Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Modified Feedback Strategies in LQ Games under Control Imperfections

Mahdis Rabbani, Navid Mojahed, Shima Nazari

TL;DR

This work analyzes how small control execution deviations in a two-player finite-horizon LQ game perturb the Nash trajectory and players' costs. It derives a first-order sensitivity result showing how opponent deviations propagate through the dynamics and cost, and then introduces a Compensated Feedback CF policy that augments the state with the disturbance signal to optimally counteract these deviations via an augmented Riccati recursion. A nonnegative gap identity guarantees that CF is at least as good as, and often strictly better than, relying on the original Nash feedback when deviations occur. The approach is validated on a numerical two-cart spring-damper system, demonstrating substantial improvement for the compensated player under actuator lag, while highlighting asymmetric effects on the opponent, with practical implications for robust multi-agent control under real-world imperfections.

Abstract

Game-theoretic approaches and Nash equilibrium have been widely applied across various engineering domains. However, practical challenges such as disturbances, delays, and actuator limitations can hinder the precise execution of Nash equilibrium strategies. This work investigates the impact of such implementation imperfections on game trajectories and players' costs in the context of a two-player finite-horizon linear quadratic (LQ) nonzero-sum game. Specifically, we analyze how small deviations by one player, measured or estimated at each stage, affect the state and cost function of the other player. To mitigate these effects, we propose an adjusted control policy that optimally compensates for the deviations under the stated information structure and can, under certain conditions, exploit them to improve performance. Rigorous mathematical analysis and proofs are provided, and the effectiveness of the proposed method is demonstrated through a representative numerical example.

Optimal Modified Feedback Strategies in LQ Games under Control Imperfections

TL;DR

This work analyzes how small control execution deviations in a two-player finite-horizon LQ game perturb the Nash trajectory and players' costs. It derives a first-order sensitivity result showing how opponent deviations propagate through the dynamics and cost, and then introduces a Compensated Feedback CF policy that augments the state with the disturbance signal to optimally counteract these deviations via an augmented Riccati recursion. A nonnegative gap identity guarantees that CF is at least as good as, and often strictly better than, relying on the original Nash feedback when deviations occur. The approach is validated on a numerical two-cart spring-damper system, demonstrating substantial improvement for the compensated player under actuator lag, while highlighting asymmetric effects on the opponent, with practical implications for robust multi-agent control under real-world imperfections.

Abstract

Game-theoretic approaches and Nash equilibrium have been widely applied across various engineering domains. However, practical challenges such as disturbances, delays, and actuator limitations can hinder the precise execution of Nash equilibrium strategies. This work investigates the impact of such implementation imperfections on game trajectories and players' costs in the context of a two-player finite-horizon linear quadratic (LQ) nonzero-sum game. Specifically, we analyze how small deviations by one player, measured or estimated at each stage, affect the state and cost function of the other player. To mitigate these effects, we propose an adjusted control policy that optimally compensates for the deviations under the stated information structure and can, under certain conditions, exploit them to improve performance. Rigorous mathematical analysis and proofs are provided, and the effectiveness of the proposed method is demonstrated through a representative numerical example.

Paper Structure

This paper contains 9 sections, 4 theorems, 45 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First-Order Sensitivity of Player 1 Cost under Opponent Deviations
  4. Control Policy Design with Disturbance Compensation
  5. Compensated Feedback under Lagged Execution Error Dynamics
  6. Specialization: Slowly-Varying Error Dynamics
  7. Numerical Example: Two Carts Coupled by a Spring--Damper
  8. Conclusion
  9. Acknowledgment

Key Result

Proposition 1

Let $\{K^\star_{1,k}, K^\star_{2,k}\}_{k=0}^{N-1}$ be the FNE gains in eq:feedback nash definition. Suppose Player 2 deviates as $u_{2,k}=u^\star_{2,k}+\Delta u_{2,k}$ while Player 1 keeps $u_{1,k}=u^\star_{1,k}=-K^\star_{1,k}x_k$. Then, for $k=1,\dots,N$, and, with $S_k := Q_1 + K_{1,k}^{\star\top} R_1 K^\star_{1,k}$, where for $j=0,\dots,N-1$. Here $\Phi(k,j)$ is as in Definition def:phi, $\D

Figures (2)

  • Figure 1: Schematic of the two-cart setup and qualitative final positions under the three cases. Case I (FNE) shows near-symmetric convergence near the origin. Case II (REF) illustrates how Player 2’s actuator lag degrades both players’ positions relative to nominal. Case III (CF) shows that the compensated policy enables Player 1 to mitigate the error and approach its nominal outcome, while Player 2 remains misaligned due to its uncompensated lag.
  • Figure 2: State trajectories $(p_1,v_1,p_2,v_2)$ and applied inputs under the three cases. The compensated feedback (CF) attenuates the effect of Player 2’s actuator lag by shaping a mismatch-aware control for Player 1.

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Definition 2
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • ...and 4 more