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On PI-control in Capacity-Limited Networks

Felix Agner, Anders Rantzer

TL;DR

This paper concerns control of a class of systems where multiple dynamically stable agents share a nonlinear and bounded control-interconnection, and shows that any equilibrium to this closed-loop system minimizes the maximum tracking error for any agent.

Abstract

This paper concerns control of a class of systems where multiple dynamically stable agents share a nonlinear and bounded control-interconnection. The agents are subject to a disturbance which is too large to reject with the available control action, making it impossible to stabilize all agents in their desired states. In this nonlinear setting, we consider two different anti-windup equipped proportional-integral control strategies and analyze their properties. We show that a fully decentralized strategy will globally, asymptotically stabilize a unique equilibrium. This equilibrium also minimizes a weighted sum of the tracking errors. We also consider a light addition to the fully decentralized strategy, where rank-1 coordination between the agents is introduced via the anti-windup action. We show that any equilibrium to this closed-loop system minimizes the maximum tracking error for any agent. A remarkable property of these results is that they rely on extremely few assumptions on the interconnection between the agents. Finally we illustrate how the considered model can be applied in a district heating setting, and demonstrate the two considered controllers in a simulation.

On PI-control in Capacity-Limited Networks

TL;DR

This paper concerns control of a class of systems where multiple dynamically stable agents share a nonlinear and bounded control-interconnection, and shows that any equilibrium to this closed-loop system minimizes the maximum tracking error for any agent.

Abstract

This paper concerns control of a class of systems where multiple dynamically stable agents share a nonlinear and bounded control-interconnection. The agents are subject to a disturbance which is too large to reject with the available control action, making it impossible to stabilize all agents in their desired states. In this nonlinear setting, we consider two different anti-windup equipped proportional-integral control strategies and analyze their properties. We show that a fully decentralized strategy will globally, asymptotically stabilize a unique equilibrium. This equilibrium also minimizes a weighted sum of the tracking errors. We also consider a light addition to the fully decentralized strategy, where rank-1 coordination between the agents is introduced via the anti-windup action. We show that any equilibrium to this closed-loop system minimizes the maximum tracking error for any agent. A remarkable property of these results is that they rely on extremely few assumptions on the interconnection between the agents. Finally we illustrate how the considered model can be applied in a district heating setting, and demonstrate the two considered controllers in a simulation.

Paper Structure

This paper contains 16 sections, 9 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Plant Description
  5. Problem Description
  6. Considered Controllers and Main Results
  7. Decentralized Control
  8. Coordinating Control
  9. Motivating Example - District Heating
  10. Numerical Example
  11. Main Proofs
  12. Additional Nonlinearity Properties
  13. Stability Proofs
  14. Optimality Proofs
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

Let Assumptions ass:assumptions on b and ass:decentralized controller tuning hold. Then the closed loop system eq:decentralized closed loop formed by the decentralized controller has a unique, globally asymptotically stable equilibrium.

Figures (3)

  • Figure 1: A district heating network. A heating plant (node 23) heats up water and pumps it out to consumers (nodes 1-22) through the supply-side network (red edges). The water subsequently returns through the return-side network (blue edges).
  • Figure 2: Outdoor temperature $T_o(t)$ used in simulation.
  • Figure 3: Resulting indoor temperatures. The three clusters of buildings, i.e., nodes 1-8, 9-14 and 15-22 are colored in red, green and blue respectively. The darker shades of each color shows the buildings further down each line.

Theorems & Definitions (19)

  • Remark 1
  • Remark 2
  • Theorem 1: Global asymptotic stability
  • Theorem 2: Equilibrium optimality
  • Theorem 3: Equilibrium optimality
  • Theorem 4: Global asymptotic stability
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 9 more