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Decentralized PI-control and Anti-windup in Resource Sharing Networks

Felix Agner, Jonas Hansson, Pauline Kergus, Anders Rantzer, Sophie Tarbouriech, Luca Zaccarian

TL;DR

The paper addresses decentralized stabilization of multiple interconnected first-order agents with an M-matrix interconnection and incrementally sector-bounded nonlinearities, under constant disturbances. It develops fully decentralized PI controllers with anti-windup, proving existence and uniqueness of a global equilibrium and global asymptotic stability under local, fully decentralized tuning rules. In the saturating case, the equilibrium is shown to minimize a weighted $\ell_1$-norm of the state mismatch when a diagonally dominant condition on $\Gamma A^{-1} B$ holds, with a district-heating numerical example illustrating performance trade-offs against coordination. The work demonstrates scalable, robust control for resource-sharing networks, with practical relevance to district heating and similar infrastructure systems.

Abstract

We consider control of multiple stable first-order \review{agents} which have a control coupling described by an M-matrix. These agents are subject to incremental sector-bounded \review{input} nonlinearities. We show that such plants can be globally asymptotically stabilized to a unique equilibrium using fully decentralized proportional-integral controllers equipped with anti-windup and subject to local tuning rules. In addition, we show that when the nonlinearities correspond to the saturation function, the closed loop asymptotically minimizes a weighted 1-norm of the agents state mismatch. The control strategy is finally compared to other state-of-the-art controllers on a numerical district heating example.

Decentralized PI-control and Anti-windup in Resource Sharing Networks

TL;DR

The paper addresses decentralized stabilization of multiple interconnected first-order agents with an M-matrix interconnection and incrementally sector-bounded nonlinearities, under constant disturbances. It develops fully decentralized PI controllers with anti-windup, proving existence and uniqueness of a global equilibrium and global asymptotic stability under local, fully decentralized tuning rules. In the saturating case, the equilibrium is shown to minimize a weighted -norm of the state mismatch when a diagonally dominant condition on holds, with a district-heating numerical example illustrating performance trade-offs against coordination. The work demonstrates scalable, robust control for resource-sharing networks, with practical relevance to district heating and similar infrastructure systems.

Abstract

We consider control of multiple stable first-order \review{agents} which have a control coupling described by an M-matrix. These agents are subject to incremental sector-bounded \review{input} nonlinearities. We show that such plants can be globally asymptotically stabilized to a unique equilibrium using fully decentralized proportional-integral controllers equipped with anti-windup and subject to local tuning rules. In addition, we show that when the nonlinearities correspond to the saturation function, the closed loop asymptotically minimizes a weighted 1-norm of the agents state mismatch. The control strategy is finally compared to other state-of-the-art controllers on a numerical district heating example.
Paper Structure (8 sections, 7 theorems, 26 equations, 1 figure, 1 table)

This paper contains 8 sections, 7 theorems, 26 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Data and Proposed Controller
  3. Main Results
  4. Numerical Example
  5. Proof of Equilibrium Existence and Uniqueness
  6. Proof of Global Asymptotic Stability
  7. Proof of Equilibrium Optimality
  8. Conclusions

Key Result

Lemma 1

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ satisfy Assumption ass: function assumption. Then $h(u) = u - f(u)$ also satisfies Assumption ass: function assumption.

Figures (1)

  • Figure 1: Temperature deviations $x$ (blue, left axis) for each strategy and the outdoor temperature $w$ (black, dotted, right axis). Around hour 100, $w$ becomes critically low and the indoor temperatures drop as the controllers saturate.

Theorems & Definitions (16)

  • Lemma 1
  • Definition 1
  • Proposition 1
  • Theorem 1: Equilibrium Existence and Uniqueness
  • Theorem 2: Global Asymptotic Stability
  • Remark 1
  • Theorem 3: Equilibrium Optimality
  • Remark 2
  • Lemma 2
  • Lemma 3
  • ...and 6 more