A System Level Approach to Controller Synthesis

Yuh-Shyang Wang; Nikolai Matni; John C. Doyle

Source-linked AI summary

A System Level Approach to Controller Synthesis

Yuh-Shyang Wang, Nikolai Matni, John C. Doyle

arXiv:1610.04815v3Oct 16, 2016eess.SYmath.OC

TL;DR

Distributed controller synthesis is difficult because large-scale cyber-physical systems impose information-sharing constraints that undermine centralized parameterization methods. The paper introduces a System Level Approach combining SLPs, SLCs, and SLS to characterize constrained stabilizing controllers convexly and support scalable localized synthesis.

Problem
Distributed cyber-physical systems use interconnected sub-controllers with local information exchange, making centralized parameterization approaches not directly applicable to constrained controller synthesis.
Method
The System Level Approach directly parameterizes closed-loop responses through SLPs, imposes structural requirements with SLCs, and formulates resulting design problems as SLS programs.
Results
The approach characterizes all achievable stable system responses and the broadest known class of constrained internally stabilizing controllers admitting a convex representation.
Takeaways & Limitations
System Level Synthesis supports constrained optimal control formulations that include many established problems and can solve localized subproblems in parallel for arbitrarily large-scale systems when the constraint is feasible.
Takeaways & Limitations
Controller sparsity may conflict with dense plants, and feasible localized constraints depend on actuator and sensor density plus information and disturbance delays.

Abstract

from arXiv · show

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.

I. INTRODUCTION

The paper motivates a system level approach for constrained controller synthesis in distributed cyber-physical systems, where communication and implementation constraints challenge centralized parameterizations. It introduces SLPs, SLCs, and SLS to characterize and convexly synthesize a broader class of structured stabilizing controllers.

Motivation: Distributed cyber-physical systems comprise interconnected sub-controllers with local sensors, actuators, and constrained information exchange.These information-sharing constraints make distributed optimal controller synthesis challenging.
Motivation: Quadratic invariance preserves convexity for a large class of structured controller problems but cannot convexly characterize localized controllers in strongly connected systems.The resulting need for global information exchange limits synthesis and implementation scalability.
System Level Approach: The system level approach directly parameterizes closed-loop maps from disturbances to control actions and states rather than only the sensor-to-actuator feedback map.Its three elements are System Level Parameterizations, System Level Constraints, and System Level Synthesis problems.
System Level Approach: SLPs permit arbitrary sets of constraints on achievable closed-loop responses, and convex SLC representations yield convex sets of constrained system responses.The corresponding controller implementation carries these response constraints into its internal structure.
Contributions: The SLP-SLC class of constrained stabilizing controllers is a strict superset of controllers convexly parameterized through quadratic invariance.SLS problems include QI-based and localized optimal control formulations as special cases.
Contributions: The paper develops SLPs for stabilizing controllers and closed-loop responses, catalogs constraints including sparsity and spatiotemporal structure, and formulates SLS optimization problems.The paper considers discrete-time LTI systems and objectives based on norms of the disturbance-to-regulated-output transfer matrix.

B. Youla Parameterization

The Youla parameterization converts centralized stabilizing controller synthesis into a convex optimization over a stable parameter, while distributed information constraints are imposed through controller subspaces. Quadratic invariance is the condition that preserves this convexity, but it limits sparsity constraints for strongly connected plants.

Centralized Youla Parameterization: The Youla parameterization uses a doubly coprime factorization of the plant to represent internally stabilizing controllers through a stable parameter Q.The controller is reconstructed from the optimized Youla parameter using the coprime factors.
Centralized Youla Parameterization: Optimizing over Q makes the model-matching formulation convex and allows convex design specifications to be incorporated into controller synthesis.After optimization, the controller is reconstructed as K = (Yr −UrQ)(Xr −VrQ)−1.
Distributed Control and QI: Distributed control models information asymmetry through communication delays, which appear as subspace constraints on the controller.The delay model assumes dedicated physical communication channels and may not apply under wireless settings.
Distributed Control and QI: A controller subspace constraint C is convexly enforceable through Q when C is quadratically invariant with respect to P22.Quadratic invariance requires KP22K ∈ C for every K ∈ C and is also necessary for this Youla-based convex enforcement.
Distributed Control and QI: Communication delays can range from instantaneous exchange to no communication, and continuous-time delay constraints may lie in H∞ rather than RH∞.These modeling choices determine the subspaces used to encode distributed information sharing.
Distributed Control and QI: Under QI, the distributed optimal control problem can be recast as a convex model-matching problem in the Youla parameter.This preserves the convex synthesis benefit of the centralized parameterization for QI controller structures.

D. QI imposes limitations on controller sparsity

The paper identifies controller sparsity as a limitation of quadratic invariance for strongly connected systems and introduces system-level parameterizations centered on achievable closed-loop responses. These responses admit affine characterizations and internally stabilizing realizations that support convex structural constraints.

