Scalable Robust Adaptive Control from the System Level Perspective

Abstract

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 fivedimensional 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 closedloop 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.Corresponding to (Const.2), let's define R(i) to be the set of subsystems from which subsystem i receives information:Definition III.1. R(i) := {j |i ∈ S(j) } Constraint 3 (Limited Computation). Every subsystem j has limited computational resources.

A System Level Approach to Discrete-Time Nonlinear Systems

“…Inspired by the recently developed system level approach to linear control theory [12], we will present a new insight on nonlinear discrete-time systems that we believe could lead to entirely new synthesis methods for nonlinear discretetime systems. The system level approach, as introduced in [12], enabled new efficient controller synthesis methods [13], [14] that allow for localized, distributed and scalable control design in large-scale systems. This is achieved by transforming constrained optimal control problems as convex optimization problems over achievable closed-loop maps that can be solved efficiently.…”

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

“…where (3a) assumes that R c (1) is the identity. For a more detailed derivation, refer to [10]. The corresponding frequencydomain equations arê…”

“…Substituting (8) into (11) and multiplying by ∆ c , then writing out ∆ c in terms of (A, B, R c , M c ), gives (10).…”

“…The solution space defined by (10), although it exists for T c ≥ T , often yields solutions that are unstable. Further, Corollary 2.1 gives a fundamental limit on the sparsity of M c .…”

“…as the objective. This is the equation error for (10), and is a heuristic for the closed-loop difference.…”