that it is possible to solve general SLS problems more efficiently by exploiting the structure of SLS constraints. In particular, we derive dynamic programming (DP) algorithms to solve SLS problems. In addition to the plain SLS without any SLCs, we extend DP to tackle infinite horizon SLS approximation and entrywise linear constraints, which form a superclass of the locality constraints. Comparing to convex program solver and naive analytical derivation, DP solves SLS 4 to 12 times faster and scales with little computation overhead. We also quantize the cost of synthesizing a controller that stabilizes the system in a finite horizon through simulations

Scalable Robust Adaptive Control from the System Level Perspective

We will present a new general framework for robust and adaptive control that allows for distributed and scalable learning and control of large systems of interconnected linear subsystems. The control method is demonstrated for a linear time-invariant system with bounded parameter uncertainties, disturbances and noise. The presented scheme continuously collects measurements to reduce the uncertainty about the system parameters and adapts dynamic robust controllers online in a stable and performance-improving way. A key enabler for our approach is choosing a time-varying dynamic controller implementation, inspired by recent work on System Level Synthesis [1]. We leverage a new robustness result for this implementation to propose a general robust adaptive control algorithm. In particular, the algorithm allows us to impose communication and delay constraints on the controller implementation and is formulated as a sequence of robust optimization problems that can be solved in a distributed manner. The proposed control methodology performs particularly well when the interconnection between systems is sparse and the dynamics of local regions of subsystems depend only on a small number of parameters. As we will show on a five-dimensional exemplary chain-system, the algorithm can utilize system structure to efficiently learn and control the entire system while respecting communication and implementation constraints. Moreover, although current theoretical results require the assumption of small initial uncertainties to guarantee robustness, we will present simulations that show good closed-loop performance even in the case of large uncertainties, which suggests that this assumption is not critical for the presented technique and future work will focus on providing less conservative guarantees

System Level Synthesis with State and Input Constraints

This paper addresses the problem of designing distributed controllers with state and input constraints in the System Level Synthesis (SLS) framework. Using robust optimization, we show how state and actuation constraints can be incorporated into the SLS structure. Moreover, we show that the dual variable associated with the constraint has the same sparsity pattern as the SLS parametrization, and therefore the computation distributes using a simple primal-dual algorithm. We provide a stability analysis for SLS design with input saturation under the Internal Model Control (IMC) framework. We show that the closed-loop system with saturation is stable if the controller has a gain less than one. In addition, a saturation compensation scheme that incorporates the saturation information is proposed which improves the naive SLS design under saturation

System Level Synthesis with State and Input Constraints

This paper addresses the problem of designing distributed controllers with state and input constraints in the System Level Synthesis (SLS) framework. Using robust optimization, we show how state and actuation constraints can be incorporated into the SLS structure. Moreover, we show that the dual variable associated with the constraint has the same sparsity pattern as the SLS parametrization, and therefore the computation distributes using a simple primal-dual algorithm. We provide a stability analysis for SLS design with input saturation under the Internal Model Control (IMC) framework. We show that the closed-loop system with saturation is stable if the controller has a gain less than one. In addition, a saturation compensation scheme that incorporates the saturation information is proposed which improves the naive SLS design under saturation

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

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

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

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