D. QI imposes limitations on controller sparsity: For strongly connected systems, controller sparsity is generally not quadratically invariant and therefore cannot be imposed convexly through the Youla parameter.This limits scalability because distributed controllers require global information exchange.
D. QI imposes limitations on controller sparsity: System responses map process and measurement disturbances to states and control actions, rather than directly parameterizing only the sensor-to-actuator feedback loop.The proposed system-level approach designs the entire closed-loop response.
D. QI imposes limitations on controller sparsity: The state-feedback feasibility conditions provide an alternative characterization of stabilizability, with a dual characterization for detectability.Feasibility of the affine constraints is equivalent to stabilizability of (A, B2), while the dual argument characterizes detectability of (A, C2).
D. QI imposes limitations on controller sparsity: The proposed controller realization preserves internal stability while allowing constraints on system responses to carry over to controller implementation.This connects response-level design with structural constraints on internal controller blocks.
D. QI imposes limitations on controller sparsity: For state feedback, stable achievable responses form an affine subspace, and every response in it is achieved by an internally stabilizing controller.The controller is K = MR^-1, implemented using the structure in Figure 2.

B. Output Feedback with D22 = 0

For strictly proper plants with D22 = 0, the paper characterizes output-feedback system responses through affine constraints and constructs internally stabilizing controllers that achieve them. The resulting realization directly exposes convex constraints on controller transfer blocks.

B. Output Feedback with D22 = 0: Output-feedback responses map state and measurement disturbances to states and control actions through {R, M, N, L}.The controller achieving the response is K = L − MR^-1N.
B. Output Feedback with D22 = 0: The affine subspace in (16) parameterizes all stable output-feedback responses achievable by internally stabilizing controllers.Feasibility is equivalent to stabilizability and detectability of (A, B2, C2).
B. Output Feedback with D22 = 0: For any response satisfying (16), K = L − MR^-1N internally stabilizes the plant and achieves x = Rδx + Nδy and u = Mδx + Lδy.The Figure 3 implementation has stable closed-loop transfer matrices from the modeled perturbations to internal variables.
B. Output Feedback with D22 = 0: The Figure 3 realization replaces constant state-space controller matrices with stable proper transfer matrices ˜R+, ˜M, ˜N, and L.Arbitrary convex constraints on these transfer matrices carry directly into the controller implementation.
B. Output Feedback with D22 = 0: General convex constraints on the controller K or its state-space realization do not themselves produce convex optimal control problems.The system-level realization instead exposes transfer blocks on which convex constraints can be imposed.

C. Specialized Implementations for Open-loop Stable Systems

For open-loop stable systems, two simplified controller realizations are presented, but their internal stability depends on plant stability and explicit treatment of controller-state perturbations.

Setting δu and δβ to zero yields the simplified realization u = Ly−MB2u, shown in Figure 4(b).
The Figure 4(b) realization is internally stable when the open-loop plant is stable, but can become unstable for unstable plants.For unstable plants, (zI−A)−1B2 is unstable and perturbation δβ can destabilize the closed loop.
An alternative realization follows from K = L(I + C2N)−1 and is likewise internally stable only when the open-loop plant is stable.This structure also requires analyzing perturbations to the controller internal state β.
For open-loop stable systems, the Figure 4(b) structure is an alternative realization of the internal model control principle applied to the Youla parameterization.The paper states that IMC is equivalent to the proposed parameterization and its simplified Figure 4(b) representation in this setting.

D. Output Feedback with D22̸ = 0

For a proper plant with D22̸ = 0, redefining the measurement produces the proposed output-feedback structure, which internally stabilizes the plant.

Defining the measurement as ȳ[t] = y[t]−D22u[t] yields the proposed controller structure for D22̸ = 0.The corresponding closed-loop transfer matrices from δu to the internal variables are modified accordingly.
The resulting Figure 5 controller structure internally stabilizes the plant.

E. System Level and Youla Parameterizations

The SL parameterization represents achievable responses through an affine kernel-space description, unlike Youla’s image-space representation, but its constraints are generally infinite dimensional.

Youla parameterization uses an image-space representation that supports efficient FIR computation but generally makes controller sparsity constraints intractable.
The SL parameterization uses a kernel-space representation in which achievable responses are specified implicitly by affine constraints.
Although the SL parameterization is convex, its affine constraints are generally infinite dimensional and therefore do not immediately yield efficient computation.

IV. SYSTEM LEVEL CONSTRAINTS

System Level Constraints impose structural or performance restrictions directly on system responses, translating convex controller constraints into convex response constraints and enabling broader constrained synthesis.

