Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Online Feedback Equilibrium Seeking

Giuseppe Belgioioso, Dominic Liao-McPherson, Mathias Hudoba de Badyn, Saverio Bolognani, Roy S. Smith, John Lygeros, Florian Dörfler

Abstract

This paper proposes a unifying design framework for dynamic feedback controllers that track solution trajectories of time-varying generalized equations, such as local minimizers of nonlinear programs or competitive equilibria (e.g., Nash) of non-cooperative games. Inspired by the feedback optimization paradigm, the core idea of the proposed approach is to re-purpose classic iterative algorithms for solving generalized equations (e.g., Josephy--Newton, forward-backward splitting) as dynamic feedback controllers by integrating online measurements of the continuous-time nonlinear plant. Sufficient conditions for closed-loop stability and robustness of the algorithm-plant cyber-physical interconnection are derived in a sampled-data setting by combining and tailoring results from (monotone) operator, fixed-point, and nonlinear systems theory. Numerical simulations on smart building automation and competitive supply-chain management are presented to support the theoretical findings.

Online Feedback Equilibrium Seeking

Abstract

This paper proposes a unifying design framework for dynamic feedback controllers that track solution trajectories of time-varying generalized equations, such as local minimizers of nonlinear programs or competitive equilibria (e.g., Nash) of non-cooperative games. Inspired by the feedback optimization paradigm, the core idea of the proposed approach is to re-purpose classic iterative algorithms for solving generalized equations (e.g., Josephy--Newton, forward-backward splitting) as dynamic feedback controllers by integrating online measurements of the continuous-time nonlinear plant. Sufficient conditions for closed-loop stability and robustness of the algorithm-plant cyber-physical interconnection are derived in a sampled-data setting by combining and tailoring results from (monotone) operator, fixed-point, and nonlinear systems theory. Numerical simulations on smart building automation and competitive supply-chain management are presented to support the theoretical findings.
Paper Structure (21 sections, 16 theorems, 100 equations, 10 figures)

This paper contains 21 sections, 16 theorems, 100 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Setting
  4. Control Strategy
  5. Closed-Loop Stability Analysis
  6. Illustrative Examples
  7. Closing the loop can lead to instability
  8. Vanishing steps do not track solutions trajectories
  9. Application Examples
  10. Temperature Regulation in Smart Buildings
  11. Competitive Supply Chain Management
  12. Conclusions
  13. Proof of Lemma \ref{['lem:gameReg']}
  14. Proof of Lemma \ref{['lmm:JN_method']}
  15. Proof of Proposition \ref{['lmm:SQNE']}
  16. ...and 6 more sections

Key Result

Theorem 1

dontchev2013euler Under Assumption ass:strong_reg, for any $w \in \mathbb{L}^{n_w}$, there exist $m$ Lipschitz continuous mappings $\bar{z}_i \in \mathbb{L}^{n_z}$, $i \in \{1,\ldots,m\}$, such that $\mathcal{S}(w) = \{\bar{z}_1,\ldots,\bar{z}_m\}$. $\square$

Figures (10)

  • Figure 1: In feedback equilibrium seeking, measurements from a countinuous-time dynamical system are incorporated into a discrete-time equilibrium seeking algorithm resulting in a coupled sampled-data cyber-physical system.
  • Figure 2: FBS controller. The GNE seeking algorithm \ref{['eq:pFB']} as a dynamic (semi) decentralized feedback controller with tunable gains $\gamma_i$'s and state $z = (u,\lambda)$.
  • Figure 3: The discrete-time interconnection \ref{['eq:discrete-time-system']} can be reconfigured into error coordinates leading to a feedback interconnection of two systems represented by $\mathcal{G}_1$, $\mathcal{G}_2$, and $\mathcal{H}$. Expressions for $\mathcal{G}_1,\mathcal{G}_2$ and $\mathcal{H}$ are given in Appendix \ref{['app:expressions']}.
  • Figure 4: Simulations of the sampled-data interconnection of the continuous-time SISO plant \ref{['eq:DI']} and the discrete-time algorithm \ref{['eq:prox_grad']}, with $\gamma=0.8$, under different choices of the sampling period $\tau$. On the left, the generated output; on the right, the correspondent control input trajectory.
  • Figure 5: Simulations of the sampled-data interconnection of the continuous-time SISO plant \ref{['eq:DI']}, subject to the additive disturbance $w(t) \!=\! 5\!\cdot\!10^{-2}t$, and the discrete-time algorithm \ref{['eq:prox_grad']}, under different choices of the step size sequence $\gamma^k$. The output tracking error diverges if the sequence vanishes as in \ref{['eq:VSS']}, while it stabilizes for constant step sizes. The oscillations are due to the combined effect of time-varying disturbance and sampled-data control.
  • ...and 5 more figures

Theorems & Definitions (35)

  • Definition 1: LISpS sontag1996new
  • Definition 2: Strong Regularityrobinson1980strongly
  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Definition 3
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 25 more