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A Set-Theoretic Robust Control Approach for Linear Quadratic Games with Unknown Counterparts

Francesco Bianchin, Robert Lefringhausen, Elisa Gaetan, Samuel Tesfazgi, Sandra Hirche

TL;DR

This work addresses robust decision-making in linear-quadratic differential games with unknown adversaries and disturbances. It tightly couples set-membership identification of adversary policies with data-driven LMI-based robust LQR to guarantee stability across all unfalsified models and achieve convergence to an $\epsilon$-Nash equilibrium. The approach provides online policy updates and demonstrated robustness in simulations of human-robot interaction and scalability to higher dimensions. The results underscore the practical impact for safety-critical, multi-agent systems where opponent strategies are uncertain and disturbances are present.

Abstract

Ensuring robust decision-making in multi-agent systems is challenging when agents have distinct, possibly conflicting objectives and lack full knowledge of each other's strategies. This is apparent in safety-critical applications such as human-robot interaction and assisted driving, where uncertainty arises not only from unknown adversary strategies but also from external disturbances. To address this, the paper proposes a robust adaptive control approach based on linear quadratic differential games. Our method allows a controlled agent to iteratively refine its belief about the adversary strategy and disturbances using a set-membership approach, while simultaneously adapting its policy to guarantee robustness against the uncertain adversary policy and improve performance over time. We formally derive theoretical guarantees on the robustness of the proposed control scheme and its convergence to $ε$-Nash strategies. The effectiveness of our approach is demonstrated in a numerical simulation.

A Set-Theoretic Robust Control Approach for Linear Quadratic Games with Unknown Counterparts

TL;DR

This work addresses robust decision-making in linear-quadratic differential games with unknown adversaries and disturbances. It tightly couples set-membership identification of adversary policies with data-driven LMI-based robust LQR to guarantee stability across all unfalsified models and achieve convergence to an -Nash equilibrium. The approach provides online policy updates and demonstrated robustness in simulations of human-robot interaction and scalability to higher dimensions. The results underscore the practical impact for safety-critical, multi-agent systems where opponent strategies are uncertain and disturbances are present.

Abstract

Ensuring robust decision-making in multi-agent systems is challenging when agents have distinct, possibly conflicting objectives and lack full knowledge of each other's strategies. This is apparent in safety-critical applications such as human-robot interaction and assisted driving, where uncertainty arises not only from unknown adversary strategies but also from external disturbances. To address this, the paper proposes a robust adaptive control approach based on linear quadratic differential games. Our method allows a controlled agent to iteratively refine its belief about the adversary strategy and disturbances using a set-membership approach, while simultaneously adapting its policy to guarantee robustness against the uncertain adversary policy and improve performance over time. We formally derive theoretical guarantees on the robustness of the proposed control scheme and its convergence to -Nash strategies. The effectiveness of our approach is demonstrated in a numerical simulation.

Paper Structure

This paper contains 10 sections, 4 theorems, 38 equations, 6 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. A Set Membership Approach to Adversary Strategy Identification
  4. Robust Response via Data-Driven LMI-based Riccati Iterations
  5. Convergence to an epsilon-Nash Equilibrium Strategy
  6. Simulation Results
  7. Convergence Properties
  8. Robustness Properties
  9. Scalability
  10. Conclusions

Key Result

Lemma 1

The set of consistent models $\Omega$ is convex and closed. It is bounded if and only if the data generating the set satisfies $\text{rank} = n_x$, where $N$ is the number of available samples and $n_x$ is the state-space dimension.

Figures (6)

  • Figure 1: Illustrative example of the effect of adding a non-redundant constraint, corresponding to a new data point, to the set description of $\Omega$. The constraint introduces a plane cut (shown in red) in the parameter space according to (\ref{['data-constr']}), reducing the polytope’s volume (thus the parameter uncertainty) and generating new vertices.
  • Figure 2: The vertices of the unfalsified strategy set $\Omega$ are shown as blue markers. These vertices correspond to extremal adversary strategies, providing a finite representation of the set. By exploiting them, stability guarantees can be enforced over the entire $\Omega$.
  • Figure 3: The evolution of the uncertainty area $A(\bm{B}_2 \bm{K}_2)$ for the estimated parameters $\bm{B}_2 \bm{K}_2$, when applying Algorithm 1 on system (\ref{['sim_example']}), is shown in blue in the main graph. This area corresponds to the polytope describing the uncertain parameters. Iterations where the uncertainty decreases are marked with gray dots, and the final settled area is shown in orange. The top-right graph illustrates the contraction of the polytopes (simple polygons) over iterations. Each dimension corresponds to one of the two estimated parameters, i.e., the non-zero entries of $\bm{B}_2 \bm{K}_2$. The red bounding box represents the initial conservative specification $\Omega^0$, gray polygons show the shrinking uncertainty sets $\Omega_k$, and the set reached at iteration 25 of Algorithm 1 is shown in orange.
  • Figure 4: The solid lines represent the evolution of the controlled agent's feedback policy, $\hat{\bm{K}}_1$, across algorithm iterations, with the first entry shown in blue and the second in orange. The corresponding optimal Nash gain values are depicted by dashed lines.
  • Figure 5: Comparison of $x_1(t)$ evolution for extremal adversary strategies. The left panel (a) shows the response under the robust solution $\hat{K}_1^{robust}$, while the right panel (b) shows the response under the least-squares estimation solution $\hat{K}_1^{LS}$. The adversarial strategy for which the least-squares-based method fails to guarantee stability is highlighted in red.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 1: engwerda
  • Lemma 1: Bisoffi2020ControllerDF
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2: Dasgupta
  • Theorem 2
  • ...and 1 more