System Level Constraints can impose structural or performance restrictions on system responses, including localized sparse subspace constraints.
The framework can tractably enforce localized constraints for general strongly connected systems, unlike prior parameterizations described by the paper.
SLPs require neither the Youla parameterization nor coprime factors; those tools are used only to establish connections with the SLA.
Any constraint on the Youla parameter can be translated into a System Level Constraint, and convexity is preserved under this translation.
The constrained controller set can be equivalently expressed through system responses, with convexity exactly matching convexity of the corresponding Youla constraint set.
The controller–Youla–response relationship is established by expressing plant outputs through process and measurement disturbances before substitution into the closed-loop mapping.

B. Quadratically Invariant Subspace Constraints

QI subspace constraints are recovered as a special case of SLCs, while localized response constraints address sparsity patterns that QI cannot convexly characterize for strongly connected systems.

QI-constrained stabilizing controllers are parameterized through the SLP using L = M(Q) within the controller subspace.
The QI corollary applies to both stable and unstable plants without requiring an initial strongly stabilizing controller or doubly coprime factorization.
For strongly connected systems, QI cannot convexly characterize localized controllers because controller sparsity constraints fail quadratic invariance.
Localized SLCs impose sparse support on system responses, enabling local controller-state and control-action computation from local measurements.The resulting system response can also decompose into local subproblems for large-scale systems.
A specified localized SLC may be infeasible because its affine-space intersection depends on actuator and sensor density, communication delay, and disturbance propagation delay.

E. FIR Constraints

FIR SLCs impose finite-horizon system responses, yielding finite-dimensional and simple filter-bank implementations while supporting spatiotemporal locality constraints.

An FIR SLC restricts the stable system response to a finite impulse-response horizon T.The controller can then be implemented using FIR filter banks.
FIR-constrained parameterizations are finite-dimensional because the affine constraints apply only to impulse-response elements through time T.
The FIR SLP has a solution when (A, B2, C2) is controllable and observable.
FIR constraints can be infeasible when the underlying system is only stabilizable and/or detectable, for which SLP relaxations may be used.
Intersecting QI, FIR, and localized SLCs produces spatiotemporal constraints combining communication delay, deadbeat response, and disturbance localization.The space-time diagram encodes spatial position vertically and time horizontally.
Feasible spatiotemporal constraints define localizability, constraining each disturbance response to be finite in both space and time under communication delays.

1) System Performance Constraints:

Performance SLCs constrain closed-loop response functionals and remain compatible with structural constraints, enabling convex multi-objective and architecture-aware controller synthesis.

If the performance functional g is convex, its sublevel-set SLC is convex and can be combined with the SLP.
Multiple performance SLCs with different system norms formulate multi-objective optimal control problems.
Closed-loop constraints can limit the effects of internal controller perturbations caused by quantization or other hardware errors.
Actuator and sensor counts can be constrained through nonzero rows and columns of controller-response transfer matrices, with convex relaxations for the resulting nonconvex constraints.
Element-wise nonnegativity constraints on response matrices provide a convex SLC for enforcing positive closed-loop systems.
Performance constraints can be combined with localized spatiotemporal structural constraints for large-scale distributed control.

B. Examples of Convex SLS

SLS combines SLPs, SLCs, and performance objectives into convex structured synthesis problems, including QI and localized formulations that can decompose across local subproblems.

Examples of Convex SLS: All distributed optimal control problems with convex Youla-domain representations are special cases of convex SLS problems.
2) Localized LQG Control: Localized SLS combines QI, sparsity, and FIR constraints to formulate localized LQG control.
2) Localized LQG Control: The localized SLS problem decomposes into independent column-wise subproblems that can be solved in parallel using local system information.
2) Localized LQG Control: For a feasible spatiotemporal SLC, localized SLS can address arbitrarily large-scale systems when sub-controllers solve their subproblems in parallel.
Examples of Convex SLS: SLS supports convex regularization to explore tradeoffs between closed-loop performance and actuator, sensor, and communication-link costs.
VI. CONCLUSION: The paper presents SLPs, SLCs, and SLS as a framework recovering many constrained optimal-control problems while admitting convex representations.

APPENDIX A STABILIZABILITY AND DETECTABILITY

The appendix characterizes stabilizability and detectability through feasibility of affine subspaces, connecting transfer-matrix constructions to classical rank tests and observer parameterizations.

The pair (A, B2) is stabilizable exactly when the affine subspace defined by (10) is non-empty.
Stabilizability constructs stable transfer matrices R and M from a stabilizing state-feedback gain, yielding a solution to equation (10a).
The converse uses right invertibility and full row rank for |z| ≥1, which is equivalent to the PBH stabilizability test.
By duality, the state-feedback analysis extends to state estimation, giving an affine-feasibility characterization of detectability for (A, C2).
The affine subspace in (37) parameterizes all detectable observers, while feasibility of (16) characterizes stabilizability and detectability of (A, B2, C2).