Optimal Modified Feedback Strategies in LQ Games under Control Imperfections

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 trajectory and the other player’s cost. To mitigate these effects, we construct a compensation law for the influenced player by augmenting the nominal game with the measurable deviation dynamics. The resulting policy is shown to be optimal within a causal affine policy class, and, for sufficiently small deviations, it locally outperforms the uncompensated equilibrium-derived feedback. Rigorous analysis and proofs are provided, and the effectiveness of the proposed approach is demonstrated through a representative numerical example.

💡 Research Summary

The paper addresses a fundamental gap in the application of Nash equilibrium strategies to real‑world multi‑agent systems: the inevitable mismatch between the commanded control inputs and the actual inputs that are executed due to actuator dynamics, communication delays, filtering, or safety layers. Focusing on a two‑player, finite‑horizon, discrete‑time linear‑quadratic (LQ) non‑zero‑sum game, the authors first derive a closed‑form sensitivity analysis that quantifies how a small deviation Δu₂,k by Player 2 propagates through the closed‑loop dynamics and perturbs Player 1’s cost. By introducing the state‑transition matrix Φ(k,j) they obtain an exact expression for the induced state perturbation Δx_k and a first‑order expansion of the cost variation ΔJ₁ with explicit weighting matrices Λ_j. This analysis reveals that the cost impact is linear in the deviation for sufficiently small errors, while higher‑order terms become relevant for larger disturbances.

Recognizing that the deviation w_k = B₂Δu₂,k can often be measured (or estimated) at each time step, the authors model its dynamics using a first‑order discrete lag: w_{k+1}=αw_k+B₂(K_{2,k+1}^⋆x_{k+1}−K_{2,k}^⋆x_k) with α∈(0,1). This captures two realistic phenomena: (i) exponential decay of a steady‑state tracking error, and (ii) a transient injection of error whenever the Nash command changes. By augmenting the original state with the disturbance, z_k=