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Communication and Control Co-design in Non-cooperative Games

Shubham Aggarwal, Tamer Başar, Dipankar Maity

TL;DR

This work investigates communication-control co-design in a two-player noncooperative (NZS) differential game, where each player combines a scheduler and a remote controller to balance control performance against communication costs under asynchronous, intermittent information exchange. The authors derive Nash estimator–controller policies, express scheduler optimization via a generalized Sylvester equation, and tackle the infinite-polynomial nature of the steady-state covariance with a truncated Neumann-series approach coupled with a projected gradient algorithm to compute Nash equilibria. They present exhaustive and iterative methods to compute Nash scheduling policies and validate the results with numerical simulations on scalar and pursuit–evasion examples, illustrating that the Nash scheduling pair is often unique and exhibits intuitive relationships with scheduling costs. The findings provide a rigorous framework for designing distributed, game-theoretic control systems under communication constraints, with practical implications for networked control, security, and multi-agent coordination in wireless settings.

Abstract

In this article, we revisit a communication-control co-design problem for a class of two-player stochastic differential games on an infinite horizon. Each 'player' represents two active decision makers, namely a scheduler and a remote controller, which cooperate to optimize over a global objective while competing with the other player. Each player's scheduler can only intermittently relay state information to its respective controller due to associated cost/constraint to communication. The scheduler's policy determines the information structure at the controller, thereby affecting the quality of the control inputs. Consequently, it leads to the classical communication-control trade-off problem. A high communication frequency improves the control performance of the player on account of a higher communication cost, and vice versa. Under suitable information structures of the players, we first compute the Nash controller policies for both players in terms of the conditional estimate of the state. Consequently, we reformulate the problem of computing Nash scheduler policies (within a class of parametrized randomized policies) into solving for the steady-state solution of a generalized Sylvester equation. Since the above-mentioned reformulation involves infinite sum of powers of the policy parameters, we provide a projected gradient descent-based algorithm to numerically compute a Nash equilibrium using a truncated polynomial approximation. Finally, we demonstrate the performance of the Nash control and scheduler policies using extensive numerical simulations.

Communication and Control Co-design in Non-cooperative Games

TL;DR

This work investigates communication-control co-design in a two-player noncooperative (NZS) differential game, where each player combines a scheduler and a remote controller to balance control performance against communication costs under asynchronous, intermittent information exchange. The authors derive Nash estimator–controller policies, express scheduler optimization via a generalized Sylvester equation, and tackle the infinite-polynomial nature of the steady-state covariance with a truncated Neumann-series approach coupled with a projected gradient algorithm to compute Nash equilibria. They present exhaustive and iterative methods to compute Nash scheduling policies and validate the results with numerical simulations on scalar and pursuit–evasion examples, illustrating that the Nash scheduling pair is often unique and exhibits intuitive relationships with scheduling costs. The findings provide a rigorous framework for designing distributed, game-theoretic control systems under communication constraints, with practical implications for networked control, security, and multi-agent coordination in wireless settings.

Abstract

In this article, we revisit a communication-control co-design problem for a class of two-player stochastic differential games on an infinite horizon. Each 'player' represents two active decision makers, namely a scheduler and a remote controller, which cooperate to optimize over a global objective while competing with the other player. Each player's scheduler can only intermittently relay state information to its respective controller due to associated cost/constraint to communication. The scheduler's policy determines the information structure at the controller, thereby affecting the quality of the control inputs. Consequently, it leads to the classical communication-control trade-off problem. A high communication frequency improves the control performance of the player on account of a higher communication cost, and vice versa. Under suitable information structures of the players, we first compute the Nash controller policies for both players in terms of the conditional estimate of the state. Consequently, we reformulate the problem of computing Nash scheduler policies (within a class of parametrized randomized policies) into solving for the steady-state solution of a generalized Sylvester equation. Since the above-mentioned reformulation involves infinite sum of powers of the policy parameters, we provide a projected gradient descent-based algorithm to numerically compute a Nash equilibrium using a truncated polynomial approximation. Finally, we demonstrate the performance of the Nash control and scheduler policies using extensive numerical simulations.

Paper Structure

This paper contains 19 sections, 4 theorems, 74 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Networked Control systems
  4. Networked games
  5. Problem Formulation
  6. Communication-constrained LQ Games
  7. Joint Control-Communication Decision-making
  8. Information Structures and Nash Equilibrium
  9. Background and Preliminaries
  10. Equilibrium Estimator-Controller Policy
  11. Error Dynamics and Objective Functions
  12. Equilibrium Scheduler Policy
  13. Discrete-time Formulation
  14. Equilibrium Scheduling Policy
  15. Algorithmic Computation of the Equilibrium Policies
  16. ...and 4 more sections

Key Result

Theorem 1

bacsar2008h Suppose Assumption Assump_1 holds. Further, let $P \succeq 0$ denote the minimal positive-semidefinite solution to the generalized algebraic Riccati equation (GARE) Then, the saddle-point control policies are unique and are given by The expected objective value of the game under this saddle-point solution is

Figures (6)

  • Figure 1: Schematic of a two-player wireless system exerting control on the state of a dynamical system.
  • Figure 2: Best response plot for each player in \ref{['exp:simpleExample']}.
  • Figure 3: Variation of ($p^*,q^*$) versus scheduling costs $\lambda_{11}$ and $\lambda_{22}$ in \ref{['exp:simpleExample']}.
  • Figure 4: Signal trajectories for each player for \ref{['exp:simpleExample']}.
  • Figure 5: Best response plot for each player in \ref{['exp:PEG']}.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • proof
  • Remark 2
  • Proposition 5.1
  • proof
  • Theorem 3: jarlebring2018krylov
  • Remark 3
  • Example 1
  • ...and 1 more