A System-Level Approach to Controller Synthesis

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. Index Terms-constrained & structured optimal control, decentralized control, large-scale systems, system level parameterization, system level constraint, system level synthesis Preliminaries & Notation: We use lower and upper case Latin letters such as x and A to denote vectors and matrices, respectively, and lower and upper case boldface Latin letters such as x and G to denote signals and transfer matrices, respectively. We use calligraphic letters such as S to denote sets. In the interest of clarity, we work with discrete time linear time invariant systems, but unless stated otherwise, all results extend naturally to the continuous time setting. We use standard definitions of the Hardy spaces H 2 and H ∞ , and denote their restriction to the set of real-rational proper transfer matrices by RH 2 and RH ∞. We use G[i] to denote the ith spectral component of a transfer function G, i.e., G(z) = ∞ i=0 1 z i G[i] for |z| > 1. Finally, we use F T to denote the space of finite impulse response (FIR) transfer matrices with horizon T , i.e., F T := {G ∈ RH ∞ | G = T i=0 1 z i G[i]}.

On the Equivalence of Youla, System-Level, and Input–Output Parameterizations

“…In [5], the authors proposed a system-level parameterization for C stab . This approach is based on the closed-loop maps from process and measurement disturbances to state and control action.…”

“…In [5], it is proved that for strictly proper 2 plant P 22 , the set of all internally stabilizing controllers can be written as…”

“…In this section, we present explicit affine mappings among Youla parameterization, SLP, and IOP. The consequences are as follows: 1) any convex system-level synthesis (SLS) introduced by [5] can be equivalently reformulated into a convex problem in terms of the Youla parameter Q or the input-output parameters Y, U, W, Z, and vice versa; 2) building on the explicit affine mappings, we show that the notion of quadratic invariance [7] allows for equivalent convex reformulations of classical distributed optimal control in either Youla parameterization, SLP, or IOP.…”

“…The Ritz approximation [3, Chapter 15] is one method for solving infinite dimensional optimization problems. Specifically, for discrete-time systems, the finite impulse response (FIR) approximation is a practical choice [5], [6].…”

“…Note that a doubly-coprime factorization of the system must be computed as a preliminary step in Youla parameterization. Recently, a system-level parameterization [5] and an input-output parameterization [6] were introduced to characterize the set of all LTI stabilizing controllers, with no need of computing a doubly-coprime factorization of the system a priori. Similar to Youla, the system-level and inputoutput parameterizations treat certain closed-loop responses as design parameters.…”