Deployment Architectures for Cyber-Physical Control Systems

Abstract

We consider the problem of how to deploy a controller to a (networked) cyber-physical system (CPS). Controlling a CPS is an involved task, and synthesizing a controller to respect sensing, actuation, and communication constraints is only part of the challenge. In addition to controller synthesis, one should also consider how the controller will work in the CPS. Put another way, the cyber layer and its interaction with the physical layer need to be taken into account. In this work, we aim to bridge the gap between theoretical controller synthesis and practical CPS deployment. We adopt the system level synthesis (SLS) framework to synthesize a state-feedback controller and provide a deployment architecture for the standard SLS controller. Furthermore, we derive a new controller realization for open-loop stable systems and introduce four different architectures for deployment, ranging from fully centralized to fully distributed. Finally, we compare the trade-offs among them in terms of robustness, memory, computation, and communication overhead. Notation and Terminology: Let RH ∞ denote the set of stable rational proper transfer matrices, and z −1 RH ∞ ⊂ RH ∞ be the subset of strictly proper stable transfer matrices. Lower-and upper-case letters (such as x and A) denote vectors and matrices respectively, while bold lower-and upper-case characters and symbols (such as u and Φ u) are reserved for signals and transfer matrices. Let A ij be the entry of A at the i th row and j th column. We define A i as the i th row and A j the j th column of A. We use Φ u [τ ] to denote the τ th spectral element of a transfer function Φ u , i.e., Φ u = ∞ τ =0 z −τ Φ u [τ ]. We briefly summarize below the major terminology: • Controller model: a (linear) map from the state vector to the control action. • Realization: a control block diagram/state space dynamics based on some controller model. • Architecture: a cyber-physical system structure built from basic components. • Synthesize: derive a controller model (and some realization). • Deploy/Implement: map a controller model (through a realization) to an architecture.

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 write A → B as a short-hand notation for "given A, we can derive B accordingly." 1 We adopt the terminology in [13] that distinguishes "realizations" from "implementations," where the former refers to the block diagrams (mathematical expressions) and the latter is reserved for the physical architecture consisting of computation, memory, and communication units.…”

“…6. The realization proposed in [13] that realizes the SLS state-feedback controller using only one convolution zΦu.…”

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

“…Explicit solutions to the general SLS problem are described in [27], and [28] uses dynamic programming to solve for SLS synthesis problems 10 times faster than using a conventional solver. [29] describes deployment architecture for SLS, and is used in the construction of the SLSpy software framework [30]. In addition to this Python implementation, a MATLABbased toolbox is also publicly available [31].…”