Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Continuous-Time Control via System Level Synthesis

Yaozhi Du, Jing Shuang Li

Abstract

This paper designs H2 and H-infinity distributed controllers with local communication and local disturbance rejection. We propose a two-step procedure: first, select closed-loop poles; then, optimize over parameterized controllers. We build on the system level synthesis (SLS) parameterization -- primarily used in the discrete-time setting -- and extend it to the general continuous-time setting. We verify our approach in simulation on a 9-node grid governed by linearized swing equations, where our distributed controllers achieve performance comparable to that of optimal centralized controllers while facilitating local disturbance rejection.

Distributed Continuous-Time Control via System Level Synthesis

Abstract

This paper designs H2 and H-infinity distributed controllers with local communication and local disturbance rejection. We propose a two-step procedure: first, select closed-loop poles; then, optimize over parameterized controllers. We build on the system level synthesis (SLS) parameterization -- primarily used in the discrete-time setting -- and extend it to the general continuous-time setting. We verify our approach in simulation on a 9-node grid governed by linearized swing equations, where our distributed controllers achieve performance comparable to that of optimal centralized controllers while facilitating local disturbance rejection.

Paper Structure

This paper contains 16 sections, 10 theorems, 56 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Local communication and control
  4. Problem statement
  5. System level synthesis
  6. Closed-loop pole selection
  7. Finite-dimensional truncation
  8. Simple pole approximation
  9. and Control via SLS
  10. State-space realization
  11. H2 control
  12. Hinf control
  13. Simulations
  14. Chain plant
  15. Grid plant
  16. ...and 1 more sections

Key Result

Theorem 1

For system equ:LTIsys evolving under causal, linear state feedback controller equ:state_feedback, the following hold: Statement 1. The affine subspaceThis subspace is nontrivial if system $(A, B)$ is stabilizable. defined by: parameterizes all closed-loop responses $\{\mathbf{\Phi}_x(s), \mathbf{\Phi}_u(s)\}$ that are achievable by an internally stabilizing controller $\mathbf{K}(s)$. Statement 2

Figures (4)

  • Figure B1: (Left) Graph topology for \ref{['eq:example']}. (Right) Grid topology for Section \ref{['sec:Simulation']}. The central node communicates with all nodes encircled by the solid red line (if $d=1$) or dashed blue line (if $d=2$).
  • Figure C1: Controller implementation \ref{['equ:controller_realization']}, in closed loop with the plant and exogeneous disturbances/noise. Exogeneous signals $\boldsymbol{\delta}_u$ and $\boldsymbol{\delta}_y$ are used to verify internal stability of the system.
  • Figure F1: Chain plant with SLS controller. A disturbance is injected at node 6 at time $t=0$; it propagates only to nodes 4, 5, 6, 7, and 8 (i.e., nodes within $d=2$ distance of node 6) before being rejected.
  • Figure F2: Grid plant with various SLS controllers. All costs are normalized by the cost of the optimal centralized controller. For this system, $d=5$ is equivalent to a centralized controller since the diameter of the graph is 5.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Lemma 2
  • proof
  • Theorem 4
  • ...and 6 more