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Extremum and Nash Equilibrium Seeking with Delays and PDEs: Designs & Applications

Tiago Roux Oliveira, Miroslav Krstić, Tamer Başar

TL;DR

Results on algorithm design and theory of ES for infinite-dimensional systems for hyperbolic and parabolic dynamics are reviewed and Nash equilibrium seeking methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization.

Abstract

The development of extremum seeking (ES) has progressed, over the past hundred years, from static maps, to finite-dimensional dynamic systems, to networks of static and dynamic agents. Extensions from ODE dynamics to maps and agents that incorporate delays or even partial differential equations (PDEs) is the next natural step in that progression through ascending research challenges. This paper reviews results on algorithm design and theory of ES for such infinite-dimensional systems. Both hyperbolic and parabolic dynamics are presented: delays or transport equations, heat-dominated equation, wave equations, and reaction-advection-diffusion equations. Nash equilibrium seeking (NES) methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization. Even heterogeneous PDE games, such as a duopoly with one parabolic and one hyperbolic agent, are considered. Several engineering applications are touched upon for illustration, including flow-traffic control for urban mobility, oil-drilling systems, deep-sea cable-actuated source seeking, additive manufacturing modeled by the Stefan PDE, biological reactors, light-source seeking with flexible-beam structures, and neuromuscular electrical stimulation.

Extremum and Nash Equilibrium Seeking with Delays and PDEs: Designs & Applications

TL;DR

Results on algorithm design and theory of ES for infinite-dimensional systems for hyperbolic and parabolic dynamics are reviewed and Nash equilibrium seeking methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization.

Abstract

The development of extremum seeking (ES) has progressed, over the past hundred years, from static maps, to finite-dimensional dynamic systems, to networks of static and dynamic agents. Extensions from ODE dynamics to maps and agents that incorporate delays or even partial differential equations (PDEs) is the next natural step in that progression through ascending research challenges. This paper reviews results on algorithm design and theory of ES for such infinite-dimensional systems. Both hyperbolic and parabolic dynamics are presented: delays or transport equations, heat-dominated equation, wave equations, and reaction-advection-diffusion equations. Nash equilibrium seeking (NES) methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization. Even heterogeneous PDE games, such as a duopoly with one parabolic and one hyperbolic agent, are considered. Several engineering applications are touched upon for illustration, including flow-traffic control for urban mobility, oil-drilling systems, deep-sea cable-actuated source seeking, additive manufacturing modeled by the Stefan PDE, biological reactors, light-source seeking with flexible-beam structures, and neuromuscular electrical stimulation.

Paper Structure

This paper contains 52 sections, 3 theorems, 229 equations, 34 figures, 1 table.

Table of Contents

  1. Introduction
  2. $N$-Player Game with Quadratic Payoffs for Nash Equilibrium Seeking
  3. Noncooperative Scenario with Delays
  4. Noncooperative Games with Heat PDEs
  5. Decentralized PDE Boundary Control using only the Known Diagonal Terms of the Hessian
  6. ISS-Like Properties for Parabolic PDE Representation
  7. Stability Analysis
  8. Heat PDEs Case
  9. Delay Case
  10. Noncooperative Duopoly with Heterogeneous Transport-Heat PDE Dynamics
  11. Quadratic Payoffs and Heterogeneous PDEs
  12. Decentralized PDE Boundary Control using only the Known Diagonal Terms of the Hessian
  13. ISS-Like Properties for Hyperbolic-Parabolic PDE Representation
  14. Stability Analysis
  15. Simulations
  16. ...and 37 more sections

Key Result

Theorem 1

Consider the closed-loop system (saco1_NC_CP14)--(saco3_NC_CP14) under Assumptions ch.13.Assumption 1. and ch13.Assumption 2., and multiple heat PDEs (eq:heat_eqn_start)--(eq:heat_eqn_end) with distinct diffusion coefficients $D_i$ for the $N$-player quadratic noncooperative game with payoff functio In particular, where $a=[a_1 \ a_2]^T$ and $\theta^*\!=\!\Theta^*$ is the unique (unknown) Nash Eq

Figures (34)

  • Figure 1: Nash equilibrium seeking schemes applied by two players ($N=2$) in a duopoly market structure with delayed players' actions.
  • Figure 2: Block diagram illustrating the Nash Equilibrium seeking strategy performed for each player. In red color, the predictor feedback used to compensate the individual delay $D_i$ for the noncooperative case. With some abuse of notation, the constants $c_i$ were chosen to denote the parameters of the filters $c_i/(s+c_i)$, but they are not necessarily the same constants which appear in the payoff functions given in (\ref{['eq:Ji']}).
  • Figure 3: Nash equilibrium seeking in a duopoly ($N=2$) noncooperative game with players acting through heat PDE dynamics.
  • Figure 4: Block diagram illustrating the Nash Equilibrium seeking strategy performed for each player ($N=2$). In magenta color are the boundary controllers used to compensate the individual heat PDEs for the noncooperative game.
  • Figure 5: Nash equilibrium seeking in a heterogeneous noncooperative game with players acting through transport-heat PDE dynamics.
  • ...and 29 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